Project Gutenberg’s Mathematical Recreations and Essays, by W. W. RouseBall
This eBook is for the use of anyone anywhere at no cost and withalmost no restrictions whatsoever. You may copy it, give it away orre-use it under the terms of the Project Gutenberg License includedwith this eBook or online at www.gutenberg.org
Title: Mathematical Recreations and Essays
Author: W. W. Rouse Ball
Release Date: October 8, 2008 [EBook #26839]
Language: English
Character set encoding: ISO-8859-1
*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS ***
First Edition, Feb. 1892. Reprinted, May, 1892.
Second Edition, 1896. Reprinted, 1905.
MATHEMATICALRECREATIONS AND ESSAYS
BY
W.W. ROUSE BALL
Fellow and Tutor of Trinity College, Cambridge.
FOURTH EDITION
London:MACMILLAN AND CO., Limited
NEW YORK: THE MACMILLAN COMPANY
[All rights reserved.]
Produced by Joshua Hutchinson, David Starner, David Wilson andthe Online Distributed Proofreading Team at http://www.pgdp.net
Transcriber’s notes
Most of the open questions discussed by the author weresettled during the twentieth century.
The author’s footnotes are labelled using printer’s marks*;footnotes showing where corrections to the text have beenmade are labelled numerically1.
Minor typographical corrections are documented in the LATEXsource.
This document is designed for two-sided printing. Consequently,the many hyperlinked cross-references are not visuallydistinguished. The document can be recompiled for morecomfortable on-screen viewing: see comments in source LATEXcode.
PREFACE TO THE FIRST EDITION.
The following pages contain an account of certain mathematical
recreations, problems, and speculations of past and present times. I
hasten to add that the conclusions are of no practical use, and most
of the results are not new. If therefore the reader proceeds further he
is at least forewarned.
At the same time I think I may assert that many of the diversions—
particularly those in the latter half of the book—are interesting, not
a few are associated with the names of distinguished mathematicians,
while hitherto several of the memoirs quoted have not been easily ac-
cessible to English readers.
The book is divided into two parts, but in both parts I have in-
cluded questions which involve advanced mathematics.
The first part consists of seven chapters, in which are included var-
ious problems and amusem*nts of the kind usually called mathematical
recreations. The questions discussed in the first of these chapters are
connected with arithmetic; those in the second with geometry; and
those in the third relate to mechanics. The fourth chapter contains
an account of some miscellaneous problems which involve both num-
ber and situation; the fifth chapter contains a concise account of magic
squares; and the sixth and seventh chapters deal with some unicursal
iii
• Project • Gutenberg • #26839 •
iv PREFACE
problems. Several of the questions mentioned in the first three chap-
ters are of a somewhat trivial character, and had they been treated in
any standard English work to which I could have referred the reader, I
should have pointed them out. In the absence of such a work, I thought
it best to insert them and trust to the judicious reader to omit them
altogether or to skim them as he feels inclined.
The second part consists of five chapters, which are mostly histori-
cal. They deal respectively with three classical problems in geometry—
namely, the duplication of the cube, the trisection of an angle, and the
quadrature of the circle—astrology, the hypotheses as to the nature of
space and mass, and a means of measuring time.
I have inserted detailed references, as far as I know, as to the sources
of the various questions and solutions given; also, wherever I have given
only the result of a theorem, I have tried to indicate authorities where
a proof may be found. In general, unless it is stated otherwise, I have
taken the references direct from the original works; but, in spite of
considerable time spent in verifying them, I dare not suppose that they
are free from all errors or misprints.
I shall be grateful for notices of additions or corrections which may
occur to any of my readers.
W.W. ROUSE BALL
Trinity College, Cambridge.
February, 1892.
• Project • Gutenberg • #26839 •
NOTE TO THE FOURTH EDITION.
In this edition I have inserted in the earlier chapters descriptions of
several additional Recreations involving elementary mathematics, and
I have added in the second part chapters on the History of the Mathe-
matical Tripos at Cambridge, Mersenne’s Numbers, and Cryptography
and Ciphers.
It is with some hesitation that I include in the book the chapters on
Astrology and Ciphers, for these subjects are only remotely connected
with Mathematics, but to afford myself some latitude I have altered
the title of the second part to Miscellaneous Essays and Problems.
W.W.R.B.
Trinity College, Cambridge.
13 May, 1905.
v
• Project • Gutenberg • #26839 •
TABLE OF CONTENTS.
PART I.
Mathematical Recreations.
Chapter I. Some Arithmetical Questions.PAGE
Elementary Questions on Numbers (Miscellaneous) . . . . . . 4Arithmetical Fallacies . . . . . . . . . . . . . . . . . . . . . . 20Bachet’s Weights Problem . . . . . . . . . . . . . . . . . . . . 27Problems in Higher Arithmetic . . . . . . . . . . . . . . . . . 29
Fermat’s Theorem on Binary Powers . . . . . . . . . . . . 31Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . 32
Chapter II. Some Geometrical Questions.
Geometrical Fallacies . . . . . . . . . . . . . . . . . . . . . . . 35Geometrical Paradoxes . . . . . . . . . . . . . . . . . . . . . . 42Colouring Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 44Physical Geography . . . . . . . . . . . . . . . . . . . . . . . 46Statical Games of Position . . . . . . . . . . . . . . . . . . . . 48
Three-in-a-row. Extension to p-in-a-row . . . . . . . . . 48Tesselation. Cross-Fours . . . . . . . . . . . . . . . . . . 50Colour-Cube Problem . . . . . . . . . . . . . . . . . . . . 51
vi
• Project • Gutenberg • #26839 •
TABLE OF CONTENTS. vii
PAGE
Dynamical Games of Position . . . . . . . . . . . . . . . . . . 52Shunting Problems . . . . . . . . . . . . . . . . . . . . . . 53Ferry-Boat Problems . . . . . . . . . . . . . . . . . . . . . 55Geodesic Problems . . . . . . . . . . . . . . . . . . . . . . 57Problems with Counters placed in a row . . . . . . . . . . 58Problems on a Chess-board with Counters or Pawns . . . . 60Guarini’s Problem . . . . . . . . . . . . . . . . . . . . . . 63
Geometrical Puzzles (rods, strings, &c.) . . . . . . . . . . . . 64Paradromic Rings . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter III. Some Mechanical Questions.
Paradoxes on Motion . . . . . . . . . . . . . . . . . . . . . . . 67Force, Inertia, Centrifugal Force . . . . . . . . . . . . . . . . . 70Work, Stability of Equilibrium, &c. . . . . . . . . . . . . . . . 72Perpetual Motion . . . . . . . . . . . . . . . . . . . . . . . . . 75Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Sailing quicker than the Wind . . . . . . . . . . . . . . . . . . 79Boat moved by a rope inside the boat . . . . . . . . . . . . . 81Results dependent on Hauksbee’s Law . . . . . . . . . . . . . 82
Cut on a tennis-ball. Spin on a cricket-ball . . . . . . . 83Flight of Birds . . . . . . . . . . . . . . . . . . . . . . . . . . 85Curiosa Physica . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter IV. Some Miscellaneous Questions.
The Fifteen Puzzle . . . . . . . . . . . . . . . . . . . . . . . . 88The Tower of Hanoı . . . . . . . . . . . . . . . . . . . . . . . 91Chinese Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 93The Eight Queens Problem . . . . . . . . . . . . . . . . . . . 97Other Problems with Queens and Chess-pieces . . . . . . . . . 102The Fifteen School-Girls Problem . . . . . . . . . . . . . . . . 103
• Project • Gutenberg • #26839 •
viii TABLE OF CONTENTS.
PAGE
Problems connected with a pack of cards . . . . . . . . . . . . 109Monge on shuffling a pack of cards . . . . . . . . . . . . . 109Arrangement by rows and columns . . . . . . . . . . . . . 111Determination of one out of 1
2n(n + 1) given couples . . . . 113
Gergonne’s Pile Problem . . . . . . . . . . . . . . . . . . . 115The Mouse Trap. Treize . . . . . . . . . . . . . . . . . . 119
Chapter V. Magic Squares.
Notes on the History of Magic Squares . . . . . . . . . . . . . 122Construction of Odd Magic Squares . . . . . . . . . . . . . . . 123
Method of De la Loubere . . . . . . . . . . . . . . . . . . . 124Method of Bachet . . . . . . . . . . . . . . . . . . . . . . . 125Method of De la Hire . . . . . . . . . . . . . . . . . . . . . 126
Construction of Even Magic Squares . . . . . . . . . . . . . . 128First Method . . . . . . . . . . . . . . . . . . . . . . . . . 129Method of De la Hire and Labosne . . . . . . . . . . . . . 132
Composite Magic Squares . . . . . . . . . . . . . . . . . . . . 134Bordered Magic Squares . . . . . . . . . . . . . . . . . . . . . 135Hyper-Magic Squares . . . . . . . . . . . . . . . . . . . . . . . 136
Pan-diagonal or Nasik Squares . . . . . . . . . . . . . . . . 136Doubly Magic Squares . . . . . . . . . . . . . . . . . . . . 137
Magic Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Magic Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Card Square . . . . . . . . . . . . . . . . . . . . . . . . . . 140Euler’s Officers Problem . . . . . . . . . . . . . . . . . . . 140Domino Squares . . . . . . . . . . . . . . . . . . . . . . . . 141Coin Squares . . . . . . . . . . . . . . . . . . . . . . . . . 141
Chapter VI. Unicursal Problems.
Euler’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 143Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Euler’s Theorems . . . . . . . . . . . . . . . . . . . . . . . 145Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
• Project • Gutenberg • #26839 •
TABLE OF CONTENTS. ix
PAGE
Mazes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Rules for completely traversing a Maze . . . . . . . . . . . 150Notes on the History of Mazes . . . . . . . . . . . . . . . . 150
Geometrical Trees . . . . . . . . . . . . . . . . . . . . . . . . 154The Hamiltonian Game . . . . . . . . . . . . . . . . . . . . . 155Knight’s Path on a Chess-Board . . . . . . . . . . . . . . . . . 158
Method of De Montmort and De Moivre . . . . . . . . . . 159Method of Euler . . . . . . . . . . . . . . . . . . . . . . . 159Method of Vandermonde . . . . . . . . . . . . . . . . . . . 163Method of Warnsdorff . . . . . . . . . . . . . . . . . . . . 164Method of Roget . . . . . . . . . . . . . . . . . . . . . . . 164Method of Moon . . . . . . . . . . . . . . . . . . . . . . . 167Method of Jaenisch . . . . . . . . . . . . . . . . . . . . . . 168Number of possible routes . . . . . . . . . . . . . . . . . . 168
Paths of other Chess-Pieces . . . . . . . . . . . . . . . . . . . 168
• Project • Gutenberg • #26839 •
x TABLE OF CONTENTS.
PART II.
Miscellaneous Essays and Problems.
Chapter VII. The Mathematical Tripos.PAGE
Medieval Course of Studies: Acts . . . . . . . . . . . . . . . . 171The Renaissance at Cambridge . . . . . . . . . . . . . . . . . 172
Rise of a Mathematical School . . . . . . . . . . . . . . . . 172Subject-Matter of Acts at different periods . . . . . . . . . . . 172Degree Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Oral Examinations always possible . . . . . . . . . . . . . . . 174Public Oral Examinations become customary, 1710–30 . . . . 175
Additional work thrown on Moderators. Stipends raised . 175Facilitates order of merit . . . . . . . . . . . . . . . . . . . 176
Scheme of Examination in 1750 . . . . . . . . . . . . . . . . . 176Right of M.A.s to take part in it . . . . . . . . . . . . . . . 176
Scheme of Examination in 1763 . . . . . . . . . . . . . . . . . 177Foundations of Smith’s Prizes, 1768 . . . . . . . . . . . . . . . 178Introduction of a Written Examination, circ. 1770 . . . . . . . 179Description of the Examination in 1772 . . . . . . . . . . . . . 179Scheme of Examination in 1779 . . . . . . . . . . . . . . . . . 182
System of Brackets . . . . . . . . . . . . . . . . . . . . . . 182Problem Papers in 1785 and 1786 . . . . . . . . . . . . . . . . 183Description of the Examination in 1791 . . . . . . . . . . . . . 184
The Poll Part of the Examination . . . . . . . . . . . . . . 185A Pass Standard introduced . . . . . . . . . . . . . . . . . . . 186Problem Papers from 1802 onwards . . . . . . . . . . . . . . . 186Description of the Examination in 1802 . . . . . . . . . . . . . 187Scheme of Reading in 1806 . . . . . . . . . . . . . . . . . . . . 189Introduction of modern analytical notation . . . . . . . . . . . 192Alterations in Schemes of Study, 1824 . . . . . . . . . . . . . 195Scheme of Examination in 1827 . . . . . . . . . . . . . . . . . 195Scheme of Examination in 1833 . . . . . . . . . . . . . . . . . 197
All the papers marked . . . . . . . . . . . . . . . . . . . . 197Scheme of Examination in 1839 . . . . . . . . . . . . . . . . . 197
• Project • Gutenberg • #26839 •
TABLE OF CONTENTS. xi
PAGE
Scheme of Examination in 1848 . . . . . . . . . . . . . . . . . 198Creation of a Board of Mathematical Studies . . . . . . . . . 198Scheme of Examination in 1873 . . . . . . . . . . . . . . . . . 199Scheme of Examination in 1882 . . . . . . . . . . . . . . . . . 200
Fall in number of students reading mathematics . . . . . . 201Origin of term Tripos . . . . . . . . . . . . . . . . . . . . . . . 201
Tripos Verses . . . . . . . . . . . . . . . . . . . . . . . . . 202
Chapter VIII. Three Geometrical Problems.
The Three Problems . . . . . . . . . . . . . . . . . . . . . . . 204The Duplication of the Cube . . . . . . . . . . . . . . . . . . 205
Legendary origin of the problem . . . . . . . . . . . . . . . 205Lemma of Hippocrates . . . . . . . . . . . . . . . . . . . . . . 206
Solutions of Archytas, Plato, Menaechmus, Apollonius, andSporus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Solutions of Vieta, Descartes, Gregory of St Vincent, andNewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
The Trisection of an Angle . . . . . . . . . . . . . . . . . . . . 210Solutions quoted by Pappus (three) . . . . . . . . . . . . . . . 210Solutions of Descartes, Newton, Clairaut, and Chasles . . . . 211The Quadrature of the Circle . . . . . . . . . . . . . . . . . . 212Incommensurability of π . . . . . . . . . . . . . . . . . . . . . 212Definitions of π . . . . . . . . . . . . . . . . . . . . . . . . . . 213Origin of symbol π . . . . . . . . . . . . . . . . . . . . . . . . 214Methods of approximating to the numerical value of π . . . . 214Geometrical methods of approximation . . . . . . . . . . . . . 214
Results of Egyptians, Babylonians, Jews . . . . . . . . . . 215Results of Archimedes and other Greek writers . . . . . . . 215Results of Roman surveyors and Gerbert . . . . . . . . . . 216Results of Indian and Eastern writers . . . . . . . . . . . . 216Results of European writers, 1200–1630 . . . . . . . . . . . 217
Theorems of Wallis and Brouncker . . . . . . . . . . . . . . . 220Analytical methods of approximation. Gregory’s series . . . . 220
Results of European writers, 1699–1873 . . . . . . . . . . . 220Geometrical approximations . . . . . . . . . . . . . . . . . . . 222Approximations by the theory of probability . . . . . . . . . . 222
• Project • Gutenberg • #26839 •
xii TABLE OF CONTENTS.
Chapter IX. Mersenne’s Numbers.PAGE
Mersenne’s Enunciation of the Theorem . . . . . . . . . . . . 224List of known results . . . . . . . . . . . . . . . . . . . . . . . 225Cases awaiting verification . . . . . . . . . . . . . . . . . . . . 225History of Investigations . . . . . . . . . . . . . . . . . . . . . 226Methods used in attacking the problem . . . . . . . . . . . . . 230
By trial of divisors of known forms . . . . . . . . . . . . . 231By indeterminate equations . . . . . . . . . . . . . . . . . 233By properties of quadratic forms . . . . . . . . . . . . . . 234By the use of a Canon Arithmeticus . . . . . . . . . . . . 234By properties of binary powers . . . . . . . . . . . . . . . 235By the use of the binary scale . . . . . . . . . . . . . . . . 235By the use of Fermat’s Theorem . . . . . . . . . . . . . . . 236
Mechanical methods of Factorizing Numbers . . . . . . . . . . 236
Chapter X. Astrology.
Astrology. Two branches: natal and horary astrology . . . . . 238Rules for casting and reading a horoscope . . . . . . . . . . . 238
Houses and their significations . . . . . . . . . . . . . . . . 238Planets and their significations . . . . . . . . . . . . . . . 240Zodiacal signs and their significations . . . . . . . . . . . . 242
Knowledge that rules were worthless . . . . . . . . . . . . . . 243Notable instances of horoscopy . . . . . . . . . . . . . . . . . 246
Lilly’s prediction of the Great Fire and Plague . . . . . . . 246Flamsteed’s guess . . . . . . . . . . . . . . . . . . . . . . . 246Cardan’s horoscope of Edward VI . . . . . . . . . . . . . . 247
Chapter XI. Cryptographs and Ciphers.
A Cryptograph. Definition. Illustration . . . . . . . . . . . . . 251A Cipher. Definition. Illustration . . . . . . . . . . . . . . . . 252Essential Features of Cryptographs and Ciphers . . . . . . . . 252Cryptographs of Three Types. Illustrations . . . . . . . . . . 253
Order of letters re-arranged . . . . . . . . . . . . . . . . . 253Use of non-significant symbols. The Grille . . . . . . . . . 256Use of broken symbols. The Scytale . . . . . . . . . . . . . 258
Ciphers. Use of arbitrary symbols unnecessary . . . . . . . . . 259
• Project • Gutenberg • #26839 •
TABLE OF CONTENTS. xiii
PAGE
Ciphers of Four Types . . . . . . . . . . . . . . . . . . . . . . 259Ciphers of the First Type. Illustrations . . . . . . . . . . . 260Ciphers of the Second Type. Illustrations . . . . . . . . . . 263Ciphers of the Third Type. Illustrations . . . . . . . . . . 265Ciphers of the Fourth Type. Illustrations . . . . . . . . . . 267
Requisites in a good Cipher . . . . . . . . . . . . . . . . . . . 268Cipher Machines . . . . . . . . . . . . . . . . . . . . . . . . . 269Historical Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . 269
Julius Caesar, Augustus . . . . . . . . . . . . . . . . . . . 269Bacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Charles I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Pepys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271De Rohan . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Marie Antoinette . . . . . . . . . . . . . . . . . . . . . . . 272The Code Dictionary . . . . . . . . . . . . . . . . . . . . . 274Poe’s Writings . . . . . . . . . . . . . . . . . . . . . . . . . 275
Chapter XII. Hyper-space.
Two subjects of speculation on Hyper-space . . . . . . . . . . 278Space of two dimensions and of one dimension . . . . . . . . . 278Space of four dimensions . . . . . . . . . . . . . . . . . . . . . 279
Existence in such a world . . . . . . . . . . . . . . . . . . 279Arguments in favour of the existence of such a world . . . 280
Non-Euclidean Geometries . . . . . . . . . . . . . . . . . . . . 284Euclid’s axioms and postulates. The parallel postulate . . . . 284Hyperbolic Geometry of two dimensions . . . . . . . . . . . . 285Elliptic Geometry of two dimensions . . . . . . . . . . . . . . 285
Elliptic, Parabolic and Hyperbolic Geometries compared . 285Non-Euclidean Geometries of three or more dimensions . . . . 287
Chapter XIII. Time and its Measurement.
Units for measuring durations (days, weeks, months, years) . . 289The Civil Calendar (Julian, Gregorian, &c.) . . . . . . . . . . 292The Ecclesiastical Calendar (date of Easter) . . . . . . . . . . 294Day of the week corresponding to a given date . . . . . . . . . 297
• Project • Gutenberg • #26839 •
xiv TABLE OF CONTENTS.
PAGE
Means of measuring Time . . . . . . . . . . . . . . . . . . . . 297Styles, Sun-dials, Sun-rings . . . . . . . . . . . . . . . . . 297Water-clocks, Sand-clocks, Graduated Candles . . . . . . . 301Clocks and Watches . . . . . . . . . . . . . . . . . . . . . 301
Watches as Compasses . . . . . . . . . . . . . . . . . . . . . . 303
Chapter XIV. Matter and Ether Theories.
Hypothesis of Continuous Matter . . . . . . . . . . . . . . . . 306Atomic Theories . . . . . . . . . . . . . . . . . . . . . . . . . 306
Popular Atomic Hypothesis . . . . . . . . . . . . . . . . . 306Boscovich’s Hypothesis . . . . . . . . . . . . . . . . . . . . 307Hypothesis of an Elastic Solid Ether. Labile Ether . . . . . 307
Dynamical Theories . . . . . . . . . . . . . . . . . . . . . . . 308The Vortex Ring Hypothesis . . . . . . . . . . . . . . . . . 308The Vortex Sponge Hypothesis . . . . . . . . . . . . . . . 309The Ether-Squirts Hypothesis . . . . . . . . . . . . . . . . 310The Electron Hypothesis . . . . . . . . . . . . . . . . . . . 311Speculations due to investigations on Radio-activity . . . . 311The Bubble Hypothesis . . . . . . . . . . . . . . . . . . . . 313
Conjectures as to the cause of Gravity . . . . . . . . . . . . . 314Conjectures to explain the finite number of species of Atoms . 318Size of the molecules of bodies . . . . . . . . . . . . . . . . . . 320
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323Notices of some works—chiefly
historico-mathematical . . . . . . . . . . . . . . . . . 335
Project Gutenberg Licensing Information . . . . . . 355
• Project • Gutenberg • #26839 •
PART I.
Mathematical Recreations.
“Les hommes ne sont jamais plus ingenieuxque dans l’invention des jeux; l’esprit s’y trouve ason aise. . . . Apres les jeux qui dependent unique-ment des nombres viennent les jeux ou entre la situ-ation. . . . Apres les jeux ou n’entrent que le nombreet la situation viendraient les jeux ou entre le mou-vement. . . . Enfin il serait a souhaiter qu’on eut uncours entier des jeux, traites mathematiquement.”(Leibnitz: letter to De Montmort, July 29, 1715.)
1
• Project • Gutenberg • #26839 •
CHAPTER I.
SOME ARITHMETICAL QUESTIONS.
The interest excited by statements of the relations between num-bers of certain forms has been often remarked. The majority of workson mathematical recreations include several such problems, which areobvious to any one acquainted with the elements of algebra, but whichto many who are ignorant of that subject possess the same kind ofcharm that some mathematicians find in the more recondite proposi-tions of higher arithmetic. I shall devote the bulk of this chapter tothese elementary problems, but I append a few remarks on one or twoquestions in the theory of numbers.
Before entering on the subject of the chapter, I may add that alarge proportion of the elementary questions mentioned here and inthe following two chapters are taken from one of two sources. The firstof these is the classical Problemes plaisans et delectables, by C.G. Ba-chet, sieur de Meziriac, of which the first edition was published in 1612and the second in 1624: it is to the edition of 1624 that the referenceshereafter given apply. Several of Bachet’s problems are taken from thewritings of Alcuin, Pacioli di Burgo, Tartaglia, or Cardan, and possi-bly some of them are of oriental origin, but I have made no attemptto add such references. The other source to which I alluded above isOzanam’s Recreations mathematiques et physiques . The greater por-tion of the original edition, published in two volumes at Paris in 1694,was a compilation from the works of Bachet, Leurechon, Mydorge, vanEtten, and Oughtred: this part is excellent, but the same cannot besaid of the additions due to Ozanam. In the Biographie Universelle al-lusion is made to subsequent editions issued in 1720, 1735, 1741, 1778,
2
• Project • Gutenberg • #26839 •
CH. I] ARITHMETICAL RECREATIONS. 3
and 1790; doubtless these references are correct, but the following edi-tions, all of which I have seen, are the only ones of which I have anyknowledge. In 1696 an edition was issued at Amsterdam. In 1723—six years after the death of Ozanam—one was issued in three volumes,with a supplementary fourth volume, containing (among other things)an appendix on puzzles: I believe that it would be difficult to find inany of the books current in England on mathematical amusem*nts asmany as a dozen puzzles which are not contained in one of these fourvolumes. Fresh editions were issued in 1741, 1750 (the second volumeof which bears the date 1749), 1770, and 1790. The edition of 1750 issaid to have been corrected by Montucla on condition that his nameshould not be associated with it; but the edition of 1790 is the earliestone in which reference is made to these corrections, though the editor isreferred to only as Monsieur M***. Montucla expunged most of whatwas actually incorrect in the older editions, and added several historicalnotes, but unfortunately his scruples prevented him from striking outthe accounts of numerous trivial experiments and truisms which over-load the work. An English translation of the original edition appearedin 1708, and I believe ran through four editions, the last of them beingpublished in Dublin in 1790. Montucla’s revision of 1790 was translatedby C. Hutton, and editions of this were issued in 1803, in 1814, and (inone volume) in 1840: my references are to the editions of 1803 and 1840.
I proceed now to enumerate some of the elementary questions con-nected with numbers which for nearly three centuries have formed alarge part of most compilations of mathematical amusem*nts. Theyare given here mainly for their historical—not for their arithmetical—interest; and perhaps a mathematician may well omit them, and passat once to the latter part of this chapter.
These questions are of the nature of tricks or puzzles and I followthe usual course and present them in that form. I may note howeverthat most of them are not worth proposing, even as tricks, unless eitherthe modus operandi is disguised or the result arrived at is differentfrom that expected; but, as I am not writing on conjuring, I refrainfrom alluding to the means of disguising the operations indicated, andgive merely a bare enumeration of the steps essential to the success ofthe method used, though I may recall the fundamental rule that notrick, however good, will bear immediate repetition, and that, if it is
• Project • Gutenberg • #26839 •
4 ARITHMETICAL RECREATIONS. [CH. I
necessary to appear to repeat it, a different method of obtaining theresult should be used.
To find a number selected by some one. There are in-numerable ways of finding a number chosen by some one, provided theresult of certain operations on it is known. I confine myself to methodstypical of those commonly used. Any one acquainted with algebra willfind no difficulty in modifying the rules here given or framing new onesof an analogous nature.
First Method*. (i) Ask the person who has chosen the number totreble it. (ii) Enquire if the product is even or odd: if it is even, requesthim to take half of it; if it is odd, request him to add unity to it andthen to take half of it. (iii) Tell him to multiply the result of the secondstep by 3. (iv) Ask how many integral times 9 divides into the latterproduct: suppose the answer to be n. (v) Then the number thought ofwas 2n or 2n + 1, according as the result of step (i) was even or odd.
The demonstration is obvious. Every even number is of the form2n, and the successive operations applied to this give (i) 6n, which iseven; (ii) 1
26n = 3n; (iii) 3× 3n = 9n; (iv) 1
99n = n; (v) 2n. Every odd
number is of the form 2n + 1, and the successive operations appliedto this give (i) 6n + 3, which is odd; (ii) 1
2(6n + 3 + 1) = 3n + 2;
(iii) 3(3n + 2) = 9n + 6; (iv) 19(9n + 6) = n + a remainder; (v) 2n + 1.
These results lead to the rule given above.Second Method †. Ask the person who has chosen the number to
perform in succession the following operations. (i) To multiply thenumber by 5. (ii) To add 6 to the product. (iii) To multiply the sumby 4. (iv) To add 9 to the product. (v) To multiply the sum by 5. Askto be told the result of the last operation: if from this product 165 issubtracted, and then the remainder is divided by 100, the quotient willbe the number thought of originally.
For let n be the number selected. Then the successive operationsapplied to it give (i) 5n; (ii) 5n + 6; (iii) 20n + 24; (iv) 20n + 33;(v) 100n + 165. Hence the rule.
Third Method ‡. Request the person who has thought of the num-ber to perform the following operations. (i) To multiply it by anynumber you like, say, a. (ii) To divide the product by any number,
* Bachet, Problemes plaisans, Lyons, 1624, problem i, p. 53.† A similar rule was given by Bachet, problem iv, p. 74.‡ Bachet, problem v, p. 80.
• Project • Gutenberg • #26839 •
CH. I] ELEMENTARY TRICKS AND PROBLEMS. 5
say, b. (iii) To multiply the quotient by c. (iv) To divide this resultby d. (v) To divide the final result by the number selected originally.(vi) To add to the result of operation (v) the number thought of atfirst. Ask for the sum so found: then, if ac/bd is subtracted from thissum, the remainder will be the number chosen originally.
For, if n was the number selected, the result of the first four op-erations is to form nac/bd; operation (v) gives ac/bd; and (vi) givesn + (ac/bd), which number is mentioned. But ac/bd is known; hence,subtracting it from the number mentioned, n is found. Of course a, b,c, d may have any numerical values it is liked to assign to them. Forexample, if a = 12, b = 4, c = 7, d = 3 it is sufficient to subtract 7from the final result in order to obtain the number originally selected.
Fourth Method*. Ask some one to select a number less than 90.(i) Request him to multiply it by 10, and to add any number he pleases,a, which is less than 10. (ii) Request him to divide the result of step (i)by 3, and to mention the remainder, say, b. (iii) Request him to multiplythe quotient obtained in step (ii) by 10, and to add any number hepleases, c, which is less than 10. (iv) Request him to divide the resultof step (iii) by 3, and to mention the remainder, say d, and the thirddigit (from the right) of the quotient; suppose this digit is e. Then,if the numbers a, b, c, d, e are known, the original number can be atonce determined. In fact, if the number is 9x + y, where x ≯ 9 andy ≯ 8, and if r is the remainder when a − b + 3(c − d) is divided by9, we have x = e, y = 9 − r.
The demonstration is not difficult. For if the selected number is9x+y, step (i) gives 90x+10y+a; (ii) let y+a = 3n+b, then the quotientobtained in step (ii) is 30x+3y+n; step (iii) gives 300x+30y+10n+c;(iv) let n + c = 3m + d, then the quotient obtained in step (iv) is100x + 10y + 3n + m, which I will denote by Q. Now the third digitin Q must be x, because, since y ≯ 8 and a ≯ 9, we have n ≯ 5; andsince n ≯ 5 and c ≯ 9, we have m ≯ 4; therefore 10y + 3n + m ≯ 99.Hence the third or hundreds digit in Q is x.
Again, from the relations y + a = 3n + b and n + c = 3m + d,we have 9m − y = a− b + 3(c − d): hence, if r is the remainder whena−b+3(c−d) is divided by 9, we have y = 9−r. [This is always true, ifwe make r positive; but if a−b+3(c−d) is negative, it is simpler to takey as equal to its numerical value; or we may prevent the occurrence of
* Educational Times, London, May 1, 1895, vol. xlviii, p. 234.
• Project • Gutenberg • #26839 •
6 ARITHMETICAL RECREATIONS. [CH. I
this case by assigning proper values to a and c.] Thus x and y are bothknown, and therefore the number selected, namely 9x + y, is known.
Fifth Method*. Ask any one to select a number less than 60.(i) Request him to divide it by 3 and mention the remainder; sup-pose it to be a. (ii) Request him to divide it by 4, and mention theremainder; suppose it to be b. (iii) Request him to divide it by 5, andmention the remainder; suppose it to be c. Then the number selectedis the remainder obtained by dividing 40a + 45b + 36c by 60.
This method can be generalized and then will apply to any numberchosen. Let a′, b′, c′, . . . be a series of numbers prime to one another,and let p be their product. Let n be any number less than p, and leta, b, c, . . . be the remainders when n is divided by a′, b′, c′, . . . respec-tively. Find a number A which is a multiple of the product b′c′d′ · · ·and which exceeds by unity a multiple of a′. Find a number B which isa multiple of a′c′d′ · · · and which exceeds by unity a multiple of b′; andsimilarly find analogous numbers C, D, . . . . Rules for the calculationof A, B, C, . . . are given in the theory of numbers, but in general, if thenumbers a′, b′, c′, . . . are small, the corresponding numbers, A, B, C, . . .can be found by inspection. I proceed to show that n is equal to theremainder when Aa + Bb + Cc + · · · is divided by p.
Let N = Aa+Bb+Cc+· · · , and let M(x) stand for a multiple of x.Now A = M(a′) + 1, therefore Aa = M(a′) + a. Hence, if the first
term in N , that is Aa, is divided by a′, the remainder is a. Again,B is a multiple of a′c′d′ · · · . Therefore Bb is exactly divisible by a′.Similarly Cc,Dd, . . . are each exactly divisible by a′. Thus every termin N , except the first, is exactly divisible by a′. Hence, if N is dividedby a′, the remainder is a. But if n is divided by a′, the remainder is a.
Therefore N − n = M(a′) .
Similarly N − n = M(b′) ,
N − n = M(c′) ,
. . . . . . . . . . . . . . . . . . . . . . .
But a′, b′, c′, . . . are prime to one another.
∴ N − n = M(a′b′c′ · · · ) = M(p) ,
that is, N = M(p) + n .
* Bachet, problem vi, p. 84: Bachet added, on p. 87, a note on the previoushistory of the problem.
• Project • Gutenberg • #26839 •
CH. I] ELEMENTARY TRICKS AND PROBLEMS. 7
Now n is less than p, hence if N is divided by p, the remainder is n.The rule given by Bachet corresponds to the case of a′ = 3, b′ = 4,
c′ = 5, p = 60, A = 40, B = 45, C = 36. If the number chosen isless than 420, we may take a′ = 3, b′ = 4, c′ = 5, d′ = 7, p = 420,A = 280, B = 105, C = 336, D = 120.
To find the result of a series of operations performedon any number (unknown to the questioner) without asking anyquestions. All rules for solving such problems ultimately dependon so arranging the operations that the number disappears from thefinal result. Four examples will suffice.
First Example*. Request some one to think of a number. Supposeit to be n. Ask him (i) to multiply it by any number you please (say)a; (ii) then to add (say) b; (iii) then to divide the sum by (say) c.(iv) Next, tell him to take a/c of the number originally chosen; and(v) to subtract this from the result of the third operation. The resultof the first three operations is (na + b)/c, and the result of operation(iv) is na/c: the difference between these is b/c, and therefore is knownto you. For example, if a = 6, b = 12, c = 4, and a/c = 11
2, then
the final result is 3.Second Example†. Ask A to take any number of counters that he
pleases: suppose that he takes n counters. (i) Ask some one else, sayB, to take p times as many, where p is any number you like to choose.(ii) Request A to give q of his counters to B, where q is any number youlike to select. (iii) Next, ask B to transfer to A a number of countersequal to p times as many counters as A has in his possession. Thenthere will remain in B’s hands q(p+1) counters: this number is knownto you; and the trick can be finished either by mentioning it or in anyother way you like.
The reason is as follows. The result of operation (ii) is that B haspn + q counters, and A has n − q counters. The result of (iii) is thatB transfers p(n− q) counters to A: hence he has left in his possession(pn + q) − p(n − q) counters, that is, he has q(p + 1).
For example, if originally A took any number of counters, then (ifyou chose p equal to 2), first you would ask B to take twice as manycounters as A had done; next (if you chose q equal to 3) you would ask
* Bachet, problem viii, p. 102.† Bachet, problem xiii, p. 123: Bachet presented the above trick in a somewhat
more general form, but one which is less effective in practice.
• Project • Gutenberg • #26839 •
8 ARITHMETICAL RECREATIONS. [CH. I
A to give 3 counters to B; and then you would ask B to give to A anumber of counters equal to twice the number then in A’s possession;after this was done you would know that B had 3(2+1), that is, 9 left.
This trick (as also some of the following problems) may be per-formed equally well with one person, in which case A may stand forhis right hand and B for his left hand.
Third Example. Ask some one to perform in succession the follow-ing operations. (i) Take any number of three digits. (ii) Form a newnumber by reversing the order of the digits. (iii) Find the difference ofthese two numbers. (iv) Form another number by reversing the order ofthe digits in this difference. (v) Add together the results of (iii) and (iv).Then the sum obtained as the result of this last operation will be 1089.
An illustration and the explanation of the rule are given below.
(i) 237 100a + 10b + c
(ii) 732 100c + 10b + a
(iii) 495 100(a− c− 1) + 90 + (10 + c− a)
(iv) 594 100(10 + c− a) + 90 + (a− c− 1)
(v) 1089 900 + 180 + 9
Fourth Example*. The following trick depends on the same prin-ciple. Ask some one to perform in succession the following operations.(i) To write down any sum of money less than £12; the number ofpounds not being the same as the number of pence. (ii) To reversethis sum, that is, to write down a sum of money obtained from it byinterchanging the numbers of pounds and pence. (iii) To find the differ-ence between the results of (i) and (ii). (iv) To reverse this difference.(v) To add together the results of (iii) and (iv). Then this sum willbe £12. 18s. 11d.
* Educational Times Reprints, 1890, vol. liii, p. 78.
• Project • Gutenberg • #26839 •
CH. I] ELEMENTARY TRICKS AND PROBLEMS. 9
For instance, take the sum £10. 17s. 5d.; we have
£. s. d.(i) . . . . . . . . . . . 10 17 5
(ii) . . . . . . . . . . . 5 17 10
(iii) . . . . . . . . . . . 4 19 7
(iv) . . . . . . . . . . . 7 19 4
(v) . . . . . . . . . . . 12 18 11
The following work explains the rule, and shows that the final resultis independent of the sum written down initially.
£. s. d.(i) . . . . . . . . . . . a b c
(ii) . . . . . . . . . . . c b a
(iii) . . . . . . . . . . . a− c− 1 19 c− a + 12
(iv) . . . . . . . . . . . c− a + 12 19 a− c− 1
(v) . . . . . . . . . . . 11 38 11
The rule can be generalized to cover any system of monetary units.
Problems involving Two Numbers. I proceed next to givea couple of examples of a class of problems which involve two numbers.
First Example*. Suppose that there are two numbers, one evenand the other odd, and that a person A is asked to select one of them,and that another person B takes the other. It is desired to knowwhether A selected the even or the odd number. Ask A to multiplyhis number by 2 (or any even number) and B to multiply his by 3(or any odd number). Request them to add the two products togetherand tell you the sum. If it is even, then originally A selected the oddnumber, but if it is odd, then originally A selected the even number.The reason is obvious.
Second Example†. The above rule was extended by Bachet to anytwo numbers, provided they were prime to one another and one of them
* Bachet, problem ix, p. 107.† Bachet, problem xi, p. 113.
• Project • Gutenberg • #26839 •
10 ARITHMETICAL RECREATIONS. [CH. I
was not itself a prime. Let the numbers be m and n, and suppose that nis exactly divisible by p. Ask A to select one of these numbers, and B totake the other. Choose a number prime to p, say q. Ask A to multiplyhis number by q, and B to multiply his number by p. Request themto add the products together and state the sum. Then A originallyselected m or n, according as this result is not or is divisible by p. Forexample, m = 7, n = 15, p = 3, q = 2.
Problems depending on the Scale of Notation. Manyof the rules for finding two or more numbers depend on the fact thatin arithmetic an integral number is denoted by a succession of digits,where each digit represents the product of that digit and a power of ten,and the number is equal to the sum of these products. For example,2017 signifies (2×103)+(0×102)+(1×10)+7; that is, the 2 represents2 thousands, i.e. the product of 2 and 103, the 0 represents 0 hundreds,i.e. the product of 0 and 102; the 1 represents 1 ten, i.e. the product of 1and 10, and the 7 represents 7 units. Thus every digit has a local value.
The application to tricks connected with numbers will be under-stood readily from three illustrative examples.
First Example*. A common conjuring trick is to ask a boy amongthe audience to throw two dice, or to select at random from a box adomino on each half of which is a number. The boy is then told torecollect the two numbers thus obtained, to choose either of them, tomultiply it by 5, to add 7 to the result, to double this result, and lastlyto add to this the other number. From the number thus obtained, theconjurer subtracts 14, and obtains a number of two digits which arethe two numbers chosen originally.
For suppose that the boy selected the numbers a and b. Each ofthese is less than ten—dice or dominoes ensuring this. The successiveoperations give (i) 5a; (ii) 5a+7; (iii) 10a+14; (iv) 10a+14+b. Hence,if 14 is subtracted from the final result, there will be left a number oftwo digits, and these digits are the numbers selected originally. Ananalogous trick might be performed in other scales of notation if it wasthought necessary to disguise the process further.
* Some similar questions were given by Bachet in problem xii, p. 117; by Oughtredin his Mathematicall Recreations (translated from or founded on van Etten’swork of 1633), London, 1653, problem xxxiv; and by Ozanam, part i, chapter x.
• Project • Gutenberg • #26839 •
CH. I] ELEMENTARY TRICKS AND PROBLEMS. 11
Second Example*. Similarly, if three numbers, say, a, b, c, arechosen, then, if each of them is less than ten, they can be found by thefollowing rule. (i) Take one of the numbers, say, a, and multiply it by2. (ii) Add 3 to the product; the result is 2a + 3. (iii) Multiply this by5, and add 7 to the product; the result is 10a + 22. (iv) To this sumadd the second number. (v) Multiply the result by 2. (vi) Add 3 to theproduct. (vii) Multiply by 5, and add the third number to the product.The result is 100a+10b+ c+235. Hence, if the final result is known, itis sufficient to subtract 235 from it, and the remainder will be a numberof three digits. These digits are the numbers chosen originally.
I have seen a similar rule applied to determine the birthday andage of some one in the audience. The result is a number of six digits,of which the first two digits give the day of the month, the middle twodigits the number of the month, and the last two digits the present age.
Third Example†. The following rule for finding a man’s age is ofthe same kind. Take the tens digit of the year of birth; (i) multiply itby 5; (ii) to the product add 2; (iii) multiply the result by 2; (iv) tothis product add the units digit of the birth-year; (v) subtract the sumfrom 110. The result is the man’s age in 1906.
The algebraic proof of the rule is obvious. Let a and b be the tensand units digits of the birth-year. The successive operations give (i) 5a;(ii) 5a + 2; (iii) 10a + 4 (iv) 10a + 4 + b; (v) 106 − (10a + b), whichis his age in 1906. The rule can be easily adapted to give the age inany specified year.
Other Problems with numbers in the denary scale. Imay mention here two or three other slight problems dependent on thecommon scale of notation, which, as far as I am aware, are unknownto most compilers of books of puzzles.
First Problem. The first of them is as follows. Take any numberof three digits: reverse the order of the digits: subtract the number soformed from the original number: then, if the last digit of the differenceis mentioned, all the digits in the difference are known.
For let a be the hundreds digit of the number chosen, b be the tensdigit, and c be the units digit. Therefore the number is 100a + 10b + c.The number obtained by reversing the digits is 100c + 10b + a. The
* Bachet gave some similar questions in problem xii, p. 117.† A similar question was given by Laisant and Perrin in their Algebre, Paris, 1892;
and in L’Illustration for July 13, 1895.
• Project • Gutenberg • #26839 •
12 ARITHMETICAL RECREATIONS. [CH. I
difference of these numbers is equal to (100a+c)−(100c+a), that is, to99(a− c). But a− c is not greater than 9, and therefore the remaindercan only be 99, 198, 297, 396, 495, 594, 693, 792, or 891—in each casethe middle digit being 9 and the digit before it (if any) being equal tothe difference between 9 and the last digit. Hence, if the last digit isknown, so is the whole of the remainder.
Second Problem. The second problem is somewhat similar and isas follows. (i) Take any number; (ii) reverse the digits; (iii) find thedifference between the number formed in (ii) and the given number;(iv) multiply this difference by any number you like to name; (v) crossout any digit except a nought; (vi) read the remainder. Then the sum ofthe digits in the remainder subtracted from the next highest multipleof nine will give the figure struck out.
This follows at once from the fact that the result of operation (iii)—and therefore also of operation (iv)—is necessarily a multiple of nine,and it is known that the sum of the digits of every multiple of nine isitself a multiple of nine.
Miscellaneous Questions. Besides these problems, properly socalled, there are numerous questions on numbers which can be solvedempirically, but which are of no special mathematical interest.
As an instance I may quote a question which attracted some at-tention in London in 1893, and may be enunciated as follows. Withthe seven digits 9, 8, 7, 6, 5, 4, 0 express three numbers whose sumis 82: each digit, being used only once, and the use of the usual no-tations for fractions being allowed. One solution is 80.69 + .74 + .5.Similar questions are with the ten digits, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, toexpress numbers whose sum is unity; a solution is 35/70 and 148/296.If the sum were 100, a solution would be 50, 49, 1/2, and 38/76. Aless straightforward question would be, with the nine digits, 9, 8, 7,6, 5, 4, 3, 2, 1, to express four numbers whose sum is 100; a solutionis 78, 15, 2
√9, and 3
√64.
Problems with a series of things which are numbered.Any collection of things which can be distinguished one from theother—especially if numbered consecutively—afford easy concrete il-lustrations of questions depending on these elementary properties ofnumbers. As examples I proceed to enumerate a few familiar tricks.The first two of these are commonly shown by the use of a watch,the last three are best exemplified by the use of a pack of playing
• Project • Gutenberg • #26839 •
CH. I] ELEMENTARY TRICKS AND PROBLEMS. 13
cards, which readily lend themselves to such illustrations, and I presentthem in these forms.
First Example*. The first of these examples is connected with thehours marked on the face of a watch. In this puzzle some one is askedto think of some hour, say, m, and then to touch a number that marksanother hour, say, n. Then if, beginning with the number touched, hetaps each successive hour marked on the face of the watch, going inthe opposite direction to that in which the hands of the watch move,and reckoning to himself the taps as m, (m + 1), &c., the (n + 12)thtap will be on the hour he thought of. For example, if he thinks of vand touches ix, then, if he taps successively ix, viii, vii, vi, . . . , goingbackwards and reckoning them respectively as 5, 6, 7, 8, . . . , the tapwhich he reckons as 21 will be on the v.
The reason of the rule is obvious, for he arrives finally at the(n + 12 − m)th hour from which he started. Now, since he goes inthe opposite direction to that in which the hands of the watch move,he has to go over (n−m) hours to reach the hour m: also it will makeno difference if in addition he goes over 12 hours, since the only effect ofthis is to take him once completely round the circle. Now (n+12−m)is always positive, since m < 12, and therefore if we make him passover (n+12−m) hours we can give the rule in a form which is equallyvalid whether m is greater or less than n.
Second Example. The following is another well-known way of in-dicating on a watch-dial an hour selected by some one. I do not knowwho first invented it. If the hour is tapped by a pencil beginning atvii and proceeding backwards round the dial to vi, v, &c., and if theperson who selected the number counts the taps, reckoning from thehour selected (thus, if he selected x, he would reckon the first tap as the11th), then the 20th tap as reckoned by him will be on the hour chosen.
For suppose he selected the nth hour. Then the 8th tap is on xiiand is reckoned by him as the (n + 8)th. The tap which he reckons as(n + 9)th is on xi, and generally the tap which he reckons as (n + p)this on the hour (20 − p). Hence, putting p − 20 − n, the tap which hereckons as 20th is on the hour n. Of course the hours indicated by thefirst seven taps are immaterial.
* Bachet, problem xx, p. 155; Oughtred, Mathematicall Recreations, London,1653, p. 28.
• Project • Gutenberg • #26839 •
14 ARITHMETICAL RECREATIONS. [CH. I
Extension. It is obvious that the same trick can be performedwith any collection of m things, such as cards or dominoes, which aredistinguishable one from the other, provided m < 20. For supposethe m things are arranged on a table in some numerical order, andthe nth thing is selected by a spectator. Then the first (19 −m) tapsare immaterial, the (20 −m)th tap must be on the mth thing and bereckoned by the spectator as the (n+20−m)th, the (20−m+1)th tapmust be on the (m−1)th thing and be reckoned as the (n+20−m+1)th,and finally the (20− n)th tap will be on the nth thing and is reckonedas the 20th tap.
Third Example. The following example rests on an extension ofthe method used in the last question; it is very simple, but I have neverseen it previously described in print. Suppose that a pack of n cardsis given to some one who is asked to select one out of the first m cardsand to remember (but not to mention) what is its number from the topof the pack (say it is actually the xth card in the pack). Then takethe pack, reverse the order of the top m cards (which can be easilyeffected by shuffling), and transfer y cards (where y < n − m) fromthe bottom to the top of the pack. The effect of this is that the cardoriginally chosen is now the (y + m − x + 1)th from the top. Returnto the spectator the pack so rearranged, and ask that the top card becounted as the (x+1)th, the next as the (x+2)th, and so on, in whichcase the card originally chosen will be the (y +m+1)th. Now y and mcan be chosen as we please, and may be varied every time the trick isperformed; thus any one unskilled in arithmetic will not readily detectthe modus operandi.
Fourth Example*. Place a card on the table, and on it place asmany other cards from the pack as with the number of pips on thecard will make a total of twelve. For example, if the card placed firston the table is the five of clubs, then seven additional cards must beplaced on it. The court cards may have any values assigned to them,but usually they are reckoned as tens. This is done again with anothercard, and thus another pile is formed. The operation may be repeatedeither only three or four times or as often as the pack will permit ofsuch piles being formed. If finally there are p such piles, and if thenumber of cards left over is r, then the sum of the number of pips onthe bottom cards of all the piles will be 13(p − 4) + r.
* A particular case of this problem was given by Bachet, problem xvii, p. 138.
• Project • Gutenberg • #26839 •
CH. I] MEDIEVAL PROBLEMS IN ARITHMETIC. 15
For, if x is the number of pips on the bottom card of a pile, thenumber of cards in that pile will be 13− x. A similar argument holdsfor each pile. Also there are 52 cards in the pack; and this must beequal to the sum of the cards in the p piles and the r cards left over.
∴ (13− x1) + (13− x2) + · · ·+ (13− xp) + r = 52 ,
∴ 13p− (x1 + x2 + · · ·+ xp) + r = 52 ,
∴ x1 + x2 + · · ·+ xp = 13p− 52 + r
= 13(p− 4) + r .
More generally, if a pack of n cards is taken, and if in each pilethe sum of the pips on the bottom card and the number of cards puton it is equal to m, then the sum of the pips on the bottom cards ofthe piles will be (m + 1)p + r − n. In an ecarte pack n = 32, and itis convenient to take m = 15.
Fifth Example. It may be noticed that cutting a pack of cardsnever alters the relative position of the cards provided that, if necessary,we regard the top card as following immediately after the bottom cardin the pack. This is used in the following trick*. Take a pack, and dealthe cards face upwards on the table, calling them one, two, three, &c. asyou put them down, and noting in your own mind the card first dealt.Ask some one to select a card and recollect its number. Turn the packover, and let it be cut (not shuffled) as often as you like. Enquire whatwas the number of the card chosen. Then, if you deal, and as soon asyou come to the original first card begin (silently) to count, reckoningthis as one, the selected card will appear at the number mentioned. Ofcourse, if all the cards are dealt before reaching this number, you mustturn the cards over and go on counting continuously.
Another similar trick is performed by handing the pack face up-wards to some one, and asking him to select a card and state its num-ber, reckoning from the top; suppose it to be the nth. Next, ask himto choose a number at which it shall appear in the pack; suppose heselects the mth. Take the pack and secretly move m−n cards from thebottom to the top (or if n is greater than m, then n−m from the topto the bottom) and of course the card will be in the required position.
* Bachet, problem xix, p. 152.
• Project • Gutenberg • #26839 •
16 ARITHMETICAL RECREATIONS. [CH. I
Medieval Problems in Arithmetic. Before leaving the sub-ject of these elementary questions, I may mention a few problems whichfor centuries have appeared in nearly every collection of mathemati-cal recreations, and therefore may claim what is almost a prescriptiveright to a place here.
First Example*. The following is a sample of one class of thesepuzzles. Three men robbed a gentleman of a vase, containing 24 ouncesof balsam. Whilst running away they met in a wood with a glass-seller,of whom in a great hurry they purchased three vessels. On reachinga place of safety they wished to divide the booty, but they found thattheir vessels contained 5, 11, and 13 ounces respectively. How couldthey divide the balsam into equal portions?
Problems like this can be worked out only by trial: there are severalsolutions, of which one is as follows.
The vessels can contain . . . . . . 24 oz. 13 oz. 11 oz. 5 oz.Their contents originally are 24 . . . 0 . . . 0 . . . 0 . . .First, make their contents . . . 0 . . . 8 . . . 11 . . . 5 . . .Second, ” ” . . . 16 . . . 8 . . . 0 . . . 0 . . .Third, ” ” . . . 16 . . . 0 . . . 8 . . . 0 . . .Fourth, ” ” . . . 3 . . . 13 . . . 8 . . . 0 . . .Fifth, ” ” . . . 3 . . . 8 . . . 8 . . . 5 . . .Sixth, ” ” . . . 8 . . . 8 . . . 8 . . . 0 . . .
Second Example†. The next of these is a not uncommon game,played by two people, say A and B. A begins by mentioning somenumber not greater than (say) six, B may add to that any number notgreater than six, A may add to that again any number not greater thansix, and so on. He wins who is the first to reach (say) 50. Obviously, if Acalls 43, then whatever B adds to that, A can win next time. Similarly,if A calls 36, B cannot prevent A’s calling 43 the next time. In thisway it is clear that the key numbers are those forming the arithmeticalprogression 43, 36, 29, 22, 15, 8, 1; and whoever plays first ought to win.
Similarly, if no number greater than m may be added at any onetime, and n is the number to be called by the victor, then the key num-
* Some similar problems were given by Bachet, appendix, problem iii, p. 206;problem ix, p. 233; by Oughtred in his Recreations, p. 22: and by Ozanam, 1803edition, vol. i, p. 174; 1840 edition, p. 79. Earlier instances occur in Tartaglia’swritings.
† Bachet, problem xxii, p. 170.
• Project • Gutenberg • #26839 •
CH. I] MEDIEVAL PROBLEMS IN ARITHMETIC. 17
bers will be those forming the arithmetical progression whose commondifference is m + 1 and whose smallest term is the remainder obtainedby dividing n by m + 1.
The same game may be played in another form by placing n coins,matches, or other objects on a table, and directing each player in turnto take away not more than m of them. Whoever takes away the lastcoin wins. Obviously the key numbers are multiples of m + 1, and thefirst player who is able to leave an exact multiple of (m + 1) coins canwin. Perhaps a better form of the game is to make that player losewho takes away the last coin, in which case each of the key numbersexceeds by unity a multiple of m + 1.
Mr Loyd has also suggested* a modification which is equivalent toplacing n counters in the form of a circle, and allowing each player insuccession to take away not more than m of them which are in unbrokensequence: m being less than n and greater than unity. In this case thesecond of the two players can always win.
Recent Extension of this Problem. The games last described arevery simple, but if we impose on the original problem the additionalrestriction that each player may not add the same number more thanthree times, the analysis becomes by no means easy. It is difficult in thiscase to say whether it is an advantage to begin or not. I have neverseen this extension described in print, and I will therefore enunciateit at length.
Suppose that each player is given eighteen cards, three of themmarked 6, three marked 5, three marked 4, three marked 3, threemarked 2, and three marked 1. They play alternately; A begins byplaying one of his cards; then B plays one of his, and so on. He winswho first plays a card which makes the sum of the points or numberson all the cards played exactly equal to 50, but he loses if he plays acard which makes this sum exceed 50. The game can be played men-tally or by noting the numbers on a piece of paper, and in practice itis unnecessary to use cards.
Thus, if they play as follows A, 4; B, 3; A, 1; B, 6; A, 3; B, 4;A, 4; B, 5; A, 4; B, 4; A, 5; the game stands at 43. B can now win, forhe may safely play 3, since A has not another 4 wherewith to follow it;and if A plays less than 4, B will win the next time. Again, if they play
* Tit-Bits, London, July 17, Aug. 7, 1897.
• Project • Gutenberg • #26839 •
18 ARITHMETICAL RECREATIONS. [CH. I
thus, A, 6; B, 3; A, 1; B, 6; A, 3; B, 4; A, 2; B, 5; A, 1; B, 5; A, 2;B, 5; A, 2; B, 3; A is now forced to play 1, and B wins by playing 1.
The game can be also played if each player is given only two cardsof each kind.
Third Example. The following medieval problem is somewhatmore elaborate. Suppose that three people, P , Q, R, select threethings, which we may denote by a, e, i, respectively, and that it isdesired to find by whom each object was selected*.
Place 24 counters on a table. Ask P to take one counter, Q to taketwo counters, and R to take three counters. Next, ask the person whoselected a to take as many counters as he has already, whoever selectede to take twice as many counters as he has already, and whoever selectedi to take four times as many counters as he has already. Note how manycounters remain on the table. There are only six ways of distributingthe three things among P , Q, and R; and the number of countersremaining on the table is different for each way. The remainders maybe 1, 2, 3, 5, 6, or 7.
Bachet summed up the results in the mnemonic line Par fer (1)Cesar (2) jadis (3) devint (5) si grand (6) prince (7). Corresponding toany remainder is a word or words containing two syllables: for instance,to the remainder 5 corresponds the word devint. The vowel in thefirst syllable indicates the thing selected by P , the vowel in the secondsyllable indicates the thing selected by Q, and of course R selected theremaining thing. Salve certa animae semita vita quies was suggestedby Oughtred† as an alternative mnemonic line.
Extension. M. Bourlet, in the course of a very kindly notice‡ ofthe second edition of this work, has given a much neater solution of theabove question, and has extended the problem to the case of n people,P0, P1, P2, . . . , Pn−1, each of whom selects one object, out of a collectionof n objects, such as dominoes or cards. It is required to know whichdomino or card was selected by each person.
Let us suppose the dominoes to be denoted or marked by the num-bers 0, 1, . . . , n − 1, instead of by vowels. Give one counter to P1, twocounters to P2, and generally k counters to Pk. Note the number ofcounters left on the table. Next ask the person who had chosen the
* Bachet, problem xxv, p. 187.† Mathematicall Recreations, London, 1653, p. 20.‡ Bulletin des sciences mathematiques, Paris, 1893, vol. xvii, pp. 105–107.
• Project • Gutenberg • #26839 •
CH. I] MEDIEVAL PROBLEMS IN ARITHMETIC. 19
domino 0 to take as many counters as he had already, and generallywhoever had chosen the domino h to take nh times as many dominoesas he had already: thus if Pk had chosen the domino numbered h, hewould take nhk counters. Note the total number of counters taken,i.e.
∑nhk. Divide it by n, then the remainder will be the number on
the domino selected by P0; divide the quotient by n, and the remainderwill be the number on the domino selected by P1; divide this quotientby n, and the remainder will be the number on the domino selectedby P2; and so on. In other words, if the number of counters taken isexpressed in the scale of notation whose radix is n, then the (h + 1)thdigit from the right will give the number on the domino selected by Ph.
Thus in Bachet’s problem with 3 people and 3 dominoes, we shouldfirst give one counter to Q, and two counters to R, while P would haveno counters; then we should ask the person who selected the dominomarked 0 or a to take as many counters as he had already, whoeverselected the domino marked 1 or e to take three times as many countersas he had already, and whoever selected the domino marked 2 or i totake nine times as many counters as he had already. By noticing theoriginal number of counters, and observing that 3 of these had beengiven to Q and R, we should know the total number taken by P , Q,and R. If this number were divided by 3, the remainder would be thenumber of the domino chosen by P ; if the quotient were divided by 3the remainder would be the number of the domino chosen by Q; andthe final quotient would be the number of the domino chosen by R.
I may add that Bachet also discussed the case when n = 4, whichhad been previously considered by Diego Palomino in 1599, but asM. Bourlet’s method is general, it is unnecessary to discuss furtherparticular cases.
Decimation. The last of these antique problems to which I referredconsists in placing men round a circle so that if every nth man is killedthe remainder shall be certain specified individuals. When decimationwas a not uncommon punishment a knowledge of this kind may havehad practical interest.
Hegesippus* says that Josephus saved his life by such a device.According to his account, after the Romans had captured Jotopat,Josephus and forty other Jews took refuge in a cave. Josephus, muchto his disgust, found that all except himself and one other man were
* De Bello Judaico, bk. iii, chaps. 16–18.
• Project • Gutenberg • #26839 •
20 ARITHMETICAL RECREATIONS. [CH. I
resolved to kill themselves, so as not to fall into the hands of theirconquerors. Fearing to show his opposition too openly he consented,but declared that the operation must be carried out in an orderly way,and suggested that they should arrange themselves round a circle andthat every third person should be killed until but one man was left,who must then commit suicide. It is alleged that he placed himself andthe other man in the 31st and 16th place respectively, with a resultwhich will be easily foreseen.
The question is usually presented in the following form. A ship,carrying as passengers fifteen Turks and fifteen Christians, encountereda storm, and the pilot declared that, in order to save the ship and crew,one-half of the passengers must be thrown into the sea. To choose thevictims the passengers were placed round a circle, and it was agreedthat every ninth man should be cast overboard, reckoning from a certainpoint. It is desired to find an arrangement by which all the Christiansshould be saved.*
Problems like this can be easily solved by counting, but it is im-possible to give a general rule. In this case, the Christians, reckoningfrom the man first counted, must occupy the places 1, 2, 3, 4, 10, 11,13, 14, 15, 17, 20, 21, 25, 28, 29. This arrangement can be recollectedby the positions of the vowels in the following doggerel rhyme,
From numbers’ aid and art, never will fame depart,
where a stands for 1, e for 2, i for 3, o for 4, and u for 5. Hence(looking only at the vowels in the verse) the order is 4 Christians, 5Turks, 2 Christians, 1 Turk, 3 Christians, 1 Turk, 1 Christian, 2 Turks,2 Christians, 3 Turks, 1 Christian, 2 Turks, 2 Christians, 1 Turk. Othersimilar mnemonic lines in French and in Latin were given by Bachetand by Ozanam respectively.
Arithmetical Fallacies. I insert next some instances ofdemonstrations† leading to arithmetical results which are obviously
* Bachet, problem xxiii, p. 174. The same problem had been previously enunci-ated by Tartaglia.
† Of the fallacies given in the text, the first, second, and third, are well known;the fourth is not new, but the earliest work in which I recollect seeing it is myAlgebra, Cambridge, 1890, p. 430; the fifth is given in G.C. Chrystal’s Algebra,
• Project • Gutenberg • #26839 •
CH. I] ARITHMETICAL FALLACIES. 21
impossible. I include algebraical proofs as well as arithmetical ones.The fallacies are so patent that in preparing the first and secondeditions I did not think such questions worth printing, but, as somecorrespondents have expressed a contrary opinion, I give them forwhat they are worth.
First fallacy. One of the oldest of these—and not a very interest-ing specimen—is as follows. Suppose that a = b, then
ab = a2 .
∴ ab− b2 = a2 − b2 .
∴ b(a− b) = (a + b)(a− b) .
∴ b = a + b .
∴ b = 2b .
∴ 1 = 2 .
Second Fallacy. Another instance, almost as puerile, is as follows.Let a and b be two unequal numbers, and let c be their arithmeticmean, hence
a + b = 2c .
∴ (a + b)(a− b) = 2c(a− b) .
∴ a2 − 2ac = b2 − 2bc .
∴ a2 − 2ac + c2 = b2 − 2bc + c2 .
∴ (a− c)2 = (b− c)2 .
∴ a = b .
Edinburgh, 1889, vol. ii, p. 159; the eighth is due to G.T. Walker, and, as faras I know, has not appeared in any other book; the ninth is due to D’Alembert;and the tenth to F. Galton. A mechanical demonstration that 1 = 2 was givenby R. Chartres in Knowledge, July, 1891. J.L.F. Bertrand pointed out that ademonstration that 1 = −1 can be also obtained from the proposition in theIntegral Calculus that, if the limits are constant, the order of integration isindifferent; hence the integral to x (from x = 0 to x = 1) of the integral toy (from y = 0 to y = 1) of a function ϕ should be equal to the integral to y(from y = 0 to y = 1) of the integral to x (from x = 0 to x = 1) of ϕ, but ifϕ = (x2 − y2)/(x2 + y2)2, this gives 1
4π = − 14π.
• Project • Gutenberg • #26839 •
22 ARITHMETICAL RECREATIONS. [CH. I
Third Fallacy. Another example, the idea of which is due to JohnBernoulli, may be stated as follows.
We have (−1)2 = 1 .
Take logarithms, ∴ 2 log(−1) = log 1 = 0 .
∴ log(−1) = 0 .
∴ −1 = e0 .
∴ −1 = 1 .
The same argument may be expressed thus. Let x be a quantitywhich satisfies the equation
ex = −1 .
Square both sides, ∴ e2x = 1 .
∴ 2x = 0 .
∴ x = 0 .
∴ ex = e0 .
But ex = −1 and e0 = 1, ∴ −1 = 1 .
Fourth Fallacy. As yet another instance, we know that
log(1 + x) = x− 12x2 + 1
3x3 − · · · .
If x = 1, the resulting series is convergent; hence we have
log 2 = 1− 12
+ 13− 1
4+ 1
5− 1
6+ 1
7− 1
8+ 1
9− · · · .
∴ 2 log 2 = 2− 1 + 23− 1
2+ 2
5− 1
3+ 2
7− 1
4+ 2
9− · · · .
Taking those terms together which have a common denominator, weobtain
2 log 2 = 1 +1
3− 1
2+
1
5+
1
7− 1
4+
1
9− · · ·
= 1− 1
2+
1
3− 1
4+
1
5− · · ·
= log 2 .
Hence 2 = 1 .
• Project • Gutenberg • #26839 •
CH. I] ARITHMETICAL FALLACIES. 23
Fifth Fallacy. This fallacy is very similar to that last given. Wehave
log 2 = 1− 12
+ 13− 1
4+ 1
5− 1
6+ · · ·
=(1 + 1
3+ 1
5+ · · ·
)−
(12
+ 14
+ 16
+ · · ·)
={(
1 + 13
+ 15
+ · · ·)
+(
12
+ 14
+ 16
+ · · ·)}
− 2(
12
+ 14
+ 16
+ · · ·)
={1 + 1
2+ 1
3+ · · ·
}−
(1 + 1
2+ 1
3+ · · ·
)= 0 .
The error in each of the foregoing examples is obvious, but thefallacies in the next examples are concealed somewhat better.
Sixth Fallacy. We can write the identity√−1 =
√−1 in the form√
−1
1=
√1
−1,
hence
√−1√1
=
√1√−1
,
therefore (√−1)2 = (
√1)2 ,
that is, −1 = 1 .
Seventh Fallacy. Again, we have√
a×√
b =√
ab .
Hence√−1×
√−1 =
√(−1)(−1) ,
therefore (√−1)2 =
√1 ,
that is, −1 = 1 .
Eighth Fallacy. The following demonstration depends on the factthat an algebraical identity is true whatever be the symbols used in it,and it will appeal only to those who are familiar with this fact.
We have, as an identity,√
x− y = i√
y − x . . . . . . . (i),
where i stands either for +√−1 or for −
√−1. Now an identity in x
and y is necessarily true whatever numbers x and y may represent.First put x = a and y = b,
∴√
a− b = i√
b− a . . . . . . . (ii).
• Project • Gutenberg • #26839 •
24 ARITHMETICAL RECREATIONS. [CH. I
Next put x = b and y = a,
∴√
b− a = i√
a− b . . . . . . (iii).
Also since (i) is an identity, it follows that in (ii) and (iii) the symbol imust be the same, that is, it represents +
√−1 or −
√−1 in both cases.
Hence, from (ii) and (iii), we have
√a− b
√b− a = i2
√b− a
√a− b ,
∴ 1 = i2 ,
that is 1 = −1 .
Ninth Fallacy. The following fallacy is due to D’Alembert*. Weknow that if the product of two numbers is equal to the product oftwo other numbers, the numbers will be in proportion, and from thedefinition of a proportion it follows that if the first term is greater thanthe second, then the third term will be greater than the fourth: thus,if ad = bc, then a : b = c : d, and if in this proportion a > b, thenc > d. Now if we put a = d = 1 and b = c = −1 we have four numberswhich satisfy the relation ad = bc and such that a > b; hence, by theproposition, c > d, that is, −1 > 1, which is absurd.
Tenth Fallacy. The mathematical theory of probability leads tovarious paradoxes: of these one specimen† will suffice. Suppose threecoins to be thrown up and the fact whether each comes down head ortail to be noticed. The probability that all three coins come down headis clearly (1
2)3, that is, is 1
8; similarly the probability that all three come
down tail is 18: hence the probability that all the coins come down alike
(i.e. either all of them heads or all of them tails) is 14. But, of three coins
thus thrown up, at least two must come down alike; now the probabilitythat the third coin comes down head is 1
2and the probability that it
comes down tail is 12, thus the probability that it comes down the same
as the other two coins is 12: hence the probability that all the coins
come down alike is 12. I leave to my readers to say whether either of
these conflicting conclusions is right and if so, which.Arithmetical Problems. To the above examples I may add the fol-
lowing questions, which I have often propounded in past years: thoughnot fallacies, they may serve to illustrate the fact that the answer to
* Opuscules mathematiques, Paris, 1761, vol. i, p. 201.† See Nature, Feb. 15, March 1, 1894, vol. xlix, pp. 365–366, 413.
• Project • Gutenberg • #26839 •
CH. I] ARITHMETICAL FALLACIES. 25
an arithmetical question is frequently different to what a hasty readermight suppose.
The first of these questions is as follows. Two clerks are engaged,one at a salary commencing at the rate of (say) £100 a year with a riseof £20 every year, the other at a salary commencing at the same rate(£100 a year) with a rise of £5 every half-year, in each case paymentsbeing made half-yearly: which has the larger income? The answer isthe latter; for in the first year the first clerk receives £100, but thesecond clerk receives £50 and £55 as his two half-yearly payments andthus receives in all £105. In the second year the first clerk receives£120, but the second clerk receives £60 and £65 as his two half-yearlypayments and thus receives in all £125. In fact the second clerk willalways receive £5 a year more than the first clerk.
As another question take the following. A man bets 1/nth of hismoney on an even chance (say tossing heads or tails with a penny): herepeats this again and again, each time betting 1/nth of all the moneythen in his possession. If, finally, the number of times he has won isequal to the number of times he has lost, has he gained or lost by thetransaction? He has, in fact, lost.
Here is another simple question to which not unfrequently I havereceived incorrect answers. One tumbler is half-full of wine, another ishalf-full of water: from the first tumbler a teaspoonful of wine is takenout and poured into the tumbler containing the water: a teaspoonfulof the mixture in the second tumbler is then transferred to the firsttumbler. As the result of this double transaction, is the quantity ofwine removed from the first tumbler greater or less than the quantityof water removed from the second tumbler? Nineteen people out oftwenty will say it is greater, but this is not the case.
Routes on a Chess-Board. A not uncommon problem can be gen-eralised as follows*. Construct a rectangular board of mn cells (or smallsquares) by ruling m + 1 vertical lines and n + 1 horizontal lines. It isrequired to know how many routes can be taken from the top left-handcorner to the bottom right-hand corner, the motion being along theruled lines and its direction being always either vertically downwardsor horizontally from left to right. The answer is (m+n)!/m!n!: thus ona square board containing 16 cells (i.e. one-quarter of a chess-board),
* The substance of the problem was given in a scholarship paper set at Cambridgeabout 30 years ago, and possibly was not new then.
• Project • Gutenberg • #26839 •
26 ARITHMETICAL RECREATIONS. [CH. I
where m = n = 4, there are 70 such routes; while on a common chess-board, where m = n = 8, there are no less than 12870 such routes. Asimilar theorem can be enunciated for a parallelopiped.
Another problem of a somewhat similar type is the determinationof the number of closed routes through mn points arranged in m rowsand n columns, following the lines of the quadrilateral net-work, andpassing once and only once through each point*.
Permutation Problems. As other simple illustrations of the verylarge number of ways in which combinations of even a few things canbe arranged, I may note that as many as 19, 958400 distinct skele-ton cubes can be formed with twelve differently coloured rods of equallength†; again there are 3, 979614, 965760 ways of arranging a set oftwenty-eight dominoes (i.e. a set from double zero to double six) ina line, with like numbers in contact‡; while there are no less than53644, 737765, 488792, 839237, 440000 possible different distributions ofhands at whist with a pack of fifty-two cards§.
Voting Problems. Here is a simple example on combinations deal-ing with the cumulative vote as affecting the representation of a mi-nority. If there are p electors each having r votes of which not morethan s may be given to one candidate, and n men are to be elected,then the least number of supporters who can secure the election of acandidate must exceed pr/(ns + r).
Exploration Problems. Another common question is concernedwith the maximum distance into a desert which could be reached froma frontier settlement by the aid of a party of n explorers, each capa-ble of carrying provisions that would last one man for a days. Theanswer is that the man who reaches the greatest distance will occupyna/(n+1) days before he returns to his starting point. If in the courseof their journey they may make depots, the longest possible journeywill occupy 1
2a(1 + 1
2+ 1
3+ · · ·+ 1/n) days. Further extensions by the
use of horses and cycles will suggest themselves.
Here I conclude my account of such of these easy problems on
* See C.F. Sainte-Marie in L’Intermediaire des mathematiciens, Paris, vol. xi,March, 1904, pp. 86–88.
† Mathematical Tripos, Cambridge, Part I, 1894.‡ Reiss in Annali di matematica, Milan, November, 1871, vol. v, pp. 63–120.§ That is (52!)/(13!)4.
• Project • Gutenberg • #26839 •
CH. I] BACHET’S WEIGHTS PROBLEM. 27
numbers or elementary algebra as seemed worth reproducing. It willbe noticed that the majority of them either are due to Bachet or werecollected by him in his classical Problemes ; but it should be addedthat besides the questions I have mentioned he enunciated, even if hedid not always solve, some other problems of greater interest. Oneinstance will suffice.
Bachet’s Weights Problem*. Among the more difficultproblems proposed by Bachet was the determination of the least num-ber of weights which would serve to weigh any integral number ofpounds from 1 lb. to 40 lbs. inclusive. Bachet gave two solutions:namely, (i) the series of weights of 1, 2, 4, 8, 16, and 32 lbs.; (ii) theseries of weights of 1, 3, 9, and 27 lbs.
If the weights may be placed in only one of the scale-pans, the firstseries gives a solution, as had been pointed out in 1556 by Tartaglia†.
Bachet, however, assumed that any weight might be placed in eitherof the scale-pans. In this case the second series gives the least possiblenumber of weights required. His reasoning is as follows. To weigh 1 lb.we must have a 1 lb. weight. To weigh 2 lbs. we must have in additioneither a 2 lb. weight or a 3 lb. weight; but, if we are confined to only onenew weight (in addition to the 1 lb. we have got already), then with noweight greater than 3 lbs. could we weigh 2 lbs.: if we use a 2 lb. weightwe then can weigh 1 lb., 2 lbs., and 3 lbs., but if we use a 3 lb. weightwe then can weigh 1 lb., (3 − 1) lbs., 3 lbs., and (3 + 1) lbs.; hencea 3 lb. weight is preferable. Similarly, to enable us to weigh 5 lbs. wemust have another weight not greater than 9 lbs., and a weight of 9 lbs.enables us to weigh every weight from 1 lb. to 13 lbs.; hence it is thebest to choose. The next weight required will be 2(1+3+9)+1 lb., thatis, will be 27 lbs.; and this enables us to weigh from 1 lb. to 40 lbs. Thusonly four weights are required, namely, 1 lb., 3 lbs., 32 lbs., and 33 lbs.
We can show similarly that the series of weights of 1, 3, 32, . . . ,3n−1 lbs. will enable us to weigh any integral number of pounds from1 lb. to (1 + 3 + 32 + · · · 3n−1) lbs., that is, to 1
2(3n − 1) lbs. This is the
least number with which the problem can be effected.To determine the arrangement of the weights to weigh any given
mass we have only to express the number of pounds in it as a numberin the ternary scale of notation, except that in finding the successive
* Bachet, Appendix, problem v, p. 215.† Trattato de’ numeri e misure, Venice, 1556, vol. ii, bk. i, chap. xvi, art. 32.
• Project • Gutenberg • #26839 •
28 ARITHMETICAL RECREATIONS. [CH. I
digits we must make every remainder either 0, 1, or −1: to effect thisa remainder 2 must be written as 3 − 1, that is, the quotient mustbe increased by unity, in which case the remainder is −1. This isexplained in most text-books on algebra.
Bachet’s argument does not prove that his result is unique or thatit gives the least possible number of weights required. These omissionshave been supplied by Major MacMahon, who has discussed the farmore difficult problem (of which Bachet’s is a particular case) of thedetermination of all possible sets of weights, not necessarily unequal,which enable us to weigh any integral number of pounds from 1 to ninclusive, (i) when the weights may be placed in only one scale-pan,and (ii) when any weight may be placed in either scale-pan. He hasinvestigated also the modifications of the results which are necessarywhen we impose either or both of the further conditions (a) that noother weighings are to be possible, and (b) that each weighing is to bepossible in only one way, that is, is to be unique*.
The method for case (i) consists in resolving 1 + x + x2 + · · ·+ xn
into factors, each factor being of the form 1 + xa + x2a + · · ·+ xma; thenumber of solutions depends on the composite character of n + 1. Themethod for case (ii) consists in resolving the expression x−n+x−n+1+· · ·+ x−1 + 1 + x + · · · + xn−1 + xn into factors, each factor being of theform x−ma + · · · + x−a + 1 + xa + · · · + xma; the number of solutionsdepends on the composite character of 2n + 1.
Bachet’s problem falls under case (ii), n = 40. MacMahon’s anal-ysis shows that there are eight such ways of factorizing x−40 + x−39 +· · ·+ 1 + x39 + x40. First, there is the expression itself in which a = 1,m = 40. Second, the expression is equal to (1 − x81)/x40(1 − x),which can be resolved into the product of (1 − x3)/x(1 − x) and(1 − x81)/x39(1 − x3); hence it can be resolved into two factors of theform given above, in one of which a = 1, m = 1, and in the other,a = 3, m = 13. Third, similarly, it can be resolved into two suchfactors, in one of which a = 1, m = 4, and in the other a = 9, m = 4.Fourth, it can be resolved into three such factors, in one of which a = 1,m = 1, in another a = 3, m = 1, and in the other, a = 9, m = 4. Fifth,it can be resolved into two such factors, in one of which a = 1, m = 13,
* See his article in the Quarterly Journal of Mathematics, 1886, vol. xxi, pp. 367–373. An account of the method is given in Nature, Dec. 4, 1890, vol. xlii,pp. 113–114.
• Project • Gutenberg • #26839 •
CH. I] PROBLEMS IN HIGHER ARITHMETIC. 29
and in the other a = 27, m = 1. Sixth, it can be resolved into threesuch factors, in one of which a = 1, m = 1, in another a = 3, m = 4,and in the other a = 27, m = 1. Seventh, it can be resolved into threesuch factors, in one of which a = 1, m = 4, in another a = 9, m = 1,and in the other a = 27, m = 1. Eighth, it can be resolved into foursuch factors, in one of which a = 1, m = 1, in another a = 3, m = 1,in another a = 9, m = 1, and in the other a = 27, m = 1.
These results show that there are eight possible sets of weightswith which any integral number of pounds from 1 to 40 can be weighedsubject to the conditions (ii), (a), and (b). If we denote p weights eachequal to w by wp, these eight solutions are 140; 1, 313; 14, 94; 1, 3, 94; 113,27; 1, 34, 27; 14, 9, 27; 1, 3, 9, 27. The last of these is Bachet’s solution:not only is it that in which the least number of weights are employed,but it is also the only unique one in which all the weights are unequal.
Problems in Higher Arithmetic. At the commencement ofthis chapter I alluded to the special interest which many mathemati-cians find in the theorems of higher arithmetic: such, for example,as that every prime of the form 4n + 1 and every power of it is ex-pressible as the sum of two squares*, and the first and second powerscan be expressed thus in only one way. For instance, 13 = 32 + 22,132 = 122 + 52, 133 = 462 + 92, and so on. Similarly 41 = 52 + 42,412 = 402 + 92, 413 = 2362 + 1152, and so on.
Propositions such as the one just quoted may be found in text-books on the theory of numbers and therefore lie outside the limitsof this work, but there are one or two questions in higher arithmeticinvolving points not yet quite cleared up which may find a place here.
Primes. The first of these is concerned with the possibility ofdetermining readily whether a given number is prime or not. Euler andGauss attached great importance to this problem, but failed to establishany conclusive test. It would seem, however, that Fermat possessedsome means of finding from its form whether a given number (at anyrate if one of certain known forms) was prime or not. Thus, in answer toMersenne who asked if he could tell without much trouble whether thenumber 100895, 598169 was a prime, Fermat wrote on April 7, 1643,that it was the product of 898423 and 112303, both of which were
* Fermat’s Diophantus, Toulouse, 1670, bk. iii, prop. 22, p. 127; or Brassinne’sPrecis, Paris, 1853, p. 65.
• Project • Gutenberg • #26839 •
30 ARITHMETICAL RECREATIONS. [CH. I
primes. I have indicated elsewhere one way by which this result can befound, and Mr F.W. Laurence has indicated another which may havebeen that used by Fermat in this particular case.
Mersenne’s Numbers*. Another illustration, confirmatory ofthe opinion that Fermat or some of his contemporaries had a test bywhich it was possible to find out whether certain numbers were prime,may be drawn from Mersenne’s Cogitata Physico-Mathematica whichwas published in 1644. In the preface to that work it is asserted that inorder that 2p − 1 may be prime, the only values of p, not greater than257, which are possible are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and257: I conjecture that the number 67 is a misprint for 61. With thiscorrection the statement appears to be true, and it has been verifiedfor all except nineteen values of p: namely, 71, 101, 103, 107, 109,137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, and257. Of these values, Mersenne asserted that p = 257 makes 2p − 1a prime, and that the other values make 2p − 1 a composite number.The demonstrations for the cases when p = 89, 127 have not beenpublished; nor have the actual factors of 2p − 1 when p = 89 been asyet determined: the discovery of these factors may be commended tothose interested in the theory of numbers.
Mersenne’s result could not be obtained empirically, and it is im-possible to suppose that it was worked out for every case; hence itwould seem that whoever first enunciated it was acquainted with cer-tain theorems in higher arithmetic which have not been re-discovered.
Perfect Numbers†. The theory of perfect numbers dependsdirectly on that of Mersenne’s Numbers. A number is said to be per-fect if it is equal to the sum of all its integral subdivisors. Thus thesubdivisors of 6 are 1, 2, and 3; the sum of these is equal to 6; hence6 is a perfect number.
It is probable that all perfect numbers are included in the formula2p−1(2p − 1), where 2p − 1 is a prime. Euclid proved that any num-ber of this form is perfect; Euler showed that the formula includes alleven perfect numbers; and there is reason to believe—though a rigiddemonstration is wanting—that an odd number cannot be perfect. If
* For references, see chapter ix below.† On the theory of perfect numbers, see bibliographical references by H. Brocard,
L’Intermediaire des mathematiciens, Paris, 1895, vol. ii, pp. 52–54; and 1905,vol. xii, p. 19.
• Project • Gutenberg • #26839 •
CH. I] GOLDBACH’S THEOREM. 31
we assume that the last of these statements is true, then every perfectnumber is of the above form. It is easy to establish that every numberincluded in this formula (except when p = 2) is congruent to unity tothe modulus 9, that is, when divided by 9 leaves a remainder 1; alsothat either the last digit is a 6 or the last two digits are 28.
Thus, if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then by Mersenne’s rulethe corresponding values of 2p − 1 are prime; they are 3, 7, 31, 127,8191, 131071, 524287, 2147483647, 2305843009213693951; and the cor-responding perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056,137438691328, 2305843008139952128, and2658455991569831744654692615953842176.
Goldbach’s Theorem. Another interesting problem in higherarithmetic is the question whether there are any even integers whichcannot be expressed as a sum of two primes. Probably there are none.The expression of all even1 integers not greater than 5000 in the formof a sum of two primes has been effected*, but a general demonstrationthat all even integers can be so expressed is wanting.
Lagrange’s Theorem†. Another theorem in higher arith-metic which, as far as I know, is still unsolved, is to the effect thatevery prime of the form 4n− 1 is the sum of a prime of the form 4n+1and of double a prime of the form 4n+1; for example, 23 = 13+2× 5.Lagrange, however, added that it was only by induction that he ar-rived at the result.
Fermat’s Theorem on Binary Powers. Fermat enrichedmathematics with a multitude of new propositions. With two excep-tions all these have been proved subsequently to be true. The first ofthese exceptions is his theorem on binary powers, in which he assertedthat all numbers of the form 2m +1, where m = 2n, are primes‡, but headded that, though he was convinced of the truth of this proposition,he could not obtain a valid demonstration.
* Transactions of the Halle Academy (Naturforschung), vol. lxxii, Halle, 1897,pp. 5–214: see also L’Intermediaire des mathematiciens, 1903, vol. x, and 1904,vol. xi.
† Nouveaux Memoires de l’Academie Royale des Sciences, Berlin, 1775, p. 356.‡ Letter of Oct. 18, 1640, Opera, Toulouse, 1679, p. 162: or Brassinne’s Precis,
p. 143.
1. ‘even’ inserted as per errata sheet
• Project • Gutenberg • #26839 •
32 ARITHMETICAL RECREATIONS. [CH. I
It may be shown that 2m + 1 is composite if m is not a power of2, but of course it does not follow that 2m + 1 is a prime if m is apower of 2. As a matter of fact the theorem is not true. In 1732 Euler*
showed that if n = 5 the formula gives 4294, 967297, which is equal to641×6, 700417: curiously enough, these factors can be deduced at oncefrom Fermat’s remark on the possible factors of numbers of the form2m ± 1, from which it may be shown that the prime factors (if any) of232 + 1 must be primes of the form 64n + 1.
During the last thirty years it has been shown† that the resultingnumbers are composite when n = 6, 9, 11, 12, 18, 23, 36, and 38:the two last numbers contain many thousands of millions of digits. Ibelieve that Eisenstein asserted that the number of primes of the form2m + 1, where m = 2n, is infinite: the proof has not been published,but perhaps it might throw some light on the general theory.
Fermat’s Last Theorem. I pass now to the only other asser-tion made by Fermat which has not been proved hitherto. This, which issometimes known as Fermat’s Last Theorem, is to the effect‡ that no in-tegral values of x, y, z can be found to satisfy the equation xn+yn = zn,if n is an integer greater than 2. This proposition has acquired extraor-dinary celebrity from the fact that no general demonstration of it hasbeen given, but there is no reason to doubt that it is true.
Fermat seems to have discovered its truth first§ for the case n = 3,and then for the case n = 4. His proof for the former of these cases islost, but that for the latter is extant‖, and a similar proof for the case ofn = 3 was given by Euler¶. These proofs depend upon showing that, ifthree integral values of x, y, z can be found which satisfy the equation,
* Commentarii Academiae Scientiarum Petropolitanae, St Petersburg, 1738,vol. vi, p. 104; see also Novi Comm. Acad. Sci. Petrop., St Petersburg, 1764,vol. ix, p. 101: or Commentationes Arithmeticae Collectae, St Petersburg, 1849,vol. i, pp. 2, 357.
† For the factors and bibliographical references, see the memoir by A.J.C. Cun-ningham and A.E. Western, Transactions of the London Mathematical Society,May 14, 1903, series 2, vol. i, p. 175.
‡ Fermat’s enunciation will be found in his edition of Diophantus, Toulouse, 1670,bk. ii, qu. 8, p. 61; or Brassinne’s Precis, Paris, 1853, p. 53. For bibliographicalreferences, see L’Intermediaire des mathematiciens, 1905, vol. xii, pp. 11, 12.
§ See a letter from Fermat quoted in my History of Mathematics, London, chap-ter xv.
‖ Fermat’s Diophantus, note on p. 339; or Brassinne’s Precis, p. 127.¶ Euler’s Algebra (English trans. 1797), vol. ii, chap. xv, p. 247.
• Project • Gutenberg • #26839 •
CH. I] FERMAT’S LAST THEOREM. 33
then it will be possible to find three other and smaller integers whichalso satisfy it: in this way finally we show that the equation mustbe satisfied by three values which obviously do not satisfy it. Thusno integral solution is possible. It would seem that this method isinapplicable except when n = 3 and n = 4.
Fermat’s discovery of the general theorem was made later. An easydemonstration can be given on the assumption that every number canbe resolved into prime (complex) factors in one and only one way. Thatassumption has been made by some writers, but it is not universallytrue. It is possible that Fermat made some such supposition, thoughit is perhaps more probable that he discovered a rigorous demonstra-tion. At any rate he asserts definitely that he had a valid proof—demonstratio mirabilis sane—and the fact that every other theorem onthe subject which he stated he had proved has been subsequently veri-fied must weigh strongly in his favour; especially as in making the onestatement in his writings which is not correct he was scrupulously care-ful to add that he could not obtain a satisfactory demonstration of it.
It must be remembered that Fermat was a mathematician of quitethe first rank who had made a special study of the theory of numbers.That subject is in itself one of peculiar interest and elegance, but itsconclusions have little practical importance, and since his time it hasbeen discussed by only a few mathematicians, while even of them notmany have made it their chief study. This is the explanation of thefact that it took more than a century before some of the simpler resultswhich Fermat had enunciated were proved, and thus it is not surprisingthat a proof of the theorem which he succeeded in establishing onlytowards the close of his life should involve great difficulties.
In 1823 Legendre* obtained a proof for the case of n = 5; in 1832Lejeune Dirichlet† gave one for n = 14, and in 1840 Lame and Lebesgue‡
gave proofs for n = 7.The proposition appears to be true universally, and in 1849 Kum-
mer§, by means of ideal primes, proved it to be so for all numbersexcept those (if any) which satisfy three conditions. It is not known
* Reprinted in his Theorie des Nombres, Paris, 1830, vol. ii, pp. 361–368: see alsopp. 5, 6.
† Crelle’s Journal, 1832, vol. ix, pp. 390–393.‡ Liouville’s Journal, 1841, vol. v, pp. 195–215, 276–9, 348–9.§ References to Kummer’s Memoirs are given in Smith’s Report to the British
Association on the Theory of Numbers, London, 1860.
• Project • Gutenberg • #26839 •
34 ARITHMETICAL RECREATIONS. [CH. I
whether any number can be found to satisfy these conditions, but itseems unlikely, and it has been shown that there is no number less than100 which does so. The proof is complicated and difficult, and therecan be little doubt is based on considerations unknown to Fermat. Imay add that to prove the truth of the proposition when n is greaterthan 4, it obviously is sufficient to confine ourselves to cases where n isa prime, and the first step in Kummer’s demonstration is to show thatin such cases one of the numbers x, y, z must be divisible by n.
Naturally there has been much speculation as to how Fermat ar-rived at the result. The modern treatment of higher arithmetic isfounded on the special notation and processes introduced by Gauss,who pointed out that the theory of discrete magnitude is essentiallydifferent from that of continuous magnitude, but until the end of thelast century the theory of numbers was treated as a branch of algebra,and such proofs by Fermat as are extant involve nothing more than el-ementary geometry and algebra, and indeed some of his arguments donot involve any symbols. This has led some writers to think that Fermatused none but elementary algebraic methods. This may be so, but thefollowing remark, which I believe is not generally known, rather pointsto the opposite conclusion. He had proposed, as a problem to the En-glish mathematicians, to show that there was only one integral solutionof the equation x2 + 2 = y3: the solution evidently being x = 5, y = 3.On this he has a note* to the effect that there was no difficulty in findinga solution in rational fractions, but that he had discovered an entirelynew method—sane pulcherrima et subtilissima—which enabled him tosolve such questions in integers. It was his intention to write a work† onhis researches in the theory of numbers, but it was never completed, andwe know but little of his methods of analysis. I venture however to addmy private suspicion that continued fractions played a not unimportantpart in his researches, and as strengthening this conjecture I may notethat some of his more recondite results—such as the theorem that aprime of the form 4n+1 is expressible as the sum of two squares—maybe established with comparative ease by properties of such fractions.
* Fermat’s Diophantus, bk. vi, prop. 19, p. 320; or Brassinne’s Precis, p. 122.† Fermat’s Diophantus, bk. iv, prop. 31, p. 181; or Brassinne’s Precis, p. 82.
• Project • Gutenberg • #26839 •
CHAPTER II.
SOME GEOMETRICAL QUESTIONS.
In this chapter I propose to enumerate certain geometrical ques-tions the discussion of which will not involve necessarily any consider-able use of algebra or arithmetic. Unluckily no writer like Bachet hascollected and classified problems of this kind, and I take the followinginstances from my note-books with the feeling that they represent thesubject but imperfectly.
The first part of the chapter is devoted to questions which are ofthe nature of formal propositions: the last part contains a descriptionof various trivial puzzles and games, which the older writers wouldhave termed geometrical, but which the reader of to-day may omitwithout loss.
In accordance with the rule I laid down for myself in the preface,I exclude the detailed discussion of theorems which involve advancedmathematics. Moreover (with one possible exception) I exclude also anymention of the numerous geometrical paradoxes which depend merelyon the inability of the eye to compare correctly the dimensions of figureswhen their relative position is changed. This apparent deception doesnot involve the conscious reasoning powers, but rests on the inaccurateinterpretation by the mind of the sensations derived through the eyes,and I do not consider such paradoxes as coming within the domain ofmathematics.
Geometrical Fallacies. Most educated Englishmen are ac-quainted with the series of logical propositions in geometry associated
35
• Project • Gutenberg • #26839 •
36 SOME GEOMETRICAL QUESTIONS. [CH. II
with the name of Euclid, but it is not known so generally that thesepropositions were supplemented originally by certain exercises. Of suchexercises Euclid issued three series: two containing easy theorems orproblems, and the third consisting of geometrical fallacies, the errorsin which the student was required to find.
The collection of fallacies prepared by Euclid is lost, and traditionhas not preserved any record as to the nature of the erroneous reason-ing or conclusions; but, as an illustration of such questions, I appendtwo or three demonstrations, leading to obviously impossible results,which perhaps may amuse any one to whom they are new. I leave thediscovery of the errors to the ingenuity of my readers.
First Fallacy. To prove that a right angle is equal to an anglewhich is greater than a right angle. Let ABCD be a rectangle. From Adraw a line AE outside the rectangle, equal to AB or DC and makingan acute angle with AB, as indicated in the diagram. Bisect CB in
A
BC
D
E
O
H
K
H, and through H draw HO at right angles to CB. Bisect CE in K,and through K draw KO at right angles to CE. Since CB and CEare not parallel the lines HO and KO will meet (say) at O. Join OA,OE, OC, and OD.
The triangles ODC and OAE are equal in all respects. For, sinceKO bisects CE and is perpendicular to it, we have OC = OE. Simi-larly, since HO bisects CB and DA and is perpendicular to them, wehave OD = OA. Also, by construction, DC = AE. Therefore the threesides of the triangle ODC are equal respectively to the three sides of
• Project • Gutenberg • #26839 •
CH. II] GEOMETRICAL FALLACIES. 37
the triangle OAE. Hence, by Euc. i. 8, the triangles are equal; andtherefore the angle ODC is equal to the angle OAE.
Again, since HO bisects DA and is perpendicular to it, we havethe angle ODA equal to the angle OAD.
Hence the angle ADC (which is the difference of ODC and ODA)is equal to the angle DAE (which is the difference of OAE and OAD).But ADC is a right angle, and DAE is necessarily greater than a rightangle. Thus the result is impossible.
Second Fallacy*. To prove that a part of a line is equal to thewhole line. Let ABC be a triangle; and, to fix our ideas, let us supposethat the triangle is scalene, that the angle B is acute, and that the
B C
A
ED
angle A is greater than the angle C. From A draw AD making theangle BAD equal to the angle C, and cutting BC in D. From A drawAE perpendicular to BC.
The triangles ABC, ABD are equiangular; hence, by Euc. vi. 19,
4ABC : 4ABD = AC2 : AD2 .
Also the triangles ABC, ABD are of equal altitude: hence, byEuc. vi. 1,
4ABC : 4ABD = BC : BD ,
∴ AC2 : AD2 = BC : BD .
∴AC2
BC=
AD2
BD.
* See a note by M. Coccoz in L’Illustration, Paris, Jan. 12, 1895.
• Project • Gutenberg • #26839 •
38 SOME GEOMETRICAL QUESTIONS. [CH. II
Hence, by Euc. ii. 13,
AB2 + BC2 − 2BC ·BE
BC=
AB2 + BD2 − 2BD ·BE
BD.
∴AB2
BC+ BC − 2BE =
AB2
BD+ BD − 2BE .
∴AB2
BC−BD =
AB2
BD−BC .
∴AB2 −BC ·BD
BC=
AB2 −BC ·BD
BD.
∴ BC = BD ,
a result which is impossible.
Third Fallacy. To prove that every triangle is isosceles. Let ABCbe any triangle. Bisect BC in D, and through D draw DO perpendic-ular to BC. Bisect the angle BAC by AO.
First. If DO and AO do not meet, then they are parallel. ThereforeAO is at right angles to BC. Therefore AB = AC.
Second. If DO and AO meet, let them meet in O. Draw OEperpendicular to AC. Draw OF perpendicular to AB. Join OB, OC.
A
B CD
O
EF
Let us begin by taking the casewhere O is inside the triangle, inwhich case E falls on AC and F onBC.
The triangles AOF and AOEare equal, since the side AO is com-mon, angle OAF = angle OAE,and angle OFA = angle OEA.Hence AF = AE. Also, the trian-gles BOF and COE are equal. For
since OD bisects BC at right angles, we have OB = OC; also, sincethe triangles AOF and AOE are equal, we have OF = OE; lastly, theangles at F and E are right angles. Therefore, by Euc. i. 47 and i. 8,the triangles BOF and COE are equal. Hence FB = EC.
Therefore AF + FB = AE + EC, that is, AB = AC.
The same demonstration will cover the case where DO and AOmeet at D, as also the case where they meet outside BC but so near itthat E and F fall on AC and AB and not on AC and AB produced.
• Project • Gutenberg • #26839 •
CH. II] GEOMETRICAL FALLACIES. 39
Next take the case where DO and AO meet outside the triangle,and E and F fall on AC and AB produced. Draw OE perpendicu-lar to AC produced. Draw OF perpendicular to AB produced. JoinOB, OC.
A
B CD
O
EF
Following the same argument as before, from the equality of thetriangles AOF and AOE, we obtain AF = AE; and, from the equalityof the triangles BOF and COE, we obtain FB = EC. ThereforeAF − FB = AE − EC, that is, AB = AC.
Thus in all cases, whether or not DO and AO meet, and whetherthey meet inside or outside the triangle, we have AB = AC: andtherefore every triangle is isosceles, a result which is impossible.
Fourth Fallacy. I am indebted to Captain Turton for the followingingenious fallacy; it appeared for the first time in the third editionof this work.
On the hypothenuse, BC, of an isosceles right-angled triangle,DBC, describe an equilateral triangle ABC, the vertex A being onthe same side of the base as D is. On CA take a point H so thatCH = CD. Bisect BD in K. Join HK and let it cut CB (produced)in L. Join DL. Bisect DL at M , and through M draw MO perpendic-ular to DL. Bisect HL at N , and through N draw NO perpendicularto HL. Since DL and HL intersect, therefore MO and NO will alsointersect; moreover, since BDC is a right angle, MO and NO bothslope away from DC and therefore they will meet on the side of DLremote from A. Join OC, OD, OH, OL.
• Project • Gutenberg • #26839 •
40 SOME GEOMETRICAL QUESTIONS. [CH. II
The triangles OMD and OML are equal, hence OD = OL. Simi-larly the triangles ONL and ONH are equal, hence OL = OH. There-fore OD = OH. Now in the triangles OCD and OCH, we haveOD = OH, CD = CH (by construction), and OC common, hence (byEuc. i. 8) the angle OCD is equal to the angle OCH, which is absurd.
Fifth Fallacy*. To prove that, if two opposite sides of a quadri-lateral are equal, the other two sides must be parallel. Let ABCD bea quadrilateral such that AB is equal to DC. Bisect AD in M , andthrough M draw MO at right angles to AD. Bisect BC in N , anddraw NO at right angles to BC.
If MO and NO are parallel, then AD and BC (which are at rightangles to them) are also parallel.
If MO and NO are not parallel, let them meet in O; then O mustbe either inside the quadrilateral as in the left-hand diagram or outside
A
B C
D
N
M
O
A
B C
D
N
M
O
the quadrilateral as in the right-hand diagram. Join OA, OB, OC, OD.Since OM bisects AD and is perpendicular to it, we have OA =
OD, and the angle OAM equal to the angle ODM . SimilarlyOB = OC, and the angle OBN equal to the angle OCN . Alsoby hypothesis AB = DC, hence, by Euc. i. 8, the triangles OAB andODC are equal in all respects, and therefore the angle AOB is equalto the angle DOC.
Hence in the left-hand diagram the sum of the angles AOM , AOBis equal to the sum of the angles DOM , DOC; and in the right-hand
* Mathesis, October, 1893, series 2, vol. iii, p. 224.
• Project • Gutenberg • #26839 •
CH. II] GEOMETRICAL FALLACIES. 41
diagram the difference of the angles AOM , AOB is equal to the dif-ference of the angles DOM , DOC; and therefore in both cases theangle MOB is equal to the angle MOC, i.e. OM (or OM produced)bisects the angle BOC. But the angle NOB is equal to the angleNOC, i.e. ON bisects the angle BOC; hence OM and ON coincidein direction. Therefore AD and BC, which are perpendicular to thisdirection, must be parallel. This result is not universally true, and theabove demonstration contains a flaw.
Sixth Fallacy. The following argument is taken from a text-bookon electricity, published in 1889 by two distinguished mathematicians,in which it was presented as valid. A given vector OP of length l canbe resolved in an infinite number of ways into two vectors OM , MP ,of lengths l′, l′′, and we can make l′/l′′ have any value we please fromnothing to infinity. Suppose that the system is referred to rectangularaxes Ox, Oy; and that OP , OM , MP make respectively angles θ, θ′,θ′′ with Ox. Hence, by projection on Oy and on Ox, we have
l sin θ = l′ sin θ′ + l′′ sin θ′′ ,
l cos θ = l′ cos θ′ + l′′ cos θ′′ .
Therefore tan θ =n sin θ′ + sin θ′′
n cos θ′ + cos θ′′,
where n = l′/l′′. This result is true whatever be the value of n.But n may have any value (ex. gr. n = ∞, or n = 0), hencetan θ = tan θ′ = tan θ′′, which obviously is impossible.
Seventh Fallacy. Here is a fallacious investigation, to whichMr Chartres first called my attention, of the value of π: it is foundedon well-known quadratures. The area of the semi-ellipse bounded bythe minor axis is (in the usual notation) equal to 1
2πab. If the centre
is moved off to an indefinitely great distance along the major axis,the ellipse degenerates into a parabola, and therefore in this particularlimiting position the area is equal to two-thirds of the circ*mscribingrectangle. But the first result is true whatever be the dimensions ofthe curve.
∴ 12πab = 2
3a× 2b,
∴ π = 8/3,
a result which is obviously untrue.
• Project • Gutenberg • #26839 •
42 SOME GEOMETRICAL QUESTIONS. [CH. II
Geometrical Paradoxes. To the above examples I may addthe following questions, which, though not exactly fallacious, lead toresults which at a hasty glance appear impossible.
First Paradox. The first is a problem, sent to me by Mr Renton,to rotate a plane lamina (say, for instance, a sheet of paper) throughfour right angles so that the effect is equivalent to turning it throughonly one right angle.
If it is desired that the effect shall be equivalent to turning itthrough a right angle about a point O, the solution is as follows. De-scribe on the lamina a square OABC. Rotate the lamina successivelythrough two right angles about the diagonal OB as axis and throughtwo right angles about the side OA as axis, and the required resultwill be attained.
Second Paradox. As in arithmetic, so in geometry, the theory ofprobability lends itself to numerous paradoxes. Here is a very simpleillustration. A stick is broken at random into three pieces. It is possibleto put them together into the shape of a triangle provided the lengthof the longest piece is less than the sum of the other two pieces (cf.Euc. i. 20), that is, provided the length of the longest piece is less thanhalf the length of the stick. But the probability that a fragment ofa stick shall be half the original length of the stick is 1
2. Hence the
probability that a triangle can be constructed out of the three piecesinto which the stick is broken would appear to be 1
2. This is not true,
for actually the probability is 14.
Third Paradox. The following example illustrates how easily theeye may be deceived in demonstrations obtained by actually dissectingthe figures and re-arranging the parts. In fact proofs by superposi-tion should be regarded with considerable distrust unless they are sup-plemented by mathematical reasoning. The well-known proofs of thepropositions Euclid i. 32 and Euclid i. 47 can be so supplemented andare valid. On the other hand, as an illustration of how deceptive a non-mathematical proof may be, I here mention the familiar paradox thata square of paper, subdivided like a chessboard into 64 small squares,can be cut into four pieces which being put together form a figure con-taining 65 such small squares*. This is effected by cutting the original
* I do not know who discovered this paradox. It is given in various modern books,but I cannot find an earlier reference to it than one by Prof. G.H. Darwin,Messenger of Mathematics, 1877, vol. vi, p. 87.
• Project • Gutenberg • #26839 •
CH. II] GEOMETRICAL PARADOXES. 43
A
B
square into four pieces in the manner indicated by the thick lines inthe first figure. If these four pieces are put together in the shape of arectangle in the way shown in the second figure it will appear as if thisrectangle contains 65 of the small squares.
This phenomenon, which in my experience non-mathematiciansfind perplexing, is due to the fact that the edges of the four piecesof paper, which in the second figure lie along the diagonal AB, do notcoincide exactly in direction. In reality they include a small lozengeor diamond-shaped figure, whose area is equal to that of one of the64 small squares in the original square, but whose length AB is muchgreater than its breadth. The diagrams show that the angle betweenthe two sides of this lozenge which meet at A is tan−1 2
5− tan−1 3
8, that
is, is tan−1 146
, which is less than 114◦. To enable the eye to distinguish
so small an angle as this the dividing lines in the first figure wouldhave to be cut with extreme accuracy and the pieces placed togetherwith great care.
The paradox depends upon the relation 5 × 13 − 82 = 1. Sim-ilar results can be obtained from the formulae 13 × 34 − 212 = 1,34×89−552 = 1,. . . ; or from the formulae 52−3×8 = 1, 132−8×21 = 1,342 − 21× 55 = 1,. . . . These numbers are obtained by finding conver-gents to the continued fraction
1 +1
1 +
1
1 +
1
1 +· · · .
A similar paradox for a square of 17 cells, by which it was shownthat 289 was equal to 288, was alluded to by Ozanam* who gave alsothe diagram for dividing a rectangle of 11 by 3 into two rectangleswhose dimensions appear to be 5 by 4 and 7 by 2.
* Ozanam, 1803 edition, vol. i, p. 299.
• Project • Gutenberg • #26839 •
44 SOME GEOMETRICAL QUESTIONS. [CH. II
Turton’s Seventy-Seven Puzzle. A far better dissection puzzle wasinvented by Captain Turton. In this a piece of cardboard, 11 inches by7 inches, subdivided into 77 small equal squares, each 1 inch by 1 inch,can be cut up and re-arranged so as to give 78 such equal squares, each1 inch by 1 inch, of which 77 are arranged in a rectangle of the samedimensions as the original rectangle from one side of which projects asmall additional square. The construction is ingenious, but cannot bedescribed without the use of a model. The trick consists in utilizingthe fact that cardboard has a sensible thickness. Hence the edges ofthe cuts can be bevelled, but in the model the bevelling is so slightas to be imperceptible save on a very close scrutiny. The play thusgiven in fitting the pieces together permits the apparent production ofan additional square.
Colouring Maps. I proceed next to mention the geometricalproposition that not more than four colours are necessary in order tocolour a map of a country (divided into districts) in such a way thatno two contiguous districts shall be of the same colour. By contiguousdistricts are meant districts having a common line as part of theirboundaries: districts which touch only at points are not contiguous inthis sense.
The problem was mentioned by A.F. Mobius* in his Lectures in1840, but it was not until Francis Guthrie† communicated it to De Mor-gan about 1850 that attention was generally called to it: it is said thatthe fact had been familiar to practical map-makers for a long time pre-viously. Through De Morgan the proposition became generally known;and in 1878 Cayley‡ recalled attention to it by stating that he did notknow of any rigorous proof of it.
Probably the following argument, though not a formal demonstra-tion, will satisfy the reader that the result is true.
Let A, B, C be three contiguous districts, and let X be any otherdistrict contiguous with all of them. Then X must lie either whollyoutside the external boundary of the area ABC or wholly inside theinternal boundary, that is, it must occupy a position either like X or
* Leipzig Transactions (Math.-phys. Classe), 1885, vol. xxxvii, pp. 1–6.† Proceedings of the Royal Society of Edinburgh, July 19, 1880, vol. x, p. 728.‡ Proceedings of the London Mathematical Society, 1878, vol. ix, p. 148, and Pro-
ceedings of the Royal Geographical Society, 1879, N.S., vol. i, p. 259.
• Project • Gutenberg • #26839 •
CH. II] THE FOUR-COLOUR THEOREM.. 45
like X ′. In either case every remaining occupied area in the figure isenclosed by the boundaries of not more than three districts: hence thereis no possible way of drawing another area Y which shall be contiguouswith A, B, C, and X. In other words, it is possible to draw on aplane four areas which are contiguous, but it is not possible to drawfive such areas.
If A, B, C are not contiguous, each with the other, or if X isnot contiguous with A, B, and C, it is not necessary to colour them alldifferently, and thus the most unfavourable case is that already treated.Moreover any of the above areas may diminish to a point and finallydisappear without affecting the argument.
That we may require at least four colours is obvious from the dia-gram on this page, since in that case the areas A, B, C, and X wouldhave to be coloured differently.
A proof of the proposition involves difficulties of a high order, whichas yet have baffled all attempts to surmount them.
The argument by which the truth of the proposition was formerlysupposed to be demonstrated was given by A.B. Kempe* in 1879, but
* He sent his first demonstration across the Atlantic to the American Journal ofMathematics, 1879, vol. ii, pp. 193–200; but subsequently he communicated it
• Project • Gutenberg • #26839 •
46 SOME GEOMETRICAL QUESTIONS. [CH. II
there is a flaw* in it.In 1880, Tait published a solution† depending on the theorem that
if a closed network of lines joining an even number of points is suchthat three and only three lines meet at each point then three coloursare sufficient to colour the lines in such a way that no two lines meetingat a point are of the same colour; a closed network being supposed toexclude the case where the lines can be divided into two groups betweenwhich there is but one connecting line. His deduction therefrom thatfour colours will suffice for a map was given in the last edition of thiswork. The demonstration appeared so straightforward that at first itwas generally accepted, but it would seem that it too involves a fallacy‡.The proof however leads to the interesting corollary that four coloursmay not suffice for a map drawn on a multiply-connected surface suchas an anchor ring.
Although a proof of the theorem is still wanting, no one has suc-ceeded in constructing a plane map which requires more than four tintsto colour it, and there is no reason to doubt the correctness of the state-ment that it is not necessary to have more than four colours for anyplane map. The number of ways which such a map can be coloured withfour tints has been also considered§, but the results are not sufficientlyinteresting to require mention here.
Physical Configuration of a Country. As I have beenalluding to maps, I may here mention that the theory of the repre-sentation of the physical configuration of a country by means of linesdrawn on a map was discussed, by Cayley and Clerk Maxwell‖. They
in simplified forms to the London Mathematical Society, Transactions, 1879,vol. x, pp. 229–231, and to Nature, Feb. 26, 1880, vol. xxi, pp. 399–400.
* See articles by P.J. Heawood in the Quarterly Journal of Mathematics, London,1890, vol. xxiv, pp. 332–338; and 1897, vol. xxxi, pp. 270–285.
† Proceedings of the Royal Society of Edinburgh, July 19, 1880, vol. x, p. 729; andPhilosophical Magazine, January, 1884, series 5, vol. xvii, p. 41.
‡ See J. Peterson of Copenhagen, L’Intermediaire des mathematiciens, vol. v,1898, pp. 225–227; and vol. vi, 1899, pp. 36–38.
§ See A.C. Dixon, Messenger of Mathematics, Cambridge, 1902–3, vol. xxxii,pp. 81–83.
‖ Cayley on ‘Contour and Slope Lines,’ Philosophical Magazine, London, October,1859, series 4, vol. xviii, pp. 264–268; Collected Works, vol. iv, pp. 108–111.J. Clerk Maxwell on ‘Hills and Dales,’ Philosophical Magazine, December, 1870,series 4, vol. xl, pp. 421–427; Collected Works, vol. ii, pp. 233–240.
• Project • Gutenberg • #26839 •
CH. II] HILLS AND DALES.. 47
showed that a certain relation exists between the number of hills, dales,passes, &c. which can co-exist on the earth or on an island. I proceedto give a summary of their nomenclature and conclusions.
All places whose heights above the mean sea level are equal are onthe same level. The locus of such points on a map is indicated by acontour-line. Roughly speaking, an island is bounded by a contour-line.It is usual to draw the successive contour-lines on a map so that thedifference between the heights of any two successive lines is the same,and thus the closer the contour-lines the steeper is the slope, but theheights are measured dynamically by the amount of work to be doneto go from one level to the other and not by linear distances.
A contour-line in general will be a closed curve. This curve mayenclose a region of elevation: if two such regions meet at a point, thatpoint will be a crunode (i.e. a real double point) on the contour-linethrough it, and such a point is called a pass. The contour-line mayenclose a region of depression: if two such regions meet at a point, thatpoint will be a crunode on the contour-line through it, and such a pointis called a fork or bar. As the heights of the corresponding level surfacesbecome greater, the areas of the regions of elevation become smaller,and at last become reduced to points: these points are the summits ofthe corresponding mountains. Similarly as the level surface sinks theregions of depression contract, and at last are reduced to points: thesepoints are the bottoms (or immits) of the corresponding valleys.
Lines drawn so as to be everywhere at right angles to the contour-lines are called lines of slope. If we go up a line of slope generally weshall reach a summit, and if we go down such a line generally we shallreach a bottom: we may come however in particular cases either to apass or to a fork. Districts whose lines of slope run to the same summitare hills. Those whose lines of slope run to the same bottom are dales.A watershed is the line of slope from a summit to a pass or a fork, andit separates two dales. A watercourse is the line of slope from a passor a fork to a bottom, and it separates two hills.
If n + 1 regions of elevation or of depression meet at a point, thepoint is a multiple point on the contour-line drawn through it; such apoint is called a pass or a fork of the nth order, and must be counted asn separate passes (or forks). If one region of depression meets anotherin several places at once, one of these must be taken as a fork andthe rest as passes.
• Project • Gutenberg • #26839 •
48 SOME GEOMETRICAL QUESTIONS. [CH. II
Having now a definite geographical terminology we can apply geo-metrical propositions to the subject. Let h be the number of hills onthe earth (or an island), then there will be also h summits; let d be thenumber of dales, then there will be also d bottoms; let p be the wholenumber of passes, p1 that of single passes, p2 of double passes, and so on;let f be the whole number of forks, f1 that of single forks, f2 of doubleforks, and so on; let w be the number of watercourses, then there willbe also w watersheds. Hence, by the theorems of Cauchy and Euler,
h = 1 + p1 + 2p2 + · · · ,
d = 1 + f1 + 2f2 + · · · ,
and w = 2(p1 + f1) + 3(p2 + f2)) + · · · .
The above results can be extended to the case of a multiply-connected closed surface.
Games. Leaving now the question of formal geometrical propo-sitions, I proceed to enumerate a few games or puzzles which dependmainly on the relative position of things, but I postpone to chapter ivthe discussion of such amusem*nts of this kind as necessitate any con-siderable use of arithmetic or algebra. Some writers regard draughts,solitaire, chess and such like games as subjects for geometrical treat-ment in the same way as they treat dominoes, backgammon, and gameswith dice in connection with arithmetic: but these discussions requiretoo many artificial assumptions to correspond with the games as actu-ally played or to be interesting.
The amusem*nts to which I refer are of a more trivial description,and it is possible that a mathematician may like to omit the remainderof the chapter. In some cases it is difficult to say whether they shouldbe classified as mainly arithmetical or geometrical, but the point is ofno importance.
Statical Games of Position. Of the innumerable staticalgames involving geometry of position I shall mention only three or four.
Three-in-a-row. First, I may mention the game of three-in-a-row,of which noughts and crosses, one form of merrilees, and go-bang arewell-known examples. These games are played on a board—generally inthe form of a square containing n2 small squares or cells. The commonpractice is for one player to place a white counter or piece or to make
• Project • Gutenberg • #26839 •
CH. II] THREE-IN-A-ROW.. 49
a cross on each small square or cell which he occupies: his opponentsimilarly uses black counters or pieces or makes a nought on each squarewhich he occupies. Whoever first gets three (or any other assignednumber) of his pieces in three adjacent cells and in a straight line wins.The mathematical theory for a board of 9 cells has been worked outcompletely, and there is no difficulty in extending it to one of 16 cells:but the analysis is lengthy and not particularly interesting. Most ofthese games were known to the ancients*, and it is for that reason Imention them here.
Three-in-a-row. Extension. I may, however, add an elegant butdifficult extension which has not previously found its way, so far as Iam aware, into any book of mathematical recreations. The problem isto place n counters on a plane so as to form as many rows as possible,each of which shall contain three and only three counters†.
It is easy to arrange the counters in a number of rows equal tothe integral part of 1
8(n − 1)2. This can be effected by the following
construction. Let P be any point on a cubic. Let the tangent at P cutthe curve again in Q. Let the tangent at Q cut the curve in A. Let PAcut the curve in B, QB cut it in C, PC cut it in D, QD cut it in E, andso on. Then the counters must be placed at the points P, Q, A, B, . . . .Thus 9 counters can be placed in 8 such rows; 10 counters in 10 rows;15 counters in 24 rows; 81 counters in 800 rows; and so on.
If however the point P is a pluperfect point of the nth order onthe cubic, then Sylvester proved that the above construction gives anumber of rows equal to the integral part of 1
6(n − 1)(n − 2). Thus 9
counters can be arranged in 9 rows, which is a well-known and easypuzzle; 10 counters in 12 rows; 15 counters in 30 rows; and so on.
Even this however is an inferior limit and may be exceeded—forinstance, Sylvester stated that 9 counters can be placed in 10 rows,each containing three counters; I do not know how he placed them, butone way of so arranging them is by putting them at points whose coor-dinates are (2, 0), (2, 2), (2, 4), (4, 0), (4, 2), (4, 4), (0, 0), (3, 2), (6, 4);another way is by putting them at the points (0, 0), (0, 2), (0, 4), (2, 1),(2, 2), (2, 3), (4, 0), (4, 2), (4, 4); more generally, the angular points of
* Becq de Fouquieres, Les jeux des anciens, second edition, Paris, 1873,chap. xviii.
† Educational Times Reprints, 1868, vol. viii, p. 106; Ibid. 1886, vol. xlv, pp. 127–128.
• Project • Gutenberg • #26839 •
50 SOME GEOMETRICAL QUESTIONS. [CH. II
a regular hexagon and the three points of intersection of opposite sidesform such a group, and therefore any projection of that figure will givea solution.
Thus at present it is not possible to say what is the maximumnumber of rows of three which can be formed from n counters placedon a plane.
Extension to p-in-a-row. The problem mentioned above at oncesuggests the extension of placing n counters so as to form as many rowsas possible, each of which shall contain p and only p counters. Suchproblems can be often solved immediately by placing at infinity thepoints of intersection of some of the lines, and (if it is so desired) sub-sequently projecting the diagram thus formed so as to bring these pointsto a finite distance. One instance of such a solution is given above.
As easy examples I may give the arrangement of 16 counters in 15rows1, each containing 4 counters; and the arrangement of 19 countersin 10 rows, each containing 5 counters. A solution of the second ofthese problems can be obtained by placing counters at the 19 points ofintersection of the 10 lines x = ±a, x = ±b, y = ±a, y = ±b, y = ±x:of these points two are at infinity. The first problem I leave to theingenuity of my readers.
Tesselation. Another of these statical recreations is known as tes-selation and consists in the formation of geometrical designs or mosaicsby means of tesselated tiles.
To those who have never looked into the matter it may be surpris-ing that patterns formed by the use of square tiles (of which one-halfbounded by a diagonal is white and the other half black) should besubject to mathematical analysis. In view of the discussion of thissubject by Montucla*, Lucas†, and other writers it would be hard torefuse to call the formation of such patterns a mathematical amuse-ment, but the treatment is (perhaps necessarily) somewhat empirical,and though there are some interesting puzzles of this kind, I do notpropose to describe them here.
* See Ozanam, 1803 edition, vol. i, p. 100; 1840 edition, p. 46.† Lucas, Recreations Mathematiques, Paris, 1882–3, vol. ii, part 4: hereafter I
shall refer to this work by the name of the author.
1. ‘13’ corrected to ’15’ as per errata sheet
• Project • Gutenberg • #26839 •
CH. II] COLOUR-CUBE PROBLEM. 51
Sylvester* proposed a modified tesselation problem which consistsin forming anallagmatic squares, that is, squares such that in every rowand every column the number of changes of colour or the number ofpermanences is constant, the tiles used being square white tiles andsquare black tiles.
If more than two colours are used, the problems become increas-ingly difficult. As a simple instance of this class of problems I maymention one, sent to me by a correspondent who termed it Cross-Fours ,wherein sixteen square counters are used, the upper half of each beingyellow, red, pink, or blue, and the lower half being gold, green, black,or white, no two counters being coloured alike. Such counters can bearranged in the form of a square so that in each vertical, horizontal,and diagonal line there shall be 8 colours and no more: they can be alsoarranged so that in each of these ten lines there shall be 6 colours andno more, or 5 colours and no more, or 4 colours and no more. Puzzlesof this kind are but little known; they are however not uninstructive.
Colour-Cube Problem. As an example of a recreation analogous totesselation I will mention the colour-cube problem; I select this partlybecause it is one of the most difficult of such puzzles, but chiefly becauseit has been subjected† to mathematical analysis.
Stripped of mathematical technicalities the problem may be enun-ciated as follows. A cube has six faces, and if six colours are chosenwe can paint each face with a different colour. By permuting the or-der of the colours we can obtain thirty such cubes, no two of whichare coloured alike. Take any one of these cubes, K, then it is desiredto select eight out of the remaining twenty-nine cubes, such that theycan be arranged in the form of a cube (whose linear dimensions aredouble those of any of the separate cubes) coloured like the cube K,and placed so that where any two cubes touch each other the faces incontact are coloured alike.
Only one collection of eight cubes can be found to satisfy theseconditions. To pick out these eight cubes empirically would be outof the question, but the mathematical analysis enables us to selectthem by the following rule. Take any face of the cube K: it has four
* Ex. gr. see the Educational Times Reprints, London, 1868, vol. x, pp. 74–76,112: see also vol. xlv, p. 127; vol. lvi, pp. 97–99.
† By Major MacMahon; an abstract of his paper, read before the London Mathe-matical Society on Feb. 9, 1893, was given in Nature, Feb. 23, 1893, vol. xlvii,p. 406.
• Project • Gutenberg • #26839 •
52 SOME GEOMETRICAL QUESTIONS. [CH. II
angles, and at each angle three colours meet. By permuting the colourscyclically we can obtain from each angle two other cubes, and the eightcubes so obtained are those required.
For instance suppose that the six colours are indicated by the lettersa, b, c, d, e, f . Let the cube K be put on a table, and to fix our ideassuppose that the face coloured f is at the bottom, the face coloured ais at the top, and the faces coloured b, c, d, and e front respectivelythe east, north, west, and south points of the compass. I may denotesuch an arrangement by (f ; a; b, c, d, e). One cyclical permutation ofthe colours which meet at the north-east corner of the top face givesthe cube (f ; c; a, b, d, e), and a second cyclical permutation gives thecube (f ; b; c, a, d, e). Similarly cyclical permutations of the colourswhich meet at the north-west corner of the top face of K give the cubes(f ; d; b, a, c, e) and (f ; c; b, d, a, e). Similarly from the top south-westcorner of K we get the cubes (f ; e; b, c, a, d) and (f ; d; b, c, e, a):and from the top south-east corner we get the cubes (f ; e; a, c, d, b)and (f ; b; e, c, d, a).
The eight cubes being thus determined it is not difficult to arrangethem in the form of a cube coloured similarly to K, and subject to thecondition that faces in contact are coloured alike; in fact they can bearranged in two ways to satisfy these conditions. One such way, takingthe cubes in the numerical order given above, is to put the cubes 3, 6,8, and 2 at the SE, NE, NW, and SW corners of the bottom face; ofcourse each placed with the colour f at the bottom, while 3 and 6 havethe colour b to the east, and 2 and 8 have the colour d to the west: thecubes 7, 1, 4, and 5 will then form the SE, NE, NW, and SW corners ofthe top face; of course each placed with the colour a at the top, while 7and 1 have the colour b to the east, and 5 and 4 have the colour d to thewest. If however K is not given, then, without the aid of mathematicalanalysis, it is a difficult puzzle to arrange the eight cubes in the formof a cube coloured similarly to one of the other twenty-two cubes andsubject to the condition that faces in contact are coloured alike.
It is easy to make similar puzzles in two dimensions which are fairlydifficult; it is somewhat surprising that none are to be bought, but Ihave never seen any except those that I have made myself.
Dynamical Games of Position. Games which are playedby moving pieces on boards of various shapes—such as merrilees, foxand geese, solitaire, backgammon, draughts, and chess—present more
• Project • Gutenberg • #26839 •
CH. II] SHUNTING PROBLEMS. 53
interest. In general, however, they permit of so many movements of thepieces that any mathematical analysis of them becomes too intricate tofollow out completely. Probably this is obvious, but it may emphasizethe impossibility of discussing such games effectively if I add that it hasbeen shown that in a game of chess there may be as many as 197299ways of playing the first four moves, and nearly 72000 different positionsat the end of the first four moves (two on each side), of which 16556arise when the players move pawns only*.
Games in which the possible movements are very limited may besusceptible of mathematical treatment. One or two of these are givenin the next chapter: here I shall confine myself mainly to puzzles andsimple amusem*nts.
Shunting Problems. The first I will mention is a little puzzle whichI bought some years ago and which was described as the “Great North-ern puzzle.” It is typical of a good many problems connected with theshunting of trains, and though it rests on a most improbable hypoth-esis, I give it as a specimen of its kind.
A
B C
D E F
P Q
R
The puzzle shows a railway, DEF , with two sidings, DBA andFCA, connected at A. The portion of the rails at A which is commonto the two sidings is long enough to permit of a single wagon, like P orQ, running in or out of it; but is too short to contain the whole of anengine, like R. Hence, if an engine runs up one siding, such as DBA,it must come back the same way.
* L’Intermediaire des mathematiciens, Paris, December, 1903, vol. x, pp. 305–308: also Royal Engineers Journal, London, August–November, 1889; or BritishAssociation Transactions, 1890, p. 745.
• Project • Gutenberg • #26839 •
54 SOME GEOMETRICAL QUESTIONS. [CH. II
Initially a small block of wood, P , coloured to represent a wagon,is placed at B; a similar block, Q, is placed at C; and a longer blockof wood, R, representing an engine, is placed at E. The problem is touse the engine R to interchange the wagons P and Q, without allowingany flying shunts.
This is effected thus. (i) R pushes P into A. (ii) R returns, pushesQ up to P in A, couples Q to P , draws them both out to F , and thenpushes them to E. (iii) P is now uncoupled, R takes Q back to A, andleaves it there. (iv) R returns to P , pulls P back to C, and leaves itthere. (v) R running successively through F , D, B comes to A, drawsQ out, and leaves it at B.
A somewhat similar puzzle, on sale in the streets in 1905, is madeas follows. A loop-line BGE connects two points B and E on a railwaytrack AF , which is supposed blocked at both ends, as shown in thediagram. In the model, the track AF is 9 inches long, AB = EF = 15
6
inches, and AH = FK = BC = DE = 14
inch. On the track and loopare eight wagons, numbered successively 1 to 8, each one inch long and
A BC D
E F
G
H K
one-quarter of an inch broad, and an engine of the same dimensions.Originally the wagons are on the track from A to F and in the order1, 2, 3, 4, 5, 6, 7, 8, and the engine is on the loop. The constructionand the initial arrangement ensure that at any one time there cannotbe more than eight vehicles on the track. Also if eight vehicles are onit only the penultimate vehicle at either end can be moved on to theloop, but if less than eight are on the track then the last two vehiclesat either end can be moved on to the loop. If the points at each endof the loop-line are clear, it will hold four, but not more than four,vehicles. The object is to reverse the order of the wagons on the track,so that from A to F they will be numbered successively 8 to 1; and todo this by means which will involve as few transferences of the engineor a wagon to or from the loop as is possible.
Other shunting problems are not uncommon, but these two exam-ples will suffice.
• Project • Gutenberg • #26839 •
CH. II] FERRY-BOAT PROBLEMS. 55
Ferry-Boat Problems. Everybody is familiar with the story of theshowman who was travelling with a wolf, a goat, and a basket of cab-bages; and for obvious reasons was unable to leave the wolf alone withthe goat, or the goat alone with the cabbages. The only means oftransporting them across a river was a boat so small that he could takein it only one of them at a time. The problem is to show how thepassage could be effected*.
A similar problem, given by Alcuin, Tartaglia, and others, is asfollows†. Three beautiful ladies have for husbands three men, who areas jealous as they are young, handsome, and gallant. The party aretravelling, and find on the bank of a river, over which they have topass, a small boat which can hold no more than two persons. How canthey pass, it being agreed that, in order to avoid scandal, no womanshall be left in the society of a man unless her husband is present?
The method of transportation to be used in the above cases isobvious, and can be illustrated practically by using six court cards outof a pack. Another problem similar to the one last mentioned is the caseof n married couples who have to cross a river by means of a boat whichcan be rowed by one person and will carry n− 1 people, but not more,with the condition that no woman is to be in the society of a man unlessher husband is present. Alcuin’s problem is the case of n = 3. Let ydenote the number of passages from one bank to the other which willbe necessary. Then it has been shown that if n = 3, y = 11; if n = 4,y = 9; and if n > 4, y = 7; the demonstration presents no difficulty.
The following analogous problem is due to the late Prof. Lucas‡.To find the smallest number x of persons that a boat must be able tocarry in order that n married couples may by its aid cross a river insuch a manner that no woman shall remain in the company of any manunless her husband is present; it being assumed that the boat can berowed by one person only. Also to find the least number of passages,say y, from one bank to the other which will be required. M. Delannoyhas shown that if n = 2, then x = 2, and y = 5. If n = 3, then x = 2,and y = 11. If n = 4, then x = 3, and y = 9. If n = 5, then x = 3, andy = 11. And finally if n > 5, then x = 4, and y = 2n − 1.
M. De Fonteney has remarked that, if there was an island in the
* Ozanam, 1803 edition, vol. i, p. 171; 1840 edition, p. 77.† Bachet, Appendix, problem iv, p. 212.‡ Lucas, vol. i, pp. 15–18, 237–238.
• Project • Gutenberg • #26839 •
56 SOME GEOMETRICAL QUESTIONS. [CH. II
middle of the river, the passage might be always effected by the aid ofa boat which could carry only two persons. If there are only two oronly three couples the island is unnecessary, and the case is covered bythe preceding method. If n > 3 then the least number of passages fromland to land which will be required is 8(n − 1).
His solution is as follows. The first nine passages will be the same,no matter how many couples there may be: the result is to transferone couple to the island and one couple to the second bank. The resultof the next eight passages is to transfer one couple from the first bankto the second bank: this series of eight operations must be repeated asoften as necessary until there is left only one couple on the first bank,only one couple on the island, and all the rest on the second bank.The result of the last seven passages is to transfer all the couples tothe second bank.
The solution for the case when there are four couples may be rep-resented as follows. Let A and a, B and b, C and c, D and d, be thefour couples. The letters in the successive lines indicate the positions ofthe men and their respective wives after different passages of the boat.
First Bank Island Second BankInitially ABCD abcd . . . . . . . . . . . . . . . .After 1st passage ABCD . .cd . . . . ab. . . . . . . . . .” 2nd ” ABCD .bcd . . . . a. . . . . . . . . . .” 3rd ” ABCD . . .d . . . . abc . . . . . . . . .” 4th ” ABCD . .cd . . . . ab. . . . . . . . . .” 5th ” . . CD . .cd AB . . ab. . . . . . . . . .” 6th ” . . CD . .cd AB . . . . . . . . . . ab. .” 7th ” . . CD . .cd AB . . .b. . . . . . a. . .” 8th ” . . CD . .cd . . . . .b. . AB . . a. . .” 9th ” . . CD . .cd . B . . .b. . A . . . a. . .” 10th ” . BCD . .cd . . . . .b. . A . . . a. . .” 11th ” . BCD . . . . . . . . .bcd A . . . a. . .” 12th ” . BCD . . .d . . . . .bc . A . . . a. . .” 13th ” . . . D . . .d . BC . .bc . A . . . a. . .” 14th ” . . . D . . .d . . . . .bc . ABC . a. . .” 15th ” . . . D . . .d . . . . abc . ABC . . . . .” 16th ” . . . D . . .d . . . . .b. . ABC . a.c .” 17th ” . . . D . . .d . B . . .b. . A . C . a.c .” 18th ” . B . D . . .d . . . . .b. . A . C . a.c .
• Project • Gutenberg • #26839 •
CH. II] GEODESICS.. 57
First Bank Island Second BankAfter 19th passage . . . . . . .d . B . D .b. . A . C . a.c .
” 20th ” . . . . . . .d . . . . .b. . ABCD a.c .” 21st ” . . . . . . .d . . . . .bc . ABCD a. . .” 22nd ” . . . . . . .d . . . . . . . . ABCD abc .” 23rd ” . . . . . .cd . . . . . . . . ABCD ab. .” 24th ” . . . . . . . . . . . . . . . . ABCD abcd
Prof. G. Tarry has suggested an extension of the problem, whichstill further complicates its solution. He supposes that each husbandtravels with a harem of m wives or concubines; moreover, as Mo-hammedan women are brought up in seclusion, it is reasonable to sup-pose that they would be unable to row a boat by themselves withoutthe aid of a man. But perhaps the difficulties attendant on the trav-els of one wife may be deemed sufficient for Christians, and I contentmyself with merely mentioning the increased anxieties experienced byMohammedans in similar circ*mstances.
Geodesics. Geometrical problems connected with finding theshortest routes from one point to another on a curved surfaceare often difficult, but geodesics on a flat surface or flat surfacesare in general readily determinable.
I append an instance*, but I should have hesitated to do so had notexperience shown that some readers do not readily see the solution. Itis as follows: A room is 30 feet long, 12 feet wide, and 12 feet high. Onthe middle line of one of the smaller side walls and one foot from theceiling is a wasp. On the middle line of the opposite wall and 11 feetfrom the ceiling is a fly. The wasp catches the fly by crawling all theway to it: the fly, paralysed by fear, remaining still. The problem is tofind the shortest route that the wasp can follow.
To obtain a solution we observe that we can cut a sheet of paper sothat, when folded properly, it will make a model to scale of the room.This can be done in several ways. If, when the paper is again spreadout flat, we can join the points representing the wasp and the fly bya straight line lying wholly on the paper we shall obtain a geodesicroute between them. Thus the problem is reduced to finding the wayof cutting out the paper which gives the shortest route of the kind.
* I heard a similar question propounded at Cambridge in 1903, but the only placewhere I have seen it in print is the Daily Mail, London, February 1, 1905.
• Project • Gutenberg • #26839 •
58 SOME GEOMETRICAL QUESTIONS. [CH. II
A
B
C
D
W
F
Here is the diagram corresponding to a solution of the above ques-tion, where A represents the floor, B and D the longer side-walls, Cthe ceiling, and W and F the positions on the two smaller side-wallsoccupied initially by the wasp and fly. In the diagram the square of thedistance between W and F is (32)2+(24)2; hence the distance is 40 feet.
Problems with Counters placed in a row. Numerous dynamicalproblems and puzzles may be illustrated with a box of counters, espe-cially if there are counters of two colours. Of course coins or pawnsor cards will serve equally well. I proceed to enumerate a few of theseplayed with counters placed in a row.
First Problem with Counters. The following problem must be fa-miliar to many of my readers. Ten counters (or coins) are placed ina row. Any counter may be moved over two of those adjacent to iton the counter next beyond them. It is required to move the coun-ters according to the above rule so that they shall be arranged in fiveequidistant couples.
If we denote the counters in their initial positions by the numbers1, 2, 3, 4, 5, 6, 7, 8, 9, 10, we proceed as follows. Put 7 on 10, then 5 on2, then 3 on 8, then 1 on 4, and lastly 9 on 6. Thus they are arrangedin pairs on the places originally occupied by the counters 2, 4, 6, 8, 10.
Similarly by putting 4 on 1, then 6 on 9, then 8 on 3, then 10 on
• Project • Gutenberg • #26839 •
CH. II] PROBLEMS WITH COUNTERS.. 59
7, and lastly 2 on 5, they are arranged in pairs on the places originallyoccupied by the counters 1, 3, 5, 7, 9.
If two superposed counters are reckoned as only one, solutions anal-ogous to those given above will be obtained by putting 7 on 10, then 5on 2, then 3 on 8, then 1 on 6, and lastly 9 on 4; or by putting 4 on 1,then 6 on 9, then 8 on 3, then 10 on 5, and lastly 2 on 7*.
There is a somewhat similar game played with eight counters, butin this case the four couples finally formed are not equidistant. Herethe transformation will be effected if we move 5 on 2, then 3 on 7, then4 on 1, and lastly 6 on 8. This form of the game is applicable equallyto (8 + 2n) counters, for if we move 4 on 1 we have left on one side ofthis couple a row of (8 + 2n− 2) counters. This again can be reducedto one of (8 + 2n − 4) counters, and in this way finally we have left 8counters which can be moved in the way explained above.
A more complete generalization would be the case of n counters,where each counter might be moved over the m counters adjacent toit on to the one beyond them.
Second Problem with Counters. Another problem of a somewhatsimilar kind is due to Tait†. Place four florins (or white counters)and four halfpence (or black counters) alternately in a line in contactwith one another. It is required in four moves, each of a pair of twocontiguous pieces, without altering the relative position of the pair, toform a continuous line of four halfpence followed by four florins.
His solution is as follows. Let a florin be denoted by a and ahalfpenny by b, and let ×× denote two contiguous blank spaces. Thenthe successive positions of the pieces may be represented thus:
Initially . . . . . . . . . . . . . . . . . × × a b a b a b a b .After the first move . . . . . b a a b a b a × × b .After the second move . . b a a b × × a a b b .After the third move . . . . b × × b a a a a b b .After the fourth move . . . b b b b a a a a × ×.
The operation is conducted according to the following rule. Sup-pose the pieces to be arranged originally in circular order, with twocontiguous blank spaces, then we always move to the blank space for
* Note by J. Fitzpatrick to a French translation of the third edition of this work,Paris, 1898.
† Philosophical Magazine, London, January, 1884, series 5, vol. xvii, p. 39.
• Project • Gutenberg • #26839 •
60 SOME GEOMETRICAL QUESTIONS. [CH. II
the time being that pair of coins which occupies the places next butone and next but two to the blank space on one assigned side of it.
A similar problem with 2n counters—n of them being white andn black—will at once suggest itself, and, if n is greater than 4, it canbe solved in n moves. I have however failed to find a simple rule whichcovers all cases alike, but solutions, due to M. Delannoy, have beengiven* for the four cases where n is of the form 4m, 4m + 2, 4m + 1, or4m+3; in the first two cases the first 1
2n moves are of pairs of dissimilar
counters and the last 12n moves are of pairs of similar counters; in the
last two cases, the first move is similar to that given above, namely, ofthe penultimate and antepenultimate counters to the beginning of therow, the next 1
2(n − 1) moves are of pairs of dissimilar counters, and
the final 12(n − 1) moves are of similar counters.
The problem is also capable of solution if we substitute the restric-tion that at each move the pair of counters taken up must be moved toone of the two ends of the row instead of the condition that the finalarrangement is to be continuous.
Tait suggested a variation of the problem by making it a conditionthat the two coins to be moved shall also be made to interchange places;in this form it would seem that 5 moves are required; or, in the generalcase, n + 1 moves are required.
Problems on a Chess-board with Counters or Pawns. The follow-ing three problems require the use of a chess-board as well as of countersor pieces of two colours. It is more convenient to move a pawn than acounter, and if therefore I describe them as played with pawns it is onlyas a matter of convenience and not that they have any connection withchess. The first is characterized by the fact that in every position notmore than two moves are possible; in the second and third problemsnot more than four moves are possible in any position. With these lim-itations, analysis is possible. I shall not discuss the similar problemsin which more moves are possible.
First Problem with Pawns†. On a row of seven squares on a chess-board 3 white pawns (or counters), denoted in the diagram by “a”s, areplaced on the 3 squares at one end, and 3 black pawns (or counters),denoted by “b”s, are placed on the 3 squares at the other end—themiddle square being left vacant. Each piece can move only in one
* La Nature, June, 1887, p. 10.† Lucas, vol. ii, part 5, pp. 141-143.
• Project • Gutenberg • #26839 •
CH. II] PROBLEMS WITH COUNTERS OR PAWNS. 61
direction; the “a” pieces can move from left to right, and the “b” piecesfrom right to left. If the square next to a piece is unoccupied, it can
a a a b b b
move on to that; or if the square next to it is occupied by a piece ofthe opposite colour and the square beyond that is unoccupied, then itcan, like a queen in draughts, leap over that piece on to the unoccupiedsquare beyond it. The object is to get all the white pawns in the placesoccupied initially by the black pawns and vice versa.
The solution requires 15 moves. It may be effected by moving first awhite pawn, then successively two black pawns then three white pawns,then three black pawns, then three white pawns, then two black pawns,and then one white pawn. We can express this solution by saying thatif we number the cells (a term used to describe each of the small squareson a chess-board) consecutively, then initially the vacant space occupiesthe cell 4 and in the successive moves it will occupy the cells 3, 5, 6,4, 2, 1, 3, 5, 7, 6, 4, 2, 3, 5, 4. Of these moves, six are simple andnine are leaps.
Similarly if we have n white pawns at one end of a row of 2n + 1cells, and n black pawns at the other end, they can be interchanged inn(n+2) moves, by moving in succession 1 pawn, 2 pawns, 3 pawns, . . . ,n − 1 pawns, n pawns, n pawns, n pawns, n − 1 pawns, . . . , 2 pawns,and 1 pawn—all the pawns in each group being of the same colour anddifferent from that of the pawns in the group preceding it. Of thesemoves 2n are simple and n2 are leaps.
Second Problem with Pawns*. A similar game may be played ona rectangular or square board. The case of a square board containing49 cells, or small squares, will illustrate this sufficiently: in this casethe initial position is shown in the annexed diagram where the “a”sdenote the pawns or pieces of one colour, and the “b”s those of theother colour. The “a” pieces can move horizontally from left to rightor vertically down, and the “b” pieces can move horizontally from rightto left or vertically up, according to the same rules as before.
The solution reduces to the preceding case. The pieces in the mid-dle column can be interchanged in 15 moves. In the course of thesemoves every one of the seven cells in that column is at some time or
* Lucas, vol. ii, part 5, p. 144.
• Project • Gutenberg • #26839 •
62 SOME GEOMETRICAL QUESTIONS. [CH. II
a a a a b b b
a a a a b b b
a a a a b b b
a a a b b b
a a a b b b b
a a a b b b b
a a a b b b b
other vacant, and whenever that is the case the pieces in the row con-taining the vacant cell can be interchanged. To interchange the piecesin each of the seven rows will require 15 moves. Hence to interchangeall the pieces will require 15 + (7× 15) moves, that is, 120 moves.
If we place 2n(n + 1) white pawns and 2n(n + 1) black pawns ina similar way on a square board of (2n + 1)2 cells, we can transposethem in 2n(n + 1)(n + 2) moves: of these 4n(n + 1) are simple and2n2(n + 1) are leaps.
Third Problem with Pawns. The following analogous, thoughsomewhat more complicated, game was I believe originally publishedin the first edition of this work: but I find that it has been since widely
a b c
d e f
g h ∗ H G
F E D
C B A
distributed in connexion with an advertisem*nt and probably now iswell-known. On a square board of 25 cells, place eight white pawns orcounters on the cells denoted by small letters in the annexed diagram,and eight black pawns or counters on the cells denoted by capital let-ters: the cell marked with an asterisk (∗) being left blank. Each pawncan move according to the laws already explained—the white pawnsbeing able to move only horizontally from left to right or verticallydownwards, and the black pawns being able to move only horizontallyfrom right to left or vertically upwards. The object is to get all the
• Project • Gutenberg • #26839 •
CH. II] PROBLEMS WITH CHESS-PIECES 63
white pawns in the places initially occupied by the black pawns andvice versa. No moves outside the dark line are permitted.
Since there is only one cell on the board which is unoccupied, andsince no diagonal moves and no backward moves are permitted, it fol-lows that at each move not more than two pieces of either colour arecapable of moving. There are however a very large number of solu-tions. The following empirical solution in forty-eight moves is one wayof effecting the transfer—the letters indicating the cells from which thepieces are successively moved:
h H ∗ f F E H G ∗ c b h g d f F C ∗ h H B A C ∗c a b h H ∗ c f F D G H B C ∗ g h e f F ∗ h H ∗ .
It will be noticed that the first twenty-four moves lead to a symmet-rical position, and that the next twenty-three moves can be at onceobtained by writing the first twenty-three moves in reverse order andinterchanging small and capital letters.
Probably, were it worth the trouble, the mathematical theory ofgames such as that just described might be worked out by the useof Vandermonde’s notation, described later in chapter vi, or by theanalogous method employed in the theory of the game of solitaire*. Ibelieve that this has not been done, and I do not think it would repaythe labour involved.
Problems on a Chess-board with Chess-pieces. There are severalmathematical recreations with chess-pieces, other than pawns, some-what similar to those given above. One of these, on the determinationof the ways in which eight queens can be placed on a board so that noqueen can take any other, is given later in chapter iv. Another, on thepath to be followed by a knight which is moved on a chess-board so thatit shall occupy every cell once and only once, is given in chapter vi.Here I will mention one of the simplest of such problems, which is in-teresting from the fact that it is given in Guarini’s manuscript writtenin 1512; it was quoted by Lucas, but so far as I know has not beenotherwise published.
Guarini’s Problem. On a board of nine cells, such as that drawnbelow, the two white knights are placed on the two top corner cells
* On the theory of the solitaire, see Reiss, ‘Beitrage zur Theorie des Solitar-Spiels,’Crelle’s Journal, Berlin, 1858, vol. liv, pp. 344–379; and Lucas, vol. i, part v,pp. 89–141.
• Project • Gutenberg • #26839 •
64 SOME GEOMETRICAL QUESTIONS. [CH. II
a C d
D B
b A c
(a, d), and the two black knights on the two bottom corner cells (b, c):the other cells are left vacant. It is required to move the knights sothat the white knights shall occupy the cells b and c, while the blackshall occupy the cells a and d.
The solution is tolerably obvious. First, move the pieces from a toA, from b to B, from c to C, and from d to D. Next, move the piecesfrom A to d, from B to a, from C to b, and from D to c. The effect ofthese eight moves is the same as if the original square had been rotatedthrough one right angle. Repeat the above process, that is, move thepieces successively from a to A, from b to B, from c to C, from d toD; from A to d, from B to a, from C to b, and from D to c. Therequired result is then attained.
Geometrical Puzzles with Rods, etc. Another species ofgeometrical puzzles, to which here I will do no more than allude, aremade of steel rods, or of wire, or of wire and string. Numbers of theseare often sold in the streets of London for a penny each, and someof them afford ingenious problems in the geometry of position. Mostof them could hardly be discussed without the aid of diagrams, butthey are inexpensive to construct, and in fact innumerable puzzles ongeometry of position can be made with a couple of stout sticks and aball of string, or even with only a box of matches: several examplesare given in the appendix to the fourth volume of the 1723 edition ofOzanam’s work. I will mention, as an easy example, analogous to onegroup of the string puzzles, that any one can take off his waistcoat(which may be unbuttoned) without taking off his coat, and withoutpulling the waistcoat over the head like a jersey.
This last feat may serve to show the difficulty of mentally realizingthe effect of geometrical alterations in a figure unless they are of thesimplest character.
Paradromic Rings. The fact just stated is illustrated by thefamiliar experiment of making paradromic rings by cutting a paper
• Project • Gutenberg • #26839 •
CH. II] PARADROMIC RINGS. 65
ring prepared in the following manner.Take a strip of paper or piece of tape, say, for convenience, an inch
or two wide and at least nine or ten inches long, rule a line in themiddle down the length AB of the strip, gum one end over the otherend B, and we get a ring like a section of a cylinder. If this ring is cutby a pair of scissors along the ruled line we obtain two rings exactlylike the first, except that they are only half the width. Next supposethat the end A is twisted through two right angles before it is gummedto B (the result of which is that the back of the strip at A is gummedover the front of the strip at B), then a cut along the line will produceonly one ring. Next suppose that the end A is twisted once completelyround (i.e. through four right angles) before it is gummed to B, thena similar cut produces two interlaced rings. If any of my readers thinkthat these results could be predicted off-hand, it may be interesting tothem to see if they can predict correctly the effect of again cutting therings formed in the second and third experiments down their middlelines in a manner similar to that above described.
The theory is due to J.B. Listing* who discussed the case whenthe end A receives m half-twists, that is, is twisted through mπ, beforeit is gummed to B.
If m is even we obtain a surface which has two sides and two edges,which are termed paradromic. If the ring is cut along a line midwaybetween the edges, we obtain two rings, each of which has m half-twists,and which are linked together 1
2m times.
If m is odd we obtain a surface having only one side and one edge.If this ring is cut along its mid-line, we obtain only one ring, but it has2m half-twists, and if m is greater than unity it is knotted.
* Vorstudien zur Topologie, Die Studien, Gottingen, 1847, part x.
• Project • Gutenberg • #26839 •
CHAPTER III.
SOME MECHANICAL QUESTIONS.
I proceed now to enumerate a few questions connected with me-chanics which lead to results that seem to me interesting from a his-torical point of view or paradoxical. Problems in mechanics generallyinvolve more difficulties than problems in arithmetic, algebra, or geome-try, and the explanations of some phenomena—such as those connectedwith the flight of birds—are still incomplete, while the explanations ofmany others of an interesting character are too difficult to find a placein a non-technical work. Here, however, I shall confine myself to ques-tions which, like those treated in the two preceding chapters, are ofan elementary, not to say trivial, character; and the conclusions arewell-known to mathematicians.
I assume that the reader is acquainted with the fundamental ideasof kinematics and dynamics, and is familiar with the three Newtonianlaws; namely, first that a body will continue in its state of rest or ofuniform motion in a straight line unless compelled to change that stateby some external force: second, that the change of momentum perunit of time is proportional to the external force and takes place inthe direction of it: and third, that the action of one body on anotheris equal in magnitude but opposite in direction to the reaction of thesecond body on the first. The first and second laws state the principlesrequired for solving any question on the motion of a particle under theaction of given forces. The third law supplies the additional principlerequired for the solution of problems in which two or more particlesinfluence one another.
66
• Project • Gutenberg • #26839 •
CH. III] ZENO’S PARADOXES.. 67
Motion. The difficulties connected with the idea of motionhave been for a long time a favourite subject for paradoxes, some ofwhich bring us into the realm of the philosophy of mathematics.
Zeno’s Paradoxes on Motion. One of the earliest of these is theremark of Zeno to the effect that since an arrow cannot move whereit is not, and since also it cannot move where it is (i.e. in the space itexactly fills), it follows that it cannot move at all. The answer that thevery idea of the motion of the arrow implies the passage from where itis to where it is not was rejected by Zeno, who seems to have thoughtthat the appearance of motion of a body was a phenomenon caused bythe successive appearances of the body at rest but in different positions.
Zeno also asserted that the idea of motion was itself inconceivable,for what moves must reach the middle of its course before it reachesthe end. Hence the assumption of motion presupposes another motion,and that in turn another, and so ad infinitum. His objection was infact analogous to the biological difficulty expressed by Swift:—
“So naturalists observe, a flea hath smaller fleas that on him prey.And these have smaller fleas to bite ’em. And so proceed ad
infinitum.”
Or as De Morgan preferred to put it
“Great fleas have little fleas upon their backs to bite ’em,And little fleas have lesser fleas, and so ad infinitum.And the great fleas themselves, in turn, have greater fleas to go
on;While these have greater still, and greater still, and so on.”
Achilles and the Tortoise. Zeno’s paradox about Achilles and thetortoise is known even more widely. The assertion was that if Achillesran ten times as fast as a tortoise, yet if the tortoise had (say) 1000yards start it could never be overtaken. To establish this, Zeno arguedthat when Achilles had gone the 1000 yards, the tortoise would still be100 yards in front of him; by the time he had covered these 100 yards,it would still be 10 yards in front of him; and so on for ever. ThusAchilles would get nearer and nearer to the tortoise but would neverovertake it. Zeno regarded this as confirming his view that the popularidea of motion is self-contradictory.
• Project • Gutenberg • #26839 •
68 MECHANICAL RECREATIONS. [CH. III
Zeno’s Paradox on Time. The fallacy of Achilles and the Tortoiseis usually explained by saying that though the time required to over-take the tortoise can be divided into an infinite number of intervals,as stated in the argument, yet these intervals get smaller and smallerin geometrical progression, and the sum of them all is a finite time:after the lapse of that time Achilles would be in front of the tortoise.Probably Zeno would have replied that this explanation rests on theassumption that space and time are infinitely divisible, propositionswhich he would not admit. He seems further to have contended thatwhile, to an accurate thinker, the notion of the infinite divisibility oftime was impossible, it was equally impossible to think of a minimummeasure of time. For suppose, he argued, that τ is the smallest con-ceivable interval, and suppose that three horizontal lines composed ofthree consecutive spans abc, a′b′c′, a′′b′′c′′ are placed so that aa′a′′, bb′b′′,cc′c′′ are vertically over one another. Imagine the second line movedas a whole one span to the right in the time τ , and simultaneouslythe third line moved as a whole one span to the left. Then b, a′, c′′
will be vertically over one another. And in this duration τ (which byhypothesis is indivisible) c′ must have passed vertically over a′′. Hencethe duration is divisible, contrary to the hypothesis.
The Paradox of Tristram Shandy. Mr Russell has enunciated* aparadox somewhat similar to that of Achilles and the Tortoise, savethat the intervals of time considered get longer and longer during thecourse of events. Tristram Shandy, as we know, took two years writingthe history of the first two days of his life, and lamented that, at thisrate, material would accumulate faster than he could deal with it, sothat he could never come to an end, however long he lived. But hadhe lived long enough, and not wearied of his task, then, even if his lifehad continued as eventfully as it began, no part of his biography wouldhave remained unwritten. For if he wrote the events of the first dayin the first year, he would write the events of the nth day in the nthyear, hence in time the events of any assigned day would be written,and therefore no part of his biography would remain unwritten. Thisargument might be put in the form of a demonstration that the partof a magnitude may be equal to the whole of it.
Questions, such as those given above, which are concerned withthe continuity and extent of space and time involve difficulties of a
* B.A.W. Russell, Principles of Mathematics, Cambridge, 1903, vol. i, p. 358.
• Project • Gutenberg • #26839 •
CH. III] ANGULAR MOTION. 69
high order.Angular Motion. A non-mathematician finds additional difficul-
ties in the idea of angular motion. For instance, here is a well-knownproposition on motion in an equiangular spiral (of which the result istrue on the ordinary conventions of mathematics) which shows that abody, moving with uniform velocity and as slowly as we please, may ina finite time whirl round a fixed point an infinite number of times.
The equiangular spiral is the trace of a point P , which moves alonga line OP , the line OP turning round a fixed point O with uniformangular velocity while the distance of P from O decreases with thetime in geometrical progression. If the radius vector rotates throughfour right angles we have one convolution of the curve. All convolutionsare similar, and the length of each convolution is a constant fraction,say 1/nth, that of the convolution immediately outside it. Inside anygiven convolution, there are an infinite number of convolutions whichget smaller and smaller as we get nearer the pole. Now suppose a pointQ to move uniformly along the spiral from any point towards the pole.If it covers the first convolution in a seconds, it will cover the next ina/n seconds, the next in a/n2 seconds, and so on, and will finally reachthe pole in (a+a/n+a/n2 +a/n3 + · · · ) seconds, that is, in an/(n−1)seconds. The velocity is uniform, and yet in a finite time, Q will havetraversed an infinite number of convolutions and therefore have circledround the pole an infinite number of times*.
Simple Relative Motion. Even if the philosophical difficulties sug-gested by Zeno are settled or evaded, the mere idea of relative motionhas been often found to present difficulties, and Zeno himself failed toexplain a simple phenomenon involving the principle. As one of theeasiest examples of this kind, I may quote the common question of howmany trains going from B to A a passenger from A to B would meetand pass on his way, assuming that the journey either way takes 41
2
hours and that the trains start from each end every hour. The answeris 9. Or again this: Take two pennies, face upwards on a table andedges in contact. Suppose that one is fixed and that the other rollson it without slipping, making one complete revolution round it andreturning to its initial position. How many revolutions round its owncentre has the rolling coin made? The answer is 2.
* The proposition is put in this form in J. Richard’s Philosophie des mathemat-iques, Paris, 1903, pp. 119–120.
• Project • Gutenberg • #26839 •
70 MECHANICAL RECREATIONS. [CH. III
Laws of Motion. I proceed next to make a few remarks on pointsconnected with the laws of motion.
The first law of motion is often said to define force, but it is in onlya qualified sense that this is true. Probably the meaning of the law isbest expressed in Clifford’s phrase, that force is “the description of acertain kind of motion”—in other words it is not an entity but merely aconvenient way of stating, without circumlocution, that a certain kindof motion is observed.
It is not difficult to show that any other interpretation lands us indifficulties. Thus some authors use the law to justify a definition thatforce is that which moves a body or changes its motion; yet the samewriters speak of a steam-engine moving a train. It would seem thenthat, according to them, a steam-engine is a force. That such state-ments are current may be fairly reckoned among mechanical paradoxes.
The idea of force is difficult to grasp. How many people, for in-stance, could predict correctly what would happen in a question as sim-ple as the following? A rope (whose weight may be neglected) hangsover a smooth pulley; it has one end fastened to a weight of 10 stone,and the other end to a sailor of weight 10 stone, the sailor and theweight hanging in the air. The sailor begins to climb up the rope; willthe weight move at all; and, if so, will it rise or fall?
It will be noted that in the first law of motion it is asserted that,unless acted on by an external force, a body in motion continues tomove (i) with uniform velocity, and (ii) in a straight line.
The tendency of a body to continue in its state of rest or of uniformmotion is called its inertia. This tendency may be used to explainvarious common phenomena and experiments. Thus, if a number ofdominoes or draughts are arranged in a vertical pile, a sharp horizontalblow on one of those near the bottom will send it out of the pile, andthose above will merely drop down to take its place—in fact they havenot time to change their relative positions before there is sufficient spacefor them to drop vertically as if they were a solid body.
This also is the principle on which depends the successful playingof “Aunt Sally,” and the performance of numerous tricks, described incollections of mathematical puzzles*.
* See Les recreations scientifiques by G. Tissandier, where several ingenious il-lustrations of inertia are given.
• Project • Gutenberg • #26839 •
CH. III] LAWS OF MOTION. 71
The statement about inertia in the first law may be taken to implythat a body set in rotation about a principal axis passing through itscentre of mass will continue to move with a uniform angular velocityand to keep its axis of rotation fixed in direction. The former of thesestatements is the assumption on which our measurement of time isbased as mentioned below in chapter xiii. The latter assists us toexplain the motion of a projectile in a resisting fluid. It affords theexplanation of why the barrel of a rifle is grooved; and why, similarly,anyone who has to throw a flat body of irregular shape (such as a card)in a given direction usually gives it a rapid rotatory motion abouta principal axis. Elegant illustrations of the fact just mentioned areafforded by a good many of the tricks of acrobats, though the fullexplanation of most of them also introduces other considerations. Thuswhen some few years ago the Japanese village at Knightsbridge was oneof the shows of London, there were some acrobats there who tossed onto the top surface of an umbrella a penny so that it alighted on itsedge, and then, by turning round the stick of the umbrella rapidly, thecoin was caused to rotate, but as the umbrella moved away underneathit the coin remained apparently stationary and standing upright, whileby diminishing or increasing the angular velocity of the umbrella thepenny was caused to run forwards or backwards. This is not a difficulttrick to execute.
The tendency of a body in motion to continue to move in a straightline is sometimes called its centrifugal force. Thus, if a train is runninground a curve, it tends to move in a straight line, and is constrained onlyby the pressure of the rails to move in the required direction. Hence itpresses on the outer rail of the curve. This pressure can be diminishedto some extent both by raising the outer rail, and by putting a guardrail, parallel and close to the inner rail, against which the wheels onthat side also will press.
An illustration of this fact occurred in a little known incident ofthe American civil war*. In the spring of 1862 a party of volunteersfrom the North made their way to the rear of the Southern armies andseized a train, intending to destroy, as they passed along it, the railwaywhich was the main line of communication between various confederatecorps and their base of operations. They were however detected andpursued. To save themselves, they stopped on a sharp curve and tore
* Capturing a Locomotive by W. Pittenger, London, 1882, p. 104.
• Project • Gutenberg • #26839 •
72 MECHANICAL RECREATIONS. [CH. III
up some rails so as to throw the engine which was following them off theline. Unluckily for themselves they were ignorant of dynamics and toreup the inner rails of the curve, an operation which did not incommodetheir pursuers.
The second law gives us the means of measuring mass, force, andtherefore work . A given agent in a given time can do only a definiteamount of work. This is illustrated by the fact that although, by meansof a rigid lever and a fixed fulcrum, any force however small may becaused to move any mass however large, yet what is gained in power islost in speed—as the popular phrase runs.
Montucla* inserted a striking illustration of this principle foundedon the well-known story of Archimedes who is said to have declaredto Hiero that, were he but given a fixed fulcrum, he could move theworld. Montucla calculated the mass of the earth and, assuming thata man could work incessantly at the rate of 116 foot-lbs. per second,which is a very high estimate, he found that it would take over threebillion centuries, i.e. 3× 1014 years, before a mass equal to that of theearth was moved as much as one inch against gravity at the surfaceof the earth: to move it one inch along a horizonal plane would takeabout 74000 centuries.
Stability of Equilibrium. It is known to all those who have readthe elements of mechanics that the centre of gravity of a body, which isresting in equilibrium under its own weight, must be vertically above itsbase: also, speaking generally, we may say that, if every small displace-ment has the effect of raising the centre of gravity, then the equilibriumis stable, that is, the body when left to itself will return to its originalposition; but, if a displacement has the effect of lowering the centre ofgravity, then for that displacement the equilibrium is unstable; while, ifevery displacement does not alter the height above some fixed plane ofthe centre of gravity, then the equilibrium is neutral. In other words,if in order to cause a displacement work has to be done against theforces acting on the body, then for that displacement the equilibriumis stable, while if the forces do work the equilibrium is unstable.
A good many of the simpler mechanical toys and tricks afford il-lustrations of this principle.
* Ozanam, 1803 edition, vol. ii, p. 18; 1840 edition, p. 202.
• Project • Gutenberg • #26839 •
CH. III] MAGIC BOTTLES. 73
Magic Bottles*. Among the most common of such toys are thesmall bottles—trays of which may be seen any day in the streets ofLondon—which keep always upright, and cannot be upset until theirowner orders them to lie down. Such a bottle is made of thin glassor varnished paper fixed to the plane surface of a solid hemisphere orsmaller segment of a sphere. Now the distance of the centre of gravityof a hom*ogeneous hemisphere from the centre of the sphere is three-eighths of the radius, and the mass of the glass or varnished paperis so small compared with the mass of the lead base that the centreof gravity of the whole bottle is still within the hemisphere. Let usdenote the centre of the hemisphere by C, and the centre of gravityof the bottle by G.
If such a bottle is placed with the hemisphere resting on a horizontalplane and GC vertical, any small displacement on the plane will tendto raise G, and thus the equilibrium is stable. This may be seen alsofrom the fact that when slightly displaced there is brought into play acouple, of which one force is the reaction of the table passing throughC and acting vertically upward, and the other the weight of the bottleacting vertically downward at G. If G is below C, this couple tends torestore the bottle to its original position.
If there is dropped into the bottle a shot or nail so heavy as toraise the centre of gravity of the whole above C, then the equilibriumis unstable, and, if any small displacement is given, the bottle fallsover on to its side.
Montucla says that in his time it was not uncommon to see boxesof tin soldiers mounted on lead hemispheres, and when the lid of thebox was taken off the whole regiment sprang to attention.
In a similar way we may explain how to balance a pencil in a verticalposition, with its point resting on the top of one’s finger, an experimentwhich is described in nearly every book of puzzles†. This is effected bytaking a penknife, of which one blade is opened through an angle of(say) 120◦, and sticking the blade in the pencil so that the handle of thepenknife is below the finger. The centre of gravity is thus brought belowthe point of support, and a small displacement given to the pencil willraise the centre of gravity of the whole: thus the equilibrium is stable.
* Ozanam, 1803 edition, vol. ii, p. 15; 1840 edition, p. 201.† Ex. gr. Oughtred, Mathematical Recreations, p. 24; Ozanam, 1803 edition, vol. ii,
p. 14; 1840 edition, p. 200.
• Project • Gutenberg • #26839 •
74 MECHANICAL RECREATIONS. [CH. III
Other similar tricks are the suspension of a bucket over the edge ofa table by a couple of sticks, and the balancing of a coin on the edge ofa wine-glass by the aid of a couple of forks*—the sticks or forks beingso placed that the centre of gravity of the whole is vertically below thepoint of support and its depth below it a maximum.
The toy representing a horseman, whose motion continually bringshim over the edge of a table into a position which seems to ensureimmediate destruction, is constructed in somewhat the same way. Awire has one end fixed to the feet of the rider; the wire is curved down-wards and backwards, and at the other end is fixed a weight. Whenthe horse is placed so that his hind legs are near the edge of the tableand his forefeet over the edge, the weight is under his hind feet. Thusthe whole toy forms a pendulum with a curved instead of a straightrod. Hence the farther it swings over the table, the higher is the cen-tre of gravity raised, and thus the toy tends to return to its originalposition of equilibrium.
An elegant modification of the prancing horse was brought out atParis in 1890 in the shape of a toy made of tin and in the figure of aman†. The legs are pivoted so as to be movable about the thighs, butwith a wire check to prevent too long a step, and the hands are fastenedto the top of a
⋂-shaped wire weighted at its ends. If the figure is placed
on a narrow sloping plank or strip of wood passing between the legs ofthe
⋂, then owing to the
⋂-shaped wire any lateral displacement of the
figure will raise its centre of gravity, and thus for any such displacementthe equilibrium is stable. Hence, if a slight lateral disturbance is given,the figure will oscillate and will rest alternately on each foot: when itis supported by one foot the other foot under its own weight movesforwards, and thus the figure will walk down the plank though with aslight reeling motion. Shortly after the publication of the third editionof this book an improved form of this toy, in the shape of a walkingelephant made in heavy metal, was issued in England, and probablyin that form it is now familiar to all who are interested in noticingstreet toys.
Columbus’s Egg. The toy known as Columbus’s egg depends onthe same principle as the magic bottle, though it leads to the converseresult. The shell of the egg is made of tin and cannot be opened. Inside
* Oughtred, p. 30; Ozanam, 1803 edition, vol. ii, p. 12; 1840 edition, p. 199.† La Nature, Paris, March, 1891.
• Project • Gutenberg • #26839 •
CH. III] CONES RUNNING UP HILL. 75
it and fastened to its base is a hollow truncated tin cone, and there isalso a loose marble inside the shell. If the egg is held properly, themarble runs inside the cone and the egg will stand on its base, butso long as the marble is outside the cone, the egg cannot be made tostand on its base.
Cones running up hill*. The experiment to make a double conerun up hill depends on the same principle as the toys above described;namely, on the tendency of a body to take a position so that its centreof gravity is as low as possible. In this case it produces the opticaleffect of a body moving by itself up a hill.
Usually the experiment is performed as follows. Arrange two sticksin the shape of a
∨, with the apex on a table and the two upper
ends resting on the top edge of a book placed on the table. Take twoequal cones fixed base to base, and place them with the curved surfacesresting on the sticks near the apex of the
∨, the common axis of the
cones being horizontal and parallel to the edge of the book. Then, ifproperly arranged, the cones will run up the plane formed by the sticks.
The explanation is obvious. The centre of gravity of the conesmoves in the vertical plane midway between the two sticks and it occu-pies a lower position as the points of contact on the sticks get fartherapart. Hence as the cone rolls up the sticks its centre of gravity de-scends.
Perpetual Motion. The idea of making a machine whichonce set going would continue to go for ever by itself has been theignis fatuus of self-taught mechanicians in much the same way as thequadrature of the circle has been of self-taught geometricians.
Now the obvious meaning of the third law of motion is that a force isonly one aspect of a stress, and that whenever a force is caused anotherequal and opposite one is brought also into existence—though it mayact upon a different body, and thus be immaterial for the particularproblem considered. The law however is capable of another interpreta-tion†, namely, that the rate at which an agent does work (that is, itsaction) is equal to the rate at which work is done against it (that is,its reaction). If it is allowable to include in the reaction the rate atwhich kinetic energy is being produced, and if work is taken to include
* Ozanam, 1803 edition, vol. ii, p. 49; 1840 edition, p. 216.† Newton’s Principia, last paragraph of the Scholium to the Laws of Motion.
• Project • Gutenberg • #26839 •
76 MECHANICAL RECREATIONS. [CH. III
that done against molecular forces, then it follows from this interpre-tation that the work done by an agent on a system is equivalent to thetotal increase of energy, that is, the power of doing work. Hence inan isolated system the total amount of energy is constant. If this isgranted, then since friction and some molecular dissipation of energycannot be wholly prevented, it must be impossible to construct in anisolated system a machine capable of perpetual motion.
I do not propose to describe in detail the various machines forproducing perpetual motion which have been suggested, but I may addthat a number of them are equivalent essentially to the one of which asection is represented in the accompanying figure.
C
It consists of two concentric vertical wheels in the same plane,and mounted on a horizontal axle through their centre, C. The spacebetween the wheels is divided into compartments by spokes inclined ata constant angle to the radii to the points whence they are drawn, andeach compartment contains a heavy bullet. Apart from these bullets,the wheels would be in equilibrium. Each bullet tends to turn thewheels round their axle, and the moment which measures this tendencyis the product of the weight of the bullet and its distance from thevertical through C.
The idea of the constructors of such machines was that, as thebullet in any compartment would roll under gravity to the lowest pointof the compartment, the bullets on the right-hand side of the diagramwould be farther from the vertical through C than those on the left.Hence the sum of the moments of the weights of the bullets on the
• Project • Gutenberg • #26839 •
CH. III] PERPETUAL MOTION. 77
right would be greater than the sum of the moments of those on theleft. Thus the wheels would turn continually in the same direction asthe hands of a watch. The fallacy in the argument is obvious.
Another large group of machines for producing perpetual motiondepended on the use of a magnet to raise a mass which was then allowedto fall under gravity. Thus, if the bob of a simple pendulum was madeof iron, it was thought that magnets fixed near the highest points whichwere reached by the bob in the swing of the pendulum would draw thebob up to the same height in each swing and thus give perpetual motion.
Of course it is only in isolated systems that the total amount ofenergy is constant, and, if a source of external energy can be obtainedfrom which energy is continually introduced into the system, perpet-ual motion is, in a sense, possible; though even here materials wouldultimately wear out. The solar heat and the tides are among the mostobvious of such sources.
There was at Paris in the latter half of the eighteenth century aclock which was an ingenious illustration of such perpetual motion*.The energy which was stored up in it to maintain the motion of thependulum was provided by the expansion of a silver rod. This expansionwas caused by the daily rise of temperature, and by means of a trainof levers it wound up the clock. There was a disconnecting apparatus,so that the contraction due to a fall of temperature produced no effect,and there was a similar arrangement to prevent overwinding. I believethat a rise of eight or nine degrees Fahrenheit was sufficient to windup the clock for twenty-four hours.
I have in my possession a watch, which produces the same effectby somewhat different means. Inside the case is a steel weight, and ifthe watch is carried in a pocket this weight rises and falls at every stepone takes, somewhat after the manner of a pedometer. The weight israised by the action of the person who has it in his pocket in taking astep, and in falling it winds up the spring of the watch. On the face isa small dial showing the number of hours for which the watch is woundup. As soon as the hand of this dial points to fifty-six hours, the trainof levers which winds up the watch disconnects automatically, so asto prevent overwinding the spring, and it reconnects again as soon asthe watch has run down eight hours. The watch is an excellent time-keeper, and a walk of about a couple of miles is sufficient to wind it
* Ozanam, 1803 edition, vol. ii, p. 105; 1840 edition, p. 238.
• Project • Gutenberg • #26839 •
78 MECHANICAL RECREATIONS. [CH. III
up for twenty-four hours.
Models. I may add here the observation, which is well knownto mathematicians, but is a perpetual source of disappointment to igno-rant inventors, that it frequently happens that an accurate model of amachine will work satisfactorily while the machine itself will not do so.
One reason for this is as follows. If all the parts of a model aremagnified in the same proportion, say m, and if thereby a line in it isincreased in the ratio m : 1, then the areas and volumes in it will beincreased respectively in the ratios m2 : 1 and m3 : 1. For example, ifthe side of a cube is doubled then a face of it will be increased in theratio 4 : 1 and its volume will be increased in the ratio 8 : 1.
Now if all the linear dimensions are increased m times, then someof the forces that act on a machine (such, for example, as the weightof part of it) will be increased m3 times, while others which depend onarea (such as the sustaining power of a beam) will be increased onlym2 times. Hence the forces that act on the machine and are broughtinto play by the various parts may be altered in different proportions,and thus the machine may be incapable of producing results similar tothose which can be produced by the model.
The same argument has been adduced in the case of animal lifeto explain why very large specimens of any particular breed or speciesare usually weak. For example, if the linear dimensions of a bird wereincreased n times, the work necessary to give the power of flight wouldhave to be increased no less than n7 times*. Again, if the linear di-mensions of a man of height 5 ft. 10 in. were increased by one-seventhhis height would become 6 ft. 8 in., but his weight would be increasedin the ratio 512 : 343 (i.e. about half as much again), while the crosssections of his legs, which would have to bear this weight, would beincreased only in the ratio 64 : 49; thus in some respects he would beless efficient than before. Of course the increased dimensions, length oflimb, or size of muscle might be of greater advantage than the relativeloss of strength; hence the problem of what are the most efficient pro-portions is not simple, but the above argument will serve to illustratethe fact that the working of a machine may not be similar to that ofa model of it.
* Helmholtz, Gesammelte Abhandlungen, Leipzig, 1881, vol. i, p. 165.
• Project • Gutenberg • #26839 •
CH. III] SAILING QUICKER THAN THE WIND. 79
Leaving now these elementary considerations I pass on to someother mechanical questions.
Sailing quicker than the Wind. As a kinematical paradoxI may allude to the possibility of sailing quicker than the wind blows ,a fact which strikes many people as curious.
The explanation* depends on the consideration of the velocity ofthe wind relative to the boat. Perhaps, however, a non-mathematicianwill find the solution simplified if I consider first the effect of the wind-pressure on the back of the sail which drives the boat forward, andsecond the resistance to motion caused by the sail being forced throughthe air.
When the wind is blowing against a plane sail the resultant pres-sure of the wind on the sail may be resolved into two components, oneperpendicular to the sail (but which in general is not a function onlyof the component velocity in that direction, though it vanishes whenthat component vanishes) and the other parallel to its plane. The lat-ter of these has no effect on the motion of the ship. The componentperpendicular to the sail tends to move the ship in that direction. Thispressure, normal to the sail, may be resolved again into two compo-nents, one in the direction of the keel of the boat, the other in thedirection of the beam of the boat. The former component drives theboat forward, the latter to leeward. It is the object of a boat-builder toconstruct the boat on lines so that the resistance of the water to motionforward shall be as small as possible, and the resistance to motion ina perpendicular direction (i.e. to leeward) shall be as large as possible;and I will assume for the moment that the former of these resistancesmay be neglected, and that the latter is so large as to render motionin that direction impossible.
Now, as the boat moves forward, the pressure of the air on thefront of the sail will tend to stop the motion. As long as its componentnormal to the sail is less than the pressure of the wind behind the sailand normal to it, the resultant of the two will be a force behind the sailand normal to it which tends to drive the boat forwards. But as thevelocity of the boat increases, a time will arrive when the pressure ofthe wind is only just able to balance the resisting force which is causedby the sail moving through the air. The velocity of the boat will not
* Ozanam, 1803 edition, vol. iii, pp. 359, 367; 1840 edition, pp. 540, 543.
• Project • Gutenberg • #26839 •
80 MECHANICAL RECREATIONS. [CH. III
B R
S
L
A
W
increase beyond this, and the motion will be then what mathematiciansdescribe as “steady.”
In the accompanying figure, let BAR represent the keel of a boat,B being the bow, and let SAL represent the sail. Suppose that thewind is blowing in the direction WA with a velocity u; and that thisdirection makes an angle θ with the keel, i.e. angle WAR = θ. Supposethat the sail is set so as to make an angle α with the keel, i.e. angleBAS = α, and therefore angle WAL = θ + α. Suppose finally that vis the velocity of the boat in the direction AB.
I have already shown that the solution of the problem depends onthe relative directions and velocities of the wind and the boat; henceto find the result reduce the boat to rest by impressing on it a velocityv in the direction BA. The resultant velocity of v parallel to BA andof u parallel to WA will be parallel to SL, if v sin α = u sin(θ + α); andin this case the resultant pressure perpendicular to the sail vanishes.
Thus, for steady motion we have v sin α = u sin(θ + α). Hence,whenever sin(θ + α) > sin α, we have v > u. Suppose, to take oneinstance, the sail to be fixed, that is, suppose α to be a constant. Thenv is a maximum if θ + α = 1
2π, that is, if θ is equal to the complement
of α. In this case we have v = u cosec α, and therefore v is greater thanu. Hence, if the wind makes the same angle α abaft the beam thatthe sail makes with the keel, the velocity of the boat will be greaterthan the velocity of the wind.
Next, suppose that the boat is running close to the wind, so thatthe wind is before the beam (see figure below), then in the same wayas before we have v sin α = u sin(θ + α), or v sin α = u sin ϕ, where
• Project • Gutenberg • #26839 •
CH. III] BOAT MOVED BY A ROPE.. 81
ϕ = angle WAS = π − θ − α. Hence v = u sin ϕ cosec α.Let w be the component velocity of the boat in the teeth of the
wind, that is, in the direction AW . Then we have w = v cos BAW =v cos(α + ϕ) = u sin ϕ cosec α cos(α + ϕ). If α is constant, this isa maximum when ϕ = 1
4π − 1
2α; and, if ϕ has this value, then
w = 12u(cosec α − 1). This formula shows that w is greater than
u, if sin α < 13. Thus, if the sails can be set so that α is less than
sin−1 13, that is, rather less than 19◦29′, and if the wind has the direc-
tion above assigned, then the component velocity of the boat in theface of the wind is greater than the velocity of the wind.
The above theory is curious, but it must be remembered that inpractice considerable allowance has to be made for the fact that noboat for use on water can be constructed in which the resistance to
B R
S
L
A
W
motion in the direction of the keel can be wholly neglected, or whichwould not drift slightly to leeward if the wind was not dead astern.Still this makes less difference than might be thought by a landsman.In the case of boats sailing on smooth ice the assumptions made aresubstantially correct, and the practical results are said to agree closelywith the theory.
Boat moved by a Rope. There is a form of boat-racing, occa-sionally used at regattas, which affords a somewhat curious illustrationof certain mechanical principles. The only thing supplied to the crew
• Project • Gutenberg • #26839 •
82 MECHANICAL RECREATIONS. [CH. III
is a coil of rope, and they have, without leaving the boat, to propel itfrom one point to another as rapidly as possible. The motion is given bytying one end of the rope to the after thwart, and giving the other enda series of violent jerks in a direction parallel to the keel. I am told thatin still water a pace of two or three miles an hour can be thus attained.
The chief cause for this result seems to be that the friction betweenthe boat and the water retards all relative motion, but is not greatenough to materially affect motion caused by a sufficiently big impulse.Hence the usual movements of the crew in the boat do not sensiblymove the centre of gravity of themselves and the boat, but this doesnot apply to an impulsive movement, and if the crew in making a jerkmove their centre of gravity towards the bow n times more rapidly thanit returns after the jerk, then the boat is impelled forwards at least ntimes more than backwards: hence on the whole the motion is forwards.
Motion of Fluids and Motion in Fluids. The theories ofmotion of fluids and motion in fluids involve considerable difficulties.Here I will mention only one or two instances—mainly illustrations ofHauksbee’s Law.
Hauksbee’s Law. When a fluid is in rapid motion the pressure isless than when it is at rest*. Thus, if a current of air is moving in a tube,the pressure on the sides of the tube is less than when the air is at rest—and the quicker the air moves the smaller is the pressure. This fact wasnoticed by Hauksbee nearly two centuries ago. In an elastic perfect fluidin which the pressure is proportional to the density, the law connectingthe pressure, p, and the steady velocity, v, is p = Πα−v2
where Π andα are constants: the establishment of corresponding formula for gaseswhere the pressure is proportional to a power of the density presentsno difficulty.
This principle is illustrated by a twopenny toy, on sale in mosttoy-shops, called the pneumatic mystery. It consists of a tube, with acup-shaped end in which rests a wooden ball. If the tube is held in avertical position, with the mouthpiece at the upper end and the cup at
* See Besant, Hydromechanics, Cambridge, 1867, art. 149, where however it isassumed that the pressure is proportional to the density. Hauksbee was theearliest writer who called attention to the problem, but I do not know who firstexplained the phenomenon; some references to it are given by Willis, CambridgePhilosophical Transactions, 1830, vol. iii, pp. 129–140.
• Project • Gutenberg • #26839 •
CH. III] SPIN ON TENNIS AND CRICKET BALLS.. 83
the lower end, then, if anyone blows hard through the tube and placesthe ball against the cup, the ball will remain suspended there. Theexplanation is that the pressure of the air below the ball is so muchgreater than the pressure of the air in the cup that the ball is held up.
The same effect may be produced by fastening to one end of a tubea piece of cardboard having a small hole in it. If a piece of paper isplaced over the hole and the experimenter blows through the tube, thepaper will not be detached from the card but will bend so as to allowthe egress of the air.
An exactly similar experiment, described in many text-books onhydromechanics, is made as follows. To one end of a straight tube aplane disc is fitted which is capable of sliding on wires projecting fromthe end of the tube. If the disc is placed at a small distance from theend, and anyone blows steadily into the tube, the disc will be drawntowards the tube instead of being blown off the wires, and will oscillateabout a position near the end of the tube.
In the same way we may make a tube by placing two books on atable with their backs parallel and an inch or so apart and laying asheet of newspaper over them. If anyone blows steadily through thetube so formed, the paper will be sucked in instead of being blown out.
The following experiment is explicable by the same argument. Onthe top of a vertical axis balance a thin horizontal rod. At each endof this rod fasten a small vertical square or sail of thin cardboard—thetwo sails being in the same plane. If anyone blows close to one of thesesquares and in a direction parallel to its plane, the square will movetowards the side on which one is blowing, and the rod with the twosails will rotate about the axis.
The experiments above described can be performed so as to illus-trate Hauksbee’s Law; but unless care is taken other causes will be alsointroduced which affect the phenomena: it is however unnecessary formy purpose to go into these details.
Cut on a Tennis-Ball. Racquet and tennis players know that if astrong cut is given to a ball it can be made to rebound off a verticalwall and then (without striking the floor or any other wall) return andhit the wall again.
This affords another illustration of Hauksbee’s Law. The expla-nation* is that the cut causes the ball to rotate rapidly about an axis
* See Magnus on ‘Die Abweichung der Geschosse’ in the Abhandlungen der
• Project • Gutenberg • #26839 •
84 MECHANICAL RECREATIONS. [CH. III
through its centre of figure, and the friction of the surface of the ballon the air produces a sort of whirlpool. This rotation is in addition toits motion of translation. Suppose the ball to be spherical and rotat-ing about an axis through its centre perpendicular to the plane of thepaper in the direction of the arrow-head, and at the same time movingthrough still air from left to right parallel to PQ. Any motion of theball perpendicular to PQ will be produced by the pressure of the airon the surface of the ball, and this pressure will, by Hauksbee’s Law,be greatest where the velocity of the air relative to the ball is least,and vice versa. To find the velocity of the air relative to the ball wemay reduce the centre of the ball to rest, and suppose a stream of airto impinge on the surface of the ball moving with a velocity equal andopposite to that of the centre of the ball. The air is not frictionless,and therefore the air in contact with the surface of the ball will be setin motion, by the rotation of the ball and will form a sort of whirlpoolrotating in the direction of the arrow-head in the figure. To find theactual velocity of this air relative to the ball we must consider how themotion due to the whirlpool is affected by the motion of the streamof air parallel to QP . The air at A in the whirlpool is moving againstthe stream of air there, and therefore its velocity is retarded: the airat B in the whirlpool is moving in the same direction as the streamof air there, and therefore its velocity is increased. Hence the relativevelocity of the air at A is less than that at B, and since the pressureof the air is greatest where the velocity is least, the pressure of the airon the surface of the ball at A is greater than on that at B, Hencethe ball is forced by this pressure in the direction from the line PQ,
P Q
A
B
which we may suppose to represent the section of the vertical wall in
Akademie der Wissenschaften, Berlin, 1852, pp. 1–23; Lord Rayleigh, ‘On theirregular flight of a tennis ball,’ Messenger of Mathematics, Cambridge, 1878,vol. vii, pp. 14–16.
• Project • Gutenberg • #26839 •
CH. III] FLIGHT OF BIRDS. 85
a racquet-court. In other words, the ball tends to move at right anglesto the line in which its centre is moving and in the direction in whichthe surface of the front of the ball is being carried by the rotation.
In the case of a lawn tennis-ball, the shape of the ball is altered bya strong cut, and this introduces additional complications.
Spin on a Cricket-Ball. The curl of a cricket-ball in its flightthrough the air, caused by a spin given by the bowler in deliveringthe ball, is explained by the same reasoning.
Thus suppose the ball is delivered in a direction lying in a verticalplane containing the two middle stumps of the wickets. A spin round ahorizontal axis parallel to the crease in a direction which the bowler’sumpire would describe as positive, namely, counter clock-wise, will, inconsequence of the friction of the air, cause it to drop, and thereforedecrease the length of the pitch. A spin in the opposite direction willcause it to rise, and therefore lengthen the pitch. A spin round avertical axis in the positive direction, as viewed from above, will makeit curl sideways in the air to the left, that is, from leg to off. A spinin the opposite direction will make it curl to the right. A spin givento the ball round the direction of motion of the centre of the ball willnot sensibly affect the motion through the air, though it would causethe ball, on hitting the ground, to break. Of course these various kindsof spin can be combined.
The questions involving the application of Hauksbee’s Law are easyas compared with many of the problems in fluid motion. The analy-sis required to attack most of these problems is beyond the scope ofthis book, but one of them may be worth mentioning even though noexplanation is given.
The Theory of the Flight of Birds. A mechanical problem of greatinterest is the explanation of the means by which birds are enabled tofly for considerable distances with no (perceptible) motion of the wings.Albatrosses, to take an instance of special difficulty, have been knownto follow for some days ships running at the rate of nine or ten knots,and sometimes for considerable periods there is no motion of the wingsor body which can be detected, while even if the bird moved its wingsit is not easy to understand how it has the muscular energy to propelitself so rapidly and for such a length of time. Of this phenomenon
• Project • Gutenberg • #26839 •
86 MECHANICAL RECREATIONS. [CH. III
various explanations* have been suggested. Notable among these areMr Maxim’s of upward air-currents, Lord Rayleigh’s of variations of thewind velocity at different heights above the ground, Dr S.P. Langley’sof the incessant occurrence of gusts of wind separated by lulls, andDr Bryan’s of vortices in the atmosphere.
It now seems reasonably certain that the second and third of thesesources of energy account for at least a portion of the observed phenom-ena. The effect of the third cause may be partially explained by notingthat the centre of gravity of the bird with extended wings is slightlybelow the aeroplane or wing surface, so that the animal forms a sortof parachute. The effect of a sudden gust of wind upon such a bodyis that the aeroplane is set in motion more rapidly than the suspendedmass, causing the structure to heel over so as to receive the wind onthe under surface of the aeroplane, and this lifts the suspended massgiving it an upward velocity. When the wind falls the greater inertia ofthe mass carries it on upwards causing the aeroplane to again presentit* under side to the air; and if while the parachute is in this positionthe wind is still blowing from the side, the suspended mass is againlifted. Thus the more the bird is blown about, the more it rises in theair; actually birds in flight are carried up by a sudden side gust of windas we should expect from this theory.
The fact that the bird is in motion tends also to keep it up, forit has been recently shown that a horizontal plane under the action ofgravity falls to the ground more slowly if it is travelling through the airwith horizontal velocity than it would do if allowed to fall vertically,hence the bird’s forward motion causes it to fall through a smallerheight between successive gusts of wind than it would do if it were atrest, Moreover it has been proved experimentally that the horsepowerrequired to support a body in horizontal flight by means of an aeroplaneis less for high than for low speeds: hence when a side-wind (that is,a wind at right angles to the bird’s course) strikes the bird, the lift isincreased in consequence of the bird’s forward velocity.
Curiosa Physica. When I was writing the first edition of these“Recreations,” I put together a chapter, following this one, on “SomePhysical Questions,” dealing with problems such as, in the Theory of
* See G.H. Bryan in the Transactions of the British Association for 1896,vol. lxvi, pp. 726-728.
• Project • Gutenberg • #26839 •
CH. III] CURIOSA PHYSICA. 87
Sound, the explanation of the fact that in some of Captain Parry’s ex-periments the report of a cannon, when fired, travelled so much morerapidly than the sound of the human voice that observers heard the re-port of the cannon when fired before that of the order to fire it*: in theKinetic Theory of Gases, the complications in our universe that mightbe produced by “Maxwell’s demon”†: in the Theory of Optics, the ex-planation of the Japanese “magic mirrors,”‡ which reflect the patternon the back of the mirror (on which the light does not fall): to whichI might add the theory of the “spectrum top,” by means of which awhite surface, on which some black lines are drawn, can be moved soas to give the impression§ that the lines are coloured (red, green, blue,slate, or drab), and the curious fact that the colours change with thedirection of rotation: it has also been recently shown that if two trainsof waves, whose lengths are in the ratio m− 1 : m + 1, be superposed,then every mth wave in the system will be big—thus the current opin-ion that every ninth wave in the open sea is bigger than the other wavesmay receive scientific confirmation. There is no lack of interesting andcurious phenomena in physics, and in some branches, notably in elec-tricity and magnetism, the difficulty is rather one of selection, but I feltthat the connection with mathematics was in general either too remoteor too technical to justify the insertion of such a collection in a workon elementary mathematical recreations, and therefore I struck out thechapter. I mention the fact now partly to express the hope that somephysicist will one day give us a collection of the kind, partly to suggestthese questions to those who are interested in such matters.
* The fact is well authenticated. Mr Earnshaw (Philosophical Transactions, Lon-don, 1860, pp. 133–148) explained it by the acceleration of a wave caused bythe formation of a kind of bore, a view accepted by Clerk Maxwell and mostphysicists, but Sir George Airy thought that the explanation was to be found inphysiology; see Airy’s Sound, second edition, London, 1871, pp. 141, 142.
† See Theory of Heat, by J. Clerk Maxwell, second edition, London, 1872, p. 308.‡ See a memoir by W.E. Ayrton and J. Perry, Proceedings of the Royal Society of
London, part i, 1879, vol. xxviii, pp. 127–148.§ See letters from Mr C.E. Benham and others in Nature, 1894–5; and a memoir
by Prof. Liveing, Cambridge Philosophical Society, November 26, 1894.
• Project • Gutenberg • #26839 •
CHAPTER IV.
SOME MISCELLANEOUS QUESTIONS.
I propose to discuss in this chapter the mathematical theory ofcertain of the more common mathematical amusem*nts and games.Some of these might have been treated in the first two chapters, but,since most of them involve mixed geometry and algebra, it is rathermore convenient to deal with them apart from the problems and puzzleswhich have been described already. This division, however, is by nomeans well defined, and the arrangement is based on convenience ratherthan on any logical distinction.
The majority of the questions here enumerated have no connectionone with another, and I jot them down almost at random.
I shall discuss in succession the Fifteen Puzzle, the Tower of Hanoı,Chinese Rings, the Eight Queens Problem, the Fifteen School-GirlsProblem, and some miscellaneous Problems connected with a pack ofcards.
The Fifteen Puzzle*. Some years ago the so-called fifteenpuzzle was on sale in all toy-shops. It consists of a shallow woodenbox—one side being marked as the top—in the form of a square, andcontains fifteen square blocks or counters numbered 1, 2, 3 . . . up to 15.The box will hold just sixteen such counters, and, as it contains onlyfifteen, they can be moved about in the box relatively to one another.Initially they are put in the box in any order, but leaving the sixteenth
* There are two articles on the subject in the American Journal of Mathematics1879, vol. ii, by Professors Woolsey Johnson and Story; but the whole theory isdeducible immediately from the proposition I give above in the text.
88
• Project • Gutenberg • #26839 •
CH. IV] THE FIFTEEN PUZZLE. 89
cell or small square empty; the puzzle is to move them so that finallythey occupy the position shown in the first of the annexed figures.
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15A B
C D
Bottom of Box
Top of Box
2 1 4 3
6 5 8 7
10 9 12 11
13 14 15
We may represent the various stages in the game by supposing thatthe blank space, occupying the sixteenth cell, is moved over the board,ending finally where it started.
The route pursued by the blank space may consist partly of tracksfollowed and again retraced, which have no effect on the arrangement,and partly of closed paths travelled round, which necessarily are cycli-cal permutations of an odd number of counters. No other motion ispossible.
Now a cyclical permutation of n letters is equivalent to n−1 simpleinterchanges; accordingly an odd cyclical permutation is equivalent toan even number of simple interchanges. Hence, if we move the countersso as to bring the blank space back into the sixteenth cell, the neworder must differ from the initial order by an even number of simpleinterchanges. If therefore the order we want to get can be obtained fromthis initial order only by an odd number of interchanges, the problemis incapable of solution; if it can be obtained by an even number, theproblem is possible.
Thus the order in the second of the diagrams given on the currentpage is deducible from that in the first diagram by six interchanges;namely, by interchanging the counters 1 and 2, 3 and 4, 5 and 6, 7 and8, 9 and 10, 11 and 12. Hence the one can be deduced from the otherby moving the counters about in the box.
If however in the second diagram the order of the last three countershad been 13, 15, 14, then it would have required seven interchanges of
• Project • Gutenberg • #26839 •
90 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
counters to bring them into the order given in the first diagram. Hencein this case the problem would be insoluble.
The easiest way of finding the number of simple interchanges neces-sary in order to obtain one given arrangement from another is to makethe transformation by a series of cycles. For example, suppose that wetake the counters in the box in any definite order, such as taking thesuccessive rows from left to right, and suppose the original order andthe final order to be respectively
1, 13, 2, 3, 5, 7, 12, 8, 15, 6, 9, 4, 11, 10, 14,and 11, 2, 3, 4, 5, 6, 7, 1, 9, 10, 13, 12, 8, 14, 15.
We can deduce the second order from the first by 12 simple inter-changes. The simplest way of seeing this is to arrange the process inthree separate cycles as follows:—
1, 11, 8 ; 13, 2, 3, 4, 12, 7, 6, 10, 14, 15, 9 ; 5.11, 8, 1 ; 2, 3, 4, 12, 7, 6, 10, 14, 15, 9, 13 ; 5.
Thus, if in the first row of figures 11 is substituted for 1, then 8 for11, then 1 for 8, we have made a cyclical interchange of 3 numbers,which is equivalent to 2 simple interchanges (namely, interchanging 1and 11, and then 1 and 8). Thus the whole process is equivalent to onecyclical interchange of 3 numbers, another of 11 numbers, and anotherof 1 number. Hence it is equivalent to (2+10+0) simple interchanges.This is an even number, and thus one of these orders can be deducedfrom the other by moving the counters about in the box.
It is obvious that, if the initial order is the same as the requiredorder except that the last three counters are in the order 15, 14, 13,it would require one interchange to put them in the order 13, 14, 15;hence the problem is insoluble.
If however the box is turned through a right angle, so as to makeAD the top, this rotation will be equivalent to 13 simple interchanges.For, if we keep the sixteenth square always blank, then such a rotationwould change any order such as
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,to 13, 9, 5, 1, 14, 10, 6, 2, 15, 11, 7, 3, 12, 8, 4,
which is equivalent to 13 simple interchanges. Hence it will change thearrangement from one in which a solution is impossible to one whereit is possible, and vice versa.
• Project • Gutenberg • #26839 •
CH. IV] THE TOWER OF HANOI. 91
Again, even if the initial order is one which makes a solution impos-sible, yet if the first cell and not the last is left blank it will be possibleto arrange the fifteen counters in their natural order. For, if we repre-sent the blank cell by b, this will be equivalent to changing the order
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, b,b, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 :
this is a cyclical interchange of 16 things and therefore is equivalentto 15 simple interchanges. Hence it will change the arrangement fromone in which a solution is impossible to one where it is possible, andvice versa.
It is evident that the above principles are applicable equally to arectangular box containing mn cells or spaces and mn − 1 counterswhich are numbered. Of course m may be equal to n. If such a boxis turned through a right angle, and m and n are both even, it willbe equivalent to mn− 3 simple interchanges—and thus will change animpossible position to a possible one, and vice versa—but unless bothm and n are even the rotation is equivalent to only an even number ofinterchanges. Similarly, if either m or n is even, and it is impossibleto solve the problem when the last cell is left blank, then it will bepossible to solve it by leaving the first cell blank.
The problem may be made more difficult by limiting the possiblemovements by fixing bars inside the box which will prevent the move-ment of a counter transverse to their directions. We can conceive alsoof a similar cubical puzzle, but we could not work it practically exceptby sections.
The Tower of Hanoı. I may mention next the ingenious puz-zle known as the Tower of Hanoı. It was brought out in 1883 byM. Claus (Lucas).
It consists of three pegs fastened to a stand, and of eight circulardiscs of wood or cardboard each of which has a hole in the middle sothat a peg can be put through it. These discs are of different radii,and initially they are placed all on one peg, so that the biggest is atthe bottom, and the radii of the successive discs decrease as we ascend:thus the smallest disc is at the top. This arrangement is called theTower. The problem is to shift the discs from one peg to another insuch a way that a disc shall never rest on one smaller than itself, and
• Project • Gutenberg • #26839 •
92 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
finally to transfer the tower (i.e. all the discs in their proper order)from the peg on which they initially rested to one of the other pegs.
The method of effecting this is as follows. (i) If initially thereare n discs on the peg A, the first operation is to transfer graduallythe top n − 1 discs from the peg A to the peg B, leaving the peg Cvacant: suppose that this requires x separate transfers. (ii) Next, movethe bottom disc to the peg C. (iii) Then, reversing the first process,transfer gradually the n − 1 discs from B to C, which will necessitatex transfers. Hence, if it requires x transfers of simple discs to move atower of n − 1 discs, then it will require 2x + 1 separate transfers ofsingle discs to move a tower of n discs. Now with 2 discs it requires 3transfers, i.e. 22−1 transfers; hence with 3 discs the number of transfersrequired will be 2(22−1)+1, that is, 23−1. Proceeding in this way wesee that with a tower of n discs it will require 2n − 1 transfers of singlediscs to effect the complete transfer. Thus the eight discs of the puzzlewill require 255 single transfers. The result can be also obtained by thetheory of finite differences. It will be noticed that every alternate moveconsists of a transfer of the smallest disc from one peg to another, thepegs being taken in cyclical order.
M. De Parville gives an account of the origin of the toy which isa sufficiently pretty conceit to deserve repetition*. In the great templeat Benares, says he, beneath the dome which marks the centre of theworld, rests a brass-plate in which are fixed three diamond needles,each a cubit high and as thick as the body of a bee. On one of theseneedles, at the creation, God placed sixty-four discs of pure gold, thelargest disc resting on the brass plate, and the others getting smallerand smaller up to the top one. This is the Tower of Bramah. Day andnight unceasingly the priests transfer the discs from one diamond needleto another according to the fixed and immutable laws of Bramah, whichrequire that the priest must not move more than one disc at a time andthat he must place this disc on a needle so that there is no smaller discbelow it. When the sixty-four discs shall have been thus transferredfrom the needle on which at the creation God placed them to one ofthe other needles, tower, temple, and Brahmins alike will crumble intodust, and with a thunder-clap the world will vanish. Would that Englishwriters were in the habit of inventing equally interesting origins for thepuzzles they produce!
* La Nature, Paris, 1884, part i, pp. 285–286.
• Project • Gutenberg • #26839 •
CH. IV] CHINESE RINGS. 93
The number of separate transfers of single discs which the Brahminsmust make to effect the transfer of the tower is 264 − 1, that is, is18, 446744, 073709, 551615: a number which, even if the priests nevermade a mistake, would require many thousands of millions of yearsto carry out.
Chinese Rings*. A somewhat more elaborate toy, known asChinese Rings, which is on sale in most English toy-shops, is repre-sented in the accompanying figure. It consists of a number of rings
hung upon a bar in such a manner that the ring at one end (say A)can be taken off or put on the bar at pleasure; but any other ringcan be taken off or put on only when the one next to it towards A ison, and all the rest towards A are off the bar. The order of the ringscannot be changed.
Only one ring can be taken off or put on at a time. [In the toy,as usually sold, the first two rings form an exception to the rule. Boththese can be taken off or put on together. To simplify the discussionI shall assume at first that only one ring is taken off or put on at atime.] I proceed to show that, if there are n rings, then in order to
* It was described by Cardan in 1550 in his De Subtilitate, bk. xv, paragr. 2, ed.Sponius, vol. iii, p. 587; by Wallis in his Algebra, second edition, 1693, Opera,vol. ii, chap. 111, pp. 472–478; and allusion is made to it also in Ozanam’sRecreations, 1723 edition, vol. iv, p. 439.
• Project • Gutenberg • #26839 •
94 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
disconnect them from the bar, it will be necessary to take a ring off orto put a ring on either 1
3(2n+1−1) times or 1
3(2n+1−2) times according
as n is odd or even.Let the taking a ring off the bar or putting a ring on the bar be
called a step. It is usual to number the rings from the free end A. Letus suppose that we commence with the first m rings off the bar andall the rest on the bar; and suppose that then it requires x − 1 stepsto take off the next ring, that is, it requires x − 1 additional steps toarrange the rings so that the first m + 1 of them are off the bar andall the rest are on it. Before taking these steps we can take off the(m + 2)th ring and thus it will require x steps from our initial positionto remove the (m + 1)th and (m + 2)th rings.
Suppose that these x steps have been made and that thus the firstm + 2 rings are off the bar and the rest on it, and let us find howmany additional steps are now necessary to take off the (m + 3)th and(m + 4)th rings. To take these off we begin by taking off the (m + 4)thring: this requires 1 step. Before we can take off the (m+3)th we mustarrange the rings so that the (m + 2)th is on and the first m + 1 ringsare off: to effect this, (i) we must get the (m + 1)th ring on and thefirst m rings off, which requires x − 1 steps, (ii) then we must put onthe (m + 2)th ring, which requires 1 step, (iii) and lastly we must takethe (m + 1)th ring off, which requires x − 1 steps: thus this series ofmovements requires in all {2(x − 1) + 1} steps. Next we can take the(m + 3)th ring off, which requires 1 step; this leaves us with the firstm + 1 rings off, the (m + 2)th on, the (m + 3)th off and all the reston. Finally to take off the (m+2)th ring, (i) we get the (m+1)th ringon and the first m rings off, which requires x− 1 steps, (ii) we take offthe (m + 2)th ring, which requires 1 step, (iii) we take (m + 1)th ringoff, which requires x − 1 steps: thus this series of movements requires{2(x − 1) + 1} steps.
Therefore, if when the first m rings are off it requires x steps totake off the (m + 1)th and (m + 2)th rings, then the number of addi-tional steps required to take off the (m + 3)th and (m + 4)th rings is1 + {2(x − 1) + 1} + 1 + {2(x − 1) + 1}, that is, is 4x.
To find the whole number of steps necessary to take off an oddnumber of rings we proceed as follows.
To take off the first ring requires 1 step;∴ to take off the first 3 rings requires 4 additional steps;∴ ” ” 5 ” ” 42 ” ” .
• Project • Gutenberg • #26839 •
CH. IV] CHINESE RINGS. 95
In this way we see that the number of steps required to take off thefirst 2n + 1 rings is 1 + 4 + 42 + · · ·+ 4n, which is equal to 1
3(22n+2− 1).
To find the number of steps necessary to take off an even numberof rings we proceed in a similar manner.
To take off the first 2 rings requires 2 steps;∴ to take off the first 4 rings requires 2× 4 additional steps;∴ ” ” 6 ” ” 2× 42 ” ” .In this way we see that the number of steps required to take off thefirst 2n rings is 2 + (2× 4) + (2× 42) + · · ·+ (2× 4n−1), which is equalto 1
3(22n+1 − 2).If we take off or put on the first two rings in one step instead of
two separate steps, these results become respectively 22n and 22n−1−1.I give the above analysis because it is the direct solution of a prob-
lem attacked by Cardan in 1550 and by Wallis in 1693—in both casesunsuccessfully—and which at one time attracted some attention. I pro-ceed next to give another solution, more elegant though rather artificial.
This solution, which is due to M. Gros*, depends on a conventionby which any position of the rings is denoted by a certain numberexpressed in the binary scale of notation in such a way that a step isindicated by the addition or subtraction of unity.
Let the rings be indicated by circles: if a ring is on the bar, it isrepresented by a circle drawn above the bar; if the ring is off the bar, itis represented by a circle below the bar. Thus figure i below representsa set of seven rings of which the first two are off the bar, the next threeare on it, the sixth is off it, and the seventh is on it.
Denote the rings which are on the bar by the digits 1 or 0 alter-nately, reckoning from left to right, and denote a ring which is off thebar by the digit assigned to that ring on the bar which is nearest to iton the left of it, or by a 0 if there is no ring to the left of it.
Thus the three positions indicated below are denoted respectivelyby the numbers written below them. The position represented in fig-ure ii is obtained from that in figure i by putting the first ring on to thebar, while the position represented in figure iii is obtained from that infigure i by taking the fourth ring off the bar.
It follows that every position of the rings is denoted by a numberexpressed in the binary scale: moreover, since in going from left to right
* Theorie du Baguenodier, by L. Gros, Lyons, 1872. I take the account of thisfrom Lucas, vol. i, part 7.
• Project • Gutenberg • #26839 •
96 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
e e e e e e e1101000
Figure i.
e e e e e e e1101001
Figure ii.
e e e e e e e1100111
Figure iii.
every ring on the bar gives a variation (that is, 1 to 0 or 0 to 1) andevery ring off the bar gives a continuation, the effect of a step by whicha ring is taken off or put on the bar is either to subtract unity from thisnumber or to add unity to it. For example, the number denoting theposition of the rings in figure ii is obtained from the number denotingthat in figure i by adding unity to it. Similarly the number denoting theposition of the rings in figure iii is obtained from the number denotingthat in figure i by subtracting unity from it.
The position when all the seven rings are off the bar is denoted bythe number 0000000: when all of them are on the bar by the number1010101. Hence to change from one position to the other requires anumber of steps equal to the difference between these two numbers inthe binary scale. The first of these numbers is 0: the second is equalto 26 + 24 + 22 + 1, that is, to 85. Therefore 85 steps are required.In a similar way we may show that to put on a set of 2n + 1 ringsrequires (1 + 21 + 22 + . . . + 22n) steps, that is, 1
3(22n+2 − 1) steps; and
to put on a set of 2n rings requires (2 + 23 + . . . + 22n−1) steps, thatis, 1
3(2n+1 − 2) steps.
I append a table indicating the steps necessary to take off the firstfour rings from a set of five rings. The diagrams in the middle columnshow the successive position of the rings after each step. The num-ber following each diagram indicates that position, each number beingobtained from the one above it by the addition of unity. The stepswhich are bracketed together can be made in one movement, and, ifthus effected, the whole process is completed in 7 movements insteadof 10 steps: this is in accordance with the formula given above.
M. Gros asserted that it is possible to take from 64 to 80 stepsa minute, which in my experience is a rather high estimate. If weaccept the lower of these numbers, it would be possible to take off 10rings in less than 8 minutes; to take off 25 rings would require morethan 582 days, each of ten hours work; and to take off 60 rings would
• Project • Gutenberg • #26839 •
CH. IV] THE EIGHT QUEENS PROBLEM. 97
Initial positiond d d d d
10101
After 1st stepd d d d d
10110}” 2nd ”
d d d d d 10111
” 3rd ”d d d d d 11000
” 4th ”d d d d d
11001}” 5th ”
d d d d d11010
” 6th ”d d d d d 11011
” 7th ”d d d d d 11100
” 8th ”d d d d d
11101
” 9th ”d d d d d
11110}” 10th ”
d d d d d 11111
necessitate no less than 768614, 336404, 564650 steps, and would requirenearly 55000, 000000 years work—assuming of course that no mistakeswere made.
The Eight Queens Problem*. The determination of thenumber of ways in which eight queens can be placed on a chess-board—or more generally, in which n queens can be placed on aboard of n2 cells—so that no queen can take any other was proposedoriginally by Nauck in 1850.
In 1874 Dr S. Gunther† suggested a method of solution by meansof determinants. For, if each symbol represents the corresponding cellof the board, the possible solutions for a board of n2 cells are given by
* On the history of this problem see W. Ahrens, Mathematische Unterhaltun-gen und Spiele, Leipzig, 1901, chap. ix—a work issued subsequent to the thirdedition of this book.
† Grunert’s Archiv der Mathematik und Physik, 1874, vol. lvi pp. 281–292.
• Project • Gutenberg • #26839 •
98 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
those terms, if any, of the determinant∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a1 b2 c3 d4 . . . . . . . . . . . .β2 a3 b4 c5 . . . . . . . . . . . .γ3 β4 a5 b6 . . . . . . . . . . . .δ4 γ5 β6 a7 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . a2n−3 b2n−2
. . . . . . . . . . . . . . β2n−2 a2n−1
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣in which no letter and no suffix appears more than once.
The reason is obvious. Every term in a determinant contains oneand only one element out of every row and out of every column: henceany term will indicate a position on the board in which the queenscannot take one another by moves rook-wise. Again in the above de-terminant the letters and suffixes are so arranged that all the sameletters and all the same suffixes lie along bishop’s paths: hence, if weretain only those terms in each of which all the letters and all the suf-fixes are different, they will denote positions in which the queens cannottake one another by moves bishop-wise. It is clear that the signs of theterms are immaterial.
In the case of an ordinary chess-board the determinant is of the 8thorder, and therefore contains 8!, that is, 40320 terms, so that it wouldbe out of the question to use this method for the usual chess-board of64 cells or for a board of larger size unless some way of picking out therequired terms could be discovered.
A way of effecting this was suggested by Dr J.W.L. Glaisher* in1874, and as far as I am aware the theory remains as he left it. Heshowed that if all the solutions of n queens on a board of n2 cells wereknown, then all the solutions of a certain type for n + 1 queens on aboard of (n+1)2 cells could be deduced, and that all the other solutionsof n + 1 queens on a board of (n + 1)2 cells could be obtained withoutdifficulty. The method will be sufficiently illustrated by one instanceof its application.
It is easily seen that there are no solutions when n = 2 and n = 3.If n = 4 there are two terms in the determinant which give solutions,namely, b2c5γ3β6 and c3β2b6γ5. To find the solutions when n = 5,
* Philosophical Magazine, London, December, 1874, series 4, vol. xlviii, pp. 457–467.
• Project • Gutenberg • #26839 •
CH. IV] THE EIGHT QUEENS PROBLEM. 99
Glaisher proceeded thus. In this case Gunther’s determinant is∣∣∣∣∣∣∣∣∣∣∣∣
a1 b2 c3 d4 e5
β2 a3 b4 c5 d6
γ3 β4 a5 b6 c7
δ4 γ5 β6 a7 b8
ε5 δ6 γ7 β8 a9
∣∣∣∣∣∣∣∣∣∣∣∣To obtain those solutions (if any) which involve a9 it is sufficient to ap-pend a9 to such of the solutions for a board of 16 cells as do not involvea. As neither of those given above involves an a we thus get two solu-tions, namely, b2c5γ3β6a9 and c3β2b6γ5a9. The solutions which involvea1, e5 and ε5 can be written down by symmetry. The eight solutionsthus obtained are all distinct; we may call them of the first type.
The above are the only solutions which can involve elements in thecorner squares of the determinant. Hence the remaining solutions areobtainable from the determinant∣∣∣∣∣∣∣∣∣∣∣∣
0 b2 c3 d4 0β2 a3 b4 c5 d6
γ3 β4 a5 b6 c7
δ4 γ5 β6 a7 b8
0 δ6 γ7 β8 0
∣∣∣∣∣∣∣∣∣∣∣∣If, in this, we take the minor of b2 and in it replace by zero every terminvolving the letter b or the suffix 2 we shall get all solutions involvingb2. But in this case the minor at once reduces to d6a5δ4β8. We thusget one solution, namely, b2d6a5δ4β8. The solutions which involve β2,δ4, δ6, β8, b8, d6, and d4 can be obtained by symmetry. Of these eightsolutions it is easily seen that only two are distinct: these may becalled solutions of the second type.
Similarly the remaining solutions must be obtained from the de-terminant ∣∣∣∣∣∣∣∣∣∣∣∣
0 0 c3 0 00 a3 b4 c5 0γ3 β4 a5 b6 c7
0 γ5 β6 a7 00 0 γ7 0 0
∣∣∣∣∣∣∣∣∣∣∣∣If, in this, we take the minor of c3, and in it replace by zero every
term involving the letter c or the suffix 3, we shall get all the solutions
• Project • Gutenberg • #26839 •
100 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
which involve c3. But in this case the minor vanishes. Hence there isno solution involving c3, and therefore by symmetry no solutions whichinvolve γ3, γ7, or c3. Had there been any solutions involving the thirdelement in the first or last row or column of the determinant we shouldhave described them as of the third type.
Thus in all there are ten and only ten solutions, namely, eight ofthe first type, two of the second type, and none of the third type.
Similarly, if n = 6, we obtain no solutions of the first type, foursolutions of the second type, and no solutions of the third type; thatis, four solutions in all. If n = 7, we obtain sixteen solutions of thefirst type, twenty-four solutions of the second type, no solutions of thethird type, and no solutions of the fourth type; that is, forty solutionsin all. If n = 8, we obtain sixteen solutions of the first type, fifty-sixsolutions of the second type, and twenty solutions of the third type,that is, ninety-two solutions in all.
It will be noticed that all the solutions of one type are not alwaysdistinct. In general, from any solution seven others can be obtained atonce. Of these eight solutions, four consist of the initial or fundamen-tal solution and the three similar ones obtained by turning the boardthrough one, two, or three right angles; the other four are the reflex-ions of these in a mirror: but in any particular case it may happenthat the reflexions reproduce the originals, or that a rotation throughone or two right angles makes no difference. Thus on boards of 42,52, 62, 72, 82, 92, 102 cells there are respectively 1, 2, 1, 6, 12, 46, 92fundamental solutions; while altogether there are respectively 2, 10, 4,40, 92, 352, 724 solutions.
The following collection of fundamental solutions may interest thereader. The positions on the board of the queens are indicated bydigits: the first digit represents the number of the cell occupied by thequeen in the first column reckoned from one end of the column, thesecond digit the number in the second column, and so on. Thus on aboard of 42 cells the solution 3142 means that one queen is on the 3rdsquare of the first column, one on the 1st square of the second column,one on the 4th square of the third column, and one on the 2nd squareof the fourth column. If a fundamental solution gives rise to only foursolutions the number which indicates it is placed in curved brackets, ( );if it gives rise to only two solutions the number which indicates it isplaced in square brackets, [ ]; the other fundamental solutions give rise
• Project • Gutenberg • #26839 •
CH. IV] THE EIGHT QUEENS PROBLEM. 101
to eight solutions each.
On a board of 42 cells there is 1 fundamental solution: namely,[3142].
On a board of 52 cells there are 2 fundamental solutions: namely,14253, [25314].
On a board of 62 cells there is 1 fundamental solution: namely,(246135).
On a board of 72 cells there are 6 fundamental solutions: namely,1357246, 3572461, (5724613), 4613572, 3162574, (2574136).
On a board of 82 cells there are 12 fundamental solutions: namely,25713864, 57138642, 71386425, 82417536, 68241753, 36824175,64713528, 36814752, 36815724, 72418536, 26831475, (64718253).The arrangement in this order is due to Mr Oram. It will be noticedthat the 10th, 11th, and 12th solutions somewhat resemble the 4th,6th, and 7th respectively. The 6th solution is the only one in which nothree queens are in a straight line.
On a board of 92 cells there are 46 fundamental solutions; one ofthem is 248396157. On a board of 102 cells there are 92 fundamentalsolutions; these were given by Dr A. Pein*; one of them is 2468t13579,where t stands for ten. On a board of 112 cells there are 341 funda-mental solutions; these have been given by Dr T.B. Sprague†: one ofthem is 15926t37e48. I may add that for a board of n2 cells there isalways a symmetrical solution of the form 246 . . . n135 . . . (n−1), whenn = 6m or n = 6m + 4, Also Mr Oram has shown that for a boardof n2 cells, when n is a prime, cyclical arrangements of the n naturalnumbers, other than in their natural order, will give solutions; see, forinstance, the solution quoted above.
The puzzle in the form of a board of 36 squares is sold in thestreets of London for a penny, a small wooden board being ruled inthe manner shown in the diagram and having holes drilled in it at thepoints marked by dots. The object is to put six pins into the holes sothat no two are connected by a straight line.
* Aufstellung von n Koniginnen auf einem Schachbrett von n2 Feldern, Leipzig,1889.
† Proceedings of the Edinburgh Mathematical Society, vol. xvii, 1898–9, pp. 43–68.
• Project • Gutenberg • #26839 •
102 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
Other Problems with Queens. Captain Turton called my atten-tion to two other problems of a somewhat analogous character, neitherof which, as far as I know, has been published elsewhere, or solvedotherwise than empirically.
The first of these is to place eight queens on a chess-board so asto command the fewest possible squares. Thus, if queens are placed oncells 1 and 2 of the second column, on cell 2 of the sixth column, on cells1, 3, and 7 of the seventh column, and on cells 2 and 7 of the eighthcolumn, eleven cells on the board will not be in check; the same numbercan be obtained by other arrangements. Is it possible to place the eightqueens so as to leave more than eleven cells out of check? I have neversucceeded in doing so, nor in showing that it is impossible to do it.
The other problem is to place m queens (m being less than 5)on a chess-board so as to command as many cells as possible. Forinstance, four queens can be placed in several ways on the board soas to command 58 cells besides those on which the queens stand, thusleaving only 2 cells which are not commanded: ex. gr. this is effected ifthe queens are placed on cell 5 of the third column, cell 1 of the fourthcolumn, cell 6 of the seventh column, and cell 2 of the eighth column;or on cell 1 of the first column, cell 7 of the third column, cell 3 of thefifth column, and cell 5 of the seventh column. A similar problem is todetermine the minimum number and the position of queens which canbe placed on a board of n2 cells so as to occupy or command every cell.It would seem that, even with the additional restriction that no queenshall be able to take any other queen, there are no less than ninety-onetypical solutions in which five queens can be placed on a chess-board
• Project • Gutenberg • #26839 •
CH. IV] THE FIFTEEN SCHOOL-GIRLS PROBLEM. 103
so as to command every cell*.Extension to other Chess Pieces. Analogous problems may be pro-
posed with other chess-pieces. For instance, questions as to the max-imum number of knights which can be placed on a board of n2 cellsso that no knight can take any other, and the minimum number ofknights which can be placed on it so as to occupy or command everycell have been propounded†.
Similar problems have also been proposed for k kings placed on achess-board of n2 cells‡. It has been asserted that, if k = 2, the numberof ways in which two kings can be placed on a board so that they maynot occupy adjacent squares is 1
2(n− 1)(n− 2)(n2 + 3n− 2). Similarly,
if k = 3, the number of ways in which three kings can be placed ona board so that no two of them occupy adjacent squares is said to be16(n − 1)(n − 2)(n4 + 3n3 − 20n2 − 30n + 132).
The Fifteen School-Girls Problem. This problem—which was first enunciated by Mr T.P. Kirkman, and is sometimesknown as Kirkman’s problem§—consists in arranging fifteen things indifferent sets of triplets. It is usually presented in the form that aschool-mistress was in the habit of taking her girls for a daily walk.
* L’Intermediaire des mathematiciens, Paris, 1901, vol. viii, p. 88.† Ibid., March, 1896, vol. iii, p. 58; 1897, vol. iv, p. 15, 254; and 1898, vol. v,
p. 87.‡ Ibid., June, 1901, p. 140.§ It was published first in the Lady’s and Gentleman’s Diary for 1850, p. 48,
and has been the subject of numerous memoirs. Among these I may single outthe papers by A. Cayley in the Philosophical Magazine, July, 1850, series 3,vol. xxxvii, pp. 50–53; by T.P. Kirkman in the Cambridge and Dublin Math-ematical Journal, 1850, vol. v, p. 260; by R.R. Anstice, Ibid., 1852, vol. vii,pp. 279–292; by B. Pierce, Gould’s Journal, Cambridge, U.S., 1860, vol. vi,pp. 169–174; by T.P. Kirkman, Philosophical Magazine, March, 1862, series 4,vol. xxiii, pp. 198–204; by W.S.B. Woolhouse in the Lady’s Diary for 1862,pp. 84–88, and for 1863, pp. 79–90, and in the Educational Times Reprints,1867, vol. viii, pp. 76–83; by J. Power in the Quarterly Journal of Mathemat-ics, 1867, vol. viii, pp. 236–251; by A.H. Frost, Ibid., 1871, vol. xi, pp. 26–37;by E. Carpmael in the Proceedings of the London Mathematical Society, 1881,vol. xii, pp. 148–156; by Lucas in his Recreations, vol. ii, part vi; by A.C. Dixonin the Messenger of Mathematics, Cambridge, October, 1893, vol. xxiii, pp. 88–89; and by W. Burnside, Ibid., 1894, vol. xxiii, pp. 137–143. It has also, sincethe issue of my third edition, been discussed by W. Ahrens in his MathematischeUnterhaltungen und Spiele, Leipzig, 1901, chapter xiv.
• Project • Gutenberg • #26839 •
104 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
The girls were fifteen in number, and were arranged in five rows ofthree each so that each girl might have two companions. The problemis to dispose them so that for seven consecutive days no girl will walkwith any of her school-fellows more than once. More generally we mayrequire to arrange 3m girls in triplets to walk out for 1
2(3m− 1) days,
so that no girl will walk with any of her school-fellows more than once.
The theory of the formation of all such possible triplets in thecase of fifteen girls is not difficult, but the extension to 3m girls is, asyet, unsolved. I proceed to describe three methods of solution: thesemethods are analytical, but I may add that the problem can be alsoattacked by geometrical methods.
Frost’s Method. The first of these solutions is due to Mr Frost. Afull exposition of it would occupy a good deal of space, but I hope thatthe following sketch will make the process intelligible.
Denote one of the girls by k. Her companions on each day aredifferent: suppose that on Sunday they are a1 and a2, on Monday b1
and b2, and so on, and finally on Saturday g1 and g2. Hence for eachday we have one triplet, and we have to find four others, but in eachof the latter no two like letters can occur together, that is, the threeletters in any of them must be all different.
Let a stand for a1, or a2, b for b1 or b2, and so on. The suffixes 1and 2 are called complementary. Then, since the three letters in each ofthe triplets we are trying to find must be different, we must make somearrangement such as putting a with bc, de, and fg; and, if so, b may beassociated with df and eg; and c with dg and ef . Thus there are sevenpossible triads, such as abc, ade, afg, bdf , beg, cdg, and cef . Moreovereach of these may stand for any one of four triplets: for instance, thetriad bdf may stand for any of the triplets b1d1f1, b1d2f2, b2d1f2, b2d2f1.
The four triads which do not involve a must be placed in the Sundaycolumn, the four which do not involve b in the Monday column, and soon. Thus each triad will occur four times.
It only remains to insert the proper suffixes. This is done as follows.Take one triad, such as bdf , and insert a different set of suffixes eachtime that it occurs; for instance, the four sets given above. Next, theother like letters (b, d, or f as the case may be) in these four columnsmust have the complementary suffixes attached.
After this is done, the next triplet in the Sunday column will beb2eg. The triad beg occurs in four columns and includes four possi-
• Project • Gutenberg • #26839 •
CH. IV] THE FIFTEEN SCHOOL-GIRLS PROBLEM. 105
ble triplets, such as b2e1g1, b2e2g2, b1e1g2, b1e2g1. Insert these, andthen give the complementary suffixes to the other like letters in thesefour columns.
In this way the arrangement is constructed gradually, by takingone triad at a time, inserting the proper suffixes to the four tripletsincluded in it, and then the complementary suffixes in the other likeletter in the same columns.
One final arrangement, thus obtained, is as follows:
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
ka1a2 kb1b2 kc1c2 kd1d2 ke1e2 kf1f2 kg1g2
b1d1f1 a1d2e2 a1d1e1 a2b2c2 a2b1c1 a1b2c1 a1b1c2
b2e1g1 a2f2g2 a2f1g1 a1f2g1 a1f1g2 a2d2e1 a2d1e2
c1d2g2 c1d1g1 b1d2f2 b1e1g2 b2d1f2 b1e2g1 b2d2f1
c2e2f2 c2e1f1 b2e2g2 c1e2f1 c2d2g1 c2d1g2 c1e1f2
We might obtain other solutions by selecting other seven triads or bychoosing other arrangements of the suffixes in each triad (or by merelyinterchanging letters and suffixes in the above order). By these meansMr Power showed that there are no less than 15567, 552000 differentsolutions; but, since the total number of ways in which the school canwalk out for a week in triplets is (455)7, the probability that any chanceway satisfies Kirkman’s condition is very small.
Frost’s method is applicable to the case of 22n−1 girls walking outfor 22n−1− 1 days in triplets. The detailed solution for 63 girls walkingout for 31 days, which corresponds to n = 3, have been given.
Anstice’s Method. Another method of attacking the problem isdue to Mr Anstice; it is illustrated by the following elegant solution,by which from the order on Sunday we can obtain the order on thefollowing six days by a cyclical permutation. Let the girls be denotedrespectively by the letters k, a1, a2, a3, a4, a5, a6, a7, b1, b2, b3, b4, b5,b6, b7; and suppose the order on Sunday to be
ka1b1, a2a3a5, a4b3b6, a6b2b7, a7b4b5 .
Then, if the suffixes are permuted cyclically, we obtain six other ar-rangements which satisfy the conditions of the problem: the reasonbeing that in the above arrangement the difference of the suffixes of ev-ery pair of like letters—such as either the “a”s or the “b”s—in a triplet
• Project • Gutenberg • #26839 •
106 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
is different for each triplet, as also is the difference of the suffixes ofevery pair of unlike letters which are in a triplet.
Two other arrangements for Sunday, from which those for the re-maining days are obtainable by cyclical permutations can be formed.These are ka1b1, a2a3a5, a4b5b7, a6b3b4, a7b2b6; and ka1b1, a2a3a5, a4b2b6,a6b5b7, a7b3b4.
Anstice’s method is applicable to the case of 2p + 1 girls walkingout for p days in triplets so that no pair may walk together more thanonce, provided p is a prime of the form 12m + 7. In such a case heshowed how to construct a fundamental arrangement for one day fromwhich the arrangements for the remaining p − 1 days can be obtainedby cyclical permutations of suffixes. The number of such fundamentalarrangements is 3(2m + 1)(3m + 1).
The problem of 15 girls corresponds to m = 0, and the three fun-damental Anstician arrangements are given above. If m = 1 we havethe problem of 39 girls. One Anstician arrangement in this case isas follows: ka1b1, a2a8a12, a5a7a10, a6a17a18, a3b10b15, a4b3b5, a9b18b19,a11b8b14, a13b9b17, a14b12b16, a15b4b7, a16b2b11, a19b6b13. If m = 2 we havethe problem of 63 girls, of which Frost has given one solution; and so on.
Gill’s Method. Another method of attacking the problem has beensuggested to me by Mr T.H. Gill. Representing the girls by a1, a2, a3,. . . , a3m he (i) forms one triplet of the type a1am+1a2m+1, from which,by cyclical permutation of the suffixes 1, 2, . . . , 3m he obtains m tripletswhich constitute an arrangement for one day, and (ii) forms 1
2(m − 1)
other triplets such that the three differences of the suffixes are different,from which, by cyclical permutations of the suffixes, the arrangementsfor the remaining 3
2(m − 1) other days can be obtained. Thus in the
case of 15 girls, the triplet a1a6a11 gives, by cyclical permutations ofthe suffixes, an arrangement for the first day and two triplets such asa1a2a5, a1a3a9 enable us to form 30 triplets from which an arrangementfor the other six days can be found. Here is a solution thus determined.
First Day: 1. 6.11; 2. 7.12; 3. 8.13; 4. 9.14; 5.10.15 .Second Day: 1. 2. 5; 3. 4. 7; 8. 9.12; 10.11.14; 13.15. 6 .Third Day: 2. 3. 6; 4. 5. 8; 9.10.13; 11.12.15; 14. 1. 7 .Fourth Day: 5. 6. 9; 7. 8.11; 12.13. 1; 14.15. 3; 2. 4.10 .Fifth Day: 7. 9.15; 8.10. 1; 3. 5.11; 4. 6.12; 13.14. 2 .Sixth Day: 9.11. 2; 10.12. 3; 5. 7.13; 6. 8.14; 15. 1. 4 .Seventh Day: 11.13. 4; 12.14. 5; 15. 2. 8; 1. 3. 9; 6. 7.10 .
• Project • Gutenberg • #26839 •
CH. IV] THE FIFTEEN SCHOOL-GIRLS PROBLEM. 107
But, although this method gives triplets with which the problem canbe solved, the final arrangement is empirical.
A solution of the problem of 21 girls for 10 days can be got by thesame method: a1a8a15 giving 7 triplets which constitute an arrangementfor one day; and a1a2a6, a1a3a11, a1a4a10 giving 63 triplets from whichan arrangement for the other nine days can be formed. Here is thesolution thus determined.
First Day: 1. 8.15; 2. 9.16; 3.10.17; 4.11.18; 5.12.19; 6.13.20; 7.14.21 .Second Day: 1. 2. 6; 4. 5. 9; 7. 8.12; 10.11.15; 13.14.18; 16.17.21; 19.20. 3 .Third Day: 7.10.16; 8.11.17; 12.15.21; 18.19. 2; 20. 1. 9; 3. 5.13; 4. 6.14 .Fourth Day: 13.16. 1; 14.17. 2; 18.21. 6; 3. 4. 8; 5. 7.15; 9.11.19; 10.12.20 .Fifth Day: 4. 7.13; 5. 8.14; 9.12.18; 15.16.20; 17.19. 6; 21. 2.10; 1. 3.11 .Sixth Day; 1. 4.10; 2. 5.11; 6. 9.15; 12.13.17; 14.16. 3; 18.20. 7; 19.21. 8 .Seventh Day: 2. 3. 7; 5. 6.10; 8. 9.13; 11.12.16; 14.15.19; 17.18. 1; 20.21. 4 .Eighth Day: 10.13.19; 11.14.20; 15.18. 3; 21. 1. 5; 2. 4.12; 6. 8.16; 7. 9.17 .Ninth Day: 16.19. 4; 17.20. 5; 21. 3. 9; 6. 7.11; 8.10.18; 12.14. 1; 13.15. 2 .Tenth Day: 19. 1. 7; 20. 2. 8; 3. 6.12; 9.10.14; 11.13.21; 15.17. 4; 16.18. 5 .
I should be interested if any of my readers could give me a similarsolution of the analogous arrangement of 33 girls for 16 days formedfrom typical triplet suffixes like 1, 12, 23; 1, 2, 10; 1, 3, 16; 1, 4, 18; 1, 5,11; 1, 6, 13; or from other sets of triplets formed in a similar way so that(except in the first triplet) the differences of the suffixes are all different.
Walecki’s Theorem. Lastly, Walecki—quoted by Lucas—hasshown that, if a solution for the case of n girls walking out in tripletsfor 1
2(n− 1) days is known, then a solution for 3n girls walking out for
12(3n − 1) days can be deduced.
For if an arrangement of the n girls, a1, a2, . . . , an for 12(n−1) days
is known; and also one of the n girls, b1, b2, . . . , bn; and also one of then girls c1, c2, . . . , cn; then an arrangement of these 3n girls for 1
2(n− 1)
days is known. A set of n triplets for another day will be given byambm+kcm+2k where m is put equal to 1, 2, . . . , n successively. Herek may have any of the n values, 0, 1, 2, . . . , (n − 1); but, wherever asuffix is greater than n, it is to be divided by n and only the remainderretained. Hence altogether we have an arrangement for n + 1
2(n− 1)
days, i.e. for 12(3n − 1) days.
The arrangement of 3 girls for one day is obvious. Hence, byWalecki’s theorem, we can deduce at once an arrangement of 3m girlsfor 1
2(3m − 1) days. And, generally, as I have given solutions of the
problem in the case of 3n girls when n = 1, 3, 5, 7, 9, 13, 15, it follows
• Project • Gutenberg • #26839 •
108 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
that for the same values of n, we can solve the analogous arrangementsof 3m × n girls.
To the original theorem J.J. Sylvester* added the corollary that theschool of 15 girls could walk out in triplets on 13 × 7 days until everypossible triplet had walked abreast once.
The generalized problem of finding the greatest number of ways inwhich x girls walking in rows of a abreast can be arranged so that everypossible combination of b of them may walk abreast once and only oncehas been solved for various cases. Suppose that this greatest numberof ways is y. It is obvious that, if all the x girls are to walk out eachday in rows of a abreast, then x must be an exact multiple of a and thenumber of rows formed each day is x/a. If such an arrangement can bemade for z days, then we have a solution of the problem to arrange xgirls to walk out in rows of a abreast for z days so that they all go outeach day and so that every possible combination of b girls may walktogether once, and only once. In the corresponding generalization ofKirkman’s problem no companionship of girls which has occurred oncemay occur again, but it does not follow necessarily that every possiblecompanionship must occur once.
An example where the solution is obvious is if x = 2n, a = 2, b = 2,in which case y = n(2n − 1), z = 2n − 1.
If we take the case x = 15, a = 3, b = 2, we find y = 35; and ithappens that these 35 rows can be divided into 7 sets, each of whichcontains all the symbols; hence z = 7. More generally, if x = 5 × 3m,a = 3, b = 2, we find y = 3
2(x − 1)/x, z = 1
2(x − 1). It will be
noticed that in the solutions of the original fifteen school-girls problemand of Walecki’s extension of it given above every possible pair of girlswalk together once; hence we might infer that in these cases we coulddetermine z as well as y.
The results of the last paragraph were given by Kirkman† in 1850.In the same memoir he also proved that, if p is a prime, and if x = pm,a = p, b = 2, then y = (pm − 1)/(p − 1); if x = (p2 + p + 1)(p + 1)where p2 + p + 1 has no divisor less than p + 1, a = p + 1, b = 2, theny = x(x− 1)/p(p + 1); if x = p3 + p + 1, a = p + 1, b = 2, then y = x;
* Philosophical Magazine, July, 1850, series 3, vol. xxxvii, p. 52; a solution bySylvester is given in the Philosophical Magazine, May, 1861, series 4, vol. xxi,p. 371.
† Cambridge and Dublin Mathematical Journal, 1850, vol. v, pp. 255–262.
• Project • Gutenberg • #26839 •
CH. IV] PROBLEMS CONNECTED WITH A PACK OF CARDS. 109
and Sylvester’s result that if x = 15, a = 3, b = 3, y = 455, z = 91.Three years later Kirkman* solved the problem when x = 2n, a = 4,b = 3. Lastly, in 1893, Sylvester† published the solution when x = 9,a = 3, b = 3, in which case y = 84, z = 28; and stated that a similarmethod was applicable when x = 3m, a = 3, b = 3: thus 9 girls canbe arranged to walk out 28 times (say 4 times a day for a week) sothat in any day the same pair never are together more than once andso that at the end of the week each girl has been associated with everypossible pair of her schoolfellows.
In 1867 Mr S. Bills‡ showed that if x = 7, a = 3, b = 2, theny = 7: if x = 15, a = 3, b = 2, then y = 35: if x = 31, a = 3, b = 2,then y = 155: and the method by which these results are proved willgive the value of y, if x = 2n − 1, a = 3, b = 2. Shortly afterwardsMr W. Lea§ showed that if x = 11, a = 5, b = 4, then y = 66; also thatif x = 16, a = 4, b = 3, then y = 140; the latter result is a particularcase of Kirkman’s theorems. It will be noticed that these writers didnot confine their discussion to cases where x is an exact multiple of a.
Problems connected with a Pack of Cards. I mentionedin chapter i that an ordinary pack of playing cards could be used to il-lustrate many tricks depending on simple properties of numbers. Mostof these involve the relative position of the cards. The principle of solu-tion generally consists in re-arranging the pack in a particular mannerso as to bring the card into some definite position. Any such rear-rangement is a species of shuffling.
I shall treat in succession of problems connected with shuffling apack, arrangements by rows and columns, the determination of a pairout of 1
2n(n + 1) pairs, Gergonne’s pile problem, and the game known
as the mouse trap.
Shuffling a Pack. Any system of shuffling a pack of cards,if carried out consistently, leads to an arrangement which can be cal-culated; but tricks that depend on it generally require considerable
* Ibid., 1853, vol. viii, pp. 38–42.† Messenger of Mathematics, February, 1893, vol. xxii, pp. 159–160.‡ Educational Times Reprints, London, 1867, vol. viii, pp. 32–33.§ Ibid., 1868, vol. ix, pp. 35–36; and 1874, vol. xxii, pp. 74–76; see also the volume
for 1869, vol. xi, p. 97.
• Project • Gutenberg • #26839 •
110 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
technical skill.Suppose for instance that a pack of n cards is shuffled, as is not
unusual, by placing the second card on the first, the third below these,the fourth above them, and so on. The theory of this system of shufflingis due to Monge*. The following are some of the results and are notdifficult to prove directly.
If the pack contains 6p + 4 cards, the (2p + 2)th card will occupythe same position in the shuffled pack. For instance, if a complete packof 52 cards is shuffled as described above, the 18th card will remainthe 18th card.
Again, if a pack of 10p+2 cards is shuffled in this way, the (2p+1)thand the (6p × 2)th cards will interchange places. For instance, if anecarte pack of 32 cards is shuffled as described above, the 7th and the20th cards will change places.
More generally, one shuffle of a pack of 2p cards will move the cardwhich was in the x0th place to the x1th place, where x1 = 1
2(2p+x0+1)
if x0 is odd, and x1 = 12(2p − x0 + 2) if x0 is even, from which the
above results can be deduced. By repeated applications of the aboveformulae we can show that the effect of m such shuffles is to movethe card which was initially in the x0th place to the xmth place where2m+1xm = (4p + 1)(2m−1 ± 2m−2 ± · · · ± 2± 1)± 2x0 + 2m ± 1, the sign± representing an ambiguity of sign.
Again, in any pack of n cards after a certain number of shufflings,not greater than n, the cards will return to their primitive order. Thiswill always be the case as soon as the original top card occupies thatposition again. To determine the number of shuffles required for a packof 2p cards, it is sufficient to put xm = x0 and find the smallest value ofm which satisfies the resulting equation for all values of x0 from 1 to 2p.It follows that, if m is the least number which makes 4m − 1 divisibleby 4p + 1, then m shuffles will be required if either 2m + 1 or 2m − 1 isdivisible by 4p+1, otherwise 2m shuffles will be required. The number
* Monge’s investigations are printed in the Memoires de l’Academie des Sciences,Paris, 1773, pp. 390–412. Among those who have studied the subject afresh Imay in particular mention V. Bouniakowski, Bulletin physico-mathematique deSt Petersbourg, 1857, vol. xv, pp. 202–205, summarised in the Nouvelles annalesde mathematiques, 1858, pp. 66–67; T. de St Laurent, Memoires de l’Academiede Gard, 1865; L. Tanner, Educational Times Reprints, 1880, vol. xxxiii, pp. 73–75; and M.J. Bourget, Liouville’s Journal, 1882, pp. 413–434. The solutionsgiven by Prof. Tanner are simple and concise.
• Project • Gutenberg • #26839 •
CH. IV] PROBLEMS CONNECTED WITH A PACK OF CARDS. 111
for a pack of 2p+1 cards is the same as that for a pack of 2p cards. Withan ecarte pack of 32 cards, six shuffles are sufficient; with a pack of 2n
cards, n + 1 shuffles are sufficient; with a full pack of 52 cards, twelveshuffles are sufficient; with a pack of 13 cards ten shuffles are sufficient;while with a pack of 50 cards fifty shuffles are required; and so on.
Mr W.H.H. Hudson* has also shown that, whatever is the law ofshuffling, yet if it is repeated again and again on a pack of n cards, thecards will ultimately fall into their initial positions after a number ofshufflings not greater than the greatest possible l.c.m. of all numberswhose sum is n.
For suppose that any particular position is occupied after the 1st,2nd, . . . , pth shuffles by the cards A1, A2, . . . , Ap respectively, and thatinitially the position is occupied by the card A0. Suppose further thatafter the pth shuffle A0 returns to its initial position, therefore A0 = Ap.Then at the second shuffling A2 succeeds A1 by the same law by whichA1 succeeded A0 at the first; hence it follows that previous to the secondshuffling A2 must have been in the place occupied by A1 previous tothe first. Thus the cards which after the successive shuffles take theplace initially occupied by A1 are A2, A3, . . . , Ap, A1; that is, after thepth shuffle A1 has returned to the place initially occupied by it: andso for all the other cards A2, A3, . . . , Ap−1.
Hence the cards A1, A2, . . . , Ap form a cycle of p cards, one or otherof which is always in one or other of p positions in the pack, and whichgo through all their changes in p shufflings. Let the number n of thepack be divided into p, q, r, . . . such cycles, whose sum is n; then thel.c.m. of p, q, r, . . . is the utmost number of shufflings necessary beforeall the cards will be brought back to their original places.
In the case of a pack of 52 cards, the greatest l.c.m. of numberswhose sum is 52 will be found by trial to be 180180.
Arrangements by Rows and Columns. A not uncommontrick, which rests on a species of shuffling, depends on the obvious factthat if n2 cards are arranged in the form of a square of n rows, eachcontaining n cards, then any card will be defined if the row and thecolumn in which it lies are mentioned.
This information is generally elicited by first asking in which rowthe selected card lies, and noting the extreme left-hand card of that
* Educational Times Reprints, London, 1865, vol. ii, p. 105.
• Project • Gutenberg • #26839 •
112 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
row. The cards in each column are then taken up, face upwards, oneat a time beginning with the lowest card of each column and takingthe columns in their order from right to left–each card taken up beingplaced on the top of those previously taken up. The cards are thendealt out again in rows, from left to right, beginning with the top left-hand corner, and a question is put as to which row contains the card.The selected card will be that card in the row mentioned which is inthe same vertical column as the card which was originally noted.
The above is the form in which the trick is usually presented, butit is greatly improved by allowing the pack to be cut as often as is likedbefore the cards are re-dealt, and then giving one cut at the end so asto make the top card in the pack one of those originally in the top row.
The explanation is obvious. For, if 16 cards are taken, the first
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Figure i.
1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16
Figure ii.
and second arrangements may be represented by figures i and ii. Forexample, if we are told that in figure i the card is in the third row,it must be either 9, 10, 11, or 12: hence, if we know in which row offigure ii it lies, it is determined. If we allow the pack to be cut betweenthe deals, we must secure somehow that the top card is either 1, 2, 3,or 4, since that will leave the cards in each row of figure ii unalteredthough the positions of the rows will be changed.
• Project • Gutenberg • #26839 •
CH. IV] PROBLEMS CONNECTED WITH A PACK OF CARDS. 113
Determination of a selected pair of cards out of12n(n + 1) given pairs*. Another common trick is to throw twenty
cards on to a table in ten couples, and ask someone to select one couple.The cards are then taken up, and dealt out in a certain manner intofour rows each containing five cards. If the rows which contain thegiven cards are indicated, the cards selected are known at once.
This depends on the fact that the number of hom*ogeneous productsof two dimensions which can be formed out of four things is 10. Hencethe hom*ogeneous products of two dimensions formed out of four thingscan be used to define ten things.
Suppose that ten pairs of cards are placed on a table and someoneselects one couple. Take up the cards in their couples. Then the first
1 2 3 5 7
4 9 10 11 13
6 12 15 16 17
8 14 18 19 20
two cards form the first couple, the next two the second couple, andso on. Deal them out in four rows each containing five cards accordingto the scheme shown in the diagram.
The first couple (1 and 2) are in the first row. Of the next couple(3 and 4), put one in the first row and one in the second. Of the nextcouple (5 and 6), put one in the first row and one in the third, and soon, as indicated in the diagram. After filling up the first row proceedsimilarly with the second row, and so on.
Enquire in which rows the two selected cards appear. If only oneline, the mth, is mentioned as containing the cards then the requiredpair of cards are the mth and (m + 1)th cards in that line. These
* Bachet, problem xvii, avertissem*nt, p. 146 et seq.
• Project • Gutenberg • #26839 •
114 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
occupy the clue squares of that line. Next, if two lines are mentioned,then proceed as follows. Let the two lines be the pth and the qth andsuppose q > p. Then that one of the required cards which is in the qthline will be the (q−p)th card which is below the first of the clue squaresin the pth line. The other of the required cards is in the pth line andis the (q − p)th card to the right of the second of the clue squares.
Bachet’s rule, in the form in which I have given it, is applicable toa pack of n(n + 1) cards divided into couples, and dealt in n rows eachcontaining n + 1 cards; for there are 1
2n(n + 1) such couples, also there
are 12n(n + 1) hom*ogeneous products of two dimensions which can be
formed out of n things. Bachet gave the diagrams for the cases of 20,30, and 42 cards: these the reader will have no difficulty in constructingfor himself, and I have enunciated the rule for 20 cards in a form whichcovers all the cases.
I have seen the same trick performed by means of a sentence andnot by numbers. If we take the case of ten couples, then after collectingthe pairs the cards must be dealt in four rows each containing five cards,in the order indicated by the sentence Matas dedit nomen Cocis. Thissentence must be imagined as written on the table, each word formingone line, The first card is dealt on the M . The next card (which isthe pair of the first) is placed on the second m in the sentence, that is,third in the third row. The third card is placed on the a. The fourthcard (which is the pair of the third) is placed on the second a, that is,fourth in the first row. Each of the next two cards is placed on a t,and so on. Enquire in which rows the two selected cards appear. If tworows are mentioned, the two cards are on the letters common to thewords that make these rows. If only one row is mentioned, the cardsare on the two letters common to that row.
The reason is obvious: let us denote each of the first pair by an a,and similarly each of any of the other pairs by an e, i, o, c, d, m, n, s,or t respectively. Now the sentence Matas dedit nomen Cocis containsfour words each of five letters; ten letters are used, and each letter isrepeated only twice. Hence, if two of the words are mentioned, theywill have one letter in common, or, if one word is mentioned, it willhave two like letters.
To perform the same trick with any other number of cards weshould require a different sentence.
The number of hom*ogeneous products of three dimensions which
• Project • Gutenberg • #26839 •
CH. IV] PROBLEMS CONNECTED WITH A PACK OF CARDS. 115
can be formed out of four things is 20, and of these the number consist-ing of products in which three things are alike and those in which threethings are different is 8. This leads to a trick with 8 trios of things,which is similar to that last given–the cards being arranged in the orderindicated by the sentence Lanata levete livini novoto.
I believe that these arrangements by sentences are known, but Iam not aware who invented them.
Gergonne’s Pile Problem. Before discussing Gergonne’stheorem I will describe the familiar three pile problem, the theory ofwhich is included in his results.
The Three Pile Problem*. This trick is usually performed as fol-lows. Take 27 cards and deal them into three piles, face upwards. By“dealing” is to be understood that the top card is placed as the bottomcard of the first pile, the second card in the pack as the bottom cardof the second pile, the third card as the bottom card of the third pile,the fourth card on the top of the first one, and so on: moreover I as-sume that throughout the problem the cards are held in the hand faceupwards. The result can be modified to cover any other way of dealing.
Request a spectator to note a card, and remember in which pileit is. After finishing the deal, ask in which pile the card is. Take upthe three piles, placing that pile between the other two. Deal again asbefore, and repeat the question as to which pile contains the given card.Take up the three packs again, placing the pile which now contains theselected card between the other two. Deal again as before, but indealing note the middle card of each pile. Ask again for the third timein which pile the card lies, and you will know that the card was the onewhich you noted as being the middle card of that pile. The trick canbe finished then in any way that you like. The usual method—but avery clumsy one—is to take up the three piles once more, placing thenamed pile between the other two as before, when the selected cardwill be the middle one in the pack, that is, if 27 cards are used it willbe the fourteenth.
The trick is often performed with 15 cards or with 21 cards, ineither of which cases the same rule holds.
* The trick is mentioned by Bachet, problem xviii, p. 143, but his analysis of itis insufficient.
• Project • Gutenberg • #26839 •
116 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
Gergonne’s Generalization. The general theory for a pack of mm
cards was given by M. Gergonne*. Suppose the pack is arranged in mpiles, each containing mm−1 cards, and that, after the first deal, thepile indicated as containing the selected card is taken up ath; afterthe second deal, is taken up bth; and so on, and finally after the mthdeal, the pile containing the card is taken up kth. Then when thecards are collected after the mth deal the selected card will be nthfrom the top where
if m is even, n = kmm−1 − jmm−2 + · · ·+ bm− a + 1 ,if m is odd, n = kmm−1 − jmm−2 + · · · − bm + a .
For example, if a pack of 256 cards (i.e. m = 4) was given, andanyone selected a card out of it, the card could be determined by makingfour successive deals into four piles of 64 cards each, and after eachdeal asking in which pile the selected card lay. The reason is that afterthe first deal you know it is one of sixty-four cards. In the next dealthese sixty-four cards are distributed equally over the four piles, andtherefore, if you know in which pile it is, you will know that it is oneof sixteen cards. After the third deal you know it is one of four cards.After the fourth deal you know which card it is.
Moreover, if the pack of 256 cards is used, it is immaterial inwhat order the pile containing the selected card is taken up after adeal. For, if after the first deal it is taken up ath, after the secondbth, after the third cth, and after the fourth dth, the card will be the(64d − 16c + 4b − a + 1)th from the top of the pack, and thus will beknown. We need not take up the cards after the fourth deal, for thesame argument will show that it is the (64− 16c + 4b− a + 1)th in thepile then indicated as containing it. Thus if a = 3, b = 4, c = 1, d = 2,it will be the 62nd card in the pile indicated after the fourth deal ascontaining it and will be the 126th card in the pack as then collected.
In exactly the same way a pack of twenty-seven cards may be used,and three successive deals, each into three piles of nine cards, will sufficeto determine the card. If after the deals the pile indicated as containingthe given card is taken up ath, bth, and cth respectively, then the cardwill be the (9c− 3b+ a)th in the pack or will be the (9− 3b+ a)th cardin the pile indicated after the third deal as containing it.
* Gergonne’s Annales de Mathematiques, Nismes, 1813–4, vol. iv, pp. 276–283.
• Project • Gutenberg • #26839 •
CH. IV] PROBLEMS CONNECTED WITH A PACK OF CARDS. 117
The method of proof will be illustrated sufficiently by consideringthe usual case of a pack of twenty-seven cards, for which m = 3, whichare dealt into three piles each of nine cards.
Suppose that, after the first deal, the pile containing the selectedcard is taken up ath: then (i) at the top of the pack there are a − 1piles each containing nine cards; (ii) next there are 9 cards, of whichone is the selected card; and (iii) lastly there are the remaining cardsof the pack. The cards are dealt out now for the second time: in eachpile the bottom 3(a− 1) cards will be taken from (i), the next 3 cardsfrom (ii), and the remaining 9 − 3a cards from (iii).
Suppose that the pile now indicated as containing the selected cardis taken up bth: then (i) at the top of the pack are 9(b − 1) cards;(ii) next are 9 − 3a cards; (iii) next are 3 cards, of which one is theselected card; and (iv) lastly are the remaining cards of the pack. Thecards are dealt out now for the third time: in each pile the bottom3(b− 1) cards will be taken from (i), the next 3− a cards will be takenfrom (ii), the next card will be one of the three cards in (iii), and theremaining 8 − 3b + a cards are from (iv).
Hence, after this deal, as soon as the pile is indicated, it is knownthat the card is the (9 − 3b + a)th from the top of that pile. If theprocess is continued by taking up this pile as cth, then the selectedcard will come out in the place 9(c− 1)+ (8− 3b+ a)+ 1 from the top,that is, will come out as the (9c − 3b + a)th card.
Since, after the third deal, the position of the card in the pile thenindicated is known, it is easy to notice the card, in which case the trickcan be finished in some way more effective than dealing again.
If we put the pile indicated always in the middle of the pack wehave a = 2, b = 2, c = 2, hence n = 9c − 3b + a = 14, which is theform in which the trick is usually presented, as was explained aboveon page 115.
I have shown that if a, b, c are known, then n is determined. Wemay modify the rule so as to make the selected card come out in anyassigned position, say the nth. In this case we have to find values ofa, b, c which will satisfy the equation n = 9c − 3b + a, where a, b, ccan have only the values 1, 2, or 3.
Hence, if we divide n by 3 and the remainder is 1 or 2, this remain-der will be a; but, if the remainder is 0, we must decrease the quotientby unity so that the remainder is 3, and this remainder will be a. In
• Project • Gutenberg • #26839 •
118 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
other words a is the smallest positive number (exclusive of zero) whichmust be subtracted from n to make the difference a multiple of 3.
Next let p be this multiple, i.e. p is the next lowest integer to n/3:then 3p = 9c−3b, therefore p = 3c− b. Hence b is the smallest positivenumber (exclusive of zero) which must be added to p to make the suma multiple of 3, and c is that multiple.
A couple of illustrations will make this clear. Suppose we wish thecard to come out 22nd from the top, therefore 22 = 9c − 3b + a. Thesmallest number which must be subtracted from 22 to leave a multipleof 3 is 1, therefore a = 1. Hence 22 = 9c− 3b + 1, therefore 7 = 3c− b.The smallest number which must be added to 7 to make a multipleof 3 is 2, therefore b = 2. Hence 7 = 3c − 2, therefore c = 3. Thusa = 1, b = 2, c = 3.
Again, suppose the card is to come out 21st. Hence 21 = 9c−3b+a.Therefore a is the smallest number which subtracted from 21 makes amultiple of 3, therefore a = 3. Hence 6 = 3c − b. Therefore b is thesmallest number which added to 6 makes a multiple of 3, thereforeb = 3. Hence 9 = 3c, therefore c = 3. Thus a = 3, b = 3, c = 3.
If any difficulty is experienced in this work, we can proceed thus.Let a = x + 1, b = 3 − y, c = z + 1; then x, y, z may have onlythe values 0, 1, or 2. In this case Gergonne’s equation takes the form9z + 3y + x = n− 1. Hence, if n− 1 is expressed in the ternary scale ofnotation, x, y, z will be determined, and therefore a, b, c will be known.
The rule in the case of a pack of mm cards is exactly similar. Wewant to make the card come out in a given place. Hence, in Gergonne’sformula, we are given n and we have to find a, b, . . . , k. We can ef-fect this by dividing n continually by m, with the convention that theremainder are to be alternately positive and negative and that theirnumerical values are to be not greater than m or less than unity.
An analogous theorem with a pack of lm cards can be constructed.C.T. Hudson and L.E. Dickson* have discussed the general case wheresuch a pack is dealt n times, each time into l piles of m cards; and theyhave shown how the piles must be taken up in order that after the nthdeal the selected card may be rth from the top.
The principle will be sufficiently illustrated by one example treatedin a manner analogous to the cases already discussed. For instance,
* Educational Times Reprints, 1868, vol. ix, pp. 89–91; and Bulletin of the Amer-ican Mathematical Society, New York, April, 1895, vol. i, pp. 184–186.
• Project • Gutenberg • #26839 •
CH. IV] PROBLEMS CONNECTED WITH A PACK OF CARDS. 119
suppose that an ecarte pack of 32 cards is dealt into 4 piles each of 8cards, and that the pile which contains some selected card is picked upath. Suppose that on dealing again into four piles, one pile is indicatedas containing the selected card, the selected card cannot be one of thebottom 2(a − 1) cards, or of the top 8 − 2a cards, but must be oneof the intermediate 2 cards, and the trick can be finished in any way,as for instance by the common conjuring ambiguity of asking someoneto choose one of them, leaving it doubtful whether the one he takes isto be rejected or retained.
The Mouse Trap. I will conclude this chapter with the baremention of another game of cards, known as the mouse trap, the dis-cussion of which involves some rather difficult algebraic analysis.
It is played as follows. A set of cards, marked with the numbers1, 2, 3, . . . , n, is dealt in any order, face upwards, in the form of a circle.The player begins at any card and counts round the circle always inthe same direction. If the kth card has the number k on it—whichevent is called a hit—the player takes up the card and begins countingafresh. According to Cayley, the player wins if he thus takes up allthe cards, and the cards win if at any time the player counts up to nwithout being able to take up a card.
For example, if a pack of only four cards is used and these cardscome in the order, 3214, then the player would obtain the second card2 as a hit, next he would obtain 1 as a hit, but if he went on forever he would not obtain another hit. On the other hand, if the cardsin the pack were initially in the order 1423, the player would obtainsuccessively all four cards in the order 1, 2, 3, 4.
The problem may be stated as the determination of what hits andhow many hits can be made with a given number of cards; and whatpermutations will give a certain number of hits in a certain order.
Cayley* showed that there are 9 arrangements of a pack of fourcards in which no hit will be made, 7 arrangements in which only onehit will be made, 3 arrangements in which only two hits will be made,and 5 arrangements in which four hits will be made.
Prof. Steen† has investigated the general theory for a pack of ncards. He has shown how to determine the number of arrangements in
* Quarterly Journal of Mathematics, 1878, vol. xv, pp. 8–10.† Ibid., vol. xv, pp. 230–241.
• Project • Gutenberg • #26839 •
120 MISCELLANEOUS MATHEMATICAL RECREATIONS. [CH. IV
which x is the first hit [Arts. 3–5]; the number of arrangements in which1 is the first hit and x is the second hit [Art. 6]; and the number ofarrangements in which 2 is the first hit and x the second hit [Arts. 7–8];but beyond this point the theory has not been carried. It is obviousthat, if there are n − 1 hits, the nth hit will necessarily follow.
Treize. The French game of treize is very similar. It is playedwith a full pack of fifty-two cards (knave, queen, and king counting as11, 12, and 13 respectively). The dealer calls out 1, 2, 3, . . . , 13, as hedeals the 1st, 2nd, 3rd, . . . , 13th cards respectively. At the beginningof a deal the dealer offers to lay or take certain odds that he will makea hit in the thirteen cards next dealt.
One of the innumerable forms of patience is played in a similar way.
• Project • Gutenberg • #26839 •
CHAPTER V.
MAGIC SQUARES.
A magic square consists of a number of integers arranged in theform of a square, so that the sum of the numbers in every row, inevery column, and in each diagonal is the same. If the integers are theconsecutive numbers from 1 to n2 the square is said to be of the nthorder, and it is easily seen that in this case the sum of the numbers inany row, column, or diagonal is equal to 1
2n(n2 + 1): this number may
be denoted by N . I confine my account to such magic squares, that is,to squares formed with consecutive integers, from 1 upwards.
Thus the first 16 integers, arranged in either of the forms given infigures i and ii below, form a magic square of the fourth order, the sum
1 15 14 4
12 6 7 9
8 10 11 5
13 3 2 16
Figure i.
15 10 3 6
4 5 16 9
14 11 2 7
1 8 13 12
Figure ii.
of the numbers in any row, column, or diagonal being 34. Similarlyfigures iii and v on page 124, figure viii on page 126, and figures xii andxiii on page 136, show magic squares of the fifth order; and figure xi onpage 133 shows a magic square of the sixth order; and figures xiv andxv on pages 137, 138, show magic squares of the eighth order.
121
• Project • Gutenberg • #26839 •
122 MAGIC SQUARES. [CH. V
The formation of these squares is an old amusem*nt, and in timeswhen mystical philosophical ideas were associated with particular num-bers it was natural that such arrangements should be deemed to possessmagical properties. Magic squares of an odd order were constructed inIndia before the Christian era according to a law of formation whichis explained hereafter. Their introduction into Europe appears to havebeen due to Moschopulus, who lived at Constantinople in the earlypart of the fifteenth century, and enunciated two methods for makingsuch squares. The majority of the medieval astrologers and physicianswere much impressed by such arrangements. In particular the famousCornelius Agrippa (1486–1535) constructed magic squares of the orders3, 4, 5, 6, 7, 8, 9, which were associated respectively with the sevenastrological “planets”: namely, Saturn, Jupiter, Mars, the Sun, Venus,Mercury, and the Moon. He taught that a square of one cell, in whichunity was inserted, represented the unity and eternity of God; whilethe fact that a square of the second order could not be constructedillustrated the imperfection of the four elements, air, earth, fire, andwater; and later writers added that it was symbolic of original sin. Amagic square engraved on a silver plate was sometimes prescribed as acharm against the plague, and one, namely, that represented in figure ion page 121, is drawn in the picture of Melancholy, painted about theyear 1500 by Albert Durer. Such charms are still worn in the East.
The development of the theory has been due mainly to Frenchmathematicians. Bachet gave a rule for the construction of any squareof an odd order in a form substantially equivalent to one of the rulesgiven by Moschopulus. The formation of magic squares, especiallyof even squares, was considered by Frenicle and Fermat. The theorywas continued by Poignard, De la Hire, Sauveur, D’Ons-en-bray, andDes Ourmes. Ozanam included in his work an essay on magic squareswhich was amplified by Montucla. From this and from De la Hire’smemoirs the larger part of the materials for this chapter are derived.Like most algebraical problems, the construction of magic squares at-tracted the attention of Euler, but he did not advance the generaltheory. In 1837 an elaborate work on the subject was compiled byB. Violle, which is useful as containing numerous illustrations. I givethe references in a footnote*.
* Bachet, Problemes plaisans, Lyons, 1624, problem xxi, p. 161; Frenicle, DiversOuvrages de Mathematique par Messieurs de l’Academie des Sciences, Paris,
• Project • Gutenberg • #26839 •
CH. V] MAGIC SQUARES OF AN ODD ORDER.. 123
I shall confine myself to establishing rules for the construction ofsquares subject to no conditions beyond those given in the definition.Rules sufficient for this purpose are contained in the works to whichI have just referred and on which I have based this sketch; some ex-tensions and developments will be found in the memoirs mentionedbelow*. I shall commence by giving rules for the construction of asquare of an odd order, and then shall proceed to similar rules for oneof an even order.
It will be convenient to use the following terms. The spaces orsmall squares occupied by the numbers are called cells. The diagonalfrom the top left-hand cell to the bottom right-hand cell is called theleading diagonal or left diagonal. The diagonal from the top right-handcell to the bottom left-hand cell is called the right diagonal.
Magic Squares of an odd order. I proceed to give methodsfor constructing odd magic squares, but for simplicity I shall apply themto the formation of squares of the fifth order, though exactly similar
1693, pp. 423–483; with an appendix (pp. 484–507), containing diagrams ofall the possible magic squares of the fourth order, 880 in number: Fermat,Opera Mathematica, Toulouse, 1679, pp. 173–178; or Brassinne’s Precis, Paris,1853, pp. 146–149: Poignard, Traite des Quarres Sublimes, Brussels, 1704: Dela Hire, Memoires de l’Academie des Sciences for 1705, Paris, 1706, part i,pp. 127–171; part ii, pp. 364–382: Sauveur, Construction des Quarres Mag-iques, Paris, 1710: D’Ons-en-bray, Memoires de l’Academie des Sciences for1750, Paris, 1754, pp. 241–271: Des Ourmes, Memoires de Mathematique et dePhysique (French Academy), Paris, 1763, vol. iv, pp. 196–241: Ozanam andMontucla, Recreations, part i, chapter xii: Euler, Commentationes Arithmeti-cae Collectae, St Petersburg, 1849, vol. ii, pp. 593–602: Violle, Traite Completdes Carres Magiques, 3 vols, Paris, 1837–8. A sketch of the history of thesubject is given in chap. iv of S. Gunther’s Geschichte der mathematischen Wis-senschaften, Leipzig, 1876. See also W. Ahrens, Mathematische Unterhaltungenund Spiele, Leipzig, 1901, chapter xii.
* In England the subject has been studied by R. Moon, Cambridge MathematicalJournal, 1845, vol. iv, pp. 209–214; H. Holditch, Quarterly Journal of Math-ematics, London, 1864, vol. vi, pp. 181–189; W.H. Thompson, Ibid., 1870,vol. x, pp. 186–202; J. Horner, Ibid., 1871, vol. xi, pp. 57–65, 123–132, 213–224; S.M. Drach, Messenger of Mathematics, Cambridge, 1873, vol. ii, pp. 169–174, 187; A.H. Frost, Quarterly Journal of Mathematics, London, 1878, vol. xv,pp. 34–49, 93–123, 366–368, in which the results of previous memoirs are in-cluded: there are also some pamphlets and articles on it of a more popularcharacter. Of recent Continental works on the subject I have no complete bib-liography, and probably it is better to omit all rather than give an imperfectlist.
• Project • Gutenberg • #26839 •
124 MAGIC SQUARES. [CH. V
proofs will apply equally to any odd square.De la Loubere’s Method*. If the reader will look at figure iii he
will see one way in which such a square containing 25 cells can beconstructed. The middle cell in the top row is occupied by 1. Thesuccessive numbers are placed in their natural order in a diagonal line
17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9
De la Loubere’s Method.Figure iii.
15 + 2 20 + 4 0 + 1 5 + 3 10 + 5
20 + 3 0 + 5 5 + 2 10 + 4 15 + 1
0 + 4 5 + 1 10 + 3 15 + 5 20 + 2
5 + 5 10 + 2 15 + 4 20 + 1 0 + 3
10 + 1 15 + 3 20 + 5 0 + 2 5 + 4
De la Loubere’s Method.Figure iv.
23 6 19 2 15
10 18 1 14 22
17 5 13 21 9
4 12 25 8 16
11 24 7 20 3
Bachet’s Method.Figure v.
which slopes upwards to the right, except that (i) when the top rowis reached the next number is written in the bottom row as if it cameimmediately above the top row; (ii) when the right-hand column isreached, the next number is written in the left-hand column, as if itimmediately succeeded the right-hand column; and (iii) when a cellwhich has been filled up already, or when the top right-hand square isreached, the path of the series drops to the row vertically below it andthen continues to mount again. Probably a glance at the diagram infigure iii will make this clear.
The reason why such a square is magic can be explained best byexpressing the numbers in the scale of notation whose radix is 5 (or n,if the magic square is of the order n), except that 5 is allowed to appearas a unit-digit and 0 is not allowed to appear as a unit-digit. The resultis shown in figure iv. From that figure it will be seen that the methodof construction ensures that every row and every column shall containone and only one of each of the unit-digits 1, 2, 3, 4, 5, the sum ofwhich is 15; and also one and only one of each of the radix-digits 0, 5,10, 15, 20, the sum of which is 50. Hence, as far as rows and columnsare concerned, the square is magic. Moreover if the square is odd, eachof the diagonals will contain one and only one of each of the unit-digits
* De la Loubere, Du Royaume de Siam (Eng. Trans.), London, 1693, vol. ii,pp. 227–247. De la Loubere was the envoy of Louis XIV to Siam in 1687–8, andthere learnt this method.
• Project • Gutenberg • #26839 •
CH. V] MAGIC SQUARES OF AN ODD ORDER.. 125
1, 2, 3, 4, 5. Also the leading diagonal will contain one and only oneof the radix-digits 0, 5, 10, 15, 20, the sum of which is 50; and if, as isthe case in the square drawn above, the number 10 is the radix-digitto be added to the unit-digits in the right diagonal, then the sum ofthe radix-digits in that diagonal is also 50. Hence the two diagonalsalso possess the magical property.
And generally if a magic square of an odd order n is constructed byDe la Loubere’s method, every row and every column must contain oneand only one of each of the unit-digits 1, 2, 3 . . . , n; and also one andonly one of each of the radix-digits 0, n, 2n, . . . , n(n− 1). Hence, as faras rows and columns are concerned, the square is magic. Moreover eachdiagonal will either contain one and only one of the unit-digits or willcontain n unit-digits each equal to 1
2(n + 1). It will also either contain
one and only one of the radix-digits or will contain n radix-digits eachequal to 1
2n(n−1). Hence the two diagonals will also possess the magical
property. Thus the square will be magic.I may notice here that, if we place 1 in any cell and fill up the square
by De la Loubere’s rule, we shall obtain a square that is magic in rowsand in columns, but it will not in general be magic in its diagonals.
It is evident that other squares can be derived from De la Loubere’ssquare by permuting the symbols properly. For instance, in figure iv,we may permute the symbols 1, 2, 3, 4, 5 in 5! ways, and we maypermute the symbols 0, 5, 15, 20 in 4! ways. Any one of these 5!arrangements combined with any one of these 4! arrangements will givea magic square. Hence we can obtain 2880 magic squares of the fifthorder of this kind, though only 720 of them are really distinct. Othersquares can however be deduced, for it may be noted that from anymagic square, whether even or odd, other magic squares of the sameorder can be formed by the mere interchange of the row and the columnwhich intersect in a cell on a diagonal with the row and the columnwhich intersect in the complementary cell of the same diagonal.
Bachet’s Method*. Another method, very similar to that of Dela Loubere, for constructing an odd magic square is as follows. Webegin by placing 1 in the cell above the middle one (that is, in a squareof the fifth order in the cell occupied by the number 7 in figure iii),and then we write the successive numbers in a diagonal line slopingupwards to the right, subject to the condition that when the cases (i)
* Bachet, Problem xxi, p. 161.
• Project • Gutenberg • #26839 •
126 MAGIC SQUARES. [CH. V
and (ii) mentioned under De la Loubere’s method occur the rules theregiven are followed, but when the case (iii) occurs the path of the seriesrises two rows, i.e. it is continued from one cell to the cell next but onevertically above it, if this cell is above the top row the path continuesfrom the corresponding cell in one of the bottom two rows followingthe analogy of rule (i) in De la Loubere’s method. Such a square isdelineated in figure v on page 124. Bachet’s method leads ultimatelyto this arrangement; except that the rules are altered so as to makethe line slope downwards. This method also gives 720 magic squaresof the fifth order.
De la Hire’s Method*. I shall now give another rule for the forma-tion of odd magic squares. To form an odd magic square of the ordern by this method, we begin by constructing two subsidiary squares,one of the unit-digits, 1, 2, . . . , n, and the other of multiples of theradix, namely, 0, n, 2n, . . . , (n − 1)n. We then form the magic squareby adding together the numbers in the corresponding cells in the twosubsidiary squares.
De la Hire gave several ways of constructing such subsidiarysquares. I select the following method (props. x and xiv of his memoir)as being the simplest, but I shall apply it to form a square of onlythe fifth order. It leads to the same results as the second of the tworules given by Moschopulus.
The first of the subsidiary squares (figure vi, below), is constructedthus. First, 3 is put in the top left-hand corner, and then the numbers1, 2, 4, 5 are written in the other cells of the top line (in any order).Next, the number in each cell of the top line is repeated in all the cells
3 4 1 5 2
2 3 4 1 5
5 2 3 4 1
1 5 2 3 4
4 1 5 2 3
First Subsidiary Square
Figure vi.
15 0 20 5 10
0 20 5 10 15
20 5 10 15 0
5 10 15 0 20
10 15 0 20 5
Second Subsidiary Square
Figure vii.
18 4 21 10 12
2 23 9 11 20
25 7 13 19 1
6 15 17 3 24
14 16 5 22 8
Resulting Magic Square
Figure viii.
which lie in a diagonal line sloping downwards to the right (see figure vi)
* Memoires de l’Academie des Sciences for 1705, part i, pp. 127–171.
• Project • Gutenberg • #26839 •
CH. V] MAGIC SQUARES OF AN ODD ORDER.. 127
according to the rule (ii) in De la Loubere’s method. The cells filled bythe same number form a broken diagonal. It follows that every row andevery column contains one and only one 1, one and only one 2, and soon. Hence the sum of the numbers in every row and in every columnis equal to 15; also, since we placed 3, which is the average of thesenumbers, in the top left-hand corner, the sum of the numbers in theleft diagonal is 15; and, since the right diagonal contains one and onlyone of each of the numbers 1, 2, 3, 4, and 5, the sum of the numbersin that diagonal also is 15.
The second of the subsidiary squares (figure vii) is constructed ina similar way with the numbers 0, 5, 10, 15, and 20, except that themean number 10 is placed in the top right-hand corner; and the brokendiagonals formed of the same numbers all slope downwards to the left.It follows that every row and every column in figure vii contains oneand only one 0, one and only one 5, and so on; hence the sum of thenumbers in every row and every column is equal to 50. Also the sumof the numbers in each diagonal is equal to 50.
If now we add together the numbers in the corresponding cells ofthese two squares, we shall obtain 25 numbers such that the sum ofthe numbers in every row, every column, and each diagonal is equalto 15 + 50, that is, to 65. This is represented in figure viii. Moreover,no two cells in that figure contain the same number. For instance, thenumbers 21 to 25 can occur only in those five cells which in figure viiare occupied by the number 20, but the corresponding cells in figure vicontain respectively the numbers 1, 2, 3, 4, and 5; and thus in figure viiieach of the numbers from 21 to 25 occurs once and only once. De la Hirepreferred to have the cells in the subsidiary squares which are filled bythe same number connected by a knight’s move and not by a bishop’smove; and usually his rule is enunciated in that form.
By permuting the numbers 1, 2, 4, 5 in figure vi we get 4! other ar-rangements, each of which combined with that in figure vii would givea magic square. Similarly by permuting the numbers 0, 5, 15, 20 infigure vii we obtain 4! other squares, each of which might be combinedwith any of the 4! arrangements deduced from figure vi. Hence alto-gether we can obtain in this way 576 magic squares of the fifth order.
There is yet another method of constructing odd squares which isdue to Poignard, and was improved by De la Hire in the memoir alreadycited. I shall not discuss it, because, though for certain assigned values
• Project • Gutenberg • #26839 •
128 MAGIC SQUARES. [CH. V
of n it is simpler than the methods which I have given, it depends onthe form of n, and particularly on the number of prime factors of n. Inthe case of a square of the fifth order, this gives an even larger numberof magic squares than the methods of De la Loubere, Bachet, and Dela Hire. I may also add that it has been shown that magic squareswhose order is a prime number can be constructed by a rule similarto De la Loubere’s, except that we begin by placing 1 in the bottomleft-hand cell, and the subsequent consecutive numbers fill cells forminga knight’s path on the square and not a bishop’s path. A square of thefifth order of this kind is given in figure xiii on page 136. There are2880 magic squares of the fifth order of this kind.
De la Hire showed that, apart from mere inversions, there were57600 magic squares of the fifth order which could be formed by themethods he enumerated. Taking account of other methods, it wouldseem that the total number of magic squares of the fifth order is verylarge, perhaps exceeding 500000.
Magic squares of an even order. The above methods areinapplicable to squares of an even order. I proceed to give two methodsfor constructing any even magic square of an order higher than two.
It will be convenient to use the following terms. Two rows whichare equidistant, the one from the top, the other from the bottom, aresaid to be complementary. Two columns which are equidistant, the onefrom the left-hand side, the other from the right-hand side, are said tobe complementary. Two cells in the same row, but in complementarycolumns, are said to be horizontally related. Two cells in the samecolumn, but in complementary rows, are said to be vertically related.Two cells in complementary rows and columns are said to be skewlyrelated ; thus, if the cell b is horizontally related to the cell a, and thecell d is vertically related to the cell a, then the cells b and d are skewlyrelated; in such a case if the cell c is vertically related to the cell b,it will be horizontally related to the cell d, and the cells a and c areskewly related: the cells a, b, c, d constitute an associated group, and ifthe square is divided into four equal quarters, one cell of an associatedgroup is in each quarter.
A horizontal interchange consists in the interchange of the numbersin two horizontally related cells. A vertical interchange consists inthe interchange of the numbers in two vertically related cells. A skewinterchange consists in the interchange of the numbers in two skewly
• Project • Gutenberg • #26839 •
CH. V] MAGIC SQUARES OF AN EVEN ORDER.. 129
related cells. A cross interchange consists in the change of the numbersin any cell and in its horizontally related cell with the numbers in thecells skewly related to them; hence, it is equivalent to two verticalinterchanges and two horizontal interchanges.
First Method*. This method is the simplest with which I am ac-quainted, and I believe, at any rate as far as concerns singly-evensquares, was published for the first time in 1893.
Begin by filling the cells of the square with the numbers 1, 2, . . . , n2
in their natural order commencing (say) with the top left-hand corner,writing the numbers in each row from left to right, and taking therows in succession from the top. I will commence by proving that acertain number of horizontal and vertical interchanges in such a squaremust make it magic, and will then give a rule by which the cells whosenumbers are to be interchanged can be at once picked out.
First, we may notice that the sum of the numbers in each diagonalis equal to N , where N = 1
2n(n2 + 1); hence the diagonals are already
magic, and will remain so if the numbers therein are not altered.Next, consider the rows. The sum of the numbers in the xth
row from the top is N − 12n2(n − 2x + 1). The sum of the numbers
in the complementary row, that is, the xth row from the bottom, isN + 1
2n2(n− 2x+1). Also the number in any cell in the xth row is less
than the number in the cell vertically related to it by n(n − 2x + 1).Hence, if in these two rows we make 1
2n interchanges of the numbers
which are situated in vertically related cells, then we increase the sumof the numbers in the xth row by 1
2n × n(n − 2x + 1), and therefore
make that row magic; while we decrease the sum of the numbers inthe complementary row by the same number, and therefore make thatrow magic. Hence, if in every pair of complementary rows we make12n interchanges of the numbers situated in vertically related cells, the
square will be made magic in rows. But, in order that the diagonalsmay remain magic, either we must leave both the diagonal numbers inany row unaltered, or we must change both of them with those in thecells vertically related to them.
The square is now magic in diagonals and in rows, and it remainsto make it magic in columns. Taking the original arrangement of thenumbers (in their natural order) we might have made the square magic
* See an article in the Messenger of Mathematics, Cambridge, September, 1893,vol. xxiii, pp. 65–69.
• Project • Gutenberg • #26839 •
130 MAGIC SQUARES. [CH. V
in columns in a similar way to that in which we made it magic in rows.The sum of the numbers originally in the yth column from the left-handside is N − 1
2n(n− 2y + 1). The sum of the numbers originally in the
complementary column, that is, the yth column from the right-handside, is N + 1
2n(n − 2y + 1). Also the number originally in any cell
in the yth column was less than the number in the cell horizontallyrelated to it by n−2y+1. Hence, if in these two columns we had made12n interchanges of the numbers situated in horizontally related cells,
we should have made the sum of the numbers in each column equal toN . If we had done this in succession for every pair of complementarycolumns, we should have made the square magic in columns. But, asbefore, in order that the diagonals might remain magic, either we musthave left both the diagonal numbers in any column unaltered, or wemust have changed both of them with those in the cells horizontallyrelated to them.
It remains to show that the vertical and horizontal interchanges,which have been considered in the last two paragraphs, can be madeindependently, that is, that we can make these interchanges of thenumbers in complementary columns in such a manner as will not affectthe numbers already interchanged in complementary rows. This willrequire that in every column there shall be exactly 1
2n interchanges
of the numbers in vertically related cells, and that in every row thereshall be exactly 1
2n interchanges of the numbers in horizontally related
cells. I proceed to show how we can always ensure this, if n is greaterthan 2. I continue to suppose that the cells are initially filled withthe numbers 1, 2, . . . , n2 in their natural order, and that we work fromthat arrangement.
A doubly-even square is one where n is of the form 4m. If the squareis divided into four equal quarters, the first quarter will contain 2mcolumns and 2m rows. In each of these columns take m cells so arrangedthat there are also m cells in each row, and change the numbers in these2m2 cells and the 6m2 cells associated with them by a cross interchange.The result is equivalent to 2m interchanges in every row and in everycolumn, and therefore renders the square magic.
One way of selecting the 2m2 cells in the first quarter is to dividethe whole square into sixteen subsidiary squares each containing m2
cells, which we may represent by the diagram below, and then we maytake either the cells in the a squares or those in the b squares; thus, if
• Project • Gutenberg • #26839 •
CH. V] MAGIC SQUARES OF AN EVEN ORDER.. 131
a b b a
b a a b
b a a b
a b b a
every number in the eight a squares is interchanged with the numberskewly related to it the resulting square is magic. A magic square of theeighth order, constructed in this way, is shown in figure xv on page 138.
Another way of selecting the 2m2 cells in the first quarter wouldbe to take the first m cells in the first column, the cells 2 to m + 1 inthe second column, and so on, the cells m + 1 to 2m in the (m + 1)thcolumn, the cells m+2 to 2m and the first cell in the (m+2)th column,and so on, and finally the 2mth cell and the cells 1 to m − 1 in the2mth column.
A singly-even square is one where n is of the form 2(2m+1). If thesquare is divided into four equal quarters, the first quarter will contain2m+1 columns and 2m+1 rows. In each of these columns take m cellsso arranged that there are also m cells in each row: as, for instance, bytaking the first m cells in the first column, the cells 2 to m + 1 in thesecond column, and so on, the cells m + 2 to 2m + 1 in the (m + 2)thcolumn, the cells m + 3 to 2m + 1 and the first cell in the (m + 3)thcolumn, and so on, and finally the (2m + 1)th cell and the cells 1 tom − 1 in the (2m + 1)th column. Next change the numbers in thesem(2m+1) cells and the 3m(2m+1) cells associated with them by crossinterchanges. The result is equivalent to 2m interchanges in every rowand in every column. In order to make the square magic we must have12n, that is, 2m+1 such interchanges in every row and in every column,
that is, we must have one more interchange in every row and in everycolumn. This presents no difficulty, for instance, in the arrangementindicated above the numbers in the (2m + 1)th cell of the first column,in the first cell of the second column, in the second cell of the thirdcolumn, and so on, to the 2mth cell in the (2m + 1)th column may beinterchanged with the numbers in their vertically related cells; this willmake all the rows magic. Next, the numbers in the 2mth cell of the firstcolumn, in the (2m + 1)th cell of the second column, in the first cell ofthe third column, in the second cell of the fourth column, and so on, tothe (2m−1)th cell of the (2m+1)th column may be interchanged with
• Project • Gutenberg • #26839 •
132 MAGIC SQUARES. [CH. V
those in the cells horizontally related to them; and this will make thecolumns magic without affecting the magical properties of the rows.
It will be observed that we have implicitly assumed that m is notzero, that is, that n is greater than 2; also it would seem that, if m = 1and therefore n = 6, then the numbers in the diagonal cells must beincluded in those to which the cross interchange is applied, but, if n > 6,this is not necessary, though it may be convenient.
The construction of odd magic squares and of doubly-even magicsquares is very easy. But though the rule given above for singly-evensquares is not difficult, it is tedious of application. It is unfortunatethat no more obvious rule—such, for instance, as one for bordering adoubly-even square—can be suggested for writing down instantly andwithout thought singly-even magic squares.
De la Hire’s Method*. I now proceed to give another way due toDe la Hire, of constructing any even magic square of an order higherthan two.
In the same manner as in his rule for making odd magic squares,we begin by constructing two subsidiary squares, one of the unit-digits,1, 2, 3, . . . , n, and the other of the radix-digits 0, n, 2n, . . . , (n − 1)n.We then form the magic square by adding together the numbers inthe corresponding cells in the two subsidiary squares. Following theanalogy of the notation used above, two numbers which are equidistantfrom the ends of the series 1, 2, 3, . . . , n are said to be complementary.Similarly numbers which are equidistant from the ends of the series0, n, 2n, . . . , (n − 1)n are said to be complementary.
For simplicity I shall apply this method to construct a magic squareof only the sixth order, though an exactly similar method will apply toany even square of an order higher than the second.
The first of the subsidiary squares (figure ix) is constructed as fol-lows. First, the cells in the leading diagonal are filled with the numbers1, 2, 3, 4, 5, 6 placed in any order whatever that puts complementarynumbers in complementary positions (ex. gr. in the order 2, 6, 3, 4,1, 5, or in their natural order 1, 2, 3, 4, 5, 6). Second, the cells ver-tically related to these are filled respectively with the same numbers.
* The rule is due to De la Hire (part 2 of his memoir) and is given by Montuclain his edition of Ozanam’s work: I have used the modified enunciation of itinserted in Labosne’s edition of Bachet’s Problemes, as it saves the introductionof a third subsidiary square. I do not know to whom the modification is due.
• Project • Gutenberg • #26839 •
CH. V] MAGIC SQUARES OF AN EVEN ORDER.. 133
1 5 4 3 2 6
6 2 4 3 5 1
6 5 3 4 2 1
1 5 3 4 2 6
6 2 3 4 5 1
1 2 4 3 5 6
First Subsidiary Square
Figure ix.
0 30 30 0 30 0
24 6 24 24 6 6
18 18 12 12 12 18
12 12 18 18 18 12
6 24 6 6 24 24
30 0 0 30 0 30
Second Subsidiary Square
Figure x.
1 35 34 3 32 6
30 8 28 27 11 7
24 23 15 16 14 19
13 17 21 22 20 18
12 26 9 10 29 25
31 2 4 33 5 36
Resulting Magic Square
Figure xi.
Third, each of the remaining cells in the first vertical column is filledeither with the same number as that already in two of them or withthe complementary number (ex. gr. in figure ix with a “1” or a “6”)in any way, provided that there are an equal number of each of thesenumbers in the column, and subject also to the provisoes mentioned inthe next paragraph but one. Fourth, the cells horizontally related tothose in the first column are filled with the complementary numbers.Fifth, the remaining cells in the second and third columns are filledin an analogous way to that in which those in the first column werefilled: and then the cells horizontally related to them are filled with thecomplementary numbers. The square so formed is necessarily magic inrows, columns, and diagonals.
The second of the subsidiary squares (figure x) is constructed as fol-lows. First, the cells in the leading diagonal are filled with the numbers0, 6, 12, 18, 24, 30 placed in any order whatever that puts comple-mentary numbers in complementary positions. Second, the cells hori-zontally related to them are filled respectively with the same numbers.Third, each of the remaining cells in the first horizontal row is filledeither with the same number as that already in two of them or withthe complementary number (ex. gr. in figure x with a “0” or a “30”) inany way, provided (i) that there are an equal number of each of thesenumbers in the row, and (ii) that if any cell in the first row of figure ixand its vertically related cell are filled with complementary numbers,then the corresponding cell in the first row of figure x and its hori-zontally related cell must be occupied by the same number*. Fourth,the cells vertically related to those in the first row are filled with the
* The insertion of this step evades the necessity of constructing (as Montucla did)a third subsidiary square.
• Project • Gutenberg • #26839 •
134 MAGIC SQUARES. [CH. V
complementary numbers. Fifth, the remaining cells in the second andthe third rows are filled in an analogous way to that in which those inthe first row were filled: and then the cells vertically related to themare filled with the complementary numbers. The square so formed isnecessarily magic in rows, columns, and diagonals.
It remains to show that proviso (ii) in the third step described in thelast paragraph can be satisfied always. In a doubly even square, that is,one in which n is divisible by 4, we need not have any complementarynumbers in vertically related cells in the first subsidiary square unlesswe please, but even if we like to insert them they will not interfere withthe satisfaction of this proviso. In the case of a singly even square, thatis, one in which n is divisible by 2, but not by 4, we cannot satisfy theproviso if any horizontal row in the first square has all its verticallyrelated squares, other than the two squares in the diagonals, filled withcomplementary numbers. Thus in the case of a singly even square itwill be necessary in constructing the first square to take care in thethird step that in every row at least one cell which is not in a diagonalshall have its vertically related cell filled with the same number as itself:this is always possible if n is greater than 2.
The required magic square will be constructed if in each cell weplace the sum of the numbers in the corresponding cells of the sub-sidiary squares, figures ix and x. The result of this is given in figure xi.The square is evidently magic. Also every number from 1 to 36 occursonce and only once, for the numbers from 1 to 6 and from 31 to 36 canoccur only in the top or the bottom rows, and the method of construc-tion ensures that the same number cannot occur twice. Similarly thenumbers from 7 to 12 and from 25 to 30 occupy two other rows, andno number can occur twice; and so on. The square in figure i on page121 may be constructed by the above rules; and the reader will haveno difficulty in applying them to any other even square.
Other Methods for Constructing any Magic Square.The above methods appear to me to be the simplest which have beenproposed. There are however two other methods , of less generality, towhich I will allude briefly in passing. Both depend on the principlethat, if every number in a magic square is multiplied by some constant,and a constant is added to the product, the square will remain magic.
The first method applies only to such squares as can be divided intosmaller magic squares of some order higher than two. It depends on the
• Project • Gutenberg • #26839 •
CH. V] COMPOSITE AND BORDERED SQUARES.. 135
fact that, if we know how to construct magic squares of the mth and nthorders, we can construct one of the mnth order. For example, a squareof 81 cells may be considered as composed of 9 smaller squares eachcontaining 9 cells, and by filling the cells in each of these small squaresin the same relative order and taking the small squares themselves inthe same order, the square can be constructed easily. Such squares arecalled composite magic squares.
The second method, which was introduced by Frenicle, consists insurrounding a magic square with a border . Thus in figure xii on thefollowing page the inner square is magic, and it is surrounded witha border in such a way that the whole square is also magic. In thismanner from the magic square of the 3rd order we can build up suc-cessively squares of the orders 5, 7, 9, &c., that is, any odd magicsquare. Similarly from the magic square of the 4th order we can buildup successively any higher even magic square.
If we construct a magic square of the first n2 numbers by borderinga magic square of (n− 2)2 numbers, the usual process is to reserve forthe 4(n− 1) numbers in the border the first 2(n− 1) natural numbersand the last 2(n − 1) numbers. Now the sum of the numbers in eachline of a square of the order (n− 2) is 1
2(n− 2){(n− 2)2 + 1}, and the
average is 12{(n− 2)2 +1}. Similarly the average number in a square of
the nth order is 12(n2 +1). The difference of these is 2(n−1). We begin
then by taking any magic square of the order (n − 2), and we add toevery number in it 2(n− 1); this makes the average number 1
2(n2 + 1).
The numbers reserved for the border occur in pairs, n2 and 1, n2−1and 2, n2−2 and 3, &c., such that the average of each pair is 1
2(n2 +1),
and they must be bordered on the square so that these numbers areopposite to one another. Thus the bordered square will be necessarilymagic, provided that the sum of the numbers in two adjacent sides ofthe external border is correct. The arrangement of the numbers in theborders will be somewhat facilitated if the number n2 + 1 − p (whichhas to be placed opposite to the number p) is denoted by p, but it isnot worth while going into further details here.
It will illustrate sufficiently the general method if I explain how thesquare in figure xii is constructed. A magic square of the third order isformed by De la Loubere’s rule, and to every number in it 8 is added:the result is the inner square in figure xii. The numbers not used are25 and 1, 24 and 2, 23 and 3, 22 and 4, 21 and 5, 20 and 6, 19 and
• Project • Gutenberg • #26839 •
136 MAGIC SQUARES. [CH. V
7, 18 and 8. The sum of each pair is 26, and obviously they must beplaced at opposite ends of any line.
I believe that with a little patience a magic square of any ordercan be thus built up, and of course it will have the property that,if each border is successively stripped off, the square will still remainmagic. Some examples are given by Violle. This is the method ofconstruction commonly adopted by self-taught mathematicians, many
1 2 19 20 23
18 16 9 14 8
21 11 13 15 5
22 12 17 10 4
3 24 7 6 25
Bordered Magic Square.Figure xii.
7 20 3 11 24
13 21 9 17 5
19 2 15 23 6
25 8 16 4 12
1 14 22 10 18
Nasik Magic Square.Figure xiii.
of whom seem to think that the empirical formation of such squaresis a valuable discovery.
There are magic circles, rectangles, crosses, diamonds, stars, andother figures: also magic cubes, cylinders, and spheres. The theory ofthe construction of such figures is of no value, and I cannot spare thespace to describe rules for forming them.
Hyper-Magic Squares. In recent times attention has beenmainly concentrated on the formation of magic squares with the impo-sition of additional conditions; some of the resulting problems involvemathematical difficulties of a high order.
Nasik Squares. In one species of hyper-magic squares the squaresare formed so that the sums of the numbers along all the rows andcolumns, both diagonals, and all the broken diagonals are the same. InEngland these are called nasik squares or pan-diagonal magic squares :in France carres diaboliques or carres magiquement magiques. Thesesquares were mentioned by De la Hire, Sauveur, and Euler; but thetheory is mainly due to Mr A.H. Frost, who has expounded it in thememoirs mentioned in the footnote on page 123, and to M. Frolow,who treated it in two memoirs, St Petersburg, 1884, and Paris, 1886.Of course a nasik square can be divided by a vertical or horizontal cut
• Project • Gutenberg • #26839 •
CH. V] MAGIC PENCILS. 137
and the pieces interchanged without affecting the magical property. Byone vertical and one horizontal transposition of this kind any numbercan be moved to any specified cell.
A nasik square of the fourth order is represented in figure ii onpage 121, and one of the fifth order is represented in figure xiii on thepreceding page. Nasik squares of the order 6n±1 can be constructed byrules analogous to those given by De la Loubere, except that a knight’sand not a bishop’s move must be used in connecting cells filled byconsecutive numbers and that for orders higher than five special rulesfor going from the cell occupied by the number kn to that occupied bythe number kn + 1 have to be laid down.
Doubly-Magic Squares. In another species of hyper-magic squaresthe problem is to construct a magic square of the nth order in such away that if the number in each cell is replaced by its mth power the
5 31 35 60 57 34 8 30
19 9 53 46 47 56 18 12
16 22 42 39 52 61 27 1
63 37 25 24 3 14 44 50
26 4 64 49 38 43 13 23
41 51 15 2 21 28 62 40
54 48 20 11 10 17 55 45
36 58 6 29 32 7 33 59
A Doubly-Magic Square.Figure xiv.
resulting square shall also be magic. Here for example (see figure xiv)is a magic square* of the eighth order, the sum of the numbers in eachline being equal to 260, so constructed that if the number in each cellis replaced by its square the resulting square is also magic (the sum ofthe numbers in each line being equal to 11180).
Magic Pencils. Hitherto I have concerned myself with num-bers arranged in lines. By reciprocating the figures composed of the
* See M. Coccoz in L’Illustration, May 29, 1897.
• Project • Gutenberg • #26839 •
138 MAGIC SQUARES. [CH. V
points on which the numbers are placed we obtain a collection of linesforming pencils, and, if these lines be numbered to correspond with thepoints, the pencils will be magic*. Thus, in a magic square of the nthorder, we arrange n2 consecutive numbers to form 2n + 2 lines, eachcontaining n numbers so that the sum of the numbers in each line is thesame. Reciprocally we can arrange n2 lines, numbered consecutively toform 2n + 2 pencils, each containing n lines, so that in each pencil thesum of the numbers designating the lines is the same.
For instance, figure xv represents a magic square of 64 consecutive
1 2 62 61 60 59 7 8
9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58 6 5 4 3 63 64
Figure xv.
numbers arranged to form 18 lines, each of 8 numbers. Reciprocally,figure xvi represents 64 lines arranged to form 18 pencils, each of 8 lines.The method of construction is fairly obvious. The eight-rayed pencil,vertex O, is cut by two parallels perpendicular to the axis of the pencil,and all the points of intersection are joined cross-wise. This gives 8pencils, with vertices A, B, . . . , H; 8 pencils, with vertices A′, . . . H ′;one pencil with its vertex at O; and one pencil with its vertex on theaxis of the last-named pencil.
The sum of the numbers in each of the 18 lines in figure xv isthe same. To make figure xvi correspond to this we must number thelines in the pencil A from left to right, 1, 9, . . . , 57, following the orderof the numbers in the first column of the square: the lines in pencil Bmust be numbered similarly to correspond to the numbers in the secondcolumn of the square, and so on. To prevent confusion in the figure I
* See Magic-Reciprocals by G. Frankenstein, Cincinnati, 1875.
• Project • Gutenberg • #26839 •
CH. V] MAGIC PENCILS. 139
O
A B C D E F G H
A' H'
Figure xvi.
have not inserted the numbers, but it will be seen that the method ofconstruction ensures that the sum of the 8 numbers which designatethe lines in each of these 18 pencils is the same.
We can proceed a step further, if the resulting figure is cut bytwo other parallel lines perpendicular to the axis, and if the points oftheir intersection with the cross-joins be joined cross-wise, these newcross-joins will intersect on the axis of the original pencil or on lines
• Project • Gutenberg • #26839 •
140 MAGIC SQUARES. [CH. V
perpendicular to it. The whole figure will now give 83 lines, arranged in244 pencils each of 8 rays, and will be the reciprocal of a magic cube ofthe 8th order. If we reciprocate back again we obtain a representationin a plane of a magic cube.
Magic Square Puzzles. Many empirical problems, closelyrelated to magic squares, will suggest themselves; but most of them aremore correctly described as ingenious puzzles than as mathematicalrecreations. The following will serve as specimens.
Magic Card Square*. The first of these is the familiar problem ofplacing the sixteen court cards (taken out of a pack) in the form of asquare so that no row, no column, and neither of the diagonals shallcontain more than one card of each suit and one card of each rank.The solution presents no difficulty, and is indicated in figure xviii onthe next page.
Euler’s Officers Problem†. A similar problem, proposed by Eulerin 1779, consists in arranging, if it be possible, thirty-six officers takenfrom six regiments—the officers being in six groups, each consisting ofsix officers of equal rank, one drawn from each regiment; say officers ofrank a, b, c, d, e, f , drawn from the 1st, 2nd, 3rd, 4th, 5th, and 6thregiments—in a solid square formation of six by six, so that each rowand each file shall contain one and only one officer of each rank andone and only one officer from each regiment. The problem is insoluble.
Extension of Euler’s Problem. More generally we may investigatethe arrangement on a chess-board, containing n2 cells, of n2 counters(the counters being divided into n groups, each group consisting of ncounters of the same colour numbered consecutively 1, 2, . . . , n) so thateach row and each column shall contain no two counters of the samecolour or marked with the same number.
For instance, if n = 3, with three red counters a1, a2, a3, threewhite counters b1, b2, b3, and three black counters c1, c2, c3, we cansatisfy the conditions by arranging them as in figure xvii on the facingpage. If n = 4, then with counters a1, a2, a3, a4; b1, b2, b3, b4; c1, c2,
* Ozanam, 1723 edition, vol. iv, p. 434.† Euler’s Commentationes Arithmeticae, St Petersburg, 1849, vol ii, pp. 302–361.
See also a paper by G. Tarry in the Comptes rendus of the French Associationfor the Advancement of Science, Paris, 1900, vol. ii, pp. 170–203; and variousnotes in L’Intermediaire des mathematiciens, Paris, vol. iii, 1896, pp. 17, 90;vol. v, 1898, pp. 83, 176, 252, vol. vi, 1899, p. 251; vol. vii, 1900, pp. 14, 311.
• Project • Gutenberg • #26839 •
CH. V] MAGIC PUZZLES.. 141
c3, c4; d1, d2, d3, d4, we can arrange them as in figure xviii below. Asolution when n = 5 is indicated in figure xix.
a1 b2 c3
b2 c1 a2
c3 a3 b1
Figure xvii.
a1 b2 c3 d4
c4 d3 a2 b1
d2 c1 b4 a3
b3 a4 d1 c2
Figure xviii.
a1 b2 c3 d4 e5
b5 c1 d2 e3 a4
c4 d5 e1 a2 b3
d3 e4 a5 b1 c2
e2 a3 b4 c5 d1
Figure xix.
The problem is soluble if n is odd or if n is of the form 4m. Ifsolutions when n = a and when n = b are known, a solution whenn = ab can be written down at once. The theory is closely connectedwith that of magic squares and need not be here discussed further.
Magic Domino Squares. Analogous problems can be made withdominoes. An ordinary set of dominoes, ranging from double zero todouble six, contains 28 dominoes. Each domino is a rectangle formedby fixing two small square blocks together side by side: of these 56blocks, eight are blank, on each of eight of them is one pip, on each ofanother eight of them are two pips, and so on. It is required to arrangethe dominoes so that the 56 blocks form a square of 7 by 7 borderedby one line of 7 blank squares and so that the sum of the pips in eachrow, each column, and in the two diagonals of the square is equal to24. A solution* is given on the following page.
Similarly, a set of dominoes, ranging from double zero to double n,contains 1
2(n + 1)(n + 2) dominoes and therefore (n + 1)(n + 2) blocks.
Can these dominoes be arranged in the form of a square of (n + 1)2
cells, bordered by a row of blanks, so that the sum of the pips in eachrow, each column, and in the two diagonals of the square is equal to12n(n + 2)?
Magic Coin Squares†. There are somewhat similar questions con-cerned with coins. Here is one applicable to a square of the third orderdivided into nine cells, as in figure xvii above. If a five-shilling pieceis placed in the middle cell c1 and a florin in the cell below it, namely,in a3 it is required to place the fewest possible current English coins in
* See L’Illustration, July 10, 1897.† See The Strand Magazine, London, December, 1896, pp. 720, 721.
• Project • Gutenberg • #26839 •
142 MAGIC SQUARES. [CH. V
Magic Domino Square.
the remaining seven cells so that in each cell there is at least one coin,so that the total value of the coins in every cell is different, and so thatthe sum of the values of the coins in each row, column, and diagonal isfifteen shillings: it will be found that thirteen additional coins will suf-fice. A similar problem is to place ten current English postage stamps,all but two being different, in the nine cells so that the sum of thevalues of the stamps in each row, column, and diagonal is ninepence.
• Project • Gutenberg • #26839 •
CHAPTER VI.
UNICURSAL PROBLEMS.
I propose to consider in this chapter some problems which ariseout of the theory of unicursal curves. I shall commence with Euler’sProblem and Theorems, and shall apply the results briefly to the theo-ries of Mazes and Geometrical Trees. The reciprocal unicursal problemsof the Hamilton Game and the Knight’s Path on a Chess-board will bediscussed in the latter half of the chapter.
Euler’s Problem. Euler’s problem has its origin in a memoir*
presented by him in 1736 to the St Petersburg Academy, in which hesolved a question then under discussion as to whether it was possibleto take a walk in the town of Konigsberg in such a way as to crossevery bridge in it once and only once.
The town is built near the mouth of the river Pregel, which theretakes the form indicated on the following page and includes the islandof Kneiphof. In 1759 there were (and according to Baedeker there arestill) seven bridges in the positions shown in the diagram, and it iseasily seen that with such an arrangement the problem is insoluble.Euler however did not confine himself to the case of Konigsberg, butdiscussed the general problem of any number of islands connected inany way by bridges. It is evident that the question will not be affected if
* Solutio problematis ad Geometriam situs pertinentis, Commentarii AcademiaeScientiarum Petropolitanae for 1736, St Petersburg, 1741, vol. viii, pp. 128–140.This has been translated into French by M. Ch. Henry; see Lucas, vol. i, part 2,pp. 21–33.
143
• Project • Gutenberg • #26839 •
144 UNICURSAL PROBLEMS. [CH. VI
we suppose the islands to diminish to points and the bridges to lengthenout. In this way we ultimately obtain a geometrical figure or network.In the Konigsberg problem this figure is of the shape indicated below,the areas being represented by the points A, B, C, D, and the bridgesbeing represented by the lines l, m, n, p, q, r, s.
A
B
C
D
s
r
l
q p
m n
Euler’s problem consists therefore in finding whether a given ge-ometrical figure can be described by a point moving so as to traverseevery line in it once and only once. A more general question is to deter-mine how many strokes are necessary to describe such a figure so thatno line is traversed twice: this is covered by the rules hereafter given.The figure may be either in three or in two dimensions, and it may berepresented by lines, straight, curved, or tortuous, joining a number ofgiven points, or a model may be constructed by taking a number ofrods or pieces of string furnished at each end with a hook so as to allowof any number of them being connected together at one point.
The theory of such figures is included as a particular case in the
• Project • Gutenberg • #26839 •
CH. VI] EULER’S PROBLEM. 145
propositions proved by Listing in his Topologie*. I shall, however, adopthere the methods of Euler, and I shall begin by giving some definitions,as it will enable me to put the argument in a more concise form.
A node (or isle) is a point to or from which lines are drawn. Abranch (or bridge or path) is a line connecting two consecutive nodes.An end (or hook) is the point at each termination of a branch. Theorder of a node is the number of branches which meet at it. A node towhich only one branch is drawn is a free node or a free end. A node atwhich an even number of branches meet is an even node: evidently thepresence of a node of the second order is immaterial. A node at whichan odd number of branches meet is an odd node. A figure is closed ifit has no free end: such a figure is often called a closed network.
A route consists of a number of branches taken in consecutive orderand so that no branch is traversed twice. A closed route terminates atthe point from which it started. A figure is described unicursally whenthe whole of it is traversed in one route.
The following are Euler’s results. (i) In a closed network the num-ber of odd nodes is even. (ii) A figure which has no odd node can bedescribed unicursally, in a re-entrant route, by a moving point whichstarts from any point on it. (iii) A figure which has two and only twoodd notes can be described unicursally by a moving point which startsfrom one of the odd nodes and finishes at the other. (iv) A figure whichhas more than two odd nodes cannot be described completely in oneroute; to which Listing added the corollary that a figure which has 2nodd nodes, and no more, can be described completely in n separateroutes. I now proceed to prove these theorems.
First. The number of odd nodes in a closed network is even.
Suppose the number of branches to be b. Therefore the number ofhooks is 2b. Let kn be the number of nodes of the nth order. Sincea node of the nth order is one at which n branches meet, there are nhooks there. Also since the figure is closed, n cannot be less than 2.
∴ 2k2 + 3k3 + 4k4 + · · · + nkn + · · · = 2b .Hence 3k3 + 5k5 + · · · is even.
∴ k3 + k5 + · · · is even.
* Die Studien, Gottingen, 1847, part x. See also Tait on Listing’s Topologie,Philosophical Magazine, London, January, 1884, series 5, vol. xvii, pp. 30–46.
• Project • Gutenberg • #26839 •
146 UNICURSAL PROBLEMS. [CH. VI
Second. A figure which has no odd node can be described unicur-sally in a re-entrant route.
Since the route is to be re-entrant it will make no difference whereit commences. Suppose that we start from a node A. Every time ourroute takes us through a node we use up one hook in entering it and onein leaving it. There are no odd nodes, therefore the number of hooksat every node is even: hence, if we reach any node except A, we shallalways find a hook which will take us into a branch previously untra-versed. Hence the route will take us finally to the node A from whichwe started. If there are more than two hooks at A, we can continue theroute over one of the branches from A previously untraversed, but inthe same way as before we shall finally come back to A.
It remains to show that we can arrange our route so as to makeit cover all the branches. Suppose each branch of the network to berepresented by a string with a hook at each end, and that at each nodeall the hooks there are fastened together. The number of hooks ateach node is even, and if they are unfastened they can be re-coupledtogether in pairs, the arrangement of the pairs being immaterial. Thewhole network will then form one or more closed curves, since now eachnode consists merely of two ends hooked together.
If this random coupling gives us one single curve then the proposi-tion is proved; for starting at any point we shall go along every branchand come back to the initial point. But if this random coupling pro-duces anywhere an isolated loop, L, then where it touches some otherloop, M , say at the node P , unfasten the four hooks there (viz. two ofthe loop L and two of the loop M) and re-couple them in any otherorder: then the loop L will become a part of the loop M . In this way,by altering the couplings, we can transform gradually all the separateloops into parts of only one loop.
For example, take the case of three isles, A, B, C, each connectedwith both the others by two bridges. The most unfavourable way of re-coupling the ends at A, B, C would be to make ABA, ACA, and BCBseparate loops. The loops ABA and ACA are separate and touch at A;hence we should re-couple the hooks at A so as to combine ABA andACA into one loop ABACA. Similarly, by re-arranging the couplingsof the four hooks at B, we can combine the loop BCB with ABACAand thus make only one loop.
I infer from Euler’s language that he had attempted to solve the
• Project • Gutenberg • #26839 •
CH. VI] EULER’S PROBLEM. 147
B C
A
problem of giving a practical rule which would enable one to describesuch a figure unicursally without knowledge of its form, but that in thishe was unsuccessful. He however added that any geometrical figure canbe described completely in a single route provided each part of it isdescribed twice and only twice, for, if we suppose that every branch isduplicated, there will be no odd nodes and the figure is unicursal. Inthis case any figure can be described completely without knowing itsform: rules to effect this are given below.
Third. A figure which has two and only two odd nodes can bedescribed unicursally by a point which starts from one of the odd nodesand finishes at the other odd node.
This at once reduces to the second theorem. Let A and Z be thetwo odd nodes. First, suppose that Z is not a free end. We can, ofcourse, take a route from A to Z; if we imagine the branches in thisroute to be eliminated, it will remove one hook from A and make iteven, will remove two hooks from every node intermediate between Aand Z and therefore leave each of them even, and will remove one hookfrom Z and therefore will make it even. All the remaining network isnow even: hence, by Euler’s second proposition, it can be describedunicursally, and, if the route begins at Z, it will end at Z. Hence,if these two routes are taken in succession, the whole figure will bedescribed unicursally, beginning at A and ending at Z. Second, if Zis a free end, then we must travel from Z to some node, Y , at whichmore than two branches meet. Then a route from A to Y which coversthe whole figure exclusive of the path from Y to Z can be determinedas before and must be finished by travelling from Y to Z.
Fourth. A figure having 2n odd nodes, and no more, can be de-scribed completely in n separate routes.
If any route starts at an odd node, and if it is continued until it
• Project • Gutenberg • #26839 •
148 UNICURSAL PROBLEMS. [CH. VI
reaches a node where no fresh path is open to it, this latter node mustbe an odd one. For every time we enter an even node there is necessarilya way out of it; and similarly every time we go through an odd nodewe use up one hook in entering and one hook in leaving, but wheneverwe reach it as the end of our route we use only one hook. If this routeis suppressed there will remain a figure with 2n− 2 odd nodes. Hencen such routes will leave one or more networks with only even nodes.But each of these must have some node common to one of the routesalready taken and therefore can be described as a part of that route.Hence the complete passage will require n and not more than n routes.It follows, as stated by Euler, that, if there are more than two oddnodes, the figure cannot be traversed completely in one route.
The Konigsberg bridges lead to a network with four odd nodes;hence, by Euler’s fourth proposition, it cannot be described unicur-sally in a single journey, though it can be traversed completely in twoseparate routes.
The first and second diagrams figured below contain only evennodes, and therefore each of them can be described unicursally. The
first of these—a re-entrant pentagon—was one of the Pythagorean sym-bols. The other is the so-called sign-manual of Mohammed, said to havebeen originally traced in the sand by the point of his scimetar with-out taking the scimetar off the ground or retracing any part of thefigure—which, as it contains only even nodes, is possible. The thirddiagram is taken from Tait’s article: it contains only two odd nodes,and therefore can be described unicursally if we start from one of themand finish at the other.
As other examples I may note that the geometrical figure formedby taking a (2n + 1)gon and joining every angular point with every
• Project • Gutenberg • #26839 •
CH. VI] MAZES AND LABYRINTHS.. 149
other angular point is unicursal. On the other hand a chess-board,divided as usual by straight lines into 64 cells, has 28 odd nodes and53 even nodes: hence it would require 14 separate pen-strokes to traceout all the boundaries without going over any more than once. Again,the diagram on page 102 has 20 odd nodes and therefore would require10 separate pen-strokes to trace it out.
It is well known that a curve which has as many nodes as is con-sistent with its degree is unicursal.
Mazes. Everyone has read of the labyrinth of Minos in Creteand of Rosamund’s Bower. A few modern mazes exist here and there—notably one, which is a very poor specimen of its kind, at HamptonCourt—and in one of these, or at any rate on a drawing of one, most ofus have threaded our way to the interior. I proceed now to consider themanner in which any such construction may be completely traversedeven by one who is ignorant of its plan.
The theory of the description of mazes is included in Euler’s theo-rems given above. The paths in the maze are what previously we havetermed branches, and the places where two or more paths meet arenodes. The entrance to the maze, the end of a blind alley, and thecentre of the maze are free ends and therefore odd nodes.
If the only odd nodes are the entrance to the maze and the centreof it–which will necessitate the absence of all blind alleys–the maze canbe described unicursally. This follows from Euler’s third proposition.Again, no matter how many odd nodes there may be in a maze, we canalways find a route which will take us from the entrance to the centrewithout retracing our steps, though such a route will take us throughonly a part of the maze. But in neither of the cases mentioned in thisparagraph can the route be determined without a plan of the maze.
A plan is not necessary, however, if we make use of Euler’s sugges-tion, and suppose that every path in the maze is duplicated. In thiscase we can give definite rules for the complete description of the wholeof any maze, even if we are entirely ignorant of its plan. Of course towalk twice over every path in a labyrinth is not the shortest way ofarriving at the centre, but, if it is performed correctly, the whole mazeis traversed, the arrival at the centre at some point in the course of theroute is certain, and it is impossible to lose one’s way.
• Project • Gutenberg • #26839 •
150 UNICURSAL PROBLEMS. [CH. VI
I need hardly explain why the complete description of such a du-plicated maze is possible, for now every node is even, and hence, byEuler’s second proposition, if we begin at the entrance we can traversethe whole maze; in so doing we shall at some point arrive at the cen-tre, and finally shall emerge at the point from which we started. Thisdescription will require us to go over every path in the maze twice,and as a matter of fact the two passages along any path will be alwaysmade in opposite directions.
If a maze is traced on paper, the way to the centre is generallyobvious, but in an actual labyrinth it is not so easy to find the correctroute unless the plan is known. In order to make sure of describinga maze without knowing its plan it is necessary to have some meansof marking the paths which we traverse and the direction in which wehave traversed them—for example, by drawing an arrow at the entranceand end of every path traversed, or better perhaps by marking the wallon the right-hand side, in which case a path may not be entered whenthere is a mark on each side of it. If we can do this, and if when a nodeis reached, we take, if it be possible, some path not previously used,or, if no other path is available, we enter on a path already traversedonce only, we shall completely traverse any maze in two dimensions*.Of course a path must not be traversed twice in the same direction, apath already traversed twice (namely, once in each direction) must notbe entered, and at the end of a blind alley it is necessary to turn backalong the path by which it was reached.
I think most people would understand by a maze a series of inter-lacing paths through which some route can be obtained leading to aspace or building at the centre of the maze. I believe that few, if any,mazes of this type existed in classical or medieval times.
One class of what the ancients called mazes or labyrinths seemsto have comprised any complicated building with numerous vaults andpassages†. Such a building might be termed a labyrinth, but it is not
* See Le probleme des labyrinthes by G. Tarry, Nouvelles Annales de mathemat-iques, May, 1895, series 3, vol. xiv.
† For instance, see the descriptions of the labyrinth at Lake Moeris given byHerodotus, bk. ii, c. 148; Strabo, bk. xvii, c. 1, art. 37; Diodorus, bk. i, cc. 61,66; and Pliny, Hist. Nat., bk. xxxvi, c. 13, arts. 84–89. On these and otherreferences see A. Wiedemann, Herodots zweites Buch, Leipzig, 1890, p. 522 etseq. See also Virgil, Aeneid, bk. v, c. v, 588; Ovid, Met., bk. viii, c. 5, 159;Strabo, bk. viii, c. 6.
• Project • Gutenberg • #26839 •
CH. VI] MAZES AND LABYRINTHS.. 151
what is usually understood by the word. The above rules would enableanyone to traverse the whole of any structure of this kind. I do notknow if there are any accounts or descriptions of Rosamund’s Bowerother than those by Drayton, Bromton, and Knyghton: in the opinionof some, these imply that the bower was merely a house, the passagesin which were confusing and ill-arranged.
Another class of ancient mazes consisted of a tortuous path confinedto a small area of ground and leading to a place or shrine in the centre*.This is a maze in which there is no chance of taking a wrong turning;but, as the whole area can be occupied by the windings of one path,the distance to be traversed from the entrance to the centre may beconsiderable, even though the piece of ground covered by the mazeis but small.
The traditional form of the labyrinth constructed for the Minotauris a specimen of this class. It was delineated on the reverses of the coinsof Cnossus, specimens of which are not uncommon; one form of it isindicated in the accompanying diagram (figure i). The design really isthe same as that drawn in figure ii, as can be easily seen by bendinground a circle the rectangular figure there given.
Mr Inwards has suggested† that this design on the coins of Cnossusmay be a survival from that on a token given by the priests as a clue to
Figure i. Figure ii.
the right path in the labyrinth there. Taking the circular form of thedesign shown above he supposed each circular wall to be replaced bytwo equidistant walls separated by a path, and thus obtained a maze
* On ancient and medieval labyrinths—particularly of this kind—see an article byMr E. Trollope in The Archaeological Journal, 1858, vol. xv, pp. 216–235, fromwhich much of the historical information given above is derived
† Knowledge, London, October, 1892.
• Project • Gutenberg • #26839 •
152 UNICURSAL PROBLEMS. [CH. VI
to which the original design would serve as the key. The route thusindicated may be at once obtained by noticing that when a node isreached (i.e. a point where there is a choice of paths) the path to betaken is that which is next but one to that by which the node wasapproached. This maze may be also threaded by the simple rule ofalways following the wall on the right-hand side or always that on theleft-hand side. The labyrinth may be somewhat improved by erecting afew additional barriers, without affecting the applicability of the aboverules, but it cannot be made really difficult. This makes a pretty toy,but though the conjecture on which it is founded is ingenious it mustbe regarded as exceedingly improbable. Another suggestion is that thecurved line on the reverse of the coins indicated the form of the ropeheld by those taking part in some rhythmic dance; while others considerthat the form was gradually evolved from the widely prevalent svastika.
Copies of the maze of Cnossus were frequently engraved on Greekand Roman gems; similar but more elaborate designs are found in nu-merous Roman mosaic pavements*. A copy of the Cretan labyrinthwas embroidered on many of the state robes of the later Emperors,and, apparently thence, was copied on to the walls and floors of vari-ous churches†. At a later time in Italy and in France these mural andpavement decorations were developed into scrolls of great complexity,but consisting, as far as I know, always of a single line. Some of thebest specimens now extant are on the walls of the cathedrals at Lucca,Aix in Provence, and Poitiers; and on the floors of the churches of SantaMaria in Trastevere at Rome, San Vitale at Ravenna, Notre Dame atSt Omer, and the cathedral at Chartres. It is possible that they wereused to represent the journey through life as a kind of pilgrim’s progress.
In England these mazes were usually, perhaps always, cut in theturf adjacent to some religious house or hermitage: and there are someslight reasons for thinking that, when traversed as a religious exercise,a pater or ave had to be repeated at every turning. After the Renais-sance, such labyrinths were frequently termed Troy-towns or Julian’sbowers. Some of the best specimens, which are still extant, are those atRockliff Marshes, Cumberland; Asenby, Yorkshire; Alkborough, Lin-colnshire; Wing, Rutlandshire; Boughton-Green, Northamptonshire;Comberton, Cambridgeshire; Saffron Walden, Essex; and Chilcombe,
* See ex. gr. Breton’s Pompeia, p. 303.† Ozanam, Graphia aureae urbis Romae, pp. 92, 178.
• Project • Gutenberg • #26839 •
CH. VI] MAZES AND LABYRINTHS.. 153
near Winchester.
The modern maze seems to have been introduced—probably fromItaly—during the Renaissance, and many of the palaces and largehouses built in England during the Tudor and the Stuart periodshad labyrinths attached to them. Those adjoining the royal palacesat Southwark, Greenwich, and Hampton Court were particularly wellknown from their vicinity to the capital. The last of these was designedby London and Wise in 1690, for William III, who had a fancy for suchconceits: a plan of it is given in various guide-books. For the majority
Maze at Hampton Court.
of the sight-seers who enter, it is sufficiently elaborate; but it is anindifferent construction, for it can be described completely by alwaysfollowing the hedge on one side (either the right hand or the left hand),and no node is of an order higher than three.
Unless at some point the route to the centre forks and subsequentlythe two forks reunite, forming a loop in which the centre of the mazeis situated, the centre can be reached by the rule just given, namely,by following the wall on one side—either on the right hand or on theleft hand. No labyrinth is worthy of the name of a puzzle which canbe threaded in this way. Assuming that the path forks as describedabove, the more numerous the nodes and the higher their order themore difficult will be the maze, and the difficulty might be increasedconsiderably by using bridges and tunnels so as to construct a labyrinthin three dimensions. In an ordinary garden and on a small piece ofground, often of an inconvenient shape, it is not easy to make a mazewhich fulfils these conditions. Here on the following page is a plan ofone which I put up in my own garden on a plot of ground which would
• Project • Gutenberg • #26839 •
154 UNICURSAL PROBLEMS. [CH. VI
not allow of more than 36 by 23 paths, but it will be noticed that noneof the nodes are of a high order.
Geometrical Trees. Euler’s original investigations were con-fined to a closed network. In the problem of the maze it was assumedthat there might be any number of blind alleys in it, the ends of whichformed free nodes. We may now progress one step farther, and sup-pose that the network or closed part of the figure diminishes to a point.This last arrangement is known as a tree. The number of unicursaldescriptions necessary to completely describe a tree is called the baseof the ramification.
We can illustrate the possible form of these trees by rods, having ahook at each end. Starting with one such rod, we can attach at eitherend one or more similar rods. Again, on any free hook we can attachone or more similar rods, and so on. Every free hook, and also everypoint where two or more rods meet, are what hitherto we have callednodes. The rods are what hitherto we have termed branches or paths.
The theory of trees—which already plays a somewhat importantpart in certain branches of modern analysis, and possibly may containthe key to certain chemical and biological theories—originated in amemoir by Cayley*, written in 1856. The discussion of the theory has
* Philosophical Magazine, March, 1857, series 4, vol. xiii, pp. 172–176; or Collected
• Project • Gutenberg • #26839 •
CH. VI] THE HAMILTONIAN GAME. 155
been analytical rather than geometrical. I content myself with notingthe following results.
The number of trees with n given nodes is nn−2. If An is thenumber of trees with n branches, and Bn the number of trees with nfree branches which are bifurcations at least, then
(1− x)−1(1− x2)−A1(1− x3)−A2 · · · = 1 + A1x + A2x2 + A3x
3 + · · · ,
(1− x)−1(1− x2)−B2(1− x3)−B3 · · · = 1 + x + 2B2x2 + 2B3x
3 + · · · .
Using these formulae we can find successively the values of A1, A2, . . . ,and B1, B2, . . . . The values of An when n = 2, 3, 4, 5, 6, 7, are 2, 4,9, 20, 48, 115; and of Bn are 1, 2, 5, 12, 33, 90.
I turn next to consider some problems where it is desired to find aroute which will pass once and only once through each node of a givengeometrical figure. This is the reciprocal of the problem treated in thefirst part of this chapter, and is a far more difficult question. I am notaware that the general theory has been considered by mathematicians,though two special cases—namely, the Hamiltonian (or Icosian) Gameand the Knight’s Path on a Chess-Board—have been treated in somedetail; and I confine myself to a discussion of these.
The Hamiltonian Game. The Hamiltonian Game consists inthe determination of a route along the edges of a regular dodecahedronwhich will pass once and only once through every angular point. SirWilliam Hamilton*, who invented this game—if game is the right termfor it—denoted the twenty angular points on the solid by letters whichstand for various towns. The thirty edges constitute the only possible
Works, Cambridge, 1890, vol. iii, no. 203, pp. 242–346: see also the paper ondouble partitions, Philosophical Magazine, November, 1860, series 4, vol. xx,pp. 337–341. On the number of trees with a given number of nodes, see theQuarterly Journal of Mathematics, London, 1889, vol. xxiii, pp. 376–378. Theconnection with chemistry was first pointed out in Cayley’s paper on isomers,Philosophical Magazine, June, 1874, series 4, vol. xlvii, pp. 444–447, and wastreated more fully in his report on trees to the British Association in 1875,Reports, pp. 257–305.
* See Quarterly Journal of Mathematics, London, 1862, vol. v, p. 305; or Philo-sophical Magazine, January, 1884, series 5, vol. xvii, p. 42; also Lucas, vol. ii,part vii.
• Project • Gutenberg • #26839 •
156 UNICURSAL PROBLEMS. [CH. VI
paths. The inconvenience of using a solid is considerable, and thedodecahedron may be represented conveniently in perspective by a flatboard marked as shown in the first of the annexed diagrams. Thesecond and third diagrams will answer our purpose equally well andare easier to draw.
A G
J
L
N
P
Q R
S
T
BC
DE
F H
KM
O
UA
BC
DE
F
G
H
JK
L
MN
OP
Q R
S
T
U A
B
C
D
E
F
U
The first problem is go “all round the world,” that is, starting fromany town, to go to every other town once and only once and to returnto the initial town; the order of the n towns to be first visited beingassigned, where n is not greater than five.
Hamilton’s rule for effecting this was given at the meeting in 1857of the British Association at Dublin. At each angular point there arethree and only three edges. Hence, if we approach a point by one edge,the only routes open to us are one to the right, denoted by r, and oneto the left, denoted by l. It will be found that the operations indicatedon opposite sides of the following equalities are equivalent,
lr2l = rlr, rl2r = lrl, lr3l = r2, rl3r = l2 .
Also the operation l5 or r5 brings us back to the initial point: we mayrepresent this by the equations
l5 = 1, r5 = 1 .
To solve the problem for a figure having twenty angular pointswe must deduce a relation involving twenty successive operations, thetotal effect of which is equal to unity. By repeated use of the relation
• Project • Gutenberg • #26839 •
CH. VI] THE HAMILTONIAN GAME. 157
l2 = rl3r we see that
1 = l5 = l2l3 = (rl3r)l3 = {rl3}2 = {r(rl3r)l}2
= {r2l3rl}2 = {r2(rl3r)lrl}2 = {r3l3rlrl}2 .
Therefore {r3l3(rl)2}2 = 1 . . . . . . . (i),
and similarly {l3r3(lr)2}2 = 1 . . . . . . . (ii).
Hence on a dodecahedron either of the operations
r r r l l l r l r l r r r l l l r l r l . . . . . . . (i),
l l l r r r l r l r l l l r r r l r l r . . . . . . . (ii),
indicates a route which takes the traveller through every town, Thearrangement is cyclical, and the route can be commenced at any pointin the series of operations by transferring the proper number of lettersfrom one end to the other. The point at which we begin is determinedby the order of certain towns which is given initially.
Thus, suppose that we are told that we start from F and thensuccessively go to B, A, U , and T , and we want to find a route fromT through all the remaining towns which will end at F . If we think ofourselves as coming into F from G, the path FB would be indicatedby l, but if we think of ourselves as coming into F from E, the pathFB would be indicated by r. The path from B to A is indicated by l,and so on. Hence our first paths are indicated either by l l l r or byr l l r. The latter operation does not occur either in (i) or in (ii), andtherefore does not fall within our solutions. The former operation maybe regarded either as the 1st, 2nd, 3rd, and 4th steps of (ii), or as the4th, 5th, 6th, and 7th steps of (i). Each of these leads to a route whichsatisfies the problem. These routes are
F B A U T P O N C D E J K L M Q R S H G F ,
and F B A U T S R K L M Q P O N C D E J H G F .
It is convenient to make a mark or to put down a counter at eachcorner as soon as it is reached, and this will prevent our passing throughthe same town twice.
A similar game may be played with other solids provided that ateach angular point three and only three edges meet. Of such solids atetrahedron and a cube are the simplest instances, but the reader can
• Project • Gutenberg • #26839 •
158 UNICURSAL PROBLEMS. [CH. VI
make for himself any number of plane figures representing such solidssimilar to those drawn on page 156. Some of these were indicated byHamilton. In all such cases we must obtain from the formulae analogousto those given above cyclical relations like (i) or (ii) there given. Thesolution will then follow the lines indicated above. This method maybe used to form a rule for describing any maze in which no node is ofan order higher than three.
For solids having angular points where more than three edgesmeet—such as the octahedron where at each angular point four edgesmeet, or the icosahedron where at each angular point five edges meet—we should at each point have more than two routes open to us; hence(unless we suppress some of the edges) the symbolical notation wouldhave to be extended before it could be applied to these solids. I offerthe suggestion to anyone who is desirous of inventing a new game.
Another and a very elegant solution of the Hamiltonian dodecahe-dron problem has been given by M. Hermary. It consists in unfoldingthe dodecahedron into its twelve pentagons, each of which is attachedto the preceding one by only one of its sides; but the solution is geo-metrical, and not directly applicable to more complicated solids.
Hamilton suggested as another problem to start from any town,to go to certain specified towns in an assigned order, then to go toevery other town once and only once, and to end the journey at somegiven town. He also suggested the consideration of the way in which acertain number of towns should be blocked so that there was no passagethrough them, in order to produce certain effects. These problems havenot, so far as I know, been subjected to mathematical analysis.
Knight’s Path on a Chess-Board. Another geometricalproblem on which a great deal of ingenuity has been expended, and ofa kind somewhat similar to the Hamiltonian game, consists in movinga knight on a chess-board in such a manner that it shall move suc-cessively on to every cell* once and only once. The literature on thissubject is so extensive† that I make no pretence to give a full account ofthe various methods for solving the problem, and I shall content myself
* The 64 small squares into which a chess-board is divided are termed cells.† For a bibliography see A. van der Linde, Geschichte und Literatur des
Schachspiels, Berlin, 1874, vol. ii, pp. 101–111. On the problem see a memoirby P. Volpicelli in Atti della Reale Accademia dei Lincei, Rome, 1872, vol. xxv,pp. 87–162: also Applications de l’Analyse Mathematique au Jeu des echecs,
• Project • Gutenberg • #26839 •
CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 159
by putting together a few notes on some of the solutions I have comeacross, particularly on those due to De Moivre, Euler, Vandermonde,Warnsdorff, and Roget.
On a board containing an even number of cells the path may ormay not be re-entrant, but on a board containing an odd number ofcells it cannot be re-entrant. For, if a knight begins on a white cell, itsfirst move must take it to a black cell, the next to a white cell, and soon. Hence, if its path passes through all the cells, then on a board ofan odd number of cells the last move must leave it on a cell of the samecolour as that on which it started, and therefore these cells cannot beconnected by one move.
The earliest solutions of which I have any knowledge are thosegiven about the end of the seventeenth century by De Montmort andDe Moivre*. They apply to the ordinary chess-board of 64 cells, and de-pend on dividing (mentally) the board into an inner square containingsixteen cells surrounded by an outer ring of cells two deep. If initiallythe knight is placed on a cell in the outer ring, it moves round that ringalways in the same direction so as to fill it up completely—only goinginto the inner square when absolutely necessary. When the outer ringis filled up the order of the moves required for filling the remaining cellspresents but little difficulty. If initially the knight is placed on the innersquare the process must be reversed. The method can be applied tosquare and rectangular boards of all sizes. It is illustrated sufficientlyby De Moivre’s solution which is given on the following page, where thenumbers indicate the order in which the cells are occupied successively.I place by its side a somewhat similar re-entrant solution, due to Euler,for a board of 36 cells. If a chess-board is used it is convenient to placea counter on each cell as the knight leaves it.
The next serious attempt to deal with the subject was made by Eu-ler† in 1759: it was due to a suggestion made by L. Bertrand of Geneva,who subsequently (in 1778) issued an account of it. This method is ap-plicable to boards of any shape and size, but in general the solutions
by C.F. de Jaenisch, 3 vols., St Petersburg, 1862–3; and General Parmentier,Association Francaise pour l’avancement des Sciences, 1891, 1892, 1894.
* I do not know where they were published originally; they were quoted byOzanam and Montucla, see Ozanam, 1803 edition, vol. i, p. 178; 1840 edition,p. 80.
† Memoires de Berlin for 1759, Berlin, 1766, pp. 310–337; or CommentationesArithmeticae Collectae, St Petersburg, 1849, vol. i, pp. 337–355.
• Project • Gutenberg • #26839 •
160 UNICURSAL PROBLEMS. [CH. VI
34 49 22 11 36 39 24 1
21 10 35 50 23 12 37 40
48 33 62 57 38 25 2 13
9 20 51 54 63 60 41 26
32 47 58 61 56 53 14 3
19 8 55 52 59 64 27 42
46 31 6 17 44 29 4 15
7 18 45 30 5 16 43 28
De Moivre’s Solution.
30 21 6 15 28 19
7 16 29 20 5 14
22 31 8 35 18 27
9 36 17 26 13 4
32 23 2 11 34 25
1 10 33 24 3 12
Euler’s Thirty-six Cell Solution.
to which it leads are not symmetrical and their mutual connexion isnot apparent.
Euler commenced by moving the knight at random over the boarduntil it has no move open to it. With care this will leave only a fewcells not traversed: denote them by a, b, . . . . His method consists inestablishing certain rules by which these vacant cells can be interpolatedinto various parts of the circuit, and also by which the circuit can bemade re-entrant.
The following example, mentioned by Legendre as one of excep-tional difficulty, illustrates the method. Suppose that we have formed
55 58 29 40 27 44 19 22
60 39 56 43 30 21 26 45
57 54 59 28 41 18 23 20
38 51 42 31 8 25 46 17
53 32 37 a 47 16 9 24
50 3 52 33 36 7 12 15
1 34 5 48 b 14 c 10
4 49 2 35 6 11 d 13
Figure i.
22 25 50 39 52 35 60 57
27 40 23 36 49 58 53 34
24 21 26 51 38 61 56 59
41 28 37 48 3 54 33 62
20 47 42 13 32 63 4 55
29 16 19 46 43 2 7 10
18 45 14 31 12 9 64 5
15 30 17 44 1 6 11 8
Figure ii.
Example of Euler’s Method.
• Project • Gutenberg • #26839 •
CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 161
the route given in figure i; namely, 1, 2, 3, . . . , 59, 60; and that thereare four cells left untraversed, namely, a, b, c, d.
We begin by making the path 1 to 60 re-entrant. The cell 1 com-mands a cell p, where p is 32, 52, or 2. The cell 60 commands a cellq, where q is 29, 59, or 51. Then, if any of these values of p and qdiffer by unity, we can make the route re-entrant. This is the case hereif p = 52, q = 51. Thus the cells 1, 2, 3, . . . , 51; 60, 59, . . . , 52 form are-entrant route of 60 moves. Hence, if we replace the numbers 60, 59,. . . , 52 by 52, 53, . . . , 60, the steps will be numbered consecutively. Irecommend the reader who wishes to follow the subsequent details ofEuler’s argument to construct this square on a piece of paper beforeproceeding further.
Next, we add the cells a, b, d to this route. In the new diagram of60 cells formed as above the cell a commands the cells there numbered51, 53, 41, 25, 7, 5, and 3. It is indifferent which of these we select:suppose we take 51. Then we must make 51 the last cell of the route of60 cells, so that we can continue with a, b, d. Hence, if the reader willadd 9 to every number on the diagram he has constructed, and thenreplace 61, 62, . . . , 69 by 1, 2, . . . , 9, he will have a route which startsfrom the cell occupied originally by 60, the 60th move is on to the celloccupied originally by 51, and the 61st, 62nd, 63rd moves will be onthe cells a, b, d respectively.
It remains to introduce the cell c. Since c commands the cell nownumbered 25, and 63 commands the cell now numbered 24, this can beeffected in the same way as the first route was made re-entrant. In factthe cells numbered 1, 2, . . . , 24; 63, 62, . . . , 25, c form a knight’s path.Hence we must replace 63, 62, . . . , 25 by the numbers 25, 26, . . . , 63,and then we can fill up c with 64. We have now a route which coversthe whole board.
Lastly, it remains to make this route re-entrant. First, we mustget the cells 1 and 64 near one another. This can be effected thus.Take one of the cells commanded by 1, such as 28, then 28 commands1 and 27. Hence the cells 64, 63, . . . , 28; 1, 2, . . . , 27 form a route; andthis will be represented in the diagram if we replace the cells numbered1, 2, . . . , 27 by 27, 26, . . . , 1.
The cell now occupied by 1 commands the cells 26, 38, 54, 12, 14,16, 28; and the cell occupied by 64 commands the cells 13, 43, 63,55. The cells 13 and 14 are consecutive, and therefore the cells 64, 63,
• Project • Gutenberg • #26839 •
162 UNICURSAL PROBLEMS. [CH. VI
. . . , 14; 1, 2, . . . , 13 form a route. Hence we must replace the numbers1, 2, . . . , 13 by 13, 12, . . . , 1, and we obtain a re-entrant route coveringthe whole board, which is represented in the second of the diagramsgiven on page 160. Euler showed how seven other re-entrant routes canbe deduced from any given re-entrant route.
It is not difficult to apply the method so as to form a route whichbegins on one given cell and ends on any other given cell.
Euler next investigated how his method could be modified so as toallow of the imposition of additional restrictions.
An interesting example of this kind is where the first 32 moves areconfined to one half of the board. One solution of this is delineatedbelow. The order of the first 32 moves can be determined by Euler’s
58 43 60 37 52 41 62 35
49 46 57 42 61 36 53 40
44 59 48 51 38 55 34 63
47 50 45 56 33 64 39 54
22 7 32 1 24 13 18 15
31 2 23 6 19 16 27 12
8 21 4 29 10 25 14 17
3 30 9 20 5 28 11 26
Euler’s Half-board Solution.
50 45 62 41 60 39 54 35
63 42 51 48 53 36 57 38
46 49 44 61 40 59 34 55
43 64 47 52 33 56 37 58
26 5 24 1 20 15 32 11
23 2 27 8 29 12 17 14
6 25 4 21 16 19 10 31
3 22 7 28 9 30 13 18
Roget’s Half-board Solution.
method. It is obvious that, if to the number of each such move weadd 32, we shall have a corresponding set of moves from 33 to 64 whichwould cover the other half of the board; but in general the cell numbered33 will not be a knight’s move from that numbered 32, nor will 64 bea knight’s move from 1.
Euler however proceeded to show how the first 32 moves might bedetermined so that, if the half of the board containing the correspond-ing moves from 33 to 64 was twisted through two right angles, the tworoutes would become united and re-entrant. If x and y are the num-bers of a cell reckoned from two consecutive sides of the board, we maycall the cell whose distances are respectively x and y from the oppositesides a complementary cell. Thus the cells (x, y) and (9− x, 9− y) arecomplementary, where x and y denote respectively the column and row
• Project • Gutenberg • #26839 •
CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 163
occupied by the cell. Then in Euler’s solution the numbers in comple-mentary cells differ by 32: for instance, the cell (3, 7) is complementaryto the cell (6, 2), the one is occupied by 57, the other by 25.
Roget’s method, which is described later, can be also applied togive half-board solutions. The result is indicated on the facing page.The close of Euler’s memoir is devoted to showing how the methodcould be applied to crosses and other rectangular figures. I may notein particular his elegant re-entrant symmetrical solution for a squareof 100 cells.
The next attempt of any special interest is due to Vandermonde*,who reduced the problem to arithmetic. His idea was to cover theboard by two or more independent routes taken at random, and thento connect the routes. He defined the position of a cell by a fractionx/y, whose numerator x is the number of the cell from one side of theboard, and whose denominator y is its number from the adjacent side ofthe board; this is equivalent to saying that x and y are the co-ordinatesof a cell. In a series of fractions denoting a knight’s path, the differencesbetween the numerators of two consecutive fractions can be only oneor two, while the corresponding difference between their denominatorsmust be two or one respectively. Also x and y cannot be less than 1 orgreater than 8. The notation is convenient, but Vandermonde appliedit merely to obtain a particular solution of the problem for a boardof 64 cells: the method by which he effected this is analogous to thatestablished by Euler, but it is applicable only to squares of an evenorder. The route that he arrives at is defined in his notation by thefollowing fractions.
55, 4
3, 2
4, 4
5, 5
3, 7
4, 8
2, 6
1, 7
3, 8
1, 6
2, 8
3, 7
1, 5
2, 6
4, 8
5, 7
7, 5
8, 6
6, 5
4, 4
6, 2
5, 1
7, 3
8, 2
6,
18, 3
7, 1
6, 2
8, 4
7, 3
5, 1
4, 2
2, 4
1, 3
3, 1
2, 3
1, 2
3, 1
1, 3
2, 1
3, 2
1, 4
2, 3
4, 1
5, 2
7, 4
8, 3
6, 4
4, 5
6,
75, 8
7, 6
8, 7
6, 8
8, 6
7, 8
6, 7
8, 5
7, 6
5, 8
4, 7
2, 5
1, 6
3.
The path is re-entrant but unsymmetrical. Had he transferred thefirst three fractions to the end of this series he would have obtainedtwo symmetrical circuits of thirty-two moves joined unsymmetrically,and might have been enabled to advance further in the problem. Van-dermonde also considered the case of a route in a cube.
* L’Histoire de l’Academie des Sciences for 1771, Paris, 1774, pp. 566-574.
• Project • Gutenberg • #26839 •
164 UNICURSAL PROBLEMS. [CH. VI
In 1773 Collini* proposed the exclusive use of symmetrical routesarranged without reference to the initial cell, but connected in such amanner as to permit of our starting from it. This is the foundationof the modern manner of attacking the problem. The method was re-invented in 1825 by Pratt†, and in 1840 by Roget, and has been subse-quently employed by various writers. Neither Collini nor Pratt showedskill in using this method. The rule given by Roget is described later.
One of the most ingenious of the solutions of the knight’s path isthat given in 1823 by Warnsdorff‡. His rule is that the knight must bemoved always to one of the cells from which it will command the fewestsquares not already traversed. The solution is not symmetrical and notre-entrant; moreover it is difficult to trace practically. The rule has notbeen proved to be true, but no exception to it is known: apparently itapplies also to all rectangular boards which can be covered completelyby a knight. It is somewhat curious that in most cases a single falsestep, except in the last three or four moves, will not affect the result.
Warnsdorff added that when, by the rule, two or more cells are opento the knight, it may be moved to either or any of them indifferently.This is not so, and with great ingenuity two or three cases of failurehave been constructed, but it would require exceptionally bad luck tohappen accidentally on such a route.
The above methods have been applied to boards of various shapes,especially to boards in the form of rectangles, crosses, and circles§.
All the more recent investigations impose additional restrictions:such as to require that the route shall be re-entrant, or more generallythat it shall begin and terminate on given cells.
The best complete solution with which I am acquainted—and onewhich I believe is not generally known—is due to Roget‖. It dividesthe whole route into four circuits, which can be combined so as toenable us to begin on any cell and terminate on any other cell of a
* Solution du Probleme du Cavalier au Jeu des echecs, Mannheim, 1773.† Studies of Chess, sixth edition, London, 1825.‡ Des Rosselsprunges einfachste und allgemeinste Losung, Schmalkalden, 1823:
see Jaenisch, vol. ii, pp. 56–61, 273–289.§ See ex. gr. T. Ciccolini’s work Del Cavallo degli Scacchi, Paris, 1836.‖ Philosophical Magazine, April, 1840, series 3, vol. xvi, pp. 305–309; see also
the Quarterly Journal of Mathematics for 1877, vol. xiv, pp. 354–359. Somesolutions, founded on Roget’s method, are given in the Leisure Hour, Sept. 13,1873, pp. 587–590; see also Ibid., Dec. 20, 1873, pp. 813–815.
• Project • Gutenberg • #26839 •
CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 165
different colour. Hence, if we like to select this last cell at a knight’smove from the initial cell, we obtain a re-entrant route. On the otherhand, the rule is applicable only to square boards containing (4n)2 cells:for example, it could not be used on the board of the French jeu desdames, which contains 100 cells.
Roget began by dividing the board of 64 cells into four quarters.Each quarter contains 16 cells, and these 16 cells can be arranged in4 groups, each group consisting of 4 cells which form a closed knight’spath. All the cells in each such path are denoted by the same letterl, e, a, or p, as the case nay be. The path of 4 cells indicated by theconsonants l and the path indicated by the consonants p are diamond-shaped: the paths indicated respectively by the vowels e and a aresquare-shaped, as may be seen by looking at one of the four quartersin figure i below.
l e a p l e a p
a p l e a p l e
e l p a e l p a
p a e l p a e l
l e a p l e a p
a p l e a p l e
e l p a e l p a
p a e l p a e l
Roget’s Solution (i).
34 51 32 15 38 53 18 3
31 14 35 52 17 2 39 54
50 33 16 29 56 37 4 19
13 30 49 36 1 20 55 40
48 63 28 9 44 57 22 5
27 12 45 64 21 8 41 58
62 47 10 25 60 43 6 23
11 26 61 46 7 24 59 42
Roget’s Solution (ii).
Now all the 16 cells on a complete chess-board which are markedwith the same letter can be combined into one circuit, and whereverthe circuit begins we can make it end on any other cell in the circuit,provided it is of a different colour to the initial cell. If it is indifferent onwhat cell the circuit terminates we may make the circuit re-entrant, andin this case we can make the direction of motion round each group (of4 cells) the same. For example, all the cells marked p can be arrangedin the circuit indicated by the successive numbers 1 to 16 in figure iiabove. Similarly all the cells marked a can be combined into the circuitindicated by the numbers 17 to 23; all the l cells into the circuit 33 to48; and all the e cells into the circuit 49 to 64. Each of the circuits
• Project • Gutenberg • #26839 •
166 UNICURSAL PROBLEMS. [CH. VI
indicated above is symmetrical and re-entrant. The consonant and thevowel circuits are said to be of opposite kinds.
The general problem will be solved if we can combine the fourcircuits into a route which will start from any given cell, and terminateon the 64th move on any other given cell of a different colour. To effectthis Roget gave the two following rules.
First. If the initial cell and the final cell are denoted the one by aconsonant and the other by a vowel, take alternately circuits indicatedby consonants and vowels, beginning with the circuit of 16 cells indi-cated by the letter of the initial cell and concluding with the circuitindicated by the letter of the final cell.
Second. If the initial cell and the final cell are denoted both byconsonants or both by vowels, first select a cell, Y , in the same circuit asthe final cell, Z, and one move from it, next select a cell, X, belongingto one of the opposite circuits and one move from Y . This is alwayspossible. Then, leaving out the cells Z and Y , it always will be possible,by the rule already given, to travel from the initial cell to the cell X in62 moves, and thence to move to the final cell on the 64th move.
In both cases however it must be noticed that the cells in eachof the first three circuits will have to be taken in such an order thatthe circuit does not terminate on a corner, and it may be desirablealso that it should not terminate on any of the border cells. This willnecessitate some caution. As far as is consistent with these restrictionsit is convenient to make these circuits re-entrant, and to take them andevery group in them in the same direction of rotation.
As an example, suppose that we are to begin on the cell numbered1 in figure ii on the previous page, which is one of those in a p circuit,and to terminate on the cell numbered 64, which is one of those in ane circuit. This falls under the first rule: hence first we take the 16cells marked p, next the 16 cells marked a, then the 16 cells marked l,and lastly the 16 cells marked e. One way of effecting this is shown inthe diagram. Since the cell 64 is a knight’s move from the initial cellthe route is re-entrant. Also each of the four circuits in the diagramis symmetrical, re-entrant, and taken in the same direction, and theonly point where there is any apparent breach in the uniformity ofthe movement is in the passage from the cell numbered 32 to thatnumbered 33.
A rule for re-entrant routes, similar to that of Roget, has been
• Project • Gutenberg • #26839 •
CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 167
given by various subsequent writers, especially by De Polignac* and byLaquiere†, who have stated it at much greater length. Neither of theseauthors seems to have been aware of Roget’s theorems. De Polignac,like Roget, illustrates the rule by assigning letters to the various squaresin the way explained above, and asserts that a similar rule is applicableto all even squares.
Roget’s method can be also applied to two half-boards, as indicatedin the figure given above on page 162.
Another way of dividing the board into closed circuits which canbe connected was given in 1843 by Moon‡. He divided the board into a
a b c d a b c d
c d a b c d a b
b a A B C D d c
d c C D A B b a
a b B A D C c d
c d D C B A a b
b a d c b a d c
d c b a d c b a
Moon’s Solution.
63 22 15 40 1 42 59 18
14 39 64 21 60 17 2 43
37 62 23 16 41 4 19 58
24 13 38 61 20 57 44 3
11 36 25 52 29 46 5 56
26 51 12 33 8 55 30 45
35 10 49 28 53 32 47 6
50 27 34 9 48 7 54 31
Jaenisch’s Solution.
central square containing 42 cells and a surrounding annulus (see figureon this page). The annulus may be divided into four closed circuits,each containing 12 cells: these are marked respectively with the lettersa, b, c, d. The central square may be divided similarly into four closedcircuits, each containing 4 cells, denoted by the letters A, B, C, D.We can connect these routes as follows. If we are moving outwardsfrom the central square to the annulus we can go from a cell A eitherto b or to c or to d (but not to a) and similarly for the other letters.If we are moving inwards from the annulus to the central square wemust go from a to D, or d to A, or b to C, or c to B, as the case maybe. Thus if the initial cell is on a, we might take either of the cycles
* Comptes Rendus, April, 1861; and Bulletin de la Societe Mathematique deFrance, 1881, vol. ix, pp. 17–24.
† Bulletin de la Societe Mathematique de France, 1880, vol. viii, pp. 82–102, 132–158.
‡ Cambridge Mathematical Journal, 1843, vol. iii, pp. 233–236.
• Project • Gutenberg • #26839 •
168 UNICURSAL PROBLEMS. [CH. VI
a D b C d A c B, or a D c B d A b C. By following these rules wealways can connect the routes into one path, but in general it will notbe re-entrant. It is convenient to take the cells in each circuit in oneand the same direction, but a circuit in the outer annulus must not endin a corner cell, and to avoid this we may have to alter the directionin which a circuit is taken.
Moon’s rule can be modified to cover the case of any doubly evensquare board, and the path can be made to begin and end on any twogiven squares, but I do not propose to go further into details.
The method which Jaenisch gives as the most fundamental is notvery different from that of Roget. It leads to eight forms, similar to thatin the diagram printed on the previous page, in which the sum of thenumbers in every column and every row is 260; but although symmetri-cal it is not in my opinion so easy to reproduce as that given by Roget.
It is as yet impossible to say how many solutions of the problemexist. Legendre* mentioned the question, but Minding† was the earliestwriter to attempt to answer it. More recent investigations have shownthat on the one hand the number of possible routes is less‡ than thenumber of combinations of 168 things taken 63 at a time, and on theother hand is greater than 31, 054144—since this latter number is thenumber of re-entrant paths of a particular type§.
There are many similar problems in which it is required to deter-mine routes by which a piece moving according to certain laws (ex. gr. achess-piece such as a king, knight, &c.) can travel from a given cell overa board so as to occupy successively all the cells, or certain specifiedcells, once and only once, and terminate its route in a given cell.
Euler’s method can be applied to find routes of this kind: for in-stance, he applied it to find a re-entrant route by which a piece thatmoved two cells forward like a castle and then one cell like a bishopwould occupy in succession all the black cells on the board. As anotherinstance, a castle, placed on a chess-board of n2 cells, can always bemoved in such a manner that it shall move successively on to everycell once and only once; moreover, starting on any cell, its path can bemade to terminate, if n be even, on any other cell of a different colour,
* Theorie des Nombres, Paris, 2nd edition, 1830, vol. ii, p. 165.† Cambridge and Dublin Mathematical Journal, 1852, vol. vii, pp. 147–156; and
Crelle’s Journal, 1853, vol. xliv, pp. 73–82.‡ Jaenisch, vol. ii, p. 268.§ Bulletin de la Societe Mathematique de France, 1881, vol. ix, pp. 1–17.
• Project • Gutenberg • #26839 •
CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 169
and, if n be odd, on any other cell of the same colour*. But it willsuffice to have discussed the classical problem of the determination ofa knight’s path on an ordinary chess-board, and I need not enter onthe discussion of other similar problems.
* L’Intermediaire des mathematiciens, Paris, July, 1901, p. 153.
• Project • Gutenberg • #26839 •
PART II.
Miscellaneous Essays and Problems.
“No man of science should think it a waste oftime to learn something of the history of his ownsubject; nor is the investigation of laborious meth-ods now fallen into disuse, or of errors once com-monly accepted the least valuable of mental disci-plines.”
“The most worthless book of a bygone day isa record worthy of preservation. Like a telescopicstar, its obscurity may render it unavailable formost purposes; but it serves, in hands which knowhow to use it, to determine the places of more im-portant bodies.”
(De Morgan.)
170
• Project • Gutenberg • #26839 •
CHAPTER VII.
THE MATHEMATICAL TRIPOS.
The Mathematical Tripos has played so prominent a part in thehistory of education at Cambridge and of mathematics in England, thata sketch of its development* may be interesting to general readers.
So far as mathematics is concerned the history of the Universitybefore Newton may be summed up very briefly. The University wasfounded towards the end of the twelfth century. Throughout the mid-dle ages the studies were organised on lines similar to those at Parisand Oxford. To qualify for a degree it was necessary to perform var-ious exercises, and especially to keep a number of acts or to opposeacts kept by other students. An act consisted in effect of a debate inLatin, thrown, at any rate in later times, into syllogistic form. It wascommenced by one student, the respondent, stating some proposition,often propounded in the form of a thesis, which was attacked by oneor more opponents, the discussion being controlled by a graduate. Theteaching was largely in the hands of young graduates—every master ofarts being compelled to reside and teach for at least one year—thoughno doubt Colleges and private hostels supplemented this instruction inthe case of their own students.
* The following pages are mostly summarised from my History of the Studyof Mathematics at Cambridge, Cambridge 1889. The subject is also treatedin Whewell’s Liberal Education, Cambridge, three parts, 1845, 1850, 1853;Wordsworth’s Scholae Academicae, Cambridge, 1877; my own Origin and His-tory of the Mathematical Tripos, Cambridge, 1880; and Dr Glaisher’s Presiden-tial Address to the London Mathematical Society, Transactions, vol xviii, 1886,pp. 4–38.
171
• Project • Gutenberg • #26839 •
172 THE MATHEMATICAL TRIPOS. [CH. VII
The Reformation in England was mainly the work of Cambridgedivines, and in the University the Renaissance was warmly welcomed.In spite of the disorder and confusion of the Tudor period, new studiesand a system of professional instruction were introduced. Probably thescience (as distinct from the art) of mathematics, save so far as involvedin the quadrivium, was still an exotic study, but it was not wholly ne-glected. Tonstall, subsequently the most eminent English arithmeticianof his time, migrated, perhaps about 1495, from Balliol College, Ox-ford, to King’s Hall, Cambridge, and in 1530 the University appointed amathematical lecturer in the person of Paynell of Pembroke Hall. Mostof the subsequent English mathematicians of the Tudor period seem tohave been educated at Cambridge; of these I may mention Record, whomigrated, probably about 1535, from Oxford, Dee, Digges, Blundeville,Buckley, Billingsley, Hill, Bedwell, Hood, Richard and John Harvey,Edward Wright, Briggs, and Oughtred. The Elizabethan statutes re-stricted liberty of thought and action in many ways, but, in spite of thecivil and religious disturbances of the early half of the 17th century themathematical school continued to grow. Horrox, Seth Ward, Foster,Rooke, Gilbert Clerke, Pell, Wallis, Barrow, Dacres, and Morland maybe cited as prominent Cambridge mathematicians of the time.
Newton’s mathematical career dates from 1665; his reputation,abilities, and influence attracted general attention to the subject. Hecreated a school of mathematics and mathematical physics, amongthe earliest members of which I note the names of Laughton, SamuelClarke, Craig, Flamsteed, Whiston, Saunderson, Jurin, Taylor, Cotes,and Robert Smith. Since then Cambridge has been regarded, as ina special sense, the home of English mathematicians, and from 1706onwards we have fairly complete accounts of the course of reading andwork of mathematical students there.
Until less than a century ago the form of the method of qualifyingfor a degree remained substantially unaltered, but the subject-matterof the discussions varied from time to time with the prevalent studiesof the place.
After the Renaissance some of the statutable exercises were “hud-dled,” that is, were reduced to a mere form. To huddle an act, theproctor generally asked some question such as Quid est nomen to whichthe answer usually expected was Nescio. In these exercises considerablelicense was allowed, particularly if there were any play on the words
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 173
involved. For example, T. Brasse, of Trinity, was accosted with thequestion, Quid est aes? to which he answered, Nescio nisi finis ex-aminationis. It should be added that retorts such as these were onlyallowed in the pretence exercises, and a candidate who in the actualexamination was asked to give a definition of happiness and repliedan exemption from Payne—that being the name of the moderator thenpresiding—was plucked for want of discrimination in time and place. Inearlier years even the farce of huddling seems to have been unnecessary,for it was said in 1675 that it was not uncommon for the proctors to take“cautions for the performance of the statutable exercises, and acceptthe forfeit of the money so deposited in lieu of their performance.”
In medieval times acts had been usually kept on some scholasticquestion or on a proposition taken from the Sentences. About the endof the fifteenth century religious questions, such as the interpretationof Biblical texts, began to be introduced, some fifty or sixty years laterthe favourite subjects were drawn either from dogmatic theology orfrom philosophy. In the seventeenth century the questions were usuallyphilosophical, but in the eighteenth century, under the influence of theNewtonian school, a large proportion of them were mathematical.
Further details about these exercises and specimens of acts kept inthe 18th century are given in my History of Mathematics at Cambridge.Here I will only say that they provided an admirable training in theart of presenting an argument, and in dialectical skill in attack anddefence. The mental strain in a contested act was severe. De Morgan,describing his act kept in 1826, wrote*,
I was badgered for two hours with arguments given and answered inLatin,—or what we call Latin—against Newton’s first section, Lagrange’sderived functions, and Locke on innate principles. And though I tookoff everything, and was pronounced by the moderator to have disputedmagno honore, I never had such a strain of thought in my life. For theinferior opponents were made as sharp as their betters by their tutors,who kept lists of queer objections drawn from all quarters.
Had the language of the discussions been changed to English, as wasrepeatedly urged from 1774 onwards, these exercises might have beenretained with advantage, but the barbarous Latin and the syllogisticform in which they were carried on prejudiced their retention.
About 1830 a custom grew up for the respondent and opponents tomeet previously and arrange their arguments together. The discussions
* Budget of Paradoxes, by A. De Morgan, London, 1872, p. 305.
• Project • Gutenberg • #26839 •
174 THE MATHEMATICAL TRIPOS. [CH. VII
then became an elaborate farce, and were a mere public performanceof what had been already rehearsed. Accordingly the moderators of1839 took the responsibility of abandoning them. This action wassingularly high-handed, since a report of May 30, 1838, had recom-mended that they should be continued, and there was no reason whythey should not have been reformed and retained as a useful featurein the scheme of study.
On the result of the acts a list of those qualified to receive degreeswas drawn up. This list was not arranged strictly in order of merit,because the proctors could insert names anywhere in it, but by thebeginning of the 18th century this power had become restricted to theright reserved to the vice-chancellor, the senior regent, and each proctorto place in the list one candidate anywhere he liked—a right which con-tinued to exist till 1828, though it was not exercised after 1797. Subjectto the granting of these honorary degrees, this final list was arrangedin order of merit into wranglers and senior optimes, junior optimes,and poll-men. The bachelors on receiving their degrees took seniorityaccording to their order on this list. The title wrangler is derived fromthese contentious discussions; the title optime from the customary com-pliment given by the moderator to a successful disputant, Domine. . . ,optime disputasti, or even optime quidem disputasti , and the title ofpoll-man from the description of this class as οÉ πολλοÐ.
The final exercises for the B.A. degree were never huddled, anduntil 1839 were carried out strictly. University officials were responsi-ble for approving the subject-matter of these acts. Stupid men offeredsome irrefutable truism, but the ambitious student courted reputationby affirming some paradox. Probably all honour men kept acts, butpoll-men were deemed to comply with the regulations by keeping op-ponencies. The proctors were responsible for presiding at these acts, orseeing that competent graduates did so. In and after 1649 two examin-ers were specially appointed for this purpose. In 1680* these examinerswere appointed by the Senate with the title of moderator, and with thejoint stipend of four shillings for everyone graduating as B.A. duringtheir year of office. In 1688 the joint stipend of the moderators was fixedat £40 a year. The moderators, like the proctors, were nominated bythe Colleges in rotation.
From the earliest times the proctors had the power of questioning
* See Grace of October 25, 1680.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 175
a candidate at the end of a disputation, and probably all candidatesfor a degree attended the public schools on certain days to give an op-portunity to the proctors, or any master that liked, to examine them*,though the opportunity was not always used. Different candidates at-tended on different days. Probably such examinations were conductedin Latin. But soon after 1710† the moderators or proctors began thecustom of summoning on one day in January all candidates whom theyproposed to question. The examination was held in public, and fromit the Senate-House Examination arose. The examination at this timedid not last more than one day, and was, there can be no doubt, partlyon philosophy and partly on mathematics. It is believed that it wasalways conducted in English, and it is likely that its rapid developmentwas largely due to this.
This introduction of a regular oral examination seems to have beenlargely due to the fact that when, in 1710, George I gave the Ely libraryto the University, it was decided to assign for its reception the oldSenate-House—now the Catalogue Room in the Library—and to builda new room for the meetings of the Senate. Pending the building of thenew Senate-House the books were stored in the Schools. As the Schoolswere thus rendered unavailable for keeping acts, considerable difficultywas found in arranging for all the candidates to keep the full numberof statutable exercises, and thus obtaining opportunities to comparethem one with another: hence the introduction of a supplementaryoral examination. The advantages of this examination as providing aready means of testing the knowledge and abilities of the candidateswere so patent that it was retained when the necessity for some systemof the kind had passed away, and finally it became systematized intoan organized test to which all questionists were subjected.
In 1731 the University raised the joint stipend of the moderatorsto £60 “in consideration of their additional trouble in the Lent Term.”This would seem to indicate that the Senate-House Examination hadthen taken formal shape, and perhaps that a definite scheme for itsconduct had become customary.
* Ex. gr. see De la Pryme’s account of his graduation in 1694, Surtees Society,vol. liv, 1870, p. 32.
† W. Reneu, in his letters of 1708–1710 describing the course for the B.A. degree,makes no mention of the Senate-House examination, and I think it is a reasonableinference that it had not then been established.
• Project • Gutenberg • #26839 •
176 THE MATHEMATICAL TRIPOS. [CH. VII
As long as the order of the list of those approved for degrees wassettled on the result of impressions derived from acts kept by the dif-ferent candidates at different times and on different subjects, it wasimpossible to arrange the men in strict order of merit, nor was muchimportance attached to the order. But, with the introduction of anexamination of all the candidates on one day, much closer attentionwas paid to securing a strict order of merit, and more confidence wasfelt in the published order. It seems to have been consequent on thisthat in and after 1747 the final lists were freely circulated, and it wasfurther arranged that the names of the honorary optimes should beindicated. In the lists given in the Calendars issued subsequent to 1799these names are struck out. It is only in exceptional cases that we areacquainted with the true order for the earlier tripos lists, but in a fewcases the relative positions of the candidates are known; for example,in 1680 Bentley came out as third though he was put down as sixthin the list of wranglers.
Of the detailed history of the examination until the middle of theeighteenth century we know nothing. From 1750 onwards, however, wehave more definite accounts of it. At this time, it would seem that allthe men from each College were taken together as a class, and questionspassed down by the proctors or moderators till they were answered: butthe examination remained entirely oral, and technically was regardedas subsidiary to the discussions which had been previously held in theschools. As each class contained men of very different abilities a customgrew up by which every candidate was liable to be taken aside to bequestioned by any M.A. who wished to do so, and this was regarded asan important part of the examination. The subjects were mathematicsand philosophy. The examination now continued for two days and ahalf. At the conclusion of the second day the moderators received thereports of those masters of arts who had voluntarily taken part in theexamination, and provisionally settled the final list; while the last half-day was used in revising and re-arranging the order of merit.
Richard Cumberland has left an account of the tests to which hewas subjected when he took his B.A. degree in 1751. Clearly the dis-putations still played an important part, and it is difficult to say whatweight was attached to the subsequent Senate-House examination; hisreference to it is only of a general character. After saying that he kept
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 177
two acts and two opponencies he continues*:
The last time I was called upon to keep an act in the schools I sentin three questions to the Moderator, which he withstood as being allmathematical, and required me to conform to the usage of proposing onemetaphysical question in the place of that, which I should think fit towithdraw. This was ground I never liked to take, and I appealed againsthis requisition: the act was accordingly put by till the matter of rightshould be ascertained by the statutes of the university, and in the resultof that enquiry it was given for me, and my question stood. . . . I yieldednow to advice, and paid attention to my health, till we were cited to thesenate house to be examined for our Bachelor’s degree. It was hardly evermy lot during that examination to enjoy any respite. I seemed an objectsingled out as every man’s mark, and was kept perpetually at the tableunder the process of question and answer.
It was found possible by means of the new examination to differen-tiate the better men more accurately than before; and accordingly, in1753, the first class was subdivided into two, called respectively wran-glers and senior optimes, a division which is still maintained.
The semi-official examination by M.A.s was regarded as the moreimportant part of the test, and the most eminent residents in the Uni-versity took part in it. Thus John Fenn, of Caius, 5th wrangler in1761, writes†:
On the following Monday, Tuesday, and Wednesday, we sat in theSenate-house for public examination; during this time I was officially ex-amined by the Proctors and Moderators, and had the honor of being takenout for examination by Mr Abbot, the celebrated mathematical tutor ofSt John’s College, by the eminent professor of mathematics Mr Waring,of Magdalene, and by Mr Jebb of Peterhouse, a man thoroughly versedin the academical studies.
This irregular examination by any master who chose to take part in itconstantly gave rise to accusations of partiality.
In 1763 the traditional rules for the conduct of the examinationtook more definite shape. Henceforth the examiners used the disputa-tions only as a means of classifying the men roughly. On the result oftheir “acts,” and probably partly also of their general reputation, thecandidates were divided into eight classes, each arranged in alphabet-ical order. The subsequent position of the men in the class was deter-mined solely by the Senate-House examination. The first two classes
* Memoirs of Richard Cumberland, London, 1806, pp. 78, 79.† Quoted by C. Wordsworth, Scholae Academicae, Cambridge, 1877, pp. 30–31.
• Project • Gutenberg • #26839 •
178 THE MATHEMATICAL TRIPOS. [CH. VII
comprised all who were expected to be wranglers, the next four classesincluded the other candidates for honours, and the last two classes con-sisted of poll-men only. Practically anyone placed in either of the firsttwo classes was allowed, if he wished, to take an aegrotat senior optime,and thus escape all further examination: this was called gulphing it.All the men from one College were no longer taken together, but eachclass was examined separately and viva voce; and hence, since all thestudents comprised in each class were of about equal attainments, itwas possible to make the examination more effective. Richard Watson,of Trinity, claimed that this change was made by him when acting asmoderator in 1763. He says*:
There was more room for partiality. . . then [i.e. in 1759] than there isnow; and I attribute the change, in a great degree, to an alteration whichI introduced the first year I was moderator [i.e. in 1763], and which hasbeen persevered in ever since. At the time of taking their Bachelor ofArts’ degree, the young men are examined in classes, and the classes arenow formed according to the abilities shown by individuals in the schools.By this arrangement, persons of nearly equal merits are examined in thepresence of each other, and flagrant acts of partiality cannot take place.Before I made this alteration, they were examined in classes, but theclasses consisted of members of the same College, and the best and worstwere often examined together.
It is probable that before the examination in the Senate-House begana candidate, if manifestly placed in too low a class, was allowed theprivilege of challenging the class to which he was assigned. Perhapsthis began as a matter of favour, and was only granted in exceptionalcases, but a few years later it became a right which every candidatecould exercise; and I think that it is partly to its development thatthe ultimate predominance of the tripos over the other exercises forthe degree is due.
In the same year, 1763, it was decided that the relative positionof the senior and second wranglers, namely, Paley, of Christ’s, andFrere, of Caius, was to be decided by the Senate-House examinationand not by the disputations. Henceforward distinction in the Senate-House examination was regarded as the most important honour opento undergraduates.
In 1768 Dr Smith, of Trinity College, founded prizes for mathe-matics and natural philosophy open to two commencing bachelors. The
* Anecdotes of the Life of Richard Watson by Himself, London, 1817, pp. 18, 19.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 179
examination followed immediately after the Senate-House examination,and the distinction, being much coveted, tended to emphasize the math-ematical side of the normal University education of the best men. Since1883 the prizes have been awarded on the result of dissertations*.
Until now the Senate-House examination had been oral, but aboutthis time, circ. 1770, it began to be the custom to dictate some orall of the questions and to require answers to be written. Only onequestion was dictated at a time, and a fresh one was not given outuntil some student had solved that previously read: a custom whichby causing perpetual interruptions to take down new questions musthave proved very harassing. We are perhaps apt to think that an ex-amination conducted by written papers is so natural that the customis of long continuance, but I know no record of any in Europe earlierthan the eighteenth century. Until 1830 the questions for the Smith’sPrize were dictated.
The following description of the Senate-House examination as itexisted in 1772 is given by Jebb†.
The moderators, some days before the arrival of the time prescribedby the vice-chancellor, meet for the purpose of forming the students intodivisions of six, eight, or ten, according to their performance in the schools,with a view to the ensuing examination.
Upon the first of the appointed days, at eight o’clock in the morning,the students enter the senate-house, preceded by a master of arts fromeach college, who. . . is called the “father” of the college. . .
After the proctors have called over the names, each of the moderatorssends for a division of the students: they sit with him round a table, withpens, ink, and paper, before them: he enters upon his task of examination,and does not dismiss the set till the hour is expired. This examinationhas now for some years been held in the english language.
The examination is varied according to the abilities of the students.The moderator generally begins with proposing some questions from thesix books of Euclid, plain trigonometry, and the first rules of algebra.If any person fails in an answer, the question goes to the next. Fromthe elements of mathematics, a transition is made to the four branchesof philosophy, viz. mechanics, hydrostatics, apparent astronomy, and op-tics, as explained in the works of Maclaurin, Cotes, Helsham, Hamil-ton, Rutherforth, Keill, Long, Ferguson, and Smith. If the moderatorfinds the set of questionists, under examination, capable of answering
* See Grace of October 25, 1885; and the Cambridge University Reporter, October23 and 30, 1883.
† The Works of J. Jebb, London, 1787, vol. ii, pp. 290–297.
• Project • Gutenberg • #26839 •
180 THE MATHEMATICAL TRIPOS. [CH. VII
him, he proceeds to the eleventh and twelfth books of Euclid, conic sec-tions, spherical trigonometry, the higher parts of algebra, and sir IsaacNewton’s Principia; more particularly those sections, which treat of themotion of bodies in eccentric and revolving orbits; the mutual action ofspheres, composed of particles attracting each other according to variouslaws; the theory of pulses, propagated through elastic mediums; and thestupendous fabric of the world. Having closed the philosophical examina-tion, he sometimes asks a few questions in Locke’s Essay on the humanunderstanding, Butler’s Analogy, or Clarke’s Attributes. But as the high-est academical distinctions are invariably given to the best proficientsin mathematics and natural philosophy, a very superficial knowledge inmorality and metaphysics will suffice.
When the division under examination is one of the highest classes,problems are also proposed, with which the student retires to a distantpart of the senate-house; and returns, with his solution upon paper, tothe moderator, who, at his leisure, compares it with the solutions of otherstudents, to whom the same problems have been proposed.
The extraction of roots, the arithmetic of surds, the invention of divis-ers, the resolution of quadratic, cubic, and biquadratic equations; togetherwith the doctrine of fluxions, and its application to the solution of ques-tions “de maximis et minimis,” to the finding of areas, to the rectificationof curves, the investigation of the centers of gravity and oscillation, and tothe circ*mstances of bodies, agitated, according to various laws, by cen-tripetal forces, as unfolded, and exemplified, in the fluxional treatises ofLyons, Saunderson, Simpson, Emerson, Maclaurin, and Newton, generallyform the subject matter of these problems.
When the clock strikes nine, the questionists are dismissed to break-fast: they return at half past nine, and stay till eleven; they go in againat half past one, and stay till three; and, lastly, they return at half-pastthree, and stay till five.
The hours of attendance are the same upon the subsequent day.On the third day they are finally dismissed at eleven.During the hours of attendance, every division is twice examined in
form, once by each of the moderators, who are engaged for the whole timein this employment.
As the questionists are examined in divisions of only six or eight at atime, but a small portion of the whole number is engaged, at any partic-ular hour, with the moderators; and, therefore, if there were no furtherexamination, much time would remain unemployed.
But the moderator’s inquiry into the merits of the candidates formsthe least material part of the examination.
The “fathers” of the respective colleges, zealous for the credit of thesocieties, of which they are the guardians, are incessantly employed inexamining those students, who appear most likely to contest the palm ofglory with their sons.
This part of the process is as follows:
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 181
The father of a college takes a student of a different college aside, and,sometimes for an hour and an half together, strictly examines him in everypart of mathematics and philosophy, which he professes to have read.
After he hath, from this examination, formed an accurate idea of thestudent’s abilities and acquired knowledge, he makes a report of his ab-solute or comparative merit to the moderators, and to every other fatherwho shall ask him the question.
Besides the fathers, all masters of arts, and doctors, of whatever facultythey be, have the liberty of examining whom they please; and they alsoreport the event of each trial, to every person who shall make the inquiry.
The moderators and fathers meet at breakfast, and at dinner. Fromthe variety of reports, taken in connection with their own examination, theformer are enabled, about the close of the second day, so far to settle thecomparative merits of the candidates, as to agree upon the names of four-and-twenty, who to them appear most deserving of being distinguished bymarks of academical approbation.
These four-and-twenty [wranglers and senior optimes] are recom-mended to the proctors for their private examination; and, if approvedby them, and no reason appears against such placing of them from anysubsequent inquiry, their names are set down in two divisions, accordingto that order, in which they deserve to stand; are afterwards printed; andread over upon a solemn day, in the presence of the vice-chancellor, andof the assembled university.
The names of the twelve [junior optimes], who, in the course of theexamination, appear next in desert, are also printed, and are read over, inthe presence of the vice-chancellor, and of the assembled university, upona day subsequent to the former. . .
The students, who appear to have merited neither praise nor censure,[the poll-men], pass unnoticed: while those, who have taken no pains toprepare themselves for the examination, and have appeared with discreditin the schools, are distinguished by particular tokens of disgrace.
Jebb’s statement about the number of wranglers and senior optimesis only approximate.
It may be added that it was now frankly recognized that the exam-ination was competitive*. Also that though it was open to any memberof the Senate to take part in it, yet the determination of the relativemerit of the students was entirely in the hands of the moderators†. Al-though the examination did not occupy more than three days it musthave been a severe physical trial to anyone who was delicate. It washeld in winter and in the Senate-House. That building was then noted
* “Emulation, which is the principle upon which the plan is constructed.” TheWorks of J. Jebb, London, 1787, vol. iii, p. 261.
† The Works of J. Jebb, London, 1787, vol. iii, p. 272.
• Project • Gutenberg • #26839 •
182 THE MATHEMATICAL TRIPOS. [CH. VII
for its draughts, and was not warmed in any way: and according totradition, on one occasion the candidates on entering in the morningfound the ink in the pots on their desks frozen.
The University was not altogether satisfied* with the scheme inforce, and in 1779† the scheme of examination was amended in variousrespects. In particular the examination was extended to four days, athird day being given up entirely to natural religion, moral philosophy,and Locke. It was further announced‡ that a candidate would not re-ceive credit for advanced subjects unless he had satisfied the examinersin Euclid and elementary Natural Philosophy.
A system of brackets or “classes quam minimae” was now intro-duced. Under this system the examiners issued on the morning of thefourth day a provisional list of men who had obtained honours, withthe names of those of about equal merit bracketed, and that day wasdevoted to arranging the names in each bracket in order of merit: theexaminers being given explicit authority to invite the assistance of oth-ers in this work. Whether at this time a candidate could request to bere-examined with the view of being moved from one bracket to anotheris uncertain, but later this also was allowed.
Under the scheme of 1779 also the number of examiners was in-creased to four, the moderators of one year becoming, as a matterof course, the examiners of the next. Thus of the four examiners ineach year, two had taken part in the examination of the previous year,and the continuity of the system of examination was maintained. Thenames of the moderators appear on the tripos lists, but the names ofthe examiners were not printed on the lists till some years later.
The right of any M.A. to take part in the examination was notaffected, though henceforth it was exercised more sparingly, and I be-lieve was not insisted on after 1785. But it became a regular customfor the moderators to invite particular M.A.s to examine and comparespecified candidates. Milner, of Queens’, was constantly asked to assistin this way.
It was not long before it became an established custom that acandidate, who was dissatisfied with the class in which he had beenplaced as the result of his disputations, might challenge it before the
* See Graces of July 5, 1773, and of February 17, 1774.† See Graces of March 19, 20, 1779.‡ Notice issued by the Vice-Chancellor, dated May 19, 1779.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 183
examination began. This power seems to have been used but rarely; itwas, however, a recognition of the fact that a place in the tripos listwas to be determined by the Senate-House examination alone, and theexaminers soon acquired the habit of settling the preliminary classeswithout exclusive reference to the previous disputations.
The earliest papers actually set in the Senate-House, and now ex-tant, are two problem papers set in 1785 and 1786 by W. Hodson, ofTrinity, then a proctor. The autograph copies from which he gave outthe questions were luckily preserved, and are in the library* of TrinityCollege. They must be almost the last problem papers which were dic-tated, instead of being printed and given as a whole to the candidates.
The problem paper in 1786 was as follows:
1. To determine the velocity with which a Body must be thrown, in a direc-tion parallel to the Horizon, so as to become a secondary planet to the Earth; asalso to describe a parabola, and never return.
2. To demonstrate, supposing the force to vary as1
D2, how far a body must
fall both within and without the Circle to acquire the Velocity with which a bodyrevolves in a Circle.
3. Suppose a body to be turned (sic) upwards with the Velocity with whichit revolves in an Ellipse, how high will it ascend? The same is asked supposingit to move in a parabola.
4. Suppose a force varying first as1
D3, secondly in a greater ratio than
1D2
but less than1
D3, and thirdly in a less ratio than
1D2
, in each of these Cases todetermine whether at all, and where the body parting from the higher Apsid willcome to the lower.
5. To determine in what situation of the moon’s Apsid they go most forwards,and in what situation of her Nodes the Nodes go most backwards, and why?
6. In the cubic equation x3 + qx + r = 0 which wants the second term;supposing x = a + b and 3ab = −q, to determine the value of x.
7. To find the fluxion of xr × (yn + zm)1/q.
8. To find the fluent ofax
a + x.
9. To find the fluxion of the mth power of the Logarithm of x.10. Of right-angled Triangles containing a given Area to find that whereof
the sum of the two legs AB + BC shall be the least possible. [This and the twofollowing questions are illustrated by diagrams. The angle at B is the right angle.]
11. To find the Surface of the Cone ABC. [The cone is a right one on acircular base.]
12. To rectify the arc DB of the semicircle DBV .
* The Challis Manuscripts, iii, 61.
• Project • Gutenberg • #26839 •
184 THE MATHEMATICAL TRIPOS. [CH. VII
In cases of equality in the Senate-House examination the acts werestill taken into account in settling the tripos order: and in 1786 whenthe second, third, and fourth wranglers came out equal in the exami-nation a memorandum was published that the second place was givento that candidate who dialectis magis est versatus, and the third placeto that one who in scholis sophistarum melius disputavit.
There seem to have been considerable intervals in the examinationby the moderators, and the examinations by the extraneous examin-ers took place in these intervals. Those candidates who at any timewere not being examined occupied themselves with amusem*nts, pro-vided they were not too boisterous and obvious: probably dice andcards played a large part in them. Gunning in an amusing account ofhis examination in 1788 talks of games with a teetotum* in which hetook part on the Wednesday (when Locke and Paley formed the sub-jects of examination), but “which was carried on with great spirit. . . byconsiderable numbers during the whole of the examination.”
About this time, 1790, the custom of printing the problem paperswas introduced, but until 1828 the other papers continued to be dic-tated. Since 1827 all the papers have been printed.
I insert here the following letter† from William Gooch, of Caius,in which he describes his examination in the Senate-House in 1791. Itmust be remembered that it is the letter of an undergraduate addressedto his father and mother, and was not intended either for preservationor publication: a fact which certainly does not detract from its value.
Monday 14 aft. 12.
We have been examin’d this Morning in pure Mathematics & I’vehitherto kept just about even with Peaco*ck which is much more than Iexpected. We are going at 1 o’clock to be examin’d till 3 in Philosophy.
From 1 till 7 I did more than Peaco*ck; But who did most at Modera-tor’s Rooms this Evening from 7 till 9, I don’t know yet;—but I did abovethree times as much as the Senr Wrangler last year, yet I’m afraid not somuch as Peaco*ck.
Between One & three o’Clock I wrote up 9 sheets of Scribbling Paperso you may suppose I was pretty fully employ’d.
Tuesday Night.I’ve been shamefully us’d by Lax to-day;—Tho’ his anxiety for Peaco*ck
must (of course) be very great, I never suspected that his Partially (sic)wd get the better of his Justice. I had entertain’d too high an opinion of
* H. Gunning, Reminiscences, second edition, London, 1855, vol. i, p. 82.† C. Wordsworth, Scholae Academicae, Cambridge, 1877, pp. 322–23.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 185
him to suppose it.—he gave Peaco*ck a long private Examination & thencame to me (I hop’d) on the same subject, but ’twas only to Bully meas much as he could,—whatever I said (tho’ right) he tried to convertinto Nonsense by seeming to misunderstand me. However I don’t entirelydispair of being first, tho’ you see Lax seems determin’d that I shall not.—Ihad no Idea (before I went into the Senate-House) of being able to contendat all with Peaco*ck.
Wednesday evening.Peaco*ck & I are still in perfect Equilibrio & the Examiners themselves
can give no guess yet who is likely to be first;—a New Examiner (Wood ofSt. John’s, who is reckon’d the first Mathematician in the University, forWaring doesn’t reside) was call’d solely to examine Peaco*ck & me only.—but by this new Plan nothing is yet determin’d.—So Wood is to examineus again to-morrow morning.
Thursday evening.Peaco*ck is declar’d first & I second,—Smith of this Coll. is either 8th
or 9th & Lucas is either 10th or 11th.—Poor Quiz Carver is one of the οÉ
πολλοÐ;—I’m perfectly satisfied that the Senior Wranglership is Peaco*ck’sdue, but certainly not so very indisputably as Lax pleases to representit—I understand that he asserts ’twas 5 to 4 in Peaco*ck’s favor. NowPeaco*ck & I have explain’d to each other how we went on, & can proveindisputably that it wasn’t 20 to 19 in his favor;—I cannot therefore bedispleas’d for being plac’d second, tho’ I’m provov’d (sic) with Lax forhis false report (so much beneath the Character of a Gentleman.)—
N.B. it is my very particular Request that you don’t mention Lax’sbehaviour to me to any one.
Such was the form ultimately taken by the Senate-House exami-nation, a form which it substantially retained without alteration fornearly half-a-century. It soon became the sole test by which candidateswere judged. The University was not obliged to grant a degree to any-one who performed the statutable exercises, and it was open to theUniversity to refuse to pass a supplicat for the B.A. degree unless thecandidate had presented himself for the Senate-House examination. In1790 James Blackburn, of Trinity, a questionist of exceptional abilities,was informed that in spite of his good disputations he would not beallowed a degree unless he also satisfied the examiners in the tripos. Heaccordingly solved one “very hard problem,” though in consequence ofa dispute with the authorities he refused to attempt any more.*
It will be recollected that the examination was now compulsory onall candidates pursuing the normal course for the B.A. degree. In 1791
* Gunning, Reminiscences, second edition, London, 1855, vol. i, p. 182.
• Project • Gutenberg • #26839 •
186 THE MATHEMATICAL TRIPOS. [CH. VII
the University laid down rules* for its conduct, so far as it concernedpoll-men, decreeing that those who passed were to be classified in fourdivisions or classes, the names in each class to be arranged alphabet-ically, but not to be printed on the official tripos lists. The classes inthe final lists must be distinguished from the eight preliminary classesissued before the commencement of the examination. The men in thefirst six preliminary classes were expected to take honours; those in theseventh and eighth preliminary classes were prima facie poll-men.
In 1799 the moderators announced† that for the future they wouldrequire every candidate to show a competent knowledge of the firstbook of Euclid, arithmetic, vulgar and decimal fractions, simple andquadratic equations, and Locke and Paley. Paley’s works seem to beheld in esteem by modern divines, and his Evidences, though not hisPhilosophy, still remains (1905) one of the subjects of the Previous Ex-amination, but his contemporaries thought less highly of his writings,or at any rate of his Philosophy. Thus Best is quoted by Wordsworth‡
as saying of Paley’s Philosophy, “The tutors of Cambridge no doubtneutralize by their judicious remarks, when they read it to their pupils,all that is pernicious in its principles”: so also Richard Watson, Bishopof Llandaff, in his anecdotal autobiography§, says, in describing theSenate-House examination in which Paley was senior wrangler, thatPaley was afterwards known to the world by many excellent produc-tions, “though there are some. . . principles in his philosophy which Iby no means approve.”
In 1800 the moderators extended to all men in the first four pre-liminary classes the privilege of being allowed to attempt the problempapers: hitherto this privilege had been confined to candidates placedin the first two classes. Until 1828 the problem papers were set in theevenings, and in the rooms of the moderator.
The University Calendars date from 1796, and from 1802 to 1882inclusive contain the printed tripos papers of the previous January. Thepapers from 1801 to 1820 and from 1838 to 1849 inclusive were alsopublished in separate volumes, which are to be found in most publiclibraries. No problems were ever set to the men in the seventh and
* See Grace of April 8, 1791.† Communicated by the moderators to fathers of Colleges on January 18, 1799,
and agreed to by the latter.‡ C. Wordsworth, Scholae Academicae, Cambridge, 1877, p. 123.§ Anecdotes of the Life of R. Watson, London, 1817, p. 19.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 187
eighth preliminary classes, which contained the poll-men. None of thebookwork papers of this time are now extant, but it is believed that theycontained but few riders. Many of the so-called problems were reallypieces of bookwork or easy riders: it must however be rememberedthat the text-books then in circulation were inferior and incomplete ascompared with modern ones.
The Calendar of 1802 contains a diffuse account of the examination.It commences as follows:
On the Monday morning, a little before eight o’clock, the students,generally about a hundred, enter the Senate-House, preceded by a masterof arts, who on this occasion is styled the father of the College to whichhe belongs. On two pillars at the entrance of the Senate-House are hungthe classes and a paper denoting the hours of examination of those whoare thought most competent to contend for honours. Immediately afterthe University clock has struck eight, the names are called over, and theabsentees, being marked, are subject to certain fines. The classes to beexamined are called out, and proceed to their appointed tables, where theyfind pens, ink, and paper provided in great abundance. In this manner,with the utmost order and regularity, two-thirds of the young men areset to work within less than five minutes after the clock has struck eight.There are three chief tables, at which six examiners preside. At the first,the senior moderator of the present year and the junior moderator of thepreceding year. At the second, the junior moderator of the present andthe senior moderator of the preceding year. At the third, two moderatorsof the year previous to the two last, or two examiners appointed by theSenate. The two first tables are chiefly allotted to the six first classes; thethird, or largest, to the οÉ πολλοÐ.
The young men hear the propositions or questions delivered by theexaminers; they instantly apply themselves; demonstrate, prove, work outand write down, fairly and legibly (otherwise their labour is of little avail)the answers required. All is silence; nothing heard save the voice of theexaminers; or the gentle request of some one, who may wish a repetition ofthe enunciation. It requires every person to use the utmost dispatch; foras soon as ever the examiners perceive anyone to have finished his paperand subscribed his name to it another question is immediately given. . .
The examiners are not seated, but keep moving round the tables, bothto judge how matters proceed and to deliver their questions at properintervals. The examination, which embraces arithmetic, algebra, flux-ions, the doctrine of infinitesimals and increments, geometry, trigonom-etry, mechanics, hydrostatics, optics, and astronomy, in all their variousgradations, is varied according to circ*mstances: no one can anticipate aquestion, for in the course of five minutes he may be dragged from Euclidto Newton, from the humble arithmetic of Bonnycastle to the abstruseanalytics of Waring. While this examination is proceeding at the three
• Project • Gutenberg • #26839 •
188 THE MATHEMATICAL TRIPOS. [CH. VII
tables between the hours of eight and nine, printed problems are deliveredto each person of the first and second classes; these he takes with him toany window he pleases, where there are pens, ink, and paper prepared forhis operations.
The examination began at eight. At nine o’clock the papers had tobe given up, and half-an-hour was allowed for breakfast. At half-pastnine the candidates came back, and were examined in the way describedabove till eleven, when the Senate-House was again cleared. An intervalof two hours then took place. At one o’clock all returned to be againexamined. At three the Senate-House was cleared for half-an-hour,and, on the return of the candidates, the examination was continuedtill five. At seven in the evening the first four classes went to the seniormoderator’s rooms to solve problems. They were finally dismissed forthe day at nine, after eight hours of examination. The work of Tuesdaywas similar to that of Monday: Wednesday was partly devoted to logicand moral philosophy. At eight o’clock on Thursday morning a firstlist was published with all candidates of about equal merits bracketed.Until nine o’clock a candidate had the right to challenge anyone abovehim to an examination to see which was the better. At nine a secondlist came out, and a candidate’s right of challenge was then confined tothe bracket immediately above his own. If he proved himself the equalof the man so challenged his name was transferred to the upper bracket.To challenge and then to fail to substantiate the claim to removal toa higher bracket was considered rather ridiculous. Revised lists werepublished at 11 a.m., 3 p.m., and 5 p.m., according to the results ofthe examination during that day. At five the whole examination ended.The proctors, moderators, and examiners then retired to a room underthe Public Library to prepare the list of honours, which was sometimessettled without much difficulty in a few hours, but sometimes not before2 a.m. or 3 a.m. the next morning. The name of the senior wranglerwas generally announced at midnight, and the rest of the list the nextmorning. In 1802 there were eighty-six candidates for honours, andthey were divided into fifteen brackets, the first and second bracketscontaining each one name only, and the third bracket four names.
It is clear from the above account that the competition fostered bythe examination had developed so much as to threaten to impair itsusefulness as guiding the studies of the men. On the other hand, therecan be no doubt that the carefully devised arrangements for obtainingan accurate order of merit stimulated the best men to throw all their
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 189
energies into the work for the examination. It is easy to point out theusual double-edged result of a strict order of merit. The problem beforethe University was to retain its advantages while checking any abusesto which it might lead.
It was the privilege of the moderators to entertain the proctorsand some of the leading resident mathematicians the night before theissue of the final list, and to communicate that list in confidence totheir guests. This pleasant custom survived till 1884. I revived thepractice in 1890 when acting as senior moderator, but it seems to havenow ceased.
In 1806 Sir Frederick Pollock was senior wrangler, and in 1869 inanswer to an appeal from De Morgan for an account of the mathemat-ical study of men at the beginning of the century he wrote a letter*
which is sufficiently interesting to bear reproduction:
I shall write in answer to your inquiry, all about my books, my studies,and my degree, and leave you to settle all about the proprieties which myletter may give rise to, as to egotism, modesty, &c. The only books I readthe first year were Wood’s Algebra (as far as quadratic equations), Bon-nycastle’s ditto, and Euclid (Simpson’s). In the second year I read Wood(beyond quadratic equations), and Wood and Vince, for what they calledthe branches. In the third year I read the Jesuit’s Newton and Vince’sFluxions; these were all the books, but there were certain mss. floatingabout which I copied—which belonged to Dealtry, second wrangler inKempthorne’s year. I have no doubt that I had read less and seen fewerbooks than any senior wrangler of about my time, or any period since;but what I knew I knew thoroughly, and it was completely at my fingers’ends. I consider that I was the last geometrical and fluxional senior wran-gler; I was not up to the differential calculus, and never acquired it. Iwent up to college with a knowledge of Euclid and algebra to quadraticequations, nothing more; and I never read any second year’s lore duringmy first year, nor any third year’s lore during my second; my forte was,that what I did know I could produce at any moment with perfect ac-curacy. I could repeat the first book of Euclid word by word and letterby letter. During my first year I was not a ‘reading ’ man (so called); Ihad no expectation of honours or a fellowship, and I attended all the lec-tures on all subjects—Harwood’s anatomical, Woollaston’s chemical, andFarish’s mechanical lectures—but the examination at the end of the firstyear revealed to me my powers. I was not only in the first class, but it wasgenerally understood I was first in the first class; neither I nor any one forme expected I should get in at all. Now, as I had taken no pains to pre-pare (taking, however, marvellous pains while the examination was going
* Memoir of A. de Morgan, London, 1882, pp. 387–392.
• Project • Gutenberg • #26839 •
190 THE MATHEMATICAL TRIPOS. [CH. VII
on), I knew better than any one else the value of my examination qualities(great rapidity and perfect accuracy); and I said to myself, ‘If you’re notan ass, you’ll be senior wrangler;’ and I took to ‘reading’ accordingly. Acurious circ*mstance occurred when the Brackets came out in the Senate-house declaring the result of the examination: I saw at the top the nameof Walter bracketed alone (as he was); in the bracket below were Fiott,Hustler , Jephson. I looked down and could not find my own name till Igot to Bolland, when my pride took fire, and I said, ‘I must have beatenthat man, so I will look up again;’ and on looking up carefully I found thenail had been passed through my name, and I was at the top bracketedalone, even above Walter. You may judge what my feelings were at thisdiscovery; it is the only instance of two such brackets, and it made myfortune—that is, made me independent, and gave me an immense collegereputation. It was said I was more than half of the examination before anyone else. The two moderators were Hornbuckle, of St John’s, and Brown(Saint Brown), of Trinity. The Johnian congratulated me. I said perhapsI might be challenged; he said, ‘Well, if you are you’re quite safe—youmay sit down and do nothing, and no one would get up to you in a wholeday.’. . .
Latterly the Cambridge examinations seem to turn upon very differentmatters from what prevailed in my time. I think a Cambridge educationhas for its object to make good members of society—not to extend scienceand make profound mathematicians. The tripos questions in the Senate-house ought not to go beyond certain limits, and geometry ought to becultivated and encouraged much more than it is.
To this De Morgan replied:
Your letter suggests much, because it gives possibility of answer. Thebranches of algebra of course mainly refer to the second part of Wood,now called the theory of equations. Waring was his guide. Turner—whom you must remember as head of Pembroke, senior wrangler of 1767—told a young man in the hearing of my informant to be sure and attendto quadratic equations. ‘It was a quadratic,’ said he, ‘made me seniorwrangler.’ It seems to me that the Cambridge revivers were Waring,Paley, Vince, Milner.
You had Dealtry’s mss. He afterwards published a very good bookon fluxions. He merged his mathematical fame in that of a ClaphamiteChristian. It is something to know that the tutor’s ms. was in vogue in1800–1806.
Now—how did you get your conic sections? How much of Newton didyou read? From Newton direct, or from tutor’s manuscript?
Surely Fiott was our old friend Dr Lee. I missed being a pupil ofHustler by a few weeks. He retired just before I went up in February1823. The echo of Hornbuckle’s answer to you about the challenge haslighted on Whewell, who, it is said, wanted to challenge Jacob, and wasanswered that he could not beat [him] if he were to write the whole day
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 191
and the other wrote nothing. I do not believe that Whewell would havelistened to any such dissuasion.
I doubt your being the last fluxional senior wrangler. So far as I know,Gipps, Langdale, Alderson, Dicey, Neale, may contest this point with you.
The answer of Sir Frederick Pollock to these questions is datedAugust 7, 1869, and is as follows.
You have put together as revivers five very different men. Woodhousewas better than Waring, who could not prove Wilson’s (Judge of C.P.)guess about the property of prime numbers; but Woodhouse (I think) didprove it, and a beautiful proof it is. Vince was a bungler, and I thinkutterly insensible of mathematical beauty.
Now for your questions. I did not get my conic sections from Vince.I copied a ms. of Dealtry. I fell in love with the cone and its sections,and everything about it. I have never forsaken my favourite pursuit; Idelighted in such problems as two spheres touching each other and alsothe inside of a hollow cone, &c. As to Newton, I read a good deal (mennow read nothing), but I read much of the notes. I detected a blunderwhich nobody seemed to be aware of. Tavel, tutor of Trinity, was not;and he argued very favourably of me in consequence. The applicationof the Principia I got from mss. The blunder was this: in calculatingthe resistance of a globe at the end of a cylinder oscillating in a resistingmedium they had forgotten to notice that there is a difference betweenthe resistance to a globe and a circle of the same diameter.
The story of Whewell and Jacob cannot be true. Whewell was a very,very considerable man, I think not a great man. I have no doubt Jacobbeat him in accuracy, but the supposed answer cannot be true; it is a mereecho of what actually passed between me and Hornbuckle on the day theTripos came out—for the truth of which I vouch. I think the examinersare taking too practical a turn; it is a waste of time to calculate actuallya longitude by the help of logarithmic tables and lunar observations. Itwould be a fault not to know how, but a greater to be handy at it.
A few minor changes in the Senate-House examinations were madein 1808*. A fifth day was added to the examination. Of the five daysthus given up to it three were devoted to mathematics, one to logic,philosophy, and religion, and one to the arrangement of the brackets.Apart from the evening paper the examination on each of the first threedays lasted six hours. Of these eighteen hours, eleven were assigned tobook-work and seven to problems. The problem papers were set from6 to 10 in the evening.
* See Graces, December 15, 1808.
• Project • Gutenberg • #26839 •
192 THE MATHEMATICAL TRIPOS. [CH. VII
A letter from Whewell dated January 19, 1816, describes his ex-amination in the Senate-House*.
Jacob. Whewell. Such is the order in which we are fixed after a week’sexamination. . . I had before been given to understand that a great dealdepended upon being able to write the greatest possible quantity in thesmallest time, but of the rapidity which was actually necessary I hadformed the most distant idea. I am upon no occasion a quick writer, andupon subjects where I could not go on without sometimes thinking a littleI soon found myself considerably behind. I was therefore surprised, andeven astonished, to find myself bracketed off, as it is called, in the secondplace; that is, on the day when a new division of the classes is made forthe purpose of having a closer examination of the respective merits of menwho come pretty near to each other, I was not classed with anybody, butplaced alone in the second bracket. The man who is at the head of thelist is of Caius College, and was always expected to be very high, thoughI do not know that anybody expected to see him so decidedly superior asto be bracketed off by himself.
The tendency to cultivate mechanical rapidity was a grave evil, andlasted long after Whewell’s time. According to rumour the highesthonours in 1845 were obtained, to the general regret of the University,by assiduous practice in writing†.
The devotion of the Cambridge school to geometrical and fluxionalmethods has led to its isolation from contemporary continental mathe-maticians. Early in the nineteenth century the evil consequence of thisbegan to be recognized; and it was felt to be little less than a scandalthat the researches of Lagrange, Laplace, and Legendre were unknownto many Cambridge mathematicians save by repute. An attempt toexplain the notation and methods of the calculus as used on the Conti-nent was made by R. Woodhouse, who stands out as the apostle of thenew movement. It is doubtful if he could have brought analytical meth-ods into vogue by himself; but his views were enthusiastically adoptedby three students, Peaco*ck, Babbage, and Herschel, who succeeded incarrying out the reforms he had suggested. They created an AnalyticalSociety which Babbage explained was formed to advocate “the princi-ples of pure d-ism as opposed to the dot-age of the University.” Thecharacter of the instruction in mathematics at the University has at alltimes largely depended on the text-books then in use, and the impor-
* S. Douglas, Life of W. Whewell, London, 1881, p. 20.† For a contemporary account of this see C.A. Bristed, Five Years in an English
University, New York, 1852, pp. 233–239.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 193
tance of good books of this class was emphasized by a traditional rulethat questions should not be set on a new subject in the tripos unlessit had been discussed in some treatise suitable and available for Cam-bridge students*. Hence the importance attached to the publication ofthe work on analytical trigonometry by Woodhouse in 1809, and of theworks on the differential calculus issued by members of the AnalyticalSociety in 1816 and 1820.
In 1817 Peaco*ck, who was moderator, introduced the symbols fordifferentiation into the papers set in the Senate-House examination.But his colleague continued to use the fluxional notation. Peaco*ckhimself wrote on March 17 of 1817 (i.e. shortly after the examination)on the subject as follows†:
I assure you. . . that I shall never cease to exert myself to the utmostin the cause of reform, and that I will never decline any office which mayincrease my power to effect it. I am nearly certain of being nominated tothe office of Moderator in the year 1818–19, and as I am an examiner invirtue of my office, for the next year I shall pursue a course even moredecided than hitherto, since I shall feel that men have been prepared forthe change, and will then be enabled to have acquired a better systemby the publication of improved elementary books. I have considerableinfluence as a lecturer, and I will not neglect it. It is by silent perseveranceonly that we can hope to reduce the many-headed monster of prejudice,and make the University answer her character as the loving mother ofgood learning and science.
In 1818 all candidates for honours, that is, all men in the first sixpreliminary classes, were allowed to attempt the problems: this changewas made by the moderators.
In 1819 G. Peaco*ck, who was again moderator, induced his col-league to adopt the new notation. It was employed in the next yearby Whewell, and in the following year by Peaco*ck again. Henceforththe calculus in its modern language and analytical methods were freelyused, new subjects were introduced, and for many years the examina-tion provided a mathematical training fairly abreast of the times.
By this time the disputations had ceased to have any immediateeffect on a man’s place in the tripos. Thus Whewell‡, writing abouthis duties as moderator in 1820, said:
* See ex. gr., the Grace of November 14, 1827, referred to below.† Proceedings of the Royal Society, London, 1859, vol. ix, pp. 538–9.‡ Whewell’s Writings and Correspondence, ed. Todhunter, London, 1876, vol. ii,
p. 36.
• Project • Gutenberg • #26839 •
194 THE MATHEMATICAL TRIPOS. [CH. VII
You would get very exaggerated ideas of the importance attached toit [an Act] if you were to trust Cumberland; I believe it was formerlymore thought of than it is now. It does not, at least immediately, produceany effect on a man’s place in the tripos, and is therefore considerablyless attended to than used to be the case, and in most years is not veryinteresting after the five or six best men: so that I look for a considerableexercise of, or rather demand for, patience on my part. The other part ofmy duty in the Senate House consists in manufacturing wranglers, senioroptimes, etc. and is, while it lasts, very laborious.
Of the examination itself in this year he wrote as follows*:
The examination in the Senate House begins to-morrow, and is ratherclose work while it lasts. We are employed from seven in the morning tillfive in the evening in giving out questions and receiving written answers tothem; and when that is over, we have to read over all the papers which wehave received in the course of the day, to determine who have done best,which is a business that in numerous years has often kept the examinersup the half of every night; but this year is not particularly numerous.In addition to all this, the examination is conducted in a building whichhappens to be a very beautiful one, with a marble floor and a highlyornamented ceiling; and as it is on the model of a Grecian temple, andas temples had no chimneys, and as a stove or a fire of any kind mightdisfigure the building, we are obliged to take the weather as it happensto be, and when it is cold we have the full benefit of it—which is likely tobe the case this year. However, it is only a few days, and we have donewith it.
A sketch of the examination in the previous year from the point of viewof an examinee was given by J.M.F. Wright†, but there is nothing ofspecial interest in it.
Sir George Airy‡ gave the following sketch of his recollections ofthe reading and studies of undergraduates of his time and of the triposof 1823, in which he had been senior wrangler:
At length arrived the Monday morning on which the examination forthe B.A. degree was to begin. . . . We were all marched in a body tothe Senate-House and placed in the hands of the Moderators. How the“candidates for honours” were separated from the οÉ πολλοÐ I do not know,I presume that the Acts and the Opponencies had something to do withit. The honour candidates were divided into six groups: and of theseNos. 1 and 2 (united), Nos. 3 and 4 (united), and Nos. 5 and 6 (united),received the questions of one Moderator. No. 1, Nos. 2 and 3 (united),
* S. Douglas, Life of Whewell, London, 1881, p. 56.† Alma Mater, London, 1827, vol. ii, pp. 58–98.‡ See Nature, vol. 35, Feb. 24, 1887, pp. 397–399.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 195
Nos. 4 and 5 (united), and No. 6, received those of the other Moderator.The Moderators were reversed on alternate days. There were no printedquestion-papers: each examiner had his bound manuscript of questions,and he read out his first question; each of the examinees who thoughthimself able proceeded to write out his answer, and then orally called out“Done.” The Moderator, as soon as he thought proper, proceeded withanother question. I think there was only one course of questions on eachday (terminating before 3 o’clock, for the Hall dinner). The examinationcontinued to Friday mid-day. On Saturday morning, about 8 o’clock, thelist of honours (manuscript) was nailed on the door of the Senate-House.
It must be remembered that for students pursuing the normalcourse the Senate-House examination still provided the only avenueto a degree. That examination involved a knowledge of the elements ofmoral philosophy and theology, an acquaintance with the rules of for-mal logic, and the power of reading and writing scholastic Latin, butmathematics was the predominant subject, and this led to a certainone-sidedness in education. The evil of this was generally recognized,and in 1822 various reforms were introduced in the University cur-riculum; in particular the Previous Examination was established forstudents in their second year, the subjects being prescribed Greek andLatin works, a Gospel, and Paley’s Evidences. Set classical books wereintroduced in the final examination of poll-men; and another honouror tripos examination was established for classical students. These al-terations came into effect in 1824; and henceforth the Senate-Houseexamination, so far as it related to mathematical students, was knownas the Mathematical Tripos.
In 1827 the scheme of examination in the Mathematical Tripos wasrevised. By regulations* which came into operation in January, 1828,another day was added, so that the examination extended over fourdays, exclusive of the day of arranging the brackets; the number ofhours of examination was twenty-three, of which seven were assignedto problems. On the first two days all the candidates had the samequestions proposed to them, inclusive of the evening problems, and theexamination on those days excluded the higher and more difficult partsof mathematics, in order, in the words of the report, “that the can-didates for honours may not be induced to pursue the more abstruseand profound mathematics, to the neglect of more elementary knowl-edge.” Accordingly, only such questions as could be solved without
* See the Grace, November 14, 1827.
• Project • Gutenberg • #26839 •
196 THE MATHEMATICAL TRIPOS. [CH. VII
the aid of the differential calculus were set on the first day, and thoseset on the second day involved only its elementary applications. Theclasses were reduced to four, determined as before by the exercises inthe schools. The regulations of 1827 definitely prescribed that all thepapers should be printed. They are also noticeable as being the lastwhich gave the examiners power to ask viva voce questions, thoughsuch questions were restricted to “propositions contained in the math-ematical works commonly in use in the University, or examples andexplanations of such propositions.” It was further recommended thatno paper should contain more questions than well-prepared studentscould be expected to answer within the time allowed for it, but that ifany candidate, before the end of the time, had answered all the ques-tions in the paper, the examiners might propose additional questionsviva voce. The power of granting honorary optime degrees now ceased;it had already fallen into abeyance. Henceforth the examination wasconducted under definite rules, and I no longer concern myself with thetraditions of the examination.
In the same year as these changes became effective the examina-tion for the poll degree was separated from the tripos with different setsof papers and a different schedule of subjects*. It was, however, stillnominally considered as forming part of the Senate-House examination,and until 1858 those who obtained a poll degree were arranged in fourclasses, described as fourth, fifth, sixth, and seventh, as if in continu-ation of the junior optimes or third class of the tripos. The year 1828therefore shews us the Senate-House examination dividing into two dis-tinct parts; one known as the mathematical tripos, the other as the pollexamination. In 18511 the classical tripos was made independent of themathematical tripos, and thus provided a separate avenue to a degree.Historically, the examination usually known as “the General” representsthe old Senate-House examination for the poll-men, but gradually it hasbeen moved to an earlier period in the normal course taken by the men.In 1852 another set of examinations, at first called “the professor’s ex-aminations,” and now somewhat modified and known as “the Specials,”was instituted for all poll-men to take before they could qualify for adegree. In 1858 the fiction that the poll-examinations were part of
* See Grace, May 21, 1828, confirming a Report of March 27, 1828.
1. ‘1850’ corrected to ‘1851’ as per errata sheet
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 197
the Senate-House examination was abandoned, and subsequently theyhave been treated as providing an independent method of obtaining thedegree: thus now the mathematical tripos is the sole representative ofthe old Senate-House examination. Since 1858 numerous other ways ofobtaining the degree have been established, and it is now possible toget it by shewing proficiency in very special, or even technical subjects.
Further changes in the mathematical tripos were introduced in1833*. The duration of the examination, before the issue of the brack-ets, was extended to five days, and the number of hours of examinationon each day was fixed at five and a-half. Seven and a-half hours wereassigned to problems. The examination on the first day was confinedto subjects that did not require the differential calculus, and only thesimplest applications of the calculus were permitted on the second andthird days. During the first four days of the examination the samepapers were set to all the candidates alike, but on the fifth day the ex-amination was conducted according to classes. No reference was madeto viva voce questions, and the preliminary classification of the bracketsonly survived in a permission to re-examine candidates if it were foundnecessary. This permissive rule remained in force till 1848, but I believethat in fact it was never used. In December, 1834, a few unimportantdetails were amended.
Mr Earnshaw, the senior moderator in 1836, informed me that hebelieved that the tripos of that year was the earliest one in which allthe papers were marked, and that in previous years the examiners hadpartly relied on their impression of the answers given.
New regulations came into force† in 1839. The examination nowlasted for six days, and continued as before for five hours and a-half eachday. Eight and a-half hours were assigned to problems. Throughoutthe whole examination the same papers were set to all candidates, andno reference was made to any preliminary classes. It was no doubt inaccordance with the spirit of these changes that the acts in the schoolsshould be abolished, but they were discontinued by the moderators of1839 without the authority of the Senate. The examination was for thefuture confined‡ to mathematics.
* See the Grace of April 6, 1832.† See Grace of May 30, 1838.‡ Under a badly-worded grace passed on May 11, 1842, on the recommendation of
a syndicate on theological studies, candidates for mathematical honours were,after 1846, required to attend the poll examination on Paley’s Moral Philosophy,
• Project • Gutenberg • #26839 •
198 THE MATHEMATICAL TRIPOS. [CH. VII
In the same year in which the new scheme came into force a pro-posal to again reopen the subject was rejected (March 6).
The difficulty of bringing professorial lectures into relation withthe needs of students has more than once been before the Univer-sity. The desirability of it was emphasized by a Syndicate in February,1843, which recommended conferences at stated intervals between themathematical professors and examiners. This report foreshadowed thecreation of a Mathematical Board, but it was rejected by the Senateon March 31.
A few years later the scheme of the examination was again recon-structed by regulations* which came into effect in 1848. The dura-tion of the examination was extended to eight days. The examinationlasted in all forty-four and a-half hours, twelve of which were devotedto problems. The first three days were assigned to specified elementarysubjects; in the papers set on these days riders were to be set as wellas bookwork, but the methods of analytical geometry and the calculuswere excluded. After the first three days there was a short interval, atthe end of which the examiners issued a list of those who had so acquit-ted themselves as to deserve mathematical honours. Only those whosenames were contained in this list were admitted to the last five daysof the examination, which was devoted to the higher parts of mathe-matics. After the conclusion of the examination the examiners, takinginto account the whole eight days, brought out the list arranged in or-der of merit. No provision was made for any rearrangement of this listcorresponding to the examination of the brackets. The arrangementsof 1848 remained in force till 1873.
In the same year as these regulations came into force, a Board ofMathematical Studies (consisting of the mathematical professors, andthe moderators and examiners for the current year and the two pre-ceding years) was constituted† by the Senate. From that time forwardtheir minutes supply a permanent record of the changes gradually in-troduced into the tripos. I do not allude to subsequent changes whichonly concern unimportant details of the examination.
In May, 1849, the Board issued a report in which, after giving a
the New Testament and Ecclesiastical History. This had not been the intentionof the Senate, and on March 14, 1855, a grace was passed making this clear.
* See Grace of May 13, 1846, confirming a report of March 23, 1846.† See Grace of October 31, 1848.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 199
review of the past and existing state of the mathematical studies inthe University, they recommended that the mathematical theories ofelectricity, magnetism, and heat should not be admitted as subjectsof examination. In the following year they issued a second report, inwhich they recommended the omission of elliptical integrals, Laplace’scoefficients, capillary attraction, and the figure of the earth consideredas heterogeneous, as well as a definite limitation of the questions inlunar and planetary theory. In making these recommendations theBoard were only giving expression to what had become the practicein the examination.
I may, in passing, mention a curious attempt which was made in1853 and1 1854 to assist candidates in judging of the relative difficultyof the questions asked. This was effected by giving to the candidates,at the same time as the examination paper, a slip of paper on whichthe marks assigned for the bookwork and rider for each question wereprinted. I mention the fact merely because these things are rapidly for-gotten and not because it is of any intrinsic value. I possess a completeset of slips which came to me from Dr Todhunter.
In 1856 there was an amusing difference of opinion between theVice-Chancellor and the moderators. The Vice-Chancellor issued anotice to say that for the convenience of the University he had directedthe tripos lists to be published at 8.0 a.m. as well as at 9.0 a.m., butwhen the University arrived at 8.0 the moderators said that they shouldnot read the list until 9.0.
Considerable changes in the scheme of examination were introducedin 1873. On December 5, 1865, the Board had recommended the ad-dition of Laplace’s coefficients and the figure of the earth consideredas heterogeneous as subjects of the examination; the report does notseem to have been brought before the Senate, but attention was calledto the fact that certain departments of mathematics and mathematicalphysics found no place in the tripos schedules, and were neglected bymost students. Accordingly a syndicate was appointed on June 6, 1867,to consider the matter, and a scheme drawn up by them was approvedin 1868* and came into effect in 1873. The new scheme of examinationwas framed on the same lines as that of 1848. The subjects in the first
* See Grace of June 2, 1868. It was carried by a majority of only five in a houseof 75.
1. ‘1853 and’ inserted as per errata sheet
• Project • Gutenberg • #26839 •
200 THE MATHEMATICAL TRIPOS. [CH. VII
three days were left unchanged, but an extra day was added, devotedto the elements of mathematical physics. The essence of the modifi-cation was the greatly extended range of subjects introduced into theschedule of subjects for the last five days, and their arrangement indivisions, the marks awarded to the five divisions being approximatelythose awarded to the three days in proportion to 2, 1, 1, 1, 2/3 to 1respectively. Under the new regulations the number of examiners wasincreased from four to five.
The assignment of marks to groups of subjects was made under theimpression that the best candidates would concentrate their abilitieson a selection of subjects from the various divisions. But it was foundthat, unless the questions were made extremely difficult, more markscould be obtained by reading superficially all the subjects in the fivedivisions than by attaining real proficiency in a few of the higher ones:while the wide range of subjects rendered it practically impossible tothoroughly cover all the ground in the time allowed. The failure was sopronounced that in 1877 another syndicate was appointed to considerthe mathematical studies and examinations of the University. Theypresented an elaborate scheme, but on May 13, 1878, some of the mostimportant parts of it were rejected and their subsequent proposals,accepted on November 21, 1878 (by 62 to 49), represented a compromisewhich pleased few members of the Senate*.
Under the new scheme which came into force in 1882 the triposwas divided into two portions: the first portion was taken at the end ofthe third year of residence, the range of subjects being practically thesame as in the regulations of 1848, and the result brought out in thecustomary order of merit. The second portion was held in the followingJanuary, and was open only to those who had been wranglers in thepreceding June. This portion was confined to higher mathematics andappealed chiefly to specialists. The result was brought out in threeclasses, each arranged in alphabetical order. The moderators and ex-aminers conducted the whole examination without any extraneous aid.
In the next year or two further amendments were made†, movingthe second part to the June of the fourth year, throwing it open to
* See Graces of May 17, 1877; May 29, 1878; and November 21, 1878: and theCambridge University Reporter, April 2, May 14, June 4, October 29, Novem-ber 12, and November 26, 1878.
† See the Graces of December 13, 1883; June 12, 1884; February 10, 1885; Octo-ber 29, 1885; and June 1, 1886.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 201
all men who had graduated in the tripos of the previous June, andtransferring the conduct of the examination in Part 2 to four examinersnominated by the Board: this put it largely under the control of theprofessors. The range of subjects of Part 2 was also greatly extended,and candidates were encouraged to select only a few of them. It wasfurther arranged that Part 1 might be taken at the end of a man’ssecond year of residence, though in that case it would not qualify for adegree. A student who availed himself of this leave could take Part 2at the end either of his third or of his fourth year as he pleased. Thetripos is still (1905) carried on under the scheme of 1886.
The general effect of these changes was to destroy the hom*ogeneityof the tripos. Objections to the new scheme were soon raised. Espe-cially, it was said—whether rightly or wrongly—that Part 1 containedtoo many technical subjects to serve as a general educational trainingfor any save mathematicians; that the distinction of a high place inthe historic list produced on its results tended to prevent the best mentaking it in their second year, though by this time they had read suf-ficiently to be able to do so; and that Part 2 was so constructed as toappeal only to professional mathematicians, and that thus the higherbranches of mathematics were neglected by all save a few specialists.
Whatever value be attached to these opinions, the number of stu-dents studying mathematics fell rapidly under the scheme of 1886. In1899 the Board proposed* further changes. These seemed to some mem-bers of the Senate to be likely to still further decrease the number ofmen who took up the subject as one of general education. At any ratethe two main proposals were rejected (February 15, 1900) by votes of151 to 130 and 161 to 129.
The curious origin of the term tripos has been repeatedly told, andan account of it may fitly close this chapter. Formerly there were threeprincipal occasions on which questionists were admitted to the titleor degree of bachelor. The first of these was the comitia priora, heldon Ash-Wednesday, for the best men in the year. The next was thecomitia posteriora, which was held a few weeks later, and at which anystudent who had distinguished himself in the quadragesimal exercisessubsequent to Ash-Wednesday had his seniority reserved to him. Lastly,there was the comitia minora, for students who had in no special waydistinguished themselves. In the fifteenth century an important part
* See Reports dated November 7, 1899, and January 20, 1900.
• Project • Gutenberg • #26839 •
202 THE MATHEMATICAL TRIPOS. [CH. VII
in the ceremony on each of these occasions was taken by a certain“ould bachilour,” who sat upon a three-legged stool or tripos before theproctors and tested the abilities of the would-be graduates by arguingsome question with the “eldest son,” who was selected from them astheir representative. To assist the latter in what was often an unequalcontest his “father,” that is, the officer of his college who was to presenthim for his degree, was allowed to come to his assistance.
Originally the ceremony was a serious one, and had a certain reli-gious character. It took place in Great St Mary’s Church, and markedthe admission of the student to a position with new responsibilities,while the season of Lent was chosen with a view to bring this intoprominence. The Puritan party objected to the observance of such ec-clesiastical ceremonies, and in the course of the sixteenth century theyintroduced much license and buffoonery into the proceedings. The partplayed by the questionist became purely formal. A serious debate stillsometimes took place between the father of the senior questionist anda regent master who represented the University; but the discussion wasprefaced by a speech by the bachelor, who came to be called Mr Triposjust as we speak of a judge as the bench, or of a rower as an oar. Ulti-mately public opinion permitted Mr Tripos to say pretty much what hepleased, so long as it was not dull and was scandalous. The speeches hedelivered or the verses he recited were generally preserved by the Reg-istrary, and were known as the tripos verses: originally they referred tothe subjects of the disputations then propounded. The earliest copiesnow extant are those for 1575.
The University officials, to whom the personal criticisms in whichthe tripos indulged were by no means pleasing, repeatedly exhorted himto remember “while exercising his privilege of humour, to be modestwithal.” In 1740, says Mr Mullinger*, “the authorities after condemn-ing the excessive license of the tripos announced that the comitia atLent would in future be conducted in the Senate-House; and all mem-bers of the University, of whatever order or degree, were forbidden toassail or mock the disputants with scurrilous jokes or unseemly witti-cisms. About the year 1747–8, the moderators initiated the practiceof printing the honour lists on the back of the sheets containing thetripos verses, and after the year 1755 this became the invariable prac-
* J.B. Mullinger, The University of Cambridge, Cambridge, vol. i, 1873, pp. 175,176.
• Project • Gutenberg • #26839 •
CH. VII] THE MATHEMATICAL TRIPOS. 203
tice. By virtue of this purely arbitrary connection these lists themselvesbecame known as the tripos; and eventually the examination itself, ofwhich they represented the results, also became known by the samedesignation.”
The tripos ceased to deliver his speech about 1750, but the issue oftripos verses continued for nearly 150 years longer. During the latterpart of this time they consisted of four sets of verses, usually in Latin,but occasionally in Greek, in which current topics in the University weretreated lightly or seriously as the writer thought fit. They were writtenfor the proctors and moderators by undergraduates or commencingbachelors, who were supposed each to receive a pair of white kid glovesin recognition of their labours. Thus gradually the word tripos changedits meaning “from a thing of wood to a man, from a man to a speech,from a speech to sets of verses, from verses to a sheet of coarse foolscappaper, from a paper to a list of names, and from a list of names to asystem of examination*.”
In 1895 the proctors and moderators, without consulting the Sen-ate, sent in no verses, and thus, in spite of widespread regret, an in-teresting custom of many centuries standing was destroyed. No doubtit may be argued that the custom had never been embodied in statuteor ordinance, and thus was not obligatory. Also it may be said thatit* continuance was not of material benefit to anybody. I do not thinkthat such arguments are conclusive, and personally I regret the disap-pearance of historic ties unless it can be shown that they cause incon-venience, which of course in this case could not be asserted.
* Wordsworth, Scholae Academicae, Cambridge, 1877, p. 21.
• Project • Gutenberg • #26839 •
CHAPTER VIII.
THREE GEOMETRICAL PROBLEMS.
Among the more interesting geometrical problems of antiquity arethree questions which attracted the special attention of the early Greekmathematicians. Our knowledge of geometry is derived from Greeksources, and thus these questions have attained a classical position inthe history of the subject. The three questions to which I refer are(i) the duplication of a cube, that is, the determination of the side of acube whose volume is double that of a given cube; (ii) the trisection ofan angle; and (iii) the squaring of a circle, that is, the determinationof a square whose area is equal to that of a given circle—each problemto be solved by a geometrical construction involving the use of straightlines and circles only, that is, by Euclidean geometry.
With the restriction last mentioned all three problems are insol-uble*. To duplicate a cube the length of whose side is a, we haveto find a line of length x, such that x3 = 2a3. Again, to trisect agiven angle, we may proceed to find the sine of the angle, say a, then,if x is the sine of an angle equal to one-third of the given angle, wehave 4x3 = 3x − a. Thus the first and second problems, when con-sidered analytically, require the solution of a cubic equation; and sincea construction by means of circles (whose equations are of the formx2 + y2 + ax + by + c = 0) and straight lines (whose equations are ofthe form αx + βy + γ = 0) cannot be equivalent to the solution of acubic equation, it is inferred that the problems are insoluble if in our
* F. Klein, Vortrage uber ausgewahlte Fragen der Elementargeometrie, Leipzig,1895.
204
• Project • Gutenberg • #26839 •
CH. VIII] THE DUPLICATION OF THE CUBE. 205
constructions we are restricted to the use of circles and right lines. Ifthe use of the conic sections is permitted, both of these questions canbe solved in many ways. The third problem is different in character,but under the same restrictions it also is insoluble.
I propose to give some of the constructions which have been pro-posed for solving the first two of these problems. To save space, I shallnot draw the necessary diagrams, and in most cases I shall not add theproofs: the latter present but little difficulty. I shall conclude with somehistorical notes on approximate solutions of the quadrature of the circle.
The Duplication of the Cube*.
The problem of the duplication of the cube was known in ancienttimes as the Delian problem, in consequence of a legend that the Delianshad consulted Plato on the subject. In one form of the story, which isrelated by Philoponus†, it is asserted that the Athenians in 430 b.c.,when suffering from the plague of eruptive typhoid fever, consulted theoracle at Delos as to how they could stop it. Apollo replied that theymust double the size of his altar which was in the form of a cube. To theunlearned suppliants nothing seemed more easy, and a new altar wasconstructed either having each of its edges double that of the old one(from which it followed that the volume was increased eight-fold) or byplacing a similar cubic altar next to the old one. Whereupon, accordingto the legend, the indignant god made the pestilence worse than before,and informed a fresh deputation that it was useless to trifle with him,as his new altar must be a cube and have a volume exactly double thatof his old one. Suspecting a mystery the Athenians applied to Plato,who referred them to the geometricians. The insertion of Plato’s nameis an obvious anachronism. Eratosthenes‡ relates a somewhat similarstory, but with Minos as the propounder of the problem.
* See Historia Problematis de Cubi Duplicatione by N.T. Reimer, Gottingen, 1798;and Historia Problematis Cubi Duplicandi by C.H. Biering, Copenhagen, 1844:also Das Delische Problem, by A. Sturm, Linz, 1895–7. Some notes on thesubject are given in my History of Mathematics.
† Philoponus ad Aristotelis Analytica Posteriora, bk. i, chap. vii.‡ Archimedis Opera cum Eutocii Commentariis, ed. Torelli, Oxford, 1792, p. 144;
ed. Heiberg, Leipzig, 1880–1, vol. iii, pp. 104–107.
• Project • Gutenberg • #26839 •
206 THREE GEOMETRICAL PROBLEMS. [CH. VIII
In an Arab work, the Greek legend was distorted into the followingextraordinarily impossible piece of history, which I cite as a curiosityof its kind. “Now in the days of Plato,” says the writer, “a plaguebroke out among the children of Israel. Then came a voice from heavento one of their prophets, saying, ‘Let the size of the cubic altar bedoubled, and the plague will cease’; so the people made another altarlike unto the former, and laid the same by its side. Nevertheless thepestilence continued to increase. And again the voice spake unto theprophet, saying, ‘They have made a second altar like unto the former,and laid it by its side, but that does not produce the duplication ofthe cube.’ Then applied they to Plato, the Grecian sage, who spake tothem, saying, ‘Ye have been neglectful of the science of geometry, andtherefore hath God chastised you, since geometry is the most sublimeof all the sciences.’ Now, the duplication of a cube depends on a rareproblem in geometry, namely. . . ”. And then follows the solution ofApollonius, which is given later.
If a is the length of the side of the given cube and x that of therequired cube, we have x3 = 2a3, that is, x : a = 3
√2 : 1. It is probable
that the Greeks were aware that the latter ratio is incommensurable, inother words, that no two integers can be found whose ratio is the sameas that of 3
√2 : 1, but it did not therefore follow that they could not
find the ratio by geometry: in fact, the side and diagonal of a squareare instances of lines whose numerical measures are incommensurable.
I proceed now to give some of the geometrical constructions whichhave been proposed for the duplication of the cube*. With one ex-ception, I confine myself to those which can be effected by the aid ofthe conic sections.
Hippocrates† (circ. 420 b.c.) was perhaps the earliest mathemati-cian who made any progress towards solving the problem. He did notgive a geometrical construction, but he reduced the question to that offinding two means between one straight line (a), and another twice aslong (2a). If these means are x and y, we have a : x = x : y = y : 2a,from which it follows that x3 = 2a3. It is in this form that the problem
* On the application to this problem of the traditional Greek methods of analysisby Hero and Philo (leading to the solution by the use of Apollonius’s circle), byNicomedes (leading to the solution by the use of the conchoid), and by Pappus(leading to the solution by the use of the cissoid), see Geometrical Analysis byJ. Leslie, Edinburgh, second edition, 1811, pp. 247–250, 453.
† Proclus, ed. Friedlein, pp. 212, 213.
• Project • Gutenberg • #26839 •
CH. VIII] THE DUPLICATION OF THE CUBE. 207
is always presented now. Formerly any process of solution by findingthese means was called a mesolabum.
One of the first solutions of the problem was that given by Archy-tas* in or about the year 400 b.c. His construction is equivalent to thefollowing. On the diameter OA of the base of a right circular cylinderdescribe a semicircle whose plane is perpendicular to the base of thecylinder. Let the plane containing this semicircle rotate round the gen-erator through O, then the surface traced out by the semicircle will cutthe cylinder in a tortuous curve. This curve will itself be cut by a rightcone, whose axis is OA and semi-vertical angle is (say) 60◦, in a pointP , such that the projection of OP on the base of the cylinder will be tothe radius of the cylinder in the ratio of the side of the required cubeto that of the given cube. Of course the proof given by Archytas is geo-metrical; and it is interesting to note that in it he shows himself familiarwith the results of the propositions Euc. iii, 18, iii, 35, and xi, 19. Toshow analytically that the construction is correct, take OA as the axisof x, and the generator of the cylinder drawn through O as axis of z,then with the usual notation, in polar coordinates, if a is the radius ofthe cylinder, we have for the equation of the surface described by thesemicircle r = 2a sin θ; for that of the cylinder r sin θ = 2a cos ϕ; andfor that of the cone sin θ cos ϕ = 1
2. These three surfaces cut in a point
such that sin3 θ = 12, and therefore (r sin θ)3 = 2a3. Hence the volume
of the cube whose side is r sin θ is twice that of the cube whose side is a.
The construction attributed to Plato† (circ. 360 b.c.) depends onthe theorem that, if CAB and DAB are two right-angled triangles,having one side, AB, common, their other sides, AD and BC, parallel,and their hypothenuses, AC and BD, at right angles, then if thesehypothenuses cut in P , we have PC : PB = PB : PA = PA : PD.Hence, if such a figure can be constructed having PD = 2PC, theproblem will be solved. It is easy to make an instrument by which thefigure can be drawn.
The next writer whose name is connected with the problem isMenaechmus‡, who in or about 340 b.c. gave two solutions of it.
In the first of these he pointed out that two parabolas having acommon vertex, axes at right angles, and such that the latus rectum
* Archimedis Opera, ed. Torelli, p. 143; ed. Heiberg, vol. iii, pp. 98–103.† Ibid., ed. Torelli, p. 135; ed. Heiberg, vol. iii, pp. 66–71.‡ Ibid., ed. Torelli, pp. 141–143; ed. Heiberg, vol. iii, pp. 92–99.
• Project • Gutenberg • #26839 •
208 THREE GEOMETRICAL PROBLEMS. [CH. VIII
of the one is double that of the other will intersect in another pointwhose abscissa (or ordinate) will give a solution. If we use analysisthis is obvious; for, if the equations of the parabolas are y2 = 2ax andx2 = ay, they intersect in a point whose abscissa is given by x3 = 2a3.It is probable that this method was suggested by the form in whichHippocrates had cast the problem: namely, to find x and y so thata : x = x : y = y : 2a, whence we have x2 = ay and y2 = 2ax.
The second solution given by Menaechmus was as follows. Describea parabola of latus rectum l. Next describe a rectangular hyperbola,the length of whose real axis is 4l, and having for its asymptotes thetangent at the vertex of the parabola and the axis of the parabola. Thenthe ordinate and the abscissa of the point of intersection of these curvesare the mean proportionals between l and 2l. This is at once obviousby analysis. The curves are x2 = ly and xy = 2l2. These cut in a pointdetermined by x3 = 2l3 and y3 = 4l3. Hence l : x = x : y = y : 2l.
The solution of Apollonius*, which was given about 220 b.c., wasas follows. The problem is to find two mean proportionals between twogiven lines. Construct a rectangle OADB, of which the adjacent sidesOA and OB are respectively equal to the two given lines. Bisect AB inC. With C as centre describe a circle cutting OA produced in a and cut-ting OB produced in b, so that aDb shall be a straight line. If this circlecan be so described, it will follow that OA : Bb = Bb : Aa = Aa : OB,that is, Bb and Aa are the two mean proportionals between OA andOB. It is impossible to construct the circle by Euclidean geometry, butApollonius gave a mechanical way of describing it.
The only other construction of antiquity to which I will refer isthat given by Diocles and Sporus†. It is as follows. Take two sides of arectangle OA, OB, equal to the two lines between which the means aresought. Suppose OA to be the greater. With centre O and radius OAdescribe a circle. Let OB produced cut the circumference in C and letAO produced cut it in D. Find a point E on BC so that if DE cutsAB produced in F and cuts the circumference in G, then FE = EG.If E can be found, then OE is the first of the means between OA andOB. Diocles invented the cissoid in order to determine E, but it canbe found equally conveniently by the aid of conics.
* Archimedis Opera, ed. Torelli, p. 137; ed. Heiberg, vol. iii, pp. 76–79. Thesolution is given in my History of Mathematics, London, 1901, p. 84.
† Ibid., ed. Torelli, pp. 138, 139, 141; ed. Heiberg, vol. iii, pp. 78–84, 90–93.
• Project • Gutenberg • #26839 •
CH. VIII] THE DUPLICATION OF THE CUBE. 209
In more modern times several other solutions have been suggested.I may allude in passing to three given by Huygens*, but I will enunciateonly those proposed respectively by Vieta, Descartes, Gregory of StVincent, and Newton.
Vieta’s construction is as follows†. Describe a circle, centre O,whose radius is equal to half the length of the larger of the two givenlines. In it draw a chord AB equal to the smaller of the two givenlines. Produce AB to E so that BE = AB. Through A draw aline AF parallel to OE. Through O draw a line DOCFG, cuttingthe circumference in D and C, cutting AF in F , and cutting BAproduced in G, so that GF = OA. If this line can be drawn thenAB : GC = GC : GA = GA : CD.
Descartes pointed out‡ that the curves x2 = ay and x2+y2 = ay+bxcut in a point (x, y) such that a : x = x : y = y : b. Of course thisis equivalent to the first solution given by Menaechmus, but Descartespreferred to use a circle rather than a second conic.
Gregory’s construction was given in the form of the following the-orem§. The hyperbola drawn through the point of intersection of twosides of a rectangle so as to have the two other sides for its asymptotesmeets the circle circ*mscribing the rectangle in a point whose distancesfrom the asymptotes are the mean proportionals between two adjacentsides of the rectangle. This is the geometrical expression of the propo-sition that the curves xy = ab and x2 + y2 = ay + bx cut in a point(x, y) such that a : x = x : y = y : b.
One of the constructions proposed by Newton is as follows‖. LetOA be the greater of two given lines. Bisect OA in B. With centre Oand radius OB describe a circle. Take a point C on the circumferenceso that BC is equal to the other of the two given lines. From O drawODE cutting AC produced in D, and BC produced in E, so that theintercept DE = OB. Then BC : OD = OD : CE = CE : OA.Hence OD and CE are two mean proportionals between any two linesBC and OA.
* Opera Varia, Leyden, 1724, pp. 393–396.† Opera Mathematica, ed. Schooten, Leyden, 1646, prop, v, pp. 242–243.‡ Geometria, bk. iii, ed. Schooten, Amsterdam, 1659, p. 91.§ Gregory of St Vincent, Opus Geometricum Quadraturae Circuli, Antwerp, 1647,
bk. vi, prop. 138, p. 602.‖ Arithmetica Universalis, Ralphson’s (second) edition, 1728, p. 242; see also
pp. 243, 245.
• Project • Gutenberg • #26839 •
210 THREE GEOMETRICAL PROBLEMS. [CH. VIII
The Trisection of an Angle*.
The trisection of an angle is the second of these classical problems,but tradition has not enshrined its origin in romance. The followingtwo constructions are among the oldest and best known of those whichhave been suggested; they are quoted by Pappus†, but I do not knowto whom they were due originally.
The first of them is as follows. Let AOB be the given angle. Fromany point P in OB draw PM perpendicular to OA. Through P drawPR parallel to OA. On MP take a point Q so that if OQ is produced tocut PR in R then QR = 2 ·OP . If this construction can be made, thenAOR = 1
3AOB. The solution depends on determining the position
of R. This was effected by a construction which may be expressedanalytically thus. Let the given angle be tan−1(b/a). Construct thehyperbola xy = ab, and the circle (x− a)2 + (y − b)2 = 4(a2 + b2). Ofthe points where they cut, let x be the abscissa which is greatest, thenPR = x − a, and tan−1(b/x) = 1
3tan−1(b/a).
The second construction is as follows. Let AOB be the given angle.Take OB = OA, and with centre O and radius OA describe a circle.Produce AO indefinitely and take a point C on it external to the circleso that if CB cuts the circumference in D then CD shall be equal toOA. Draw OE parallel to CDB. Then, if this construction can bemade, AOE = 1
3AOB. The ancients determined the position of the
point C by the aid of the conchoid: it could be also found by the useof the conic sections.
I proceed to give a few other solutions; confining myself to thoseeffected by the aid of conics.
Among the other constructions given by Pappus‡ I may quote thefollowing. Describe a hyperbola whose eccentricity is two. Let its centrebe C and its vertices A and A′. Produce CA′ to S so that A′S = CA′.On AS describe a segment of a circle to contain the given angle. Let
* On the bibliography of the subject see the supplements to L’Intermediaire desMathematiciens, Paris, May and June, 1904.
† Pappus, Mathematicae Collectiones, bk. iv, props. 32, 33 (ed. Commandino,Bonn, 1670, pp. 97–99). On the application to this problem of the traditionalGreek methods of analysis see Geometrical Analysis, by J. Leslie, Edinburgh,second edition, 1811, pp. 245–247.
‡ Pappus, bk. iv, prop. 34, pp. 99–104.
• Project • Gutenberg • #26839 •
CH. VIII] THE TRISECTION OF AN ANGLE. 211
the orthogonal bisector of AS cut this segment in O. With centre Oand radius OA or OS describe a circle. Let this circle cut the branchof the hyperbola through A′ in P . Then SOP = 1
3SOA.
In modern times one of the earliest of the solutions by a direct useof conics was suggested by Descartes, who effected it by the intersectionof a circle and a parabola. His construction* is equivalent to finding thepoints of intersection other than the origin, of the parabola y2 = 1
4x and
the circle x2+y2− 134x+4ay = 0. The ordinates of these points are given
by the equation 4y3 = 3y − a. The smaller positive root is the sine ofone-third of the angle whose sine is a. The demonstration is ingenious.
One of the solutions proposed by Newton is practically equivalentto the third one which is quoted above from Pappus. It is as follows†.Let A be the vertex of one branch of a hyperbola whose eccentricity istwo, and let S be the focus of the other branch. On AS describe thesegment of a circle containing an angle equal to the supplement of thegiven angle. Let this circle cut the S branch of the hyperbola in P .Then PAS will be equal to one-third of the given angle.
The following elegant solution is due to Clairaut‡. Let AOB be thegiven angle. Take OA = OB, and with centre O and radius OA describea circle. Join AB, and trisect it in H, K, so that AH = HK = KB.Bisect the angle AOB by OC cutting AB in L. Then AH = 2 · HL.With focus A, vertex H, and directrix OC, describe a hyperbola.Let the branch of this hyperbola which passes through H cut thecircle in P . Draw PM perpendicular to OC and produce it to cutthe circle in Q. Then by the focus and directrix property we haveAP : PM = AH : HL = 2 : 1, ∴ AP = 2 · PM = PQ. Hence, bysymmetry, AP = PQ = QR. ∴ AOP = POQ = QOR.
I may conclude by giving the solution which Chasles§ regards as themost fundamental. It is equivalent to the following proposition. If OAand OB are the bounding radii of a circular arc AB, then a rectangularhyperbola having OA for a diameter and passing through the point ofintersection of OB with the tangent to the circle at A will pass through
* Geometria, bk. iii, ed. Schooten, Amsterdam, 1659, p. 91.† Arithmetica Universalis, problem xlii, Ralphson’s (second) edition, London,
1728, p. 148; see also pp. 243–245.‡ I believe that this was first given by Clairaut, but I have mislaid my reference.
The construction occurs as an example in the Geometry of Conics, by C. Taylor,Cambridge, 1881, No. 308, p. 126.
§ Traite des sections coniques, Paris, 1865, art. 37, p. 36.
• Project • Gutenberg • #26839 •
212 THREE GEOMETRICAL PROBLEMS. [CH. VIII
one of the two points of trisection of the arc.
The Quadrature of the Circle*.
The object of the third of the classical problems was the deter-mination of a side of a square whose area should be equal to that ofa given circle.
The investigation, previous to the last two hundred years, of thisquestion was fruitful in discoveries of allied theorems, but in more re-cent times it has been abandoned by those who are able to realize whatis required. The history of this subject has been treated by compe-tent writers in such detail that I shall content myself with a very briefallusion to it.
Archimedes showed† (what possibly was known before) that theproblem is equivalent to finding the area of a right-angled trianglewhose sides are equal respectively to the perimeter of the circle andthe radius of the circle. Half the ratio of these lines is a number, usu-ally denoted by π.
That this number is incommensurable had been long suspected,and has been now demonstrated. The earliest analytical proof of it wasgiven by Lambert‡ in 1761; in 1803 Legendre§ extended the proof toshow that π2 was also incommensurable; and recently Lindemann‖ has
* See Montucla’s Histoire des Recherches sur la Quadrature du Cercle, edited byP.L. Lacroix, Paris, 1831; also various articles by A. De Morgan, and especiallyhis Budget of Paradoxes, London, 1872. A popular sketch of the subject hasbeen compiled by H. Schubert, Die Quadratur des Zirkels, Hamburg, 1889; andsince the publication of the earlier editions of these Recreations Prof. F. Rudio ofZurich has given an analysis of the arguments of Archimedes, Huygens, Lambert,and Legendre on the subject, with an introduction on the history of the problem,Leipzig, 1892.
† Archimedis Opera, ΚÔκλου µèτρησις, prop. i, ed. Torelli, pp. 203–205; ed.Heiberg, vol. i, pp. 258–261, vol. iii, pp. 269–277.
‡ Memoires de l’Academie de Berlin for 1761, Berlin, 1768, pp. 265–322.§ Legendre’s Geometry, Brewster’s translation, Edinburgh, 1824, pp. 239–245.‖ Ueber die Zahl π, Mathematische Annalen, Leipzig, 1882, vol. xx, pp. 213–
225. The proof leads to the conclusion that, if x is a root of a rational integralalgebraical equation, then ex cannot be rational: hence, if πi was the root ofsuch an equation, eπi could not be rational; but eπi is equal to −1, and thereforeis rational; hence πi cannot be the root of such an algebraical equation, andtherefore neither can π.
• Project • Gutenberg • #26839 •
CH. VIII] APPROXIMATIONS TO THE VALUE OF π. 213
shown that π cannot be the root of a rational algebraical equation.
An earlier attempt by James Gregory to give a geometrical demon-stration of this is worthy of notice. Gregory proved* that the ratioof the area of any arbitrary sector to that of the inscribed or circum-scribed polygons is not expressible by a finite number of algebraicalterms. Hence he inferred that the quadrature was impossible. Thiswas accepted by Montucla, but it is not conclusive, for it is conceiv-able that some particular sector might be squared, and this particularsector might be the whole circle.
In connection with Gregory’s proposition above cited, I may addthat Newton† proved that in any closed oval an arbitrary sectorbounded by the curve and two radii cannot be expressed in termsof the co-ordinates of the extremities of the arc by a finite number ofalgebraical terms. The argument is condensed and difficult to follow:the same reasoning would show that a closed oval curve cannot berepresented by an algebraical equation in polar co-ordinates. Fromthis proposition no conclusion as to the quadrature of the circle is tobe drawn, nor did Newton draw any. In the earlier editions of thiswork I expressed an opinion that the result presupposed a particulardefinition of the word oval, but on more careful reflection I think thatthe conclusion is valid without restriction.
With the aid of the quadratrix, or the conchoid, or the cissoid, thequadrature of the circle is easy, but the construction of those curvesassumes a knowledge of the value of π, and thus the question is begged.
I need hardly add that, if π represented merely the ratio of thecircumference of a circle to its diameter, the determination of its nu-merical value would have but slight interest. It is however a mereaccident that π is defined usually in that way, and it really representsa certain number which would enter into analysis from whatever sidethe subject was approached.
I recollect a distinguished professor explaining how different wouldbe the ordinary life of a race of beings born, as easily they might be,so that the fundamental processes of arithmetic, algebra and geometrywere different to those which seem to us so evident, but, he added, it isimpossible to conceive of a universe in which e and π should not exist.
* Vera Circuli et Hyperbolae Quadratura, Padua, 1668: this is reprinted in Huy-gens’s Opera Varia, Leyden, 1724, pp. 405–462.
† Principia, bk. i, section vi, lemma xxviii.
• Project • Gutenberg • #26839 •
214 THREE GEOMETRICAL PROBLEMS. [CH. VIII
I have quoted elsewhere an anecdote, which perhaps will bear rep-etition, that illustrates how little the usual definition of π suggests itsproperties. De Morgan was explaining to an actuary what was thechance that a certain proportion of some group of people would at theend of a given time be alive; and quoted the actuarial formula, involvingπ, which, in answer to a question, he explained stood for the ratio of thecircumference of a circle to its diameter. His acquaintance, who hadso far listened to the explanation with interest, interrupted him andexclaimed, “My dear friend, that must be a delusion, what can a circlehave to do with the number of people alive at the end of a given time?”In reality the fact that the ratio of the length of the circumference ofa circle to its diameter is the number denoted by π does not afford thebest analytical definition of π, and is only one of its properties.
The use of a single symbol to denote this number 3.14159 . . . seemsto have been introduced about the beginning of the eighteenth century.W. Jones* in 1706 represented it by π; a few years later† John Bernoullidenoted it by c; Euler in 1734 used p, and in 1736 used c; Chr. Goldbackin 1742 used π; and after the publication of Euler’s Analysis the symbolπ was generally employed.
The numerical value of π can be determined by either of two meth-ods with as close an approximation to the truth as is desired.
The first of these methods is geometrical. It consists in calculatingthe perimeters of polygons inscribed in and circ*mscribed about a cir-cle, and assuming that the circumference of the circle is intermediatebetween these perimeters‡. The approximation would be closer if theareas and not the perimeters were employed. The second and modernmethod rests on the determination of converging infinite series for π.
We may say that the π-calculators who used the first method re-garded π as equivalent to a geometrical ratio, but those who adoptedthe modern method treated it as the symbol for a certain number whichenters into numerous branches of mathematical analysis.
It may be interesting if I add here a list of some of the approxi-mations to the value of π given by various writers§. This will indicate
* Synopsis Palmariorum Matheseos, London, 1706, pp. 243, 263 et seq.† See notes by G. Enestrom in the Bibliotheca Mathematica, Stockholm, 1889,
vol. iii, p. 28; Ibid., 1890, vol. iv, p. 22.‡ The history of this method has been written by K.E.I. Selander, Historik ofver
Ludolphska Talet, Upsala, 1868.§ For the methods used in classical times and the results obtained, see the notices
• Project • Gutenberg • #26839 •
CH. VIII] APPROXIMATIONS TO THE VALUE OF π. 215
incidentally those who have studied the subject to the best advantage.
The ancient Egyptians* took 256/81 as the value of π, this is equalto 3.1605 . . . ; but the rougher approximation of 3 was used by theBabylonians† and by the Jews‡. It is not unlikely that these numberswere obtained empirically.
We come next to a long roll of Greek mathematicians who attackedthe problem. Whether the researches of the members of the IonianSchool, the Pythagoreans, Anaxagoras, Hippias, Antipho, and Brysoled to numerical approximations for the value of π is doubtful, and theirinvestigations need not detain us. The quadrature of certain lunes byHippocrates of Chios is ingenious and correct, but a value of π cannotbe thence deduced; and it seems likely that the later members of theAthenian School concentrated their efforts on other questions.
It is probable that Euclid§, the illustrious founder of the Alexand-rian School, was aware that π was greater than 3 and less than 4, buthe did not state the result explicitly.
The mathematical treatment of the subject began with Archi-medes, who proved that π is less than 31
7and greater than 310
71, that
is, it lies between 3.1428 . . . and 3.1408 . . . . He established‖ this byinscribing in a circle and circ*mscribing about it regular polygonsof 96 sides, then determining by geometry the perimeters of thesepolygons, and finally assuming that the circumference of the circlewas intermediate between these perimeters: this leads to the result
of their authors in M. Cantor’s Geschichte der Mathematik, Leipzig, vol. i, 1880.For medieval and modern approximations, see the article by A. De Morgan onthe Quadrature of the Circle in vol. xix of the Penny Cyclopaedia, London, 1841;with the additions given by B. de Haan in the Verhandelingen of Amsterdam,1858, vol. iv, p. 22: the conclusions were tabulated, corrected, and extended byDr J.W.L. Glaisher in the Messenger of Mathematics, Cambridge, 1873, vol. ii,pp. 119–128; and Ibid., 1874, vol. iii, pp. 27–46.
* Ein mathematisches Handbuch der alten Aegypter (i.e. the Rhind papyrus), byA. Eisenlohr, Leipzig, 1877, arts. 100–109, 117, 124.
† Oppert, Journal Asiatique, August, 1872, and October, 1874.‡ 1 Kings, ch. 7, ver. 23; 2 Chronicles, ch. 4, ver. 2.§ These results can be deduced from Euc. iv, 15, and iv, 8: see also book xii,
prop. 16.‖ Archimedis Opera, ΚÔκλου µèτρησις, prop. iii, ed. Torelli, Oxford, 1792, pp. 205–
216; ed. Heiberg, Leipzig, 1880, vol. i, pp. 263-271.
• Project • Gutenberg • #26839 •
216 THREE GEOMETRICAL PROBLEMS. [CH. VIII
6336/201714
< π < 14688/467312,1 from which he deduced the lim-
its given above. This method is equivalent to using the propositionsin θ < θ < tan θ, where θ = π/96: the values of sin θ and tan θ werededuced by Archimedes from those of sin 1
3π and tan 1
3π by repeated
bisections of the angle. With a polygon of n sides this process gives avalue of π correct to at least the integral part of (2 log n−1.19) places ofdecimals. The result given by Archimedes is correct to 2 places of dec-imals. His analysis leads to the conclusion that the perimeters of thesepolygons for a circle whose diameter is 4970 feet would lie between15610 feet and 15620 feet—actually it is about 15613 feet 9 inches.
Apollonius discussed these results, but his criticisms have been lost.Hero of Alexandria gave* the value 3, but he quoted† the result
22/7: possibly the former number was intended only for rough ap-proximations.
The only other Greek approximation that I need mention is thatgiven by Ptolemy‡, who asserted that π = 3◦8′30′′. This is equivalentto taking π = 3 + 8
60+ 30
3600= 3 17
120= 3.1416.
The Roman surveyors seem to have used 3, or sometimes 4, forrough calculations. For closer approximations they often employed 31
8
instead of 317, since the fractions then introduced are more convenient
in duodecimal arithmetic. On the other hand Gerbert§ recommendedthe use of 22/7.
Before coming to the medieval and modern European mathemati-cians it may be convenient to note the results arrived at in India andthe East.
Baudhayana‖ took 49/16 as the value of π.Arya-Bhata¶, circ. 530, gave 62832/20000, which is equal to 3.1416.
He showed that, if a is the side of a regular polygon of n sides inscribedin a circle of unit diameter, and if b is the side of a regular inscribedpolygon of 2n sides, then b2 = 1
2− 1
2(1 − a2)
12 . From the side of an
* Mensurae, ed. Hultsch, Berlin, 1864, p. 188.† Geometria, ed. Hultsch, Berlin, 1864, pp. 115, 136.‡ Almagest, bk. vi, chap. 7; ed. Halma, vol. i, p. 421.§ Œuvres de Gerbert, ed. Olleris, Clermont, 1867, p. 453.‖ The Sulvasutras by G. Thibaut, Asiatic Society of Bengal, 1875, arts. 26–28.¶ Lecons de calcul d’Aryabhata, by L. Rodet in the Journal Asiatique, 1879, se-
ries 7, vol. xiii, pp. 10, 21.
1. Inserted 14688/
• Project • Gutenberg • #26839 •
CH. VIII] APPROXIMATIONS TO THE VALUE OF π. 217
inscribed hexagon, he found successively the sides of polygons of 12,24, 48, 96, 192, and 384 sides. The perimeter of the last is given as equalto√
9.8694, from which his result was obtained by approximation.Brahmagupta*, circ. 650, gave
√10, which is equal to 3.1622 . . . .
He is said to have obtained this value by inscribing in a circle of unitdiameter regular polygons of 12, 24, 48, and 96 sides, and calculat-ing successively their perimeters, which he found to be
√9.65,
√9.81,√
9.86,√
9.87 respectively; and to have assumed that as the number ofsides is increased indefinitely the perimeter would approximate to
√10.
Bhaskara, circ. 1150, gave two approximations. One†—possiblycopied from Arya-Bhata, but said to have been calculated afresh byArchimedes’s method from the perimeters of regular polygons of 384sides—is 3927/1250, which is equal to 3.1416: the other‡ is 754
240, which
is equal to 3.1416, but it is uncertain whether this was not given onlyas an approximate value.
Among the Arabs the values 22/7,√
10, and 62832/20000 weregiven by Alkarisimi§, circ. 830; and no doubt were derived from Indiansources. He described the first as an approximate value, the second asused by geometricians, and the third as used by astronomers.
In Chinese works the values 3, 227, 157
50are said to occur: probably
the last two results were copied from the Arabs.Returning to European mathematicians, we have the following suc-
cessive approximations to the value of π: many of those prior to theeighteenth century having been calculated originally with the view ofdemonstrating the incorrectness of some alleged quadrature.
Leonardo of Pisa‖, in the thirteenth century, gave for π the value1440/4581
3which is equal to 3.1418 . . . . In the fifteenth century, Pur-
bach¶ gave or quoted the value 62832/20000, which is equal to 3.1416;Cusa believed that the accurate value was 3
4(√
3 +√
6) which is equalto 3.1423 . . . ; and, in 1464, Regiomontanus** is said to have given a
* Algebra. . . from Brahmegupta and Bhascara, trans. by H.T. Colebrooke, London,1817, chap. xii, art. 40, p. 308.
† Ibid., p. 87.‡ Ibid., p. 95.§ The Algebra of Mohammed ben Musa, ed. by F. Rosen, London, 1831, pp. 71–72.‖ Boncompagni’s Scritti di Leonardo, vol. ii (Practica Geometriae), Rome, 1862,
p. 90.¶ Appendix to the De Triangulis of Regiomontanus, Basle, 1541, p. 131.** In his correspondence with Cardinal Cusa, De Quadratura Circuli, Nuremberg,
• Project • Gutenberg • #26839 •
218 THREE GEOMETRICAL PROBLEMS. [CH. VIII
value equal to 3.14243.Vieta*, in 1579, showed that π was greater than 31415926535/1010,
and less than 31415926537/1010. This was deduced from the perimetersof the inscribed and circ*mscribed polygons of 6 × 216 sides, obtainedby repeated use of the formula 2 sin2 1
2θ = 1 − cos θ. He also gave† a
result equivalent to the formula
2
π=
√2
2
√(2 +
√2)
2
√{2 +
√(2 +
√2)}
2· · · .
The father of Adrian Metius‡, in 1585, gave 355/113, which is equalto 3.14159292 . . . , and is correct to 6 places of decimals. This was a cu-rious and lucky guess, for all that he proved was that π was intermediatebetween 377/120 and 333/106, whereon he jumped to the conclusionthat he should obtain the true fractional value by taking the mean ofthe numerators and the mean of the denominators of these fractions.
In 1593 Adrian Romanus§ calculated the perimeter of the inscribedregular polygon of 1073, 741824 (i.e. 230) sides, from which he deter-mined the value of π correct to 15 places of decimals.
L. van Ceulen devoted no inconsiderable part of his life to thesubject. In 1596‖ he gave the result to 20 places of decimals: this wascalculated by finding the perimeters of the inscribed and circ*mscribedregular polygons of 60 × 233 sides, obtained by the repeated use of atheorem of his discovery equivalent to the formula 1−cos A = 2 sin2 1
2A.
I possess a finely executed engraving of him of this date, with the resultprinted round a circle which is below his portrait. He died in 1610, andby his directions the result to 35 places of decimals (which was as far ashe had then calculated it) was engraved on his tombstone¶ in St Peter’s
1533, wherein he proved that Cusa’s result was wrong. I cannot quote the exactreference, but the figures are given by competent writers and I have no doubtare correct.
* Canon Mathematicus seu ad Triangula, Paris, 1579, pp. 56, 66: probably thiswork was printed for private circulation only, it is very rare.
† Vietae Opera, ed. Schooten, Leyden, 1646, p. 400.‡ Arithmeticae libri duo et Geometriae, by A. Metius, Leyden, 1626, pp. 88, 89.
[Probably issued originally in 1611.]§ Ideae Mathematicae, Antwerp, 1593: a rare work, which I have never been able
to consult.‖ Vanden Circkel, Delf, 1596, fol. 14, p. 1; or De Circulo, Leyden, 1619, p. 3.¶ The inscription is quoted by Prof. de Haan in the Messenger of Mathematics,
1874, vol. iii, p. 25.
• Project • Gutenberg • #26839 •
CH. VIII] APPROXIMATIONS TO THE VALUE OF π. 219
Church, Leyden. His posthumous arithmetic* contains the result to 32places; this was obtained by calculating the perimeter of a polygon, thenumber of whose sides is 262, i.e. 4, 611686, 018427, 387904. Van Ceulenalso compiled a table of the perimeters of various regular polygons.
Willebrord Snell†, in 1621, obtained from a polygon of 230 sides anapproximation to 34 places of decimals. This is less than the numbersgiven by van Ceulen, but Snell’s method was so superior that he ob-tained his 34 places by the use of a polygon from which van Ceulen hadobtained only 14 (or perhaps 16) places. Similarly, Snell obtained froma hexagon an approximation as correct as that for which Archimedeshad required a polygon of 96 sides, while from a polygon of 96 sides hedetermined the value of π correct to seven decimal places instead of thetwo places obtained by Archimedes. The reason is that Archimedes,having calculated the lengths of the sides of inscribed and circ*mscribedregular polygons of n sides, assumed that the length of 1/nth of theperimeter of the circle was intermediate between them; whereas Snellconstructed from the sides of these polygons two other lines which gavecloser limits for the corresponding arc. His method depends on the the-orem 3 sin θ/(2 + cos θ) < θ < (2 sin 1
3θ + tan 1
3θ), by the aid of which
a polygon of n sides gives a value of π correct to at least the integralpart of (4 log n − .2305) places of decimals, which is more than twicethe number given by the older rule. Snell’s proof of his theorem isincorrect, though the result is true.
Snell also added a table‡ of the perimeters of all regular inscribedand circ*mscribed polygons, the number of whose sides is 10×2n wheren is not greater than 19 and not less than 3. Most of these werequoted from van Ceulen, but some were recalculated. This list hasproved useful in refuting circle-squarers. A similar list was given byJames Gregory§.
In 1630 Grienberger‖, by the aid of Snell’s theorem, carried the
* De Arithmetische en Geometrische Fondamenten, Leyden, 1615, p. 163; or p. 144of the Latin translation by W. Snell, published at Leyden in 1615 under thetitle Fundamenta Arithmetica et Geometrica. This was reissued, together witha Latin translation of the Vanden Circkel, in 1619, under the title De Circulo;in which see pp. 3, 29–32, 92.
† Cyclometricus, Leyden, 1621, p. 55.‡ It is quoted by Montucla, ed. 1831, p. 70.§ Vera Circuli et Hyperbolae Quadratura, prop. 29, quoted by Huygens, Opera
Varia, Leyden, 1724, p. 447.‖ Elementa Trigonometrica, Rome, 1630, end of preface.
• Project • Gutenberg • #26839 •
220 THREE GEOMETRICAL PROBLEMS. [CH. VIII
approximation to 39 places of decimals. He was the last mathematicianwho adopted the classical method of finding the perimeters of inscribedand circ*mscribed polygons. Closer approximations serve no usefulpurpose. Proofs of the theorems used by Snell and other calculators inapplying this method were given by Huygens in a work* which may betaken as closing the history of this method.
In 1656 Wallis† proved that
π
2=
2 · 2 · 4 · 4 · 6 · 6 · · ·1 · 3 · 3 · 5 · 5 · 7 · 7 · · ·
,
and quoted a proposition given a few years earlier by ViscountBrouncker to the effect that
π
4= 1 +
12
2 +
32
2 +
52
2 +. . . ,
but neither of these theorems was used to any large extent for cal-culation.
Subsequent calculators have relied on converging infinite series, amethod that was hardly practicable prior to the invention of the calcu-lus, though Descartes‡ had indicated a geometrical process which wasequivalent to the use of such a series. The employment of infinite serieswas proposed by James Gregory§, who established the theorem thatθ = tan θ − 1
3tan3 θ + 1
5tan5 θ − · · · , the result being true only if θ lies
between −14π and 1
4π.
The first mathematician to make use of Gregory’s series for ob-taining an approximation to the value of π was Abraham Sharp‖, who,
* De Circula Magnitudine Inventa, 1654; Opera Varia, pp. 351–387. The proofsare given in G. Pirie’s Geometrical Methods of Approximating the Value of π,London, 1877, pp. 21–23.
† Arithmetica Infinitorum, Oxford, 1656, prop. 191. An analysis of the investiga-tion by Wallis was given by Cayley, Quarterly Journal of mathematics, 1889,vol. xxiii, pp. 165–169.
‡ See Euler’s paper in the Novi Commentarii Academiae Scientiarum, St Peters-burg, 1763, vol. viii, pp. 157–168.
§ See the letter to Collins, dated Feb. 15, 1671, printed in the Commercium Epis-tolicum, London, 1712, p. 25, and in the Macclesfield Collection, Correspondenceof Scientific Men of the Seventeenth Century, Oxford, 1841, vol. ii, p. 216.
‖ See Life of A. Sharp by W. Cudworth, London, 1889, p. 170. Sharp’s work isgiven in one of the preliminary discourses (p. 53 et seq.) prefixed to H. Sherwin’sMathematical Tables. The tables were issued at London in 1705: probably thediscourses were issued at the same time, though the earliest copies I have seenwere printed in 1717.
• Project • Gutenberg • #26839 •
CH. VIII] APPROXIMATIONS TO THE VALUE OF π. 221
in 1699, on the suggestion of Halley, determined it to 72 places ofdecimals (71 correct). He obtained this value by putting θ = 1
6π in
Gregory’s series.Machin*, earlier than 1706, gave the result to 100 places (all cor-
rect). He calculated it by the formula
14π = 4 tan−1 1
5− tan−1 1
239.
De Lagny†, in 1719, gave the result to 127 places of decimals (112correct), calculating it by putting θ = 1
6π in Gregory’s series.
Hutton‡, in 1776, and Euler§, in 1779, suggested the use of theformulae 1
4π = tan−1 1
2+ tan−1 1
3or 1
4π = 5 tan−1 1
7+ 2 tan−1 3
79, but
neither carried the approximation as far as had been done previously.Vega, in 1789‖, gave the value of π to 143 places of decimals (126
correct); and, in 1794¶, to 140 places (136 correct).Towards the end of the last century Baron Zach saw in the Radcliffe
Library, Oxford, a manuscript by an unknown author which gives thevalue of π to 154 places of decimals (152 correct).
In 1841 Rutherford** calculated it to 208 places of decimals (152correct), using the formula 1
4π = 4 tan−1 1
5− tan−1 1
70+ tan−1 1
99.
In 1844 Dase†† calculated it to 205 places of decimals (200 correct),using the formula 1
4π = tan−1 1
2+ tan−1 1
5+ tan−1 1
8.
In 1847 Clausen‡‡ carried the approximation to 250 places ofdecimals (248 correct), calculating it independently by the formulae14π = 2 tan−1 1
3+ tan−1 1
7and 1
4π = 4 tan−1 1
5− tan−1 1
239.
In 1853 Rutherford§§ carried his former approximation to 440 placesof decimals (all correct), and William Shanks prolonged the approxima-tion to 530 places. In the same year Shanks published an approximation
* W. Jones’s Synopsis Palmariorum, London, 1706, p. 243; and Maseres, Scrip-tores Logarithmici, London, 1796, vol. iii, pp. vii–ix, 155–164.
† Histoire de l’Academie for 1719, Paris, 1721, p. 144.‡ Philosophical Transactions, 1776, vol. lxvi, pp. 476–492.§ Nova Acta Academiae Scientiarum Petropolitanae for 1793, St Petersburg, 1798,
vol. xi, pp. 133–149: the memoir was read in 1779.‖ Nova Acta Academiae Scientiarum Petropolitanae for 1790, St Petersburg, 1795,
vol. ix, p. 41.¶ Thesaurus Logarithmorum (logarithmisch-trigonometrischer Tafeln), Leipzig,
1794, p. 633.** Philosophical Transactions, 1841, p. 283.†† Crelle’s Journal, 1844, vol. xxvii, p. 198.‡‡ Schumacher, Astronomische Nachrichten, vol. xxv, col. 207.§§ Proceedings of the Royal Society, Jan. 20, 1853, vol. vi, pp. 273-275.
• Project • Gutenberg • #26839 •
222 THREE GEOMETRICAL PROBLEMS. [CH. VIII
to 607 places*: and in 1873 he carried the approximation to 707 placesof decimals†. These were calculated from Machin’s formula.
In 1853 Richter, presumably in ignorance of what had been done inEngland, found the value of π to 333 places‡ of decimals (330 correct); in1854 he carried the approximation to 400 places§; and in 1855 carriedit to 500 places‖.
Of the series and formulae by which these approximations havebeen calculated, those used by Machin and Dase are perhaps the easiestto employ. Other series which converge rapidly are the following,
π
6=
1
2+
1
2· 1
3 · 23+
1 · 32 · 4
· 1
5 · 25+ · · · ,
and
π
4= 2 + 22 tan−1 1
28+ tan−1 1
443− 5 tan−1 1
1393− 10 tan−1 1
11018,
the latter of these is due to Mr Escott¶.As to those writers who believe that they have squared the circle
their number is legion and, in most cases, their ignorance profound, buttheir attempts are not worth discussing here. “Only prove to me that itis impossible,” said one of them, “and I will set about it immediately”;and doubtless the statement that the problem is insoluble has attractedmuch attention to it.
Among the geometrical ways of approximating to the truth thefollowing is one of the simplest. Inscribe in the given circle a square,and to three times the diameter of the circle add a fifth of a side of thesquare, the result will differ from the circumference of the circle by lessthan one-seventeen-thousandth part of it.
An approximate value of π has been obtained experimentally bythe theory of probability. On a plane a number of equidistant parallelstraight lines, distance apart a, are ruled; and a stick of length l, which
* Contributions to Mathematics, W. Shanks, London, 1853, pp. 86, 87.† Proceedings of the Royal Society, 1872–3, vol. xxi, p. 318; 1873–4, vol. xxii,
p. 45.‡ Grunert’s Archiv, vol. xxi, p. 119.§ Ibid., vol. xxiii, p. 476: the approximation given in vol. xxii, p. 473, is correct
only to 330 places.‖ Ibid., vol. xxv, p. 472; and Elbingen Anzeigen, No. 85.¶ L’Intermediaire des mathematiciens, Paris, Dec. 1896, vol. iii, p. 276.
• Project • Gutenberg • #26839 •
CH. VIII] APPROXIMATIONS TO THE VALUE OF π. 223
is less than a, is dropped on to the plane. The probability that it willfall so as to lie across one of the lines is 2l/πa. If the experiment isrepeated many hundreds of times, the ratio of the number of favourablecases to the whole number of experiments will be very nearly equal tothis fraction: hence the value of π can be found. In 1855 Mr A. Smith*
of Aberdeen made 3204 trials, and deduced π = 3.1553. A pupil of Prof.De Morgan*, from 600 trials, deduced π = 3.137. In 1864 Captain Fox†
made 1120 trials with some additional precautions, and obtained asthe mean value π = 3.1419.
Other similar methods of approximating to the value of π have beenindicated. For instance, it is known that if two numbers are writtendown at random, the probability that they will be prime to each otheris 6/π2. Thus, in one case‡ where each of 50 students wrote down 5pairs of numbers at random, 154 of the pairs were found to consist ofnumbers prime to each other. This gives 6/π2 = 154/250, from whichwe get π = 3.12.
* A. De Morgan, Budget of Paradoxes, London, 1872, pp. 171, 172 [quoted froman article by De Morgan published in 1861].
† Messenger of Mathematics, Cambridge, 1873, vol. ii, pp. 113, 114.‡ Note on π by R. Chartres. Philosophical Magazine, London, series 6, vol. xxxix,
March, 1904, p. 315.
• Project • Gutenberg • #26839 •
CHAPTER IX.
MERSENNE’S NUMBERS.
One of the unsolved riddles of higher arithmetic, to which I havealluded in Chapter I, is the discovery of the method by which Mersenneor his contemporaries determined values of p which make a number ofthe form 2p − 1 a prime. It is convenient to describe such primesas Mersenne’s Numbers . In this chapter, for shortness, I use N todenote a number of the form 2p − 1. In a memoir in the Messenger ofMathematics in 1891 I gave a brief sketch of the history of the problem.I here repeat the facts in somewhat more detail, and add a sketch ofmethods used in attacking the problem.
Mersenne’s enunciation of the results associated with his name isin the preface to his Cogitata*. The passage is as follows:
“Vbi fuerit operae pretium aduertere xxviii numeros a Petro Bungopro perfectis exhibitos, capite xxviii, libri de Numeris, non esse omnesPerfectos, quippe 20 sunt imperfecti, adeovt [adeunt?] solos octo perfectoshabeat. . . qui sunt e regione tabulae Bungi, 1, 2, 3, 4, 8, 10, 12, et 29:quique soli perfecti sunt, vt qui Bungum habuerint, errori medicinamfaciant.
Porro numeri perfecti adeo rari sunt, vt vndecim dumtaxat potuerinthactenus inueniri: hoc est, alii tres a Bongianis differentes: neque enimvllus est alius perfectus ab illis octo, nisi superes exponentem numerum62, progressionis duplae ab 1 incipientis. Nonus enim perfectus est potes-tas exponentis 68 minus 1. Decimus, potestas exponentis 128, minus 1.Vndecimus denique, potestas 258, minus 1, hoc est potestas 257, vnitatedecurtata, multiplicata per potestatem 256.
* Cogitata Physico-Mathematica, Paris, 1644, praefatio generalis, article 19.
224
• Project • Gutenberg • #26839 •
CH. IX] MERSENNE’S NUMBERS. 225
Qui vndecim alios repererit, nouerit se analysim omnem, quae fuerithactenus, superasse: memineritque interea nullum esse perfectum a 17000potestate ad 32000; & nullum potestatum interuallum tantum assignariposse, quin detur illud absque perfectis. Verbi gratia, si fuerit exponens1050000, nullus erit numerus progressionis duplae vsque ad 2090000, quiperfectis numeris seruiat, hoc est qui minor vnitate, primus existat.
Vnde clarum est quam rari sint perfecti numeri, & quam merito virisperfectis comparentur; esseque vnam ex maximis totius Matheseos diffi-cultatibus, praescriptam numerorum perfectorum multitudinum exhibere;quemadmodum & agnoscere num dati numeri 15, aut 20 caracteribus con-stantes, sint primi necne, cum nequidem saeculum integrum huic examini,quocumque modo hactenus cognito, sufficiat.
It is evident that, if p is not a prime, then N is composite, and twoor more of its factors can be written down by inspection. Hence we mayconfine ourselves to prime values of p. Mersenne, in effect, asserted thatthe only values of p, not greater than 257, which make N a prime, are1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257: I assume that the number 67is a misprint for 61. With this correction we have no reason to doubtthe truth of the statement, but it has not been definitely established.
There are 56 primes not greater than 257. The determination ofthe prime or composite character of N for the 9 cases when p is lessthan 20 presents no difficulty: in only one of them is N composite.For 2 of the remaining 47 cases (namely, when p = 23 and 37) thedecomposition of N had been given by Fermat. For 9 of them (namely,when p = 29, 43, 73, 83, 131, 179, 191, 239, 251) the factors of N weregiven by Euler. He also proved that N was prime when p = 31. Planagave the factors of N when p = 41. Landry and Le Lasseur discoveredthe factors in 10 cases (namely, when p = 47, 53, 59, 79, 97, 113, 151,211, 223, and 233), but their analysis has not been published. Seelhoffshowed that N was prime when p = 61, Cunningham gave the factorswhen p = 197, and Cole the factors when p = 67. Statements havebeen made that the composite character of N when p = 89, and itsprime character when p = 127 have been proved, but the proofs havenot been published or verified.
Thus there are 21 values of p for which Mersenne’s statement stillawaits verification. These are 71, 89, 101, 103, 107, 109, 127, 137, 139,149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, 257. For thesevalues N is (according to Mersenne) prime when p = 127, and 257, andis composite for the other values, but as explained above it is probablethat the character of N is known when p = 89 and 127.
• Project • Gutenberg • #26839 •
226 MERSENNE’S NUMBERS. [CH. IX
To put the matter in another way. According to Mersenne’s state-ment (corrected by the substitution of 61 for 67), 44 of the 56 primesless than 258 make N composite and the remaining 12 primes make Nprime. In 25 out of the 44 cases in which N is said to be composite weknow its factors, and in 19 cases the statement is still unverified. In 10out of the 12 cases in which he said that N was prime his statementhas been verified, and in 2 cases it is still unverified.
From the wording of the last clause in the above quotation it hasbeen conjectured that the result had been communicated to Mersenne,and that he published it without being aware of how it was proved. Initself this seems probable. He was a good mathematician, but not anexceptional genius. It would be strange if he established a propositionwhich has baffled Euler, Lagrange, Legendre, Gauss, Jacobi, and othermathematicians of the first rank; but if the proposition is due to Fer-mat, with whom Mersenne was in constant correspondence, the case isaltered, and not only is the absence of a demonstration explained, butwe cannot be sure that we have attacked the problem on the best lines.
The known results as to the prime or composite character of N ,and in the latter case its smallest factor, are given in the table on thefacing page. The cases that remain as yet unverified are marked withan asterisk.
Before describing the methods used for attacking the problem itwill be convenient to state in more detail when and by whom theseresults were established.
The factors (if any) of such values of N as are less than a millioncan be verified easily: they have been known for a long time, and Ineed not allude to them in detail.
The factors of N when p = 11, 23, and 37 had been indicated byFermat*, some four years prior to the publication of Mersenne’s work,in a letter dated October 18, 1640. The passage is as follows:
En la progression double, si d’un nombre quarre, generalement parlant,vous otez 2 ou 8 ou 32 &c., les nombres premiers moindres de l’unitequ’un multiple du quaternaire, qui mesureront le reste, feront l’effet requis.Comme de 25, qui est un quarre, otez 2; le reste 23 mesurera la 11e
puissance −1; otez 2 de 49, le reste 47 mesurera la 23e puissance −1.Otez 2 de 225, le reste 223 mesurera la 37e puissance −1, &c.
* Oeuvres de Fermat, Paris, vol. ii, 1894, p. 210; or Opera Mathematica, Toulouse,1679, p. 164; or Brassinne’s Precis, Paris, 1853, p. 144.
• Project • Gutenberg • #26839 •
CH. IX] MERSENNE’S NUMBERS. 227
p Value of N = 2p − 11 1 prime2 3 prime3 7 prime5 31 prime7 127 prime
11 2047 = 23× 89 composite13 8191 prime17 131071 prime19 524287 prime23 8388607 = 47× 178481 composite Fermat29 536870911 = 233× 1103× 2089 composite Euler31 2147483647 prime Euler37 137438953471 = 223× 616318177 composite Fermat41 2199023255551 = 13367× 164511353 composite Plana43 8796093022207 = 431× 9719× 2099863 composite Euler47 2351× 4513× 13264529 composite Landry53 6361× 69431× 20394401 composite Landry59 179951× 3203431780337 composite Landry61 2305843009213693951 prime Seelhoff67 ≡ 0 (193707721) composite Cole71 2361183241434822606847 ∗73 ≡ 0 (439) composite Euler79 ≡ 0 (2687) composite Le Lasseur83 ≡ 0 (167) composite Euler89 618970019642690137449562111 ∗97 ≡ 0 (11447) composite Le Lasseur
101 2535301200456458802993406410751 ∗103 10141204801825835211973625643007 ∗107 162259276829213363391578010288127 ∗109 649037107316853453566312041152511 ∗113 ≡ 0 (3391) composite Le Lasseur127 170141183460469231731687303715884105727 ∗131 ≡ 0 (263) composite Euler137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗151 ≡ 0 (18121) composite Le Lasseur157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗163 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗
• Project • Gutenberg • #26839 •
228 MERSENNE’S NUMBERS. [CH. IX
p Value of N = 2p − 1173 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗179 ≡ 0 (359) composite Euler181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗191 ≡ 0 (383) composite Euler193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗197 ≡ 0 (7487) composite Cunningham199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗211 ≡ 0 (15193) composite Le Lasseur223 ≡ 0 (18287) composite Le Lasseur227 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗233 ≡ 0 (1399) composite Le Lasseur239 ≡ 0 (479) composite Euler241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗251 ≡ 0 (503) composite Euler257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗
The factors of N when p = 29, 43, and 73 were given by Euler*
in 1732. The fact that N is composite for the values p = 83, 131,179, 191, 239, and 251 follows from a proposition enunciated, at thesame time, by Euler to the effect that, if 4n + 3 and 8n + 7 are primes,then 24n+3 − 1 ≡ 0 (mod 8n + 7). This was proved by Lagrange† inhis classical memoir of 1775. The proposition also covers the cases ofp = 11 and p = 23. This is the only general theorem on the subjectwhich is yet established.
The fact that N is prime when p = 31 was proved by Euler‡ in1771. Fermat had asserted, in the letter mentioned above, that theonly possible prime factors of 2p ± 1, when p was prime, were of theform np + 1, where n is an integer. This was proved by Euler§ in 1748,who added that, since 2p ± 1 is odd, every factor of it must be odd,and therefore if p is odd n must be even. But if p is a given numberwe can define n much more closely, and Euler showed that the prime
* Commentarii Academiae Scientiarum Petropolitanae, 1738, vol. vi, p. 103; orCommentationes Arithmeticae Collectae, vol. i, p. 2.
† Nouveaux Memoires de l’Academie des Sciences de Berlin, 1775, pp. 323–356.‡ Histoire de l’Academie des Sciences for 1772, Berlin, 1774, p. 36. See also
Legendre, Theorie des Nombres, third edition, Paris, 1830, vol. i, pp. 222–229.§ Novi Commentarii Academiae Scientiarum Petropolitanae, vol. i, p. 20; or Com-
mentationes Arithmeticae Collectae, St Petersburg, 1849, vol. i, pp. 55, 56.
• Project • Gutenberg • #26839 •
CH. IX] MERSENNE’S NUMBERS. 229
factors (if any) of 231 − 1 were necessarily primes of the form 248n + 1or 248n+63; also they must be less than
√231 − 1, that is, than 46339.
Hence it is necessary to try only forty divisors to see if 231 − 1 is aprime or composite.
The factors of N when p = 41 were given by Plana* in 1859. Heshowed that the prime factors (if any) are primes of the form 328n+1 or328n+247, and lie between 1231 and
√241 − 1, that is, 1048573. Hence
it is necessary to try only 513 divisors to see if 241 − 1 is composite:the seventeenth of these divisors gives the required factors. This isthe same method of attacking the problem which was used by Eulerin 1771, but it would be very laborious to employ it for values of pgreater than 41. Plana† added the forms of the prime divisors of N ,if p is not greater than 101.
That N is prime when p = 127 seems to have been verified byLucas‡ in 1876 and 1877. The demonstration has not been published.
The discovery of the factors of N for the values p = 47, 53, and 59is due apparently to the late F. Landry, who established theorems onthe factors (if any) of numbers of certain forms. Instead of publishinghis results he issued a challenge to all mathematicians to solve thegeneral problem. This is contained in a rare pamphlet published atParis in 1867, of which I possess a copy, in which the factors of certainnumbers are given, and on page 8 of which it is implied that he hadobtained the factors of 2p − 1 when p = 47, 53, and 59. He seemsto have communicated his results to Lucas, who quoted them in thememoir cited below§.
The factors of N when p = 79 and 113 were given first byLe Lasseur, and were quoted by Lucas in the same memoir§.
A factor of N when p = 233 was discovered later by Le Lasseur,and was quoted by Lucas in 1882‖.
* G.A.A. Plana, Memorie della Reale Accademia delle Scienze di Torino, Series 2,vol. xx, 1863, p. 130.
† Ibid., p. 137.‡ Sur la Theorie des Nombres Premiers, Turin, 1876, p. 11; and Recherches sur
les Ouvrages de Leonard de Pise, Rome, 1877, p. 26, quoted by Lieut.-ColonelA.J.C. Cunningham, Proceedings of the London Mathematical Society, Nov. 14,1895, vol. xxvii, p. 54.
§ American Journal of Mathematics, 1878, vol. i, p. 236.‖ Recreations, 1882–3, vol. i, p. 241.
• Project • Gutenberg • #26839 •
230 MERSENNE’S NUMBERS. [CH. IX
The factors of N when p = 97, 151, 211, and 223 were determinedsubsequently by Le Lasseur, and were quoted by Lucas* in 1883.
That N is prime when p = 61 had been conjectured by Landry andin 1886 a demonstration was offered by Seelhoff†. His demonstration isopen to criticism, but the fact has been verified by others‡, and maybe accepted as proved.
That N is composite when p = 89 seems to have been verifiedby Lucas§ in 1891, but the demonstration has not been published, norhave the actual factors been discovered.
That 7487 is a factor of N when p = 197 was shown by A.J.C. Cun-ningham in 1895‖.
That N is not prime when p = 67 seems to have been verified byLucas¶ in 1876 and 1877. The composite nature of1 N = 2p − 1 whenp = 67 was confirmed by E. Fauquembergue**, and was also implied byLucas†† in 1891. The factors were given by F.N. Cole‡‡ in 1903.
Bickmore in the memoir§§ cited below showed that 1433 is anotherfactor of N if p = 179; and that 1913 and 5737 are other factors ofN if p = 239.
I turn next to consider the methods by which these results canbe obtained. It is impossible to believe that the statement made by
* Recreations, vol. ii, p. 230.† P.H.H. Seelhoff, Zeitschrift fur Mathematik und Physik, 1886, vol. xxxi, p. 178.‡ See Weber-Wellstein, Encyclopaedie der Elementar-Mathematik, p. 48; and
F.N. Cole, Bulletin of the American Mathematical Society, December, 1903,p. 136.
§ Theorie des Nombres, Paris, 1891, p. 376.‖ Proceedings of the London Mathematical Society, March 14, 1895, vol. xxvi,
p. 261.¶ Sur la Theorie des Nombres Premiers, Turin, 1876, p. 11, quoted by Lieut-
Colonel A.J.C. Cunningham, Proceedings of the London Mathematical Society,Nov. 14, 1895, vol. xxvii, p. 54, and Recherches sur les Ouvrages de Leonardde Pise, Rome, 1877, p. 26.
** L’Intermediaire des mathematiciens, Paris, Sept. 1894, vol. i, p. 148.†† Theorie des Nombres, Paris, 1891, p. 376.‡‡ On the Factoring of Large Numbers, Bulletin of the American Mathematical
Society, December, 1903, pp. 134–137.§§ C.E. Bickmore, Messenger of Mathematics, Cambridge, 1895, vol. xxv, p. 19.
1. Corrected: originally N 2p = 1
• Project • Gutenberg • #26839 •
CH. IX] MERSENNE’S NUMBERS. 231
Mersenne rested on an empirical conjecture, but the riddle as to howit was discovered is still, after nearly 250 years, unsolved.
I cannot offer any solution of the riddle. But it may be interestingto indicate some ways which have been used in attacking the problem.The object is to find a prime divisor q (other than N and 1) of a numberN when N is of the form 2p−1 and p is a prime. It can be easily shownthat q must be of the form 2pt + 1. Also q must be of one of the forms8i ± 1: for N is of the form 2A2 − B2, where A is even and B odd,hence* any factor of it must be of the form 2a2− b2, that is, of the form8i±1, and 2 must be a quadratic residue of q. The theory of residues is,however, of but little use in finding factors of the cases that still awaitsolution, though the possibility some day of finding a complete series ofsolutions by properties of residues must not be neglected†. Our presentknowledge of the means of factorizing N has been summed up in thestatement‡ that a prime factor of the form 2pt+1 can be found directlyby rules due to Legendre, Gauss, and Jacobi, when t = 1, 3, 4, 8, or12; and that a factor of the form 2ptt′ + 1 can be found indirectly bya method due to Bickmore when t = 1, 3, 4, 8, or 12, and t′ is an oddinteger greater than 3. But this only indicates how little has yet beendone towards finding a general solution of the problem.
First. There is the simple but crude method of trying all possibleprime divisors q which are of the form 2pt + 1 as well as of one ofthe forms 8i ± 1.
The chief known results for the smaller factors may be summarizedby saying that a prime of this form will divide N when t = 1, if p = 11,23, 83, 131, 179, 191, 239, or 251; when t = 3, if p = 37, 73, or 233;when t = 5, if p = 43; when t = 15, if p = 113; when t = 17, if p = 79;when t = 19, if p = 29, or 197; when t = 25, if p = 47; when t = 41,if p = 223; when t = 59, if p = 97; when t = 163, if p = 41; whent = 1525, if p = 59; when t = 4, if p = 11, 29, 179, or 239; when t = 8,if p = 11; when t = 12, if p = 239; when t = 36, if p = 29, or 211; when
* Legendre, Theorie des Nombres, third edition, Paris, 1830, vol. i, § 143. In thecase of Mersenne’s numbers, B = b = 1.
† For methods of finding the residue indices of 2 see Bickmore, Messenger of Math-ematics, May, 1895, vol. xxv, pp. 15–21; see also Lieut-Colonel A.J.C. Cunning-ham on 2 as a 16-ic residue, Proceedings of the London Mathematical Society,1895–6, vol. xxvii, pp. 85–122.
‡ Transactions of the British Association for the Advancement of Science (IpswichMeeting), 1895, p. 614.
• Project • Gutenberg • #26839 •
232 MERSENNE’S NUMBERS. [CH. IX
t = 60, if p = 53, or 151; and when t = 1445580, if p = 67.
Of the 25 cases in which we know that Mersenne’s statement of thecomposite character of N is correct all save 3 can be easily verified bytrial in this way. For neglecting all values of t not exceeding, say, 60which make q either composite or not of one of the forms 8i±1 we havein each case only some 20 or so divisors to try. Of the 3 other cases inwhich Mersenne’s statement of the composite character of N has beenverified, one verification (p = 41) is due to Plana, who frankly confessedthat the result was reached “par un heureux hasard”; a second is dueto Landry (p = 59), who did not explain how he obtained the factors;and the third is due to Cole (p = 67), who established it by the use ofquadratic residues of N , involving laborious numerical work.
Of the 10 cases in which we know that Mersenne’s statement ofthe prime character of N is correct all save one may be verified bytrial in this way, for the number of possible factors is not large. Theexception is the case where p equals 61, which Seelhoff and others haveshown to be prime.
Thus practically we may say that simple empirical trials would atonce lead us to all the conclusions known except in the case of p = 41due to Plana, of p = 59 due to Landry, of p = 61 due to Seelhoff, and ofp = 67 due to Cole. In fact, save for these four results the conclusionsof all mathematicians to date could be obtained by anyone by a fewhours’ arithmetical work.
As p increases the number of factors to be tried increases so fastthat, if p is large, it would be practically impossible to apply the test toobtain large factors. This is an important point, for it has been assertedthat in the cases still awaiting verification there are no factors less than50, 000. Hence, we may take it as reasonably certain that this cannothave been the method by which the result was originally obtained; nor,as here enunciated, is it likely to give many factors not yet known. Ofcourse it is possible there may be ways by which the number of possiblevalues of t might be further limited, and if we could find them we mightthus diminish the number of possible factors to be tried, but it will beobserved that the values of N which still await verification are verylarge, for instance, when p = 257, N contains no less than 78 digits.
It is hardly necessary to add that if q is known and is not verylarge we can determine whether or not it is a factor of N without thelabour of performing the division.
• Project • Gutenberg • #26839 •
CH. IX] MERSENNE’S NUMBERS. 233
For instance, if we want to verify that q = 13367 is a factor of Nwhen p = 41, we proceed thus. Take the power of 2 nearest to q or toits square-root. We have to the modulus q
214 = 16384 ≡ 3017 ≡ 7× 431 ,∴ 228 ≡ 49(−1377) ≡ −638 ,∴ 227 ≡ −319 ,∴ 214+27 ≡ (3017)(−319) ≡ 1 ,∴ 241 ≡ 1 .
Second. We can proceed by reducing the problem to the solutionof an indeterminate equation.
It is clear that we can obtain a factor of N if we can express it asthe difference of the squares, or more generally of the nth powers, oftwo integers u and v. Further, if we can express a multiple of N , saymN , in this form, we can find a factor of mN and (with certain obviouslimitations as to the value of m) this will lead to a factor of N . It maybe also added that if m can be found so that N/m is expressible as acontinued fraction of a certain form, a certain continuant* defined bythe form of the continued fraction is a factor of N .
Since N can always be expressed as the difference of two squares,this method seems a natural one by which to attack the problem. Ifwe put
N = n2 + a = (n + b)2 − (b2 + 2bn− a),
we can make use of the known forms of u and v, and thus obtain anindeterminate equation between two variables x and y of the form
x2 = (2py + H)2 − 4(K − y)
where H and K are numbers which can be easily calculated. Integralvalues of x and y where y < K will determine values of u and v, andthus give factors of N .
* See J.G. Birch in the Messenger of Mathematics, August, 1902, vol. xxii, pp. 52–55.
• Project • Gutenberg • #26839 •
234 MERSENNE’S NUMBERS. [CH. IX
We can also attack the problem by indeterminate equations in an-other way. For the factors must be of the form 2pt+1 and 8ps+1, hence
(2pt + 1)(8ps + 1) = N ,
= 2p − 1 ,
= 2(2p−1 − 1) + 1 ,
∴ 4s + t + 8pst = (2p−1 − 1)/p ,
= (say)α + 8pβ .
Hence 4s + t = α + 8px, and st = β − x ,
where x ≯ β and t is odd. These results again lead to an indetermi-nate equation.
But, in both cases, unless p is small, the resulting equations areintractable.
Third. A not uncommon method of attacking problems such asthis, dealing with the factorization of large numbers, is through thetheory of quadratic forms*. At best this is a difficult method to use,and we have no reason to think that it would have been employed by amathematician of the seventeenth century. I here content myself withalluding to it.
Fourth. There is yet another way in which the problem might beattacked. The problem will be solved if we can find an odd prime q sothat to it as modulus 2p+y ≡ z, and 2y ≡ z, where y and z may haveany values we like to choose. If such values of q, y, and z can be found,we have 2y(2p − 1) ≡ 0. Therefore 2p = 1, that is, q is a divisor of N .
For example, to the modulus 23, we have
28 ≡ 3 ,
216 ≡ 32 .
Also 25 ≡ 32 .
Therefore 216 − 25 ≡ 0 ,
∴ 211 − 1 ≡ 0 .
Without going further into the matter we may say that the a prioridetermination of the values q, y, and z introduces us to an almost
* For a sketch of this see G.B. Mathews, Theory of Numbers, part 1, Cambridge,1891, pp. 261-271. See also F.N. Cole’s paper, On the Factoring of Large Num-bers, Bulletin of the American Mathematical Society, December, 1903, pp. 134-137; and Quadratic Partitions by A.J.C. Cunningham, London, 1904.
• Project • Gutenberg • #26839 •
CH. IX] MERSENNE’S NUMBERS. 235
untrodden field. It is just possible (though I should suppose unlikely)that the key to the riddle is to be found on methods of finding q, y, z,to satisfy the above conditions. For instance, if we could say what wasthe remainder when 2x was divided by a prime q of the form 2pt + 1,and if the remainders were the same when x = u and x = v, then tothe modulus q we should have, 2u ≡ 2v, and therefore 2u−v ≡ 1.
It should however be noted that Jacobi’s Canon Arithmeticusand the similar canon drawn up by Cunningham would, if carried farenough, enable us to solve the problem by this method. Cunningham’sCanon gives the solution of the congruence 2x ≡ R for all prime moduliless than 1000, but it is of no use in determining factors of N larger than1000. It is however possible that if R or q have certain forms such acanon might be constructed, and thus lead to a solution of the problem.
Fifth. It is noteworthy that the odd values of p specified byMersenne are primes of one of the four forms 2q ± 1 or 2q ± 3, butit is not the case that all primes of these forms make N a prime, forinstance, N is composite if p = 23 + 3 = 11 or if p = 25 − 3 = 29.
This fact has suggested to more than one mathematician the possi-bility that some test as to the prime or composite character of N whenp is of one of these forms may be discoverable. Of course this is merelya conjecture. There is however this to say for it, that we know thatFermat* had paid attention to numbers of this form.
Sixth. The number N when expressed in the binary scale, consistsof 1 repeated p times. This has suggested whether the work connectedwith the determination of factors of N might not with advantage beexpressed in the binary scale. A method based on the use of propertiesof this scale has been indicated by G. de Longchamps†, but as theregiven it would be unlikely to lead to the discovery of large divisors. Iam, however, inclined to think that greater advantages would be gainedby working in a scale whose radix was 4p or may-be 8p—the resultingnumbers being then expressed by a reasonably small number of digits.In fact when expressed in the latter scale in only one out of the 25 casesin which the factors of N are known does the smallest factor containmore than two digits.
* Ex. gr., see above, page 31.† Comptes rendus de l’Academie des Sciences, Paris, Nov. 1877, vol. lxxxv,
pp. 950–952.
• Project • Gutenberg • #26839 •
236 MERSENNE’S NUMBERS. [CH. IX
Seventh. I have reserved to the last the description of the methodwhich seems to me to be the most hopeful.
We know by Fermat’s Theorem that if x + 1 is a prime then 2x − 1is divisible by x + 1. Hence if 2pt + 1 is a prime we have, to themodulus 2pt + 1
22pt − 1 ≡ 0 ,
∴ (2p − 1)(1 + 2p + 22p + · · ·+ 2(2t−1)p) ≡ 0 .
Hence, a divisor of 2p−1 will be known, if we can find a value of t suchthat 2pt + 1 is prime and the second factor is prime to it.
This method may be used to establish Euler’s theorem of 1732. Forif we put t = 1, and if 2p+1 is a prime, we have, to the modulus 2p+1
(2p − 1)(2p + 1) ≡ 0 .
Hence 2p ≡ 1 if 2p +1 is prime to 2p+1. This is the case if p = 4m+3.Hence 2p + 1 is a factor of N if p = 11, 23, 83, 131, 179, 191, 239, and251, for in these cases 2p + 1 is prime.
The problem of Mersenne’s Numbers is a particular case of the de-termination of the factors of an−1. This has been the subject of inves-tigations by many mathematicians: an outline of their conclusions hasbeen given by Bickmore*. I ought also to add a reference to the generalmethod suggested by F.W. Lawrence† for the factorization of any highnumber: it is possible that Fermat used some method analogous to this.
Finally, I should add that machines‡ have been devised for inves-tigating whether a number is prime, but I do not know that any havebeen constructed suitable for numbers as large as those involved in thenumbers in question.
* Messenger of Mathematics, Cambridge, 1895–6, vol. xxv, pp. 1–44; also 1896–7,vol. xxvi, pp. 1–38; see also a note by Mr E.B. Escott in the Messenger, 1903–4,vol. xxxiii, p. 49.
† Ibid., 1894–5, vol. xxiv, pp. 100–109; Quarterly Journal of Mathematics, 1896,vol. xxviii, pp. 285–311; and Transactions of the London Mathematical Society,May 13, 1897, vol. xxviii, pp. 465–475.
‡ F.W. Lawrence, Quarterly Journal of Mathematics, 1896, already quoted,pp. 310–311; see also C.A. Laisant, Comptes Rendus Association Francais pourl’avancement des sciences, 1891 (Marseilles), vol. xx, pp. 165–8.
• Project • Gutenberg • #26839 •
CHAPTER X.
ASTROLOGY.
Astrologers professed to be able to foretell the future, andwithin certain limits to control it. I propose to give in this chaptera concise account of the rules they used for this purpose*.
I have not attempted to discuss the astrology of periods earlierthan the middle ages, for the technical laws of the ancient astrologyare not known with accuracy. At the same time there is no doubt that,as far back as we have any definite historical information, the art waspractised in the East; that thence it was transplanted to Egypt, Greece,and Rome; and that the medieval astrology was founded on it. It isprobable that the rules did not differ materially from those described inthis chapter†, and it may be added that the more intelligent thinkers ofthe old world recognised that the art had no valid pretences to accuracy.I may note also that the history of the development of the art ceaseswith the general acceptance of the Copernican theory, after which thepractice of astrology rapidly became a mere cloak for imposture.
* I have relied mainly on the Manual of Astrology by Raphael—whose real namewas R.C. Smith—London, 1828, to which the references to Raphael hereaftergiven apply; and on Cardan’s writings, especially his commentary on Ptolemy’swork and his Geniturarum Exempla. I am indebted also for various referencesand gossip to Whewell’s History of the Inductive Sciences; to various worksby Raphael, published in London between 1825 and 1832; and to a pamphletby M. Uhlemann, entitled Grundzuge der Astronomie und Astrologie, Leipzig,1857.
† On the influences attributed to the planets, see The Dialogue of Bardesan onFate, translated by W. Cureton in the Spicilegium Syriacum, London, 1855.
237
• Project • Gutenberg • #26839 •
238 ASTROLOGY. [CH. X
All the rules of the medieval astrology—to which I confine myself—are based on the Ptolemaic astronomy, and originate in the Tetrabiblos*
which is said, it may be falsely, to have been written by Ptolemy himself.The art was developed by numerous subsequent writers, especially byAlbohazen†, and Firmicus. The last of these collected the works of mostof his predecessors in a volume‡, which remained a standard authorityuntil the close of the sixteenth century.
I may begin by reminding the reader that though there was a fairlygeneral agreement as to the methods of procedure and interpretation—which alone I attempt to describe—yet there was no such thing as afixed code of rules or a standard text-book. It is therefore difficult toreduce the rules to any precise and definite form, and almost impossi-ble, within the limits of a chapter, to give detailed references. At thesame time the practice of the elements of the art was tolerably wellestablished and uniform, and I feel no doubt that my account, as faras it goes, is substantially correct.
There were two distinct problems with which astrologers concernedthemselves. One was the determination in general outline of the lifeand fortunes of an enquirer: this was known as natal astrology , andwas effected by the erection of a scheme of nativity. The other was themeans of answering any specific question about the individual: this wasknown as horary astrology . Both depended on the casting or erectingof a horoscope. The person for whom it was erected was known asthe native.
A horoscope was cast according to the following rules§. The spacebetween two concentric and similarly situated squares was divided intotwelve spaces, as shown in the annexed diagram. These twelve spaceswere known technically as houses ; they were numbered consecutively1, 2, . . . , 12 (see figure); and were described as the first house, the se-cond house, and so on. The dividing lines were termed cusps : the linebetween the houses 12 and 1 was called the cusp of the first house,the line between the houses 1 and 2 was called the cusp of the secondhouse, and so on, finally the line between the houses 11 and 12 wascalled the cusp of the twelfth house. Each house had also a name of its
* There is an English translation by J. Wilson, London [n.d.]; and a French trans-lation is given in Halma’s edition of Ptolemy’s works.
† De judiciis astrorum, ed. Liechtenstein, Basle, 1571.‡ Astronomicorum, eight books, Venice, 1499.§ Raphael, pp. 91–109.
• Project • Gutenberg • #26839 •
CH. X] ASTROLOGY. 239
3 4 5
2
1
12
11 10 9
6
7
8
own—thus the first house was called the ascendant house, the eighthhouse was called the house of death, and so on—but as these namesare immaterial for my purpose I shall not define them.
Next, the positions which the various astrological signs and planetshad occupied at some definite time and place (for instance, the timeand place of birth of the native, if his nativity was being cast) weremarked on the celestial sphere. This sphere was divided into twelveequal spaces by great circles drawn through the zenith, the angle be-tween any two consecutive circles being 30◦. The first circle was drawnthrough the East point, and the space between it and the next circletowards the North corresponded to the first house, and sometimes wascalled the first house. The next space, proceeding from East to North,corresponded to the second house, and so on. Each of the twelve spacesbetween these circles corresponded to one of the twelve houses, andeach of the circles to one of the cusps.
In delineating* a horoscope, it was usual to begin by inserting thezodiacal signs. A zodiacal sign extends over 30◦, and was marked onthe cusp which passed through it: by its side was written a numberindicating the distance to which its influence extended in the earlier ofthe two houses divided by the cusp. Next the position of the planets inthese signs were calculated, and each planet was marked in its proper
* Raphael, pp. 118–131.
• Project • Gutenberg • #26839 •
240 ASTROLOGY. [CH. X
house and near the cusp belonging to the zodiacal sign in which theplanet was then situated: it was followed by a number indicating itsright ascension measured from the beginning of the sign. The name ofthe native and the date for which the horoscope was cast were insertedusually in the central square. The diagram near the end of this chapteris a facsimile of the horoscope of Edward VI as cast by Cardan and willserve as an illustration of the above remarks.
We are now in a position to explain how a horoscope was read orinterpreted. Each house was associated with certain definite questionsand subjects, and the presence or absence in that house of the varioussigns and planets gave the answer to these questions or informationon these subjects.
These questions cover nearly every point on which informationwould be likely to be sought. They may be classified roughly as follows.For the answer, so far as it concerns the native, to all questions con-nected with his life and health, look in house 1; for questions connectedwith his wealth, refer to house 2; for his kindred and communicationsto him, refer to 3; for his parents and inheritances, refer to 4; for hischildren and amusem*nts, refer to 5; for his servants and illnesses, referto 6; for his marriage and amours, refer to 7; for his death, refer to 8;for his learning, religion and travels, refer to 9; for his trade and repu-tation, refer to 10; for his friends, refer to 11; and finally for questionsconnected with his enemies, refer to house 12.
I proceed to describe briefly the influences of the planets, and shallthen mention those of the zodiacal signs; I should note however thatin practice the signs were in many respects more influential than theplanets.
The astrological “planets” were seven in number, and included theSun and the Moon. They were Saturn or the Great Infortune, Jupiteror the Great Fortune, Mars or the Lesser Infortune, the Sun, Venusor the Lesser Fortune, Mercury, and the Moon: the above order beingthat of their apparent times of rotation round the earth.
Each of them had a double signification. In the first place it im-pressed certain characteristics, such as good fortune, feebleness, &c., onthe dealings of the native with the subjects connected with the housein which it was located; and in the second place it imported certainobjects into the house which would affect the dealings of the nativewith the subjects of that house.
• Project • Gutenberg • #26839 •
CH. X] ASTROLOGY. 241
To describe the exact influence of each planet in each house wouldinvolve a long explanation, but the general effect of their presence maybe indicated roughly as follows*. The presence of Saturn is malignant:that of Jupiter is propitious: that of Mars is on the whole injurious:that of the Sun indicates respectability and moderate success: thatof Venus is rather favourable: that of Mercury implies rapid practicalaction: and lastly the presence of the Moon merely faintly reflects theinfluence of the planet nearest her, and suggests rapid changes andfickleness. Besides the planets, the Moon’s nodes and some of the moreprominent fixed stars† also had certain influences.
These vague terms may be illustrated by taking a few simple cases.For example, in casting a nativity, the life, health, and general
career of the native were determined by the first or ascendant house,whence comes the expression that a man’s fortune is in the ascendant.Now the most favourable planet was Jupiter. Therefore, if at the instantof birth Jupiter was in the first house, the native might expect a long,happy, healthy life; and being born under Jupiter he would have a“jovial” disposition. On the other hand, Saturn was the most unluckyof all the planets, and was as potent as malignant. If at the instant ofbirth he was in the first house, his potency might give the native a longlife, but it would be associated with an angry and unhappy temper, aspirit covetous, revengeful, stern, and unloveable, though constant infriendship no less than in hate, which was what astrologers meant by a“saturnine” character. Similarly a native born under Mercury, that is,with Mercury in the first house, would be of a mercurial nature, whileanyone born under Mars would have a martial bent.
Moreover it was the prevalent opinion that a jovial person wouldhave his horoscope affected by Jupiter, even if that planet had notbeen in the ascendant at the time of birth. Thus the horoscope of anadult depended to some extent on his character and previous life. Itis hardly necessary to point out how easily this doctrine enabled anastrologer to make the prediction of the heavens agree with facts thatwere known or probable.
In the same way the other houses are affected. For instance, noastrologer, who believed in the art, would have wished to start on along journey when Saturn was in the ninth house or house of travels;
* Raphael, pp. 70–90; pp. 204–209.† Raphael, pp. 129–131,
• Project • Gutenberg • #26839 •
242 ASTROLOGY. [CH. X
and, if at the instant of birth Saturn was in that house, the nativealways would incur considerable risk on his journeys.
Moreover every planet was affected to some extent by its aspect(conjunction, opposition, or quadrature) to every other planet accord-ing to elaborate rules* which depended on their positions and directionsof motion: in particular the angular distance between the Sun and theMoon—sometimes known as the “part of fortune”—was regarded asspecially important, and this distance affected the whole horoscope.In general, conjunction was favourable, quadrature unfavourable, andopposition ambiguous.
Each planet not only influenced the subjects in the house in whichit was situated, but also imported certain objects into the house. ThusSaturn was associated with grandparents, paupers, beggars, labourers,sextons, and gravediggers. If, for example, he was present in the fourthhouse, the native might look for a legacy from some such person; ifhe was present in the twelfth house, the native must be careful of theconsequences of the enmity of any such person; and so on.
Similarly Jupiter was associated generally with lawyers, priests,scholars, and clothiers; but, if he was conjoined with a malignant planet,he represented knaves, cheats, and drunkards. Mars indicated soldiers(or, if in a watery sign, sailors on ships of war), masons, doctors, smiths,carpenters, cooks, and tailors; but, if afflicted with Mercury or theMoon, he denoted the presence of thieves. The Sun implied the actionof kings, goldsmiths, and coiners; but, if afflicted by a malignant planet,he denoted false pretenders. Venus imported musicians, embroiderers,and purveyors of all luxuries; but, if afflicted, prostitutes and bullies.Mercury imported astrologers, philosophers, mathematicians, states-men, merchants, travellers, men of intellect, and cultured workmen;but, if afflicted, he signified the presence of pettifoggers, attorneys,thieves, messengers, footmen, and servants. Lastly, the presence of theMoon introduced sailors and those engaged in inferior offices.
I come now to the influence and position of the zodiacal signs. Sofar as the first house was concerned, the sign of the zodiac which wasthere present was even more important than the planet or planets, forit was one of the most important indications of the durations of life.
Each sign was connected with certain parts of the body—ex. gr.Aries influenced the head, neck and shoulders—and that part of the
* Raphael, pp. 132–170.
• Project • Gutenberg • #26839 •
CH. X] ASTROLOGY. 243
body was affected according to the house in which the sign was. Fur-ther each sign was associated with certain countries and connected thesubjects of the house in which the sign was situated with those coun-tries: ex. gr. Aries was associated especially with events in England,France, Syria, Verona, Naples, &c.
The sign in the first house determined also the character and ap-pearance of the native*. Thus the character of a native born underAries (m) was passionate; under Taurus (f ) was dull and cruel; underGemini (m) was active and ingenious; under Cancer (f ) was weak andyielding; under Leo (m) was generous, resolute, and ambitious; underVirgo (f ) was sordid and mean; under Libra (m) was amorous andpleasant; under Scorpio (f ) was cold and reserved; under Sagittarius(m) was generous, active, and jolly; under Capricorn (f ) was weak andnarrow; under Aquarius (m) was honest and steady; and under Pisces(f ) was phlegmatic and effeminate.
Moreover the signs were regarded as alternately masculine and fem-inine, as indicated above by the letters m or f placed after each sign. Amasculine sign is fortunate, and all planets situated in the same househave their good influence rendered thereby more potent and their un-favourable influence mitigated. But all feminine signs are unfortunate,their direct effect is evil, and they tend to nullify all the good influenceof any planet which they afflict (i.e. with which they are connected),and to increase all its evil influences, while they also import an elementof fickleness into the house and often turn good influences into malig-nant ones. The precise effect of each sign was different on every planet.
I think the above account is sufficient to enable the reader to forma general idea of the manner in which a horoscope was cast and inter-preted, and I do not propose to enter into further details. This is the lessnecessary as the rules—especially as to the relative importance to beassigned to various planets when their influence was conflicting—wereso vague that astrologers had little difficulty in finding in the horoscopeof a client any fact about his life of which they had information or anytrait of character which they suspected him to possess.
That this vagueness was utilized by quacks is notorious, but nodoubt many an astrologer in all honesty availed himself of it, whetherconsciously or unconsciously. It must be remembered also that the ruleswere laid down at a time when men were unacquainted with any exact
* Raphael, pp. 61–69.
• Project • Gutenberg • #26839 •
244 ASTROLOGY. [CH. X
science, with the possible exception of mathematics, and further that,if astrology had been reduced to a series of inelastic rules applicableto all horoscopes, the number of failures to predict the future correctlywould have rapidly led to a recognition of the folly of the art. As it was,the failures were frequent and conspicuous enough to shake the faithof most thoughtful men. Moreover it was a matter of common remarkthat astrologers showed no greater foresight in meeting the difficultiesof life than their neighbours, while they were neither richer, wiser, norhappier for their supposed knowledge. But though such observationswere justified by reason they were often forgotten in times of difficultyand danger. A prediction of the future and the promise of definiteadvice as to the best course of action, revealed by the heavenly bodiesthemselves, appealed to the strongest desires of all men, and it was withreluctance that the futility of the advice was gradually recognized.
The objections to the scheme had been stated clearly by severalclassical writers. Cicero* pointed out that not one of the futures fore-told for Pompey, Crassus, and Caesar had been verified by their subse-quent lives, and added that the planets, being almost infinitely distant,cannot be supposed to affect us. He also alluded to the fact, which wasespecially pressed by Pliny†, that the horoscopes of twins are practi-cally identical though their careers are often very different, or as Plinyput it, every hour in every part of the world are born lords and slaves,kings and beggars.
In answer to the latter obvious criticism astrologers replied by quot-ing the anecdote of Publius Nigidius Figulus, a celebrated Roman as-trologer of the time of Julius Caesar. It is said that when an opponentof the art urged as an objection the different fates of persons born in twosuccessive instants, Nigidius bade him make two contiguous marks ona potter’s wheel, which was revolving rapidly near them. On stoppingthe wheel, the two marks were found to be far removed from each other.Nigidius received the name of Figulus, the potter, in remembrance ofthis story, but his argument, says St Augustine‡, who gives us the nar-rative, was as fragile as the ware which the wheel manufactured.
On the other hand Seneca and Tacitus may be cited as being on
* Cicero, De Divinatione, ii, 42.† Pliny, Historia Naturalis, vii, 49; xxix, 1.‡ St Augustine, De Civitate Dei, bk. v, chap. iii; Opera omnia, ed. Migne, vol. vii,
p. 143.
• Project • Gutenberg • #26839 •
CH. X] ASTROLOGY. 245
the whole favourable to the claims of astrology, though both recognizedthat it was mixed up with knavery and fraud. An instance of successfulprediction which is given by the latter of these writers* may be usedmore correctly as an illustration of how the ordinary professors of theart varied their predictions to suit their clients and themselves. Thestory deals with the first introduction of the astrologer Thrasyllus tothe emperor Tiberius. Those who were brought to Tiberius on any im-portant matter were admitted to an interview in an apartment situatedon a lofty cliff in the island of Capreae. They reached this place by anarrow path overhanging the sea, accompanied by a single freedman ofgreat bodily strength; and on their return, if the emperor had conceivedany doubts of their trustworthiness, a single blow buried the secret andits victim in the ocean. After Thrasyllus had, in this retreat, stated theresults of his art as they concerned the emperor, the latter asked theastrologer whether he had calculated how long he himself had to live.The astrologer examined the aspect of the stars, and while he did thisshowed, as the narrative states, hesitation, alarm, increasing terror,and at last declared that the present hour was for him critical, perhapsfatal. Tiberius embraced him, and told him he was right in supposinghe had been in danger but that he should escape it; and made himthenceforth a confidential counsellor. But Thrasyllus would have beenbut a sorry astrologer had he not foreseen such a question and preparedan answer which he thought fitted to the character of his patron.
A somewhat similar story is told† of Louis XI of France. He sent fora famous astrologer whose death he was meditating, and asked him toshow his skill by foretelling his own future. The astrologer replied thathis fate was uncertain, but it was so inseparably interwoven with thatof his questioner that the latter would survive him but by a few hours,whereon the superstitious monarch not only dismissed him uninjured,but took steps to secure his subsequent safety. The same anecdoteis also related of a Scotch student who, being captured by Algerianpirates, predicted to the Sultan that their fates were so involved thathe should predecease the Sultan by only a few weeks. This may havebeen good enough for a barbarian, but with a civilized monarch it
* Annales, vi, 22: quoted by Whewell, History of the Inductive Sciences, vol. i,p. 313.
† Personal Characteristics from French History, by Baron F. Rothschild, London,1896, p. 10. The story was introduced by Sir Walter Scott in Quentin Durward(chap. xv).
• Project • Gutenberg • #26839 •
246 ASTROLOGY. [CH. X
probably would in most cases be less effectual, as certainly it is lessartistic, than the answer of Thrasyllus.
I may conclude by mentioning a few notable cases of horoscopy.Among the most successful instances of horoscopy enumerated by
Raphael* is one by W. Lilly, given in his Monarchy or No Monarchy,published in 1651, in which he predicted a plague in London so ter-rible that the number of deaths should exceed the number of coffinsand graves, to be followed by “an exorbitant fire.” The prediction wasamply verified in 1665 and 1666. In fact Lilly’s success was embar-rassing, for the Committee of the House of Commons, which sat toinvestigate the causes of the fire and ultimately attributed it to thepapists, thought that he must have known more about it than he choseto declare, and on Oct. 25, 1666, summoned him before them. I mayadd that Lilly proved himself a match for his questioners.
An even more curious instance of a lucky hit is told of Flamsteed†,the first astronomer royal. It is said that an old lady who had lost someproperty wearied Flamsteed by her perpetual requests that he woulduse his observatory to discover her property for her. At last, tiredout with her importunities, he determined to show her the folly of herdemand by making a prediction, and, after she had found it false, toexplain again to her that nothing else could be expected. Accordinglyhe drew circles and squares round a point that represented her houseand filled them with all sorts of mystical symbols. Suddenly strikinghis stick into the ground he said, “Dig there and you will find it.” Theold lady dug in the spot thus indicated, and found her property; and itmay be conjectured that she believed in astrology for the rest of her life.
Perhaps the belief that the royal observatory was built for suchpurposes may be still held, for De Morgan, writing in 1850, says that“persons still send to Greenwich to have their fortunes told, and in onecase a young gentleman wrote to know who his wife was to be, andwhat fee he was to remit.”
It is easier to give instances of success in horoscopy than of failure.Not only are all ambiguous predictions esteemed to be successful, but itis notorious that prophecies which have been verified by the subsequent
* Manual of Astrology, p. 37.† The story, though in a slightly different setting, is given in The London Chron-
icle, Dec. 3, 1771, and it is there stated that Flamsteed attributed the result tothe direct action of the devil.
• Project • Gutenberg • #26839 •
CH. X] ASTROLOGY. 247
course of events are remembered and quoted, while the far more numer-ous instances in which the prophecies have been falsified are forgottenor passed over in silence.
As exceptionally well-authenticated instances of failures I may men-tion the twelve cases collected by Cardan in his Geniturarum Exempla.These are good examples because Cardan was not only the most em-inent astrologer of his time, but was a man of science, and perhaps itis not too much to say was accustomed to accurate habits of thought;moreover, as far as I can judge, he was perfectly honest in his beliefin astrology. To English readers the most interesting of these is thehoroscope of Edward VI of England, the more so as Cardan has left afull account of the affair, and has entered into the reasons of his failureto predict Edward’s death.
To show how Cardan came to be mixed up in the transaction Ishould explain that in 1552 Cardan went to Scotland to prescribe forJohn Hamilton, the archbishop of St Andrews, who was ill with asthmaand dropsy and about whose treatment the physicians had disagreed*.On his return through London, Cardan stopped with Sir John Cheke,the Professor of Greek at Cambridge, who was tutor to the youngking. Six months previously, Edward had been attacked by measles andsmall-pox which had made his health even weaker than before. Theking’s guardians were especially anxious to know how long he wouldlive, and they asked Cardan to cast Edward’s nativity with particularreference to that point.
The Italian was granted an audience in October, of which he wrotea full account in his diary, quoted in the Geniturarum Exempla. Theking, says he†, was “of a stature somewhat below the middle height,pale faced, with grey eyes, a grave aspect, decorous, and handsome. He
* Luckily they left voluminous reports on the case and the proper treatment for it.The only point on which there was a general agreement was that the phlegm, in-stead of being expectorated, collected in his Grace’s brains, and that thereby theoperations of the intellect were impeded. Cardan was celebrated for his successin lung diseases, and his remedies were fairly successful in curing the asthma.His fee was 500 crowns for travelling expenses from Pavia, 10 crowns a day, andthe right to see other patients; the archbishop actually gave him 2300 crowns inmoney and numerous presents in kind; his fees from other persons during thesame time must have amounted to about an equal sum (see Cardan’s De LibrisPropriis, ed. 1557, pp. 159–175; Consilia Medica, Opera, vol. ix, pp. 124–148;De Vita Propria, ed. 1557, pp. 138, 193 et seq.).
† I quote from Morley’s translation, vol. ii, p. 135 et seq.
• Project • Gutenberg • #26839 •
248 ASTROLOGY. [CH. X
was rather of a bad habit of body than a sufferer from fixed diseases,and had a somewhat projecting shoulder-blade.” But, he continues, hewas a boy of most extraordinary wit and promise. He was then butfifteen years old and he was already skilled in music and dialectics, amaster of Latin, English, French, and fairly proficient in Greek, Italian,
and Spanish. He “filled with the highest expectation every good andlearned man, on account of his ingenuity and suavity of manners. . . .When a royal gravity was called for, you would think that it was anold man you saw, but he was bland and companionable as becamehis years. He played upon the lyre, took concern for public affairs,was liberal of mind, and in these respects emulated his father, who,while he studied to be [too] good, managed to seem bad.” And inanother place* he describes him as “that boy of wondrous hopes.” Atthe close of the interview Cardan begged leave to dedicate to Edwarda work on which he was then engaged. Asked the subject of the work,Cardan replied that he began by showing the cause of comets. Thesubsequent conversation, if it is reported correctly, shows good senseand considerable logical skill on the part of the young king.
* De Rerum Varietate, p. 285.
• Project • Gutenberg • #26839 •
CH. X] ASTROLOGY. 249
I have reproduced on the preceding page a facsimile of Cardan’soriginal drawing of Edward’s horoscope. The horoscope was cast andread with unusual care. I need not quote the minute details givenabout Edward’s character and subsequent career, but obviously thepredictions were founded on the impressions derived from the above-mentioned interview. The conclusion about his length of life was thathe would certainly live past middle age, though after the age of 55 years3 months and 17 days various diseases would fall to his lot*.
In the following July the king died, and Cardan felt it necessaryfor his reputation to explain the cause of his error. The title of his dis-sertation is Quae post consideravi de eodem†. In effect his explanationis that a weak nativity can never be predicted from a single horoscope,and that to have ensured success he must have cast the nativity of everyone with whom Edward had come intimately into contact; and, failingthe necessary information to do so, the horoscope could be regardedonly as a probable prediction.
This was the argument usually offered to account for non-success.A better defence would have been the one urged by Raphael‡ and bySouthey§ that there might be other planets unknown to the astrologerwhich had influenced the horoscope, but I do not think that medievalastrologers assigned this reason for failure.
I have not alluded to the various adjuncts of the art, but astrologersso frequently claimed the power to be able to raise spirits that perhaps Imay be pardoned for remarking that I believe some of the more impor-tant and elaborate of these deceptions were effected not infrequently bymeans of a magic lantern, the pictures being sometimes thrown on toa mirror, and at other times on to a thick cloud of smoke which causedthe images to move and finally disappear in a fantastic way capableof many explanations‖.
I would conclude by repeating again that though the practice ofastrology was so often connected with impudent quackery, yet one oughtnot to forget that nearly every physician and man of science in medievalEurope was an astrologer. These observers did not consider that its
* Geniturarum Exempla, p. 19.† Ibid., p. 23.‡ The Familiar Astrologer, London, 1832, p. 248.§ The Doctor, chap. 92.‖ See ex. gr. the life of Cellini, chap. xiii, Roscoe’s translation, pp. 144-146. See
also Sir David Brewster’s Letters on Natural Magic.
• Project • Gutenberg • #26839 •
250 ASTROLOGY. [CH. X
rules were definitely established, and they laboriously collected muchof the astronomical evidence that was to crush their art. Thus, thoughthere never was a time when astrology was not practised by knaves,there was a period of intellectual development when it was honestlyaccepted as a difficult but a real science.
• Project • Gutenberg • #26839 •
CHAPTER XI.
CRYPTOGRAPHS AND CIPHERS.
The art of constructing cryptographs or ciphers—intelligible tothose who know the key and unintelligible to others—has been studiedfor centuries. Their usefulness on certain occasions, especially in timeof war, is obvious, while it may be a matter of great importance tothose from whom the key is concealed to discover it. But the romanceconnected with the subject, the not uncommon desire to discover asecret, and the implied challenge to the ingenuity of all from whom thekey is hidden, have attracted to the subject the attention of many towhom its utility is a matter of indifference.
The leading authorities on the subject, few of which are less thana century old, are enumerated in an article by J.E. Bailey in the ninthedition of the Encyclopaedia Britannica, and references to various his-toric ciphers are there given. My knowledge of the subject, however, islimited to ciphers which I have met with in the course of casual read-ing, and I prefer to discuss the subject as it has presented itself to me,with no attempt to make it historically complete and no reference toother authorities. In fact the theory of the subject is not sufficientlyimportant to make it worth while to try to deal with it historicallyor exhaustively.
Most writers use the words cryptograph and cipher as synonymous.I employ them, however, with different meanings, which I proceed todefine.
A cryptograph may be defined as a manner of writing in which theletters or symbols employed are used in their normal sense, but are soarranged that the communication is intelligible only to those possessing
251
• Project • Gutenberg • #26839 •
252 CRYPTOGRAPHS AND CIPHERS. [CH. XI
the key. The word is sometimes used to denote the communicationmade. A simple example is a communication in which every word isspelt backwards. Thus:
ymene deveileb ot eb gniriter troper noitisop no ssorc daor.
A cipher may be defined as a manner of writing by characters ar-bitrarily invented or by an arbitrary use of letters, words, or charactersin other than their ordinary sense, intelligible only to those possessingthe key. The word is sometimes used to denote the communicationmade. A simple example is when each letter is replaced by the onethat immediately follows it in the natural order of the alphabet, a be-ing replaced by b, b by c, and so on, and finally z by a. In this cipherthe above message would read:
fofnz cfmjfwfe up cf sfujsjoh sfqpsu qptjujpo po dsptt spbe.
In both cryptographs and ciphers the essential feature is that thecommunication may be freely given to all the world though it is un-intelligible save to those who possess the key. The key must not beaccessible to anyone, and if possible it should be known only to thoseusing the cryptograph or cipher. The art of constructing a cryptographlies in the concealment of the proper order of the essential letters orwords: the art of constructing a cipher lies in concealing what lettersor words are represented by the symbols used. In an actual commu-nication cipher symbols may be arranged cryptographically, and thusfurther hinder a reading of the message. Thus the message given abovewould read in a cryptographic cipher as
znfof efwfjmfc pu fc hojsjufs uspqfs opjujtpq op ttpsd ebps.
If the message were sent in Latin or some foreign language it wouldfurther diminish the chance of it being read by a stranger through whosehands it passed. But I may confine myself to messages in English, andfor the present to simple cryptographs and ciphers.
A communication in cryptograph or cipher must be in writing orin some permanent form. Thus to make small muscular movements—such, ex. gr., as talking on the fingers, or breathing long and short inthe Morse dot and dash system, or making use of pre-arranged signsby a fan or stick, or flashing signals by light—do not here concern us.
• Project • Gutenberg • #26839 •
CH. XI] CRYPTOGRAPHS 253
Again, the mere fact that the message is concealed or conveyed se-cretly does not make it a cryptograph or cipher. The majority of storiesdealing with secret communications are concerned with the artfulnesswith which the message is concealed or conveyed and have nothing todo with cryptographs or ciphers. Many of the ancient instances of se-cret communication are of this type*. Illustrations are to be found inmessages conveyed by pigeons, or wrapped round arrows shot over thehead of a foe, or written on the paper wrapping of a cigarette, or theuse of ink which becomes visible only when the recipient treats thepaper on which it is written by some chemical or physical process.
Again, a communication in a foreign language or in any recognizednotation like shorthand is not an instance of a cipher. A letter in Chi-nese or Polish or Russian might be often used for conveying a secretmessage from one part of England to another, but it fails to fulfil ourtest that if published to all the world it would be concealed from ev-eryone, unless submitted to some special investigation. On the otherhand, in practice, foreign languages or systems of shorthand which arebut little known may serve to conceal a communication better than aneasy cipher, for in the last case the key may be found with but littletrouble, while in the other cases, though the key may be accessible, itis probable that there are only a few who know where to look for it. Anillustration of this is afforded by the system used by Pepys in writinghis Diary which is further alluded to below.
I proceed to enumerate some of the better known types of cryp-tographs. There are at least three distinct types. The first type com-prises those in which the order of the letters is changed in some pre-arranged manner. The second type comprises those in which the con-cealment is due to the introduction of non-significant letters. The thirdtype comprises those in which the letters used are written in fragments.The types are not exclusive, and any particular cryptograph may com-prise the distinctive feature of two or all the types.
A cryptograph of the first type is one in which the successive lettersof the message are re-arranged in some pre-determined manner.
One of the most obvious cryptographs of this type is to write eachword or the message itself backwards. He would, however, be a carelessreader who could be deceived by this. Here is an instance in which the
* A long list of classical authorities for different devices used in ancient times forconcealing messages is given in Mercury by J. Wilkins, London, 1641, pp. 27–36.
• Project • Gutenberg • #26839 •
254 CRYPTOGRAPHS AND CIPHERS. [CH. XI
whole message is written backwards:
tsop yb tnes tnemeerga fo seniltuo smret ruo tpecca yeht.
In such a case it is unnecessary to indicate the division into words byleaving spaces between them, and we might divide the letters artifi-cially, as thus:
Ts opybtne stne meer gafos eniltu osmret ruot peccaye ht.
Systems of this kind which depend on altering the places of lettersor lines in some pre-arranged manner have always been common. Iquote a couple of instances* from Wilkins’s book to which I have alreadyreferred—it was a work which seems to have been studied diligentlyby many of those who took part in the civil disturbances of the 17thcentury, and gives an excellent account of some of the easier systemsof cryptographs and ciphers.
The first example I take from him is where the letters which makeup the communication are written vertically up or down. Thus themessage: The pestilence continues to increase might be written thus:
e i o t n l i ts n t i o e t ha c s n c n s ee r e u e c e p
Again, Wilkins says that the cryptograph may be yet further ob-scured by placing the letters which make up the message in any pre-arranged but discontinuate order. For instance if the message runs tofour lines we may put the first letter at the beginning of the first line,the second at the beginning of the fourth line, the third at the end ofthe first line, the fourth at the end of the fourth line, the fifth at thebeginning of the second line, the sixth at the beginning of the thirdline, the seventh at the end of the second line, the eighth at the endof the third line, and so on. Thus the message: Wee shall make anIrruption upon the Enemie, from the North, at ten of the clock this
* Mercury, or the Secret and Swift Messenger, by J. Wilkins, London, 1641,pp. 50–52.
• Project • Gutenberg • #26839 •
CH. XI] CRYPTOGRAPHS 255
night would read thus:
Wm rpeta hhs cteinpkehaih fonoih kftoe nilanoerr ocgt tthmnu rleauo mhtei nlen ettes,
where, to obscure the message further, it is divided arbitrarily intowhat appear to be words.
Another instance of a cryptograph of this type may be constructedthus. First, by writing the message in lines of some arranged length, say,for instance, each containing seventeen letters—the letters in successivelines being arranged vertically under those corresponding to them inthe upper line—and either leaving no spaces between the words or in-serting some pre-arranged letter or letters or digits between them, suchas j, q, z. The message can be then sent as a cryptograph by writ-ing the letters in order in successive vertical lines. This only comesto saying that we write successively the 1st, 18th, 35th letters of theoriginal message, and then the 2nd, 19th, 36th letters, and so on. Toconfuse the decipherer the final reading may be arbitrarily put intowhat might represent words. If, however, we know the clue number,say c, it is easy enough to read the communication. For if it dividesinto the number of letters n times with a remainder r it suffices tore-write the message in lines putting n + 1 letters in each of the firstr lines, and n letters in each of the last c − r lines, and then thecommunication can be read by reading the columns downwards. Forinstance, if the following communication, containing 270 letters, werereceived: Ahtzeipqhgesoaeouazsesewaeqtmusfdtbenzcesjteottqizyczhtzjioarhqettjrfesftnzmroomohyearziaqneornbreotlennkaerwizesjuasjodezwjzzszjbrrittjnfjlweuzroqyfohtqayeizsleopjidihaloalhpepkrheanazsrvliimosiadygtpekijscerqvvjqjqajqnyjintkaehsbhsnbgoaotqetqeuuesayqurntpebqstzamztqrj , and the clue number were 17 we should put 16 letters in each ofthe first 15 lines and 15 letters in each of the last 2 lines. The commu-nication could then be discovered by reading the columns downwards:the letters j, q, and z marking the ends of words.
Another cryptograph of this type may be constructed by arrang-ing the letters cyclically, and agreeing that the communication is to bemade by selected letters, as, for instance, every seventh, second, sev-enth, second, and so on. Thus if the communication were Ammunition
• Project • Gutenberg • #26839 •
256 CRYPTOGRAPHS AND CIPHERS. [CH. XI
too low to allow of a sortie, which consists of 32 letters, the successivesignificant letters would come in the order 7, 9, 16, 18, 25, 27, 2, 4, 13,15, 24, 28, 5, 8, 20, 22, 1, 6, 21, 26, 11, 14, 32, 10, 31, 12, 17, 23, 3,29, 30, 19—the numbers being selected as in the decimation problemgiven above on pages 19–20, and being struck out from the 32 cycle assoon as they are determined. The above communication would thenread Ttriooalmolaoonmsueoawotnliotifw . This is a good cryptograph,but it is troublesome to construct, especially if the message is long, andfor that reason is not to be recommended.
A cryptograph of the second type is one in which the message isexpressed in ordinary writing, but in it are introduced a number ofdummies or non-significant letters or digits thus concealing which ofthe letters are relevant.
One way of picking out those letters which are relevant is by the useof a perforated card of the shape of (say) a sheet of note-paper, whichwhen put over such a sheet permits only such letters as are on certainportions of it to be visible. Such a card is known as a grille. An exampleof a grille with four openings is figured below. A communication madein this way may be easily concealed from anyone who does not possess
A B
CD
a card of the same pattern. If the recipient possesses such a card hehas only to apply it in order to read the message.
The use of the grille may be rendered less easy to detect if it beused successively in different positions, for instance, with the edges ABand CD successively put along the top of the paper containing themessage. On the next page, for instance, is a message which, with theaid of the grille figured above, is at once intelligible. On applying thegrille to it with the line AB along the top HK we get the first halfof the communication, namely, 1000 rifles se. On applying the grillewith the line CD along the top HK we get the rest of the message,
• Project • Gutenberg • #26839 •
CH. XI] CRYPTOGRAPHS 257
namely, nt to L to-day. The other spaces in the paper are filled withnon-significant letters or numerals in any way we please. Of course anyone using such a grille would not divide the sheet of paper on whichthe communication was written into cells, but in the figure I have doneso in order to render the illustration clearer.
981 264 070 523 479 100
NTT ORI SON SON AHY DTC
BFS PUM OLT KFE LJO EGX
AEU QJT EGO FLE HV E WLA
FML AES REM REM ODA SSE
Y ZZ EPD QJC EKS TIM OEF
H K
We can avoid the awkward expedient of having to use a perforatedcard, which may fall into undesired hands, by introducing a certainpre-arranged number of dummies or non-significant letters or symbolsbetween those which make up the message. Thus, to take an extremecase, we might arrange that only every 101st letter should form ourcommunication, and the intervening 100 letters should be written atrandom. But such a communication would be 101 times longer thanthe message, a nearly fatal objection if it had to be written in a hurry ortelegraphed. A better method, and one which is not easily discovered bya stranger, is to arrange that (say) only every alternate second and thirdletter shall be relevant. Thus the first, third, sixth, eighth, eleventh,etc., letters are those that make up the message. Such a communicationwould be only two and a half times as long as the message, but eventhat might be a great disadvantage if time in sending the messagewas of importance. For a message written at leisure this need notmatter much, and in such a code the introduction of a sufficient numberof unnecessary letters in some pre-arranged manner gives an effectualmeans of conveying a message in secret.
We can also avoid the use of a perforated card if it be arrangedthat every nth word shall give the message, the other words being non-significant, though of course inserted as far as possible so as to make thecomplete communication run as a whole. But the difficulty of compos-ing a document of this kind and its great length render it unsuitable forany purpose except an occasional communication composed at leisure
• Project • Gutenberg • #26839 •
258 CRYPTOGRAPHS AND CIPHERS. [CH. XI
and sent in writing. This method is said to have been used by the Earlof Argyle when plotting against James II.
Similarly any system that rests on picking out certain letters in adocument, which letters form a communication in ordinary writing, isa cryptograph. Thus a communication conveyed by a newspaper, inwhich the letters making up the message are indicated by pen dots orpin pricks or in some other agreed way, is a cryptograph.
A kind of secret writing which may perhaps be considered to con-stitute a third type of cryptograph is a communication on paper whichis legible only when the paper is folded in a particular way. An exampleis a message written across the edges of a strip of paper wrapped spiral-wise round a stick called a scytale. When the paper is unwound andtaken off the stick the letters appear broken, and may seem to consistof arbitrary signs, but by wrapping the paper round a similar stick themessage can be again read. This system is said to have been used bythe Lacedemonians*. The concealment can never have been effectualagainst an intelligent reader who got possession of the paper.
The defect of the method is that the broken letters at once attractattention and suggest the system used. If the fact can be concealedthat the visible symbols are parts of letters the cryptograph would bemuch improved. As an illustration take the appended communicationwhich is said to have been given to the Young Pretender during hiswanderings after Culloden. If it be creased along the lines BB and
CC (CC being along the second line of the second score), and thenfolded over, with B inside, so that the crease C lies over the line A(which is the second line of the first score) thus leaving only the top
* For references, see Wilkins, supra, p. 38.
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 259
and bottom of the piece of paper visible, it will be found to read concealyourself, your foes look for you. I have seen what purports to be theoriginal, but of the truth of the anecdote I know nothing, and thedesirability of concealing himself must have been so patent that it washardly necessary to communicate it by a cryptograph.
I proceed next to some of the more common types of ciphers. It isimmaterial whether we invent characters to denote the various letters;or whether we employ special symbols to represent them, such as thesymbol ( for a, the symbol : for b, and so on; or whether we use theletters in a non-natural sense, such as the letter z for a, the letter yfor b, and so on. The rules for reading the cipher will be the samein each case.
In early times it was a common practice to invent arbitrary symbolsto represent the letters. If the symbols are invented for the purpose theyprovoke attention, hence it would seem that preferably we should usesymbols which are not likely to attract special notice. For instance,the symbols may be musical notes, in which case the message wouldappear as a piece of music. Geometrical figures have also been usedfor the same purpose. It is not even necessary to employ written signs.Natural objects have often been used, as in a necklace of beads, ora bouquet of flowers, where the different shaped or coloured beads ordifferent flowers stand for different letters or words. An even moresubtle form of disguising the cipher is to make the different distancesbetween consecutive knots or beads indicate the different letters.
Of all such systems we may say that a careful scrutiny shows thatdifferent symbols are being used, and as soon as the various symbolsare distinguished one from the other no additional complication is in-troduced, while for practical purposes they are more trouble to sendor receive than those written in symbols in current use. AccordinglyI confine myself to ciphers written by the use of the current lettersand numerals.
It is convenient to divide ciphers into four classes. The first classcomprises ciphers in which the same letter or word is always representedby the same symbol, and this symbol always represents the same letteror word. The second class comprises ciphers in which the same letteror word is, in some or all cases, represented by more than one symbol,and this symbol always represents the same letter or word. The thirdclass comprises ciphers in which the same symbol represents sometimes
• Project • Gutenberg • #26839 •
260 CRYPTOGRAPHS AND CIPHERS. [CH. XI
one letter or word and sometimes another. The fourth class comprisesciphers in which each letter or word is always represented by the samesymbol, but more than one letter or word may be represented by thesame symbol.
A cipher of the first type then is one in which the same letteror word is always represented by the same symbol, and this symbolalways represents the same letter or word.
Perhaps the simplest illustration of a cipher of this type is to em-ploy one language, but written as far as practical in the alphabet ofanother language. It is said that during the Indian Mutiny messagesin English, but written in Greek characters, were used freely, and suc-cessfully baffled the ingenuity of the enemy, into whose hands they fell.If this is true, the intelligence of the Hindoos must have been muchless than that with which they are usually credited. The device, how-ever, is an old one, for we are told* that Edward VI was accustomedto make notes in cipher “with Greek characters, to the end that theywho waited on him should not read them.”
A common cipher of this type is made by using the actual lettersof the alphabet, but in a non-natural sense as indicating other letters.Thus we may use each letter to represent the one immediately followingit in the natural order of the alphabet—the letters being supposed tobe cyclically arranged—a standing for b wherever it occurs, b standingfor c, and so on, and finally z standing for a. Or more generally wemay write the letters of the alphabet in a line, and under them re-writethe letters in any order we like. For instance
a b c d e f g h i j k l m n o p q r s t u v w x y zo l k m a z s q x e u f y r t h c w b v n i d g j p
In such a scheme, we must in our communication replace a by o, bby l, etc. The recipient will prepare a key by rearranging the lettersin the second line in their natural order and placing under them thecorresponding letter in the first line. Then wherever a comes in themessage he receives he will replace it by e; similarly he will replace bby s, and so on.
A cipher of this kind is not uncommonly used in military signalling,the order of the letters being given by the use of a key-word. If, forinstance, Pretoria is chosen as the key-word, we write the letters in
* Sir John Hayward, Life of Edward VI., edition of 1636, p. 20.
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 261
this order, striking out any which occur more than once, and continuewith the unused letters of the alphabet in their natural order, writingthe whole in two lines thus:
p r e t o i a b c d f g hz y x w v u s q n m l k j
Then in using the cipher p is replaced by z and vice versa, r by y, andso on. A long message in such a cipher would be easily discoverable, butit is rapidly composed by the sender and read by the receiver, and forsome purposes may be useful, especially if the discovery of the purportof the message is, after a few hours, immaterial.
A summary of the usual rules for reading ciphers of this type,whether written in English, French, German, Italian, Dutch, Latin,or Greek, was given by D.A. Conradus in 1742*; and similar rules havebeen given by various later writers. In English the letter which occursmost frequently is e. The next most common letters are said to be t,a, o, and i ; then n; then r, s, and h; then d and l ; then c, w, u, and m;then f, y, g, p, and b; then v and k ; and then x, q, j, and z. The mostcommon double letters are ee, ll, oo, and ss ; while in more than halfthe cases of a double letter at the end of a word, the letter is either lor s. Also, t and h form a common conjunction. I need not, however,go here into further details of this kind.
Assuming that the division into words is given, that non-significantsymbols are not introduced, and that the problem is not complicatedby the avoidance of the use of common words, a communication ofany considerable length can usually be read with but little difficulty.The hints given by Conradus will at once suggest certain hypothesesas to which letters stand for which. Taking one of these hypotheses wewrite the message, replacing the symbols by the letters we conjecturethat they represent and replace the others by dots. If the hypothesisis tenable, the arrangement will probably suggest some of the missingletters. If, for example, we find two words emphs-all and empht-ewhere the missing letter is represented by the same symbol, the firstword shows us that the missing letter is h, m, or t, and the second wordshews us that it must be e, h, i, or o, hence it must be h. Every freshletter so determined makes the hypothesis more probable and renders
* Gentleman’s Magazine, 1742, vol. xii, pp. 133–135, 185–186, 241–242, 473–475.See also the Collected Works of E.A. Poe in 4 volumes, vol. i, p. 30 et seq.
• Project • Gutenberg • #26839 •
262 CRYPTOGRAPHS AND CIPHERS. [CH. XI
it easier to guess what the remaining symbols represent. The chiefdifficulty is to get a working hypothesis for the first few letters—if itis the true solution, probably the puzzle will be readily solved—but tomake up a working hypothesis for even a few letters requires patience.
Ciphers of this class in which the division between the words isgiven are to be avoided. If we leave a space between such words awould-be decipherer is given an immense help. He will naturally try ifa word denoted by a single symbol can be an i or an a, while the wordsof two or three letters will often stand revealed and so provide a definitegroundwork on which he can construct the key. A long word may alsobetray the secret. For instance, if the decipherer has reason to suspectthat the message related to something connected with Birmingham,and he found that a particular word of ten letters had its second andfifth letters alike, as also its fourth and tenth letters, he would naturallysee how the key would work if the word represented Birmingham, andon this hypothesis would at once know the letters represented by eightsymbols. With reasonable luck this should suffice to enable him to tellif the hypothesis was tenable. The effect of this can be avoided byleaving no spaces between the words, but this might lead to confusionand is not to be recommended. We can also use letters which occurbut rarely, such as j, q, x, z, to separate words, and probably this isthe best method.
Ciphers of this type suggest themselves naturally to those ap-proaching the subject for the first time, and are commonly made bymerely shifting the letters a certain number of places forward. If thisis done we may decrease the risk of detection by altering the amountof shifting at short (and preferably irregular) intervals. Thus it maybe agreed that if initially we shift every letter one place forward thenwhenever we come to the letter (say) n we shall shift every letter onemore place forward. In this way the cipher changes continually, andis essentially changed to one of the third class; but even with this im-provement it is probable that an expert would decode a fairly longmessage without much difficulty.
We can have ciphers for numerals as well as for letters: such ciphersare common in many shops. Any word or sentence containing tendifferent letters will answer the purpose. Thus, an old tradesman of myacquaintance used the excellent precept Be just O Man—the first letterrepresenting 1, the second 2, and so on. In this cipher the price 10/6
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 263
would be marked bn/t. This is an instance of a cipher of the first type.
A cipher of the second type is one in which the same letter or wordis, in some or all cases, represented by more than one symbol, and thissymbol always represents the same letter or word. Such ciphers wereuncommon before the Renaissance, but the fact that to those who heldthe key they were not more difficult to write or read than ciphers ofthe first type, while the key was not so easily discovered, led to theircommon adoption in the seventeenth century.
A simple instance of such a cipher is given by the use of numeralsto denote the letters of the alphabet. Thus a may be represented by11 or by 37 or by 63, b by 12 or by 38 or by 64, and so on, and finallyz by 36 or by 62 or by 88, while we can use 89 or 90 to signify theend of a word and the numbers 91 to 99 to denote words or sentenceswhich constantly occur. Of course in practice no one would employ thenumbers in an order like this, which suggests their meaning, but it willserve to illustrate the principle. I have deliberately used numbers ofonly two digits, as the recipient can then point off the symbols usedin twos, and will know that each pair of symbols represents a letter,word, or sentence in the message. A disadvantage of this cipher is thatsince each letter is denoted by two symbols the length of the messageis doubled by putting it in cipher.
The cipher can be improved by introducing after every (say)eleventh digit a non-significant digit. If this is done the recipient ofthe message must erase every twelfth digit before he begins to readthe message. With this addition the difficulty of discovering the keyis considerably increased.
The same principle is sometimes applied with letters instead ofnumbers. For instance, if we take a word (say) of n letters, preferablyall different, and construct a table as shown below of n2 cells, eachcell is defined by two letters of the key-word. Thus, if we choose theword smoking-cap we shall have 100 cells, and each cell is determineduniquely by the two letters denoting its row and column. If we fill thesecells in order with the letters of the alphabet we shall have a systemsimilar to that explained above, where a will be denoted by ss or ogor no, and so for the other letters. The last 22 cells may be used todenote the first 22 letters of the alphabet, or better, three or four ofthem may be used after the end of a word to show that it is ended,and the rest may be used to denote words or sentences which are likely
• Project • Gutenberg • #26839 •
264 CRYPTOGRAPHS AND CIPHERS. [CH. XI
S M O K I N G C A P
S a b c d e f g h i j
M k l m n o p q r s t
O u v w x y z a b c d
K e f g h i j k l m n
I o p q r s t u v w x
N y z a b c d e f g h
G i j k l m n o p q r
C s t u v w x y z
A
P
to occur frequently.
Like the similar cipher with numbers this can be improved by in-troducing after every mth letter any single letter which it is agreedshall be non-significant. To decipher a communication so written it isnecessary to know the clue-word and the clue-number.
Here for instance is a communication written in the above cipherwith the clue-word smoking-cap, and with 7 as the clue-number: ngmksigrioicpssamckscakqignassnxmigpoasuiamnocmpaminscnogcpncisyikskamsssgnncaekknoomkhscpcmscbgpngsiawssgiggndiica1. In this sentencethe letters denoting the 79th, 80th, 81st, and 82nd cells have beenused to denote the end of a word, and no use has been made of thelast 18 cells.
Another cipher of this type is made as follows*. The sender andrecipient of the message furnish themselves with identical copies of somebook. In the cipher only numerals are used, and these numerals indicatethe locality of the letters in the book. For example, the first letter inthe communication might be indicated by 79–8–5, meaning that it isthe 5th letter in the 8th line of the 79th page. But though secrecy might
* The method is well known. It is mentioned by E.A. Poe, Collected Works,vol. iii, pp. 338–9, but is much older.
1. The original text read . . . sssgnnn. . . , which leads to gobbledegook in the deci-phered message.
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 265
be secured, it would be very tedious to prepare or decode a message,and the method is not as safe as some of those described below.
Another cipher of this type is for the sender and receiver to agreeon some common book of reference and to agree further on a numberwhich, if desired, may be communicated as part of the message. Toemploy this cipher the page of the book indicated by the given numbermust be used. The first letter in it is taken to signify a, the next b, andso on–any letter which occurs a second time or more frequently beingneglected. It may be also arranged that after n letters of the messagehave been ciphered, the next n letters shall be written in a similarcipher taken from the pth following page of the book, and so on. Thusthe possession of the code-book would be of little use to anyone whodid not also know the numbers employed. It is so easy to conceal theclue number that with ordinary prudence it would be almost impossiblefor an unauthorized person to discover a message sent in this cipher.The clue number may be communicated indirectly in many ways. Forinstance, it may be arranged that the number to be used shall be thenumber sent, plus (say) q, or that the number to be used shall be anagreed multiple of the number actually sent.
A cipher of the third type is one in which the same symbol rep-resents sometimes one letter or word and sometimes another. Usuallysuch ciphers are easily made or read by those who have the key, butare difficult to discover by those who do not possess it.
A simple example is the employment of pre-arranged numbers inshifting forward the letters that make the communication. For instance,if we agree on the clue number (say) 4276, then the first letter in thecommunication is replaced by the fourth letter which follows it in thenatural order of the alphabet: for instance, if it were an a it would bereplaced by e. The next letter is replaced by the second letter whichfollows it in the natural order of the alphabet: for instance, if it were ana it would be replaced by c. The next letter is replaced by the seventhafter it. The next by the sixth after it. The next by the fourth, andso on to the end of the message. Of course to read the message therecipient would reverse the process. If the letters of the alphabet arewritten at uniform intervals along a ruler, and another ruler similarlymarked with the digits can slide along it, the letter corresponding tothe shifting of any given number of places can be read at once.
It would be undesirable to allow the division into words to appear
• Project • Gutenberg • #26839 •
266 CRYPTOGRAPHS AND CIPHERS. [CH. XI
in the message, and either the words must be run on continuously, orpreferably the less common letters j, q, z may be used to mark thedivision of words. It is also well to conceal the number of digits inthe clue-number. This can be done and the cipher much improved byinserting after every (say) mth letter a non-significant letter.
Here for instance is a communication written in this cipher withthe clue-numbers 4276 and 7: atpznhvaxuxhiepxafwghzniyprpsikbdkzyygkqprgezuytlkobldifebzmxlpogquyitcmgxkckuexvsqkaziaggsigaytnvvsstyvuaslywgjuzmcsfctqbpwjvaepfxhibwpxiultxlavvtqzoxwkvtuvvfheqbxnpvismphzmqtuwxjykeevltif . The recipient would begin by striking out everyeighth letter. He would then shift back every letter 4, 2, 7, 6, 4, 2, &c.,places respectively, and in reading it would leave out the letters j, q,and z as only marking the ends of words.
This is an excellent cipher, and it has the additional merit of notmaterially lengthening the message. It can be rendered still more diffi-cult by arranging that either or both the clue-numbers shall be changedaccording to some definite scheme, and it may be further agreed thatthey shall change automatically every day or week.
A somewhat similar system was proposed by Wilkins*. He tooka key-word, such as prudentia, and constructed as many alphabets asthere were letters in it, each alphabet being arranged cyclically andbeginning respectively with the letters p, r, u, d, e, n, t, i, and a. Hethus got a table like the following, giving nine possible letters whichmight stand for any letter of the alphabet. Using this we may varythe cipher in successive words or letters of the communication. Thus
a b c d e f g h i k l m n o p q r s t u v w x y z
p q r s t u v w x y z a b c d e f g h i k l m n o
r s t u v w x y z a b c d e f g h i k l m n o p q
u v w x y z a b c d e f g h i k l m n o p q r s t
d e f g h i k l m n o p q r s t u v w x y z a b c
e f g h i k l m n o p q r s t u v w x y z a b c d
n o p q r s t u v w x y z a b c d e f g h i k l m
t u v w x y z a b c d e f g h i k l m n o P q r s
i k l m n o p q r s t u v w x y z a b c d e f g h
a b c d e f g h i k l m n o p q r s t u v w x y z
* Mercury, by J. Wilkins, London, 1641, pp. 59, 60.
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 267
the message The prisoners have mutinied and seized the railway stationwould, according as the cipher changes in successive words or letters,read as Hwt fhziedvhi bupy pxwmqmhg erh ervmrq max zirteig station oras Hyy svvlwnthm lehx uukzgmiq tvd gvcciq mqe frcoanr atpkcrr. I havetaken Wilkins’s key-word, but it is obvious that it would be desirable toomit a wherever it appears in it, since otherwise, if the cipher changesin successive words, some of the words may appear unaltered in thecipher, as is shown in the first of the examples given above.
A cipher of the fourth type is one in which each letter is alwaysrepresented by the same symbol, but more than one letter may berepresented by the same symbol. Such ciphers were not uncommon atthe beginning of the nineteenth century, and were usually framed bymeans of a key-sentence containing about as many letters as there areletters in the alphabet.
Thus if the key-phrase is The fox jumped over the garden gate, wewrite under it the letters of the alphabet in their usual sequence asshown below:
T h e f o x j u m p e d o v e r t h e g a r d e n g a t e .a b c d e f g h i j k l m n o p q r s t u v w x y z a b c .
Then we write the message replacing a by t or a, b by h or t, c by e, dby f, and so on. Here is such a message. M foemho nea ge eoo jmdhohgavf teg ev ume afrmeo. But it will be observed that in the cipher a mayrepresent a or u, d may represent l or w, e may represent c or k or o ors or x, g may represent t or z, h may represent b or r, o may represente or m, r may represent p or v, and t may represent a or b or q. Andthe recipient, in deciphering it, must judge as best he can which is theright meaning to be assigned to these letters when they appear.
An instance of a cipher of the fourth type is afforded by a notesent by the duch*ess de Berri to her adherents in Paris, in which sheemployed the key phrase
l e g o u v e r n e m e n t p r o v i s o i r e.a b c d e f g h i j k l m n o p q r s t u v x y.
Hence in putting her message into cipher she replaced a by l, b by e, cby g, and so on. She forgot however to supply the key to the recipientsof the message, but her friend Berryer had little difficulty in readingit by the aid of the rules I have indicated, and thence deduced thekey-phrase she had employed.
• Project • Gutenberg • #26839 •
268 CRYPTOGRAPHS AND CIPHERS. [CH. XI
Having considered various classes of cryptographs and ciphers Imay now consider what features we should regard as important inchoosing a cipher intended for practical use.
In the first place, it is obvious that the means employed should besuch as not to excite suspicion if the communication falls into unautho-rized hands. But this is a counsel of perfection, and almost impossibleto attain.
In the second place, we may say that, under modern conditions inwar, finance, or diplomacy, a cipher may be useless unless it can be tele-graphed or telephoned. If this is deemed important, it will practicallyrestrict us to the use of the 26 letters of the alphabet, the 10 numericalsymbols for the digits, to which if we like we may add a few additionalmarks such as punctuation stops, brackets, &c. The same conditionwill require that the message should not be unduly lengthened by be-ing turned into cipher. Hence any considerable use of non-significantsymbols is to be deprecated.
In the third place, the key to the cipher should be such that itcan be easily reproduced from memory. For, if the key is so elaboratethat those who use it are obliged to preserve it in some tangible andaccessible form, unauthorized persons may obtain the power of readingmessages. Hence the key should be reproducible at will. Further, itis desirable that the key should be of such a character that it (or achange of it) can be telegraphed or otherwise communicated withoutthe probability of exciting suspicion.
In the fourth place, a cipher should be capable of change at shortintervals. So that if the reading of one message in it be discoveredsubsequent messages may be undecipherable even though the systemused is unaltered.
Lastly, no ambiguity should be possible in deciphering the commu-nication. This will exclude ciphers of the fourth type.
Accordingly in choosing a good cipher we should seek for one inwhich only current letters, symbols, or words are employed; such thatit* use does not unduly lengthen the message; such that the key toit can be reproduced at will and need not be kept in a form whichmight betray the secret to an unauthorized person; such that the keyto it changes or can be changed at short intervals; and such that it isnot ambiguous. Many ciphers of the second and third types fulfil theseconditions, but it is generally desirable to avoid ciphers of the first type
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 269
unless circ*mstances permit of the free use of a code-book.
The use of instruments giving a cipher, which is or can be variedconstantly and automatically, has been often recommended. Severalhave been constructed on the lines of the well-known letter-locks*. Thepossession of the key of the instrument as well as a knowledge of theclue-word is necessary to enable anyone to read a message, but the riskof some instrument, when set, falling into unauthorized hands mustbe taken into account. Since equally good ciphers can be constructedwithout the use of mechanical devices I do not think their employmentcan be recommended.
This chapter has already run to such a length that I cannot findspace to describe more than one or two ciphers that appear in historyor fiction, but, we may say that until recently most of the historicalciphers are not difficult to read.
It is said that Julius Caesar in making secret memoranda was ac-customed to move every letter four places forward, writing d for a, efor b, &c. This would be a very easy instance of a cipher of the firsttype, but it may have been effective at that time. His nephew Augus-tus sometimes used a similar cipher, in which each letter was movedforward one place†.
Bacon proposed a cipher in which each letter was denoted by agroup of five letters consisting of A and B only. Since there are 32such groups, he had 6 symbols to spare, which he could use to separatewords or to which he could assign special meanings. A message in thiscipher would be five times as long as the original message. This maybe compared with the far superior system of the five (or four) digitcodebook system in use at the present time.
Charles I used ciphers freely in important correspondence—the ma-jority being of the second type. He was foolish enough to take a cabinetcontaining many confidential letters in cipher, to some of which their
* See, for instance, the descriptions of those devised by Sir Charles Wheatstone,given in his Scientific Papers, London, 1879, pp. 342–347; and by Capt. Baz-eries in Comptes Rendus, Association Francais pour l’avancement des sciences,vol. xx (Marseilles), 1891, p. 160, et seq.
† Of some of Caesar’s correspondence, Suetonius says (cap. 56) si quis investi-gare et persequi velit, quartam elementorum literam, id est, d pro a, et perindereliquas commutet. And of Augustus he says (cap. 88) quoties autem per notasscribit, b pro a, c pro b, ac deinceps eadem ratione, sequentes literas ponit; prox autem duplex a.
• Project • Gutenberg • #26839 •
270 CRYPTOGRAPHS AND CIPHERS. [CH. XI
readings were appended, on the field of Naseby, where they fell intothe hands of Fairfax*. The House of Commons sent them to a com-mittee presided over by a Mr Tate. It is commonly believed that theCommittee referred the papers to J. Wallis†, then Fellow of Queens’College, Cambridge, and subsequently Savilian Professor at Oxford,who discovered the key to them. At any rate the letters were read.
In these ciphers each letter was represented by a number. Theclues to some of the ciphers were provided by the King who had writ-ten over the number the letter which it represented, as shown in thefollowing quotation:
c a t o l i c k s i n F11 18 45 35 23 27 11 25 47 28 40 148 haue layed
t h i r p u r s e s t o g e t h e r45 31 27 51 33 62 50 47 7 48 45 35 21 7 46 32 7 51
f o r s u p l y of a r m e s. – – –15 35 50 a 47 62 33 23 74 k1 17 51 42 7 47. – – –
The published letters show that the King used different ciphers atdifferent times, though perhaps he used the same one in all correspon-dence with any particular person, but the general character of thosehe employed is the same. The sentence quoted above is taken from aletter from Queen Henrietta Maria of January 26, 1643. In this andanother letter a few months later a is represented by 17 or 18, b by 13,c by 11 or 12, d by 5, e by 7 or 8 or 9 or 10, f by 15 or 16, g by 21,h by 31 or 32, i by 27 or 28, k by 25, l by 23 or 24, m by 42 or 44, nby 39 or 40 or 41, o by 35 or 36 or 37 or 38, p by 33 or 34, r by 50 or51 or 52, s by 47 or 48, t by 45 or 46, u by 62 or 63, w by 58, and yby 74 or 77. Numbers of three digits were used to represent particularpeople or places. Thus 148 stood for France, 189 for the King, 260 forthe Queen, 354 for Prince Rupert, and so on. Further, there were afew special symbols, thus k1 stood for but , n1 for to, and f 1 for is.The numbers 2 to 4 and 65 to 72 were non-significant, and were to bestruck out or neglected by the recipient of the message. Each symbolis separated from that which follows it by a full-stop.
* First Report of the Royal Commission on Historical Manuscripts, 1870, pp. 2,4.
† See his letters on Cryptography, Opera, vol. iii, pp. 659–672.
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 271
The Queen seems to have found writing in cipher a great trouble.In the letter from which I have already quoted a sentence she says. . . que je suis extrement tourmantee du mal de teete qui fait que jemesteray en syfre par un autre se qui jovois fait moy mesme, and sheuses the cipher only for the particular words it was desired to conceal.Thus she writes Mr Capell nous a fait voir que sy 27, 23, &c., &c. Ifby this she saved herself trouble, she did it at the cost of rendering thecipher much easier to read.
The system used by Charles was in considerable repute during theseventeenth century, but even without extraneous help it is possible fora diligent student to discover the key if the message is fairly long. Anexcellent illustration of this fact is to be found in the writings of thelate Sir Charles Wheatstone. A paper in cipher, every page of whichwas initialled by Charles I, and countersigned by Lord Digby, was pur-chased some years ago by the British Museum. It was believed to bea state paper of importance. It consists of a series of numbers (about150 different symbols being used) without any clue to their meaning,or any indication of a division between the words employed. The taskof reading it was rendered the more difficult by the supposition, whichproved incorrect, that the document was in English; but notwithstand-ing this, Sir Charles Wheatstone discovered the key*. In this ciphera was represented by any of the numbers 12, 13, 14, 15, 16, or 17, bby 18 or 19, and so on, while some 65 special words were representedby particular numbers.
I may note in passing that Charles also used a species of shorthand,in which the letters were represented by four strokes varying in lengthand position. Essentially the system is simple, though it is troublesometo read or write.
The famous diary of Samuel Pepys is commonly said to have beenwritten in cipher, but in reality it is written in shorthand accordingto a system invented by T. Shelton†. It is however somewhat diffi-cult to read, for the vowels are usually omitted, and Pepys used somearbitrary signs for terminations, particles, and certain words—so farturning it into a cipher. Further, in certain places, when the matter
* The document, its translation, and the key used are given in Wheatstone’sScientific Papers, London, 1879, pp. 321–341.
† Tachy-graphy by T. Shelton. The earliest edition I have seen is dated 1641. Asomewhat similar system by W. Cartwright was issued by J. Rich under thetitle Semographie, London, 1644.
• Project • Gutenberg • #26839 •
272 CRYPTOGRAPHS AND CIPHERS. [CH. XI
is such that it can hardly be expressed with decency, he changed fromEnglish to a foreign language, or inserted non-significant letters. Shel-ton’s system had been forgotten when attention was first attracted tothe diary. Accordingly we may say that, to those who first tried toread it, it was written in cipher, but Pepys’s contemporaries wouldhave properly described it as being written in shorthand, though witha few modifications of his own invention.
A system of shorthand specially invented for the purpose is a truecipher. One such system in which each letter is represented either by adot or by a line of constant length was used by the Earl of Glamorgan,better known by his subsequent title as Marquis of Worcester, in 1645,as also by Charles I. in some of his private correspondence. It is a cipherof the first type and has the defects inherent in almost every cipher ofthis kind: in fact Glamorgan’s letter was deciphered, and the systemdiscovered by Mr Dircks*. Obsolete systems of shorthand† might bethus used to form an effective cipher.
It is always difficult to read a very short message in cipher, sincenecessarily the clues are few in number. When the Chevalier de Rohanwas sent to the Bastille, on suspicion of treason, there was no evidenceagainst him except what might be extracted from Monsieur Latruau-mont. The latter died without making any admission. De Rohan’sfriends had arranged with him to communicate the result of Latruau-mont’s examination, and accordingly in sending him some fresh bodylinen they wrote on one of the shirts Mg dulhxcclgu ghj yxuj, lm ct ulgcalj. For twenty-four hours de Rohan pored over the message, but, failingto read it, he admitted his guilt, and was executed November 27, 1674.
The cipher is a very simple one of the first type, but the commu-nication is so short that unless the key were known it would not beeasy to read it. Had de Rohan suspected that the second word wasprisonnier, it would have given him 7 out of the 12 letters used, andas the first and third words suggest the symbols used for l and t, hecould hardly have failed to read the message.
The cipher on the facing page is said to have been employed by
* Life of the Marquis of Worcester by H. Dircks, London, 1865. Worcester’ssystem of shorthand was described by him in his Century of Inventions, London,1663, sections 3, 4, 5.
† Various systems, including those used in classical and medieval times, are de-scribed in the History of Shorthand by T. Anderson, London, 1882.
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 273
Marie Antoinette*. I take it that it was used in the method indicated
ABAO
BP
CQ
DR
ES
FT
GV
HX
IY
LZ
MN
CDMZ
AN
BO
CP
DQ
ER
FS
GT
HV
IX
LY
EFLN
MO
AP
BQ
CR
DS
ET
FV
GX
HY
IZ
GHIN
LO
MP
AQ
BR
CS
DT
EV
FX
GY
HZ
ILHN
IO
LP
MQ
AR
BS
CT
DV
EX
FY
GZ
MNGN
HO
IP
LQ
MR
AS
BT
CV
DX
EY
FZ
OPFN
GO
HP
IQ
LR
MS
AT
BV
CX
DY
EZ
QREN
FO
GP
HQ
IR
LS
MT
AV
BX
CY
DZ
STDN
EO
FP
GQ
HR
IS
LT
MV
AX
BY
CZ
VXCN
DO
EP
FQ
GR
HS
IT
LV
MX
AY
BZ
YZBN
CO
DP
EQ
FR
GS
HT
IV
LX
MY
AZ
on page 266 above. If so, the first word in the communication wouldbe rewritten according to the scheme given in the first line, a beingreplaced by o, and vice versa, b by p, and so on. The second word wouldbe rewritten according to the scheme in the second line, and so on.
* The key is given, but without explanation, in Juniper Hall, by C. Hill, London,1904, p. 13.
• Project • Gutenberg • #26839 •
274 CRYPTOGRAPHS AND CIPHERS. [CH. XI
One of the modern systems is the five digit code-book cipher, towhich I have already alluded. According to the general belief, it is fre-quently employed in certain official communications at the present day.A code dictionary is prepared in which every word likely to be usedis printed, and the words are numbered consecutively 00000, 00001, . . .up, if necessary, to 99999. Thus each word is represented by a numberof five digits, and there are 105 such numbers available. The messageis first written down in words. Below that it is written in numbers,each word being replaced by the number corresponding to it. To eachof these numbers is added some definite prearranged clue-number—the words in the dictionary being assumed to be arranged cyclically,so that if the resulting number exceeds 105 it is denoted only by theexcess above 105. The resulting numbers are sent as a message. On re-ceipt of a message it is divided into consecutive groups of five numbers,each group representing a word. From each number is subtracted theprearranged clue-number, and then the message can be read off by thecode dictionary. When a code message is published by the Governmentreceiving it, the construction of the sentences is usually altered beforepublication, so that the key may not be discoverable by anyone in pos-session of the code-book or who has seen the cipher message. This isa rule applicable to all cryptographs and ciphers.
This is a cipher with 105 symbols, and as each symbol consists offive digits, a message of n words is denoted by 5n digits, and probablyis not longer than the message when written in the ordinary way. Sincehowever the number of words required is less than 105, the spare num-bers may be used to represent collocations of words which constantlyoccur, and if so the cipher message may be slightly shortened.
If the clue number is the same all through the message it would bepossible by not more than 105 trials to discover the message. This isnot a serious risk, but, slight though it is, it can be avoided if the cluenumber is varied; the clue number might, for instance, be 781 for thefirst three words, 791 for the next five words, 801 for the next sevenwords, and so on. Further it may be arranged that the clue numbersshall be changed every day; thus on the seventh day of the month theymight be 781, 791, &c., and on the eighth day 881, 891, &c., and so on.
This cipher can however be further improved by inserting at somestep, say after each mth digit, an unmeaning digit. For example, if,in the original message written in numbers, we insert a 9 after every
• Project • Gutenberg • #26839 •
CH. XI] CIPHERS 275
seven digits we shall get a collection of words (each represented byfive digits), most of which would have no connection with the originalmessage, and probably the number of digits used in the message itselfwould no longer be a multiple of 5. Of course the receiver has only toreverse the process in order to read the message.
It is however unnecessary to use five symbols for each word. Forif we make a similar code with the twenty-six letters of the alphabetinstead of the ten digits, four letters for each word or phrase would giveus 264, that is, 456976 possible variations. Thus the message would beshorter and the power of the code increased. Further, if we like to usethe ten digits and the twenty-six letters of the alphabet—all of whichare easily telegraphed—we could, by only using three symbols, obtain363, that is, 46656 possible words, which would be sufficient for allpractical purposes.
This code, at any rate with these modifications, is undecipher-able by strangers, but it has the disadvantages that those who use itmust always have the code dictionary available, and that it takes aconsiderable time to code or decode a communication. For practicalpurposes its use would be confined to communications which could bedeciphered at leisure in an office, It is especially suitable in the caseof communications between officials, each supplied with a competentstaff of secretaries or clerks—as from an ambassador to his chief, ora commander in the field to his war office. It is an excellent exampleof a cipher of the first type, but it is not clear that it possesses anysuperiority over some of the simple ciphers of the third type.
One of the best known writers on the subject of cryptographs andciphers is E.A. Poe, indeed probably a good many readers have madetheir first acquaintance with a cipher in his story of The Gold Bug,the interest of which turns on reading a simple cipher of the first type.In earlier times J. Tritheim of Spanheim, G. Porta of Naples, Cardan,Niceron, and J. Wilkins occupied much the same position, while when-ever ciphers were freely used skilful decipherers seem to have arisen.
Poe wrote an essay on cryptography in which he said that it may beroundly asserted that human ingenuity cannot concoct a cipher whichhuman ingenuity cannot resolve—a conclusion which is hardly justifiedby the known facts. In an earlier article he once made a similar re-mark so far as ciphers of the first class are concerned, with the impliedlimitation that only 26 symbols may be used. In this sense the obser-vation is correct. His assertion excited some attention, and numerous
• Project • Gutenberg • #26839 •
276 CRYPTOGRAPHS AND CIPHERS. [CH. XI
communications in cipher were sent to him. More than one of his cor-respondents did not play the game fairly, not only employing foreignlanguages, but using several different ciphers in the same communica-tion. Nevertheless he resolved all except one; and he proved that thislast was a fraud, being merely a jargon of random characters, havingno meaning whatever.
• Project • Gutenberg • #26839 •
CHAPTER XII.
HYPER-SPACE*.
I propose to devote the remaining pages to the consideration,from the point of view of a mathematician, of certain properties ofspace, time, and matter, and to a sketch of some hypotheses as to theirnature. I shall not discuss the metaphysical theories that profess to
* On the possibility of the existence of space of more than three dimensions seeC.H. Hinton, Scientific Romances, London, 1886, a most interesting work, fromwhich I have derived much assistance in compiling the earlier part of this chapter;his later work, The Fourth Dimension, London, 1904, may be also consulted.See also G.F. Rodwell, Nature, May 1, 1873, vol. viii, pp. 8, 9; and E.A. Abbott,Flatland, London, 1884.
The theory of Non-Euclidean geometry is due primarily to Lobatschewsky,Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin, 1840 (orig-inally given in a lecture in 1826); to Gauss (ex. gr. letters to Schumacher, May 17,1831, July 12, 1831, and Nov. 28, 1846, printed in Gauss’s collected works); andto J. Bolyai, Appendix to the first volume of his father’s Tentamen, Maros-Vasarkely, 1832; though the subject had been discussed by J. Saccheri as longago as 1733: its development was mainly the work of G.F.B. Riemann, Ueber dieHypothesen welche der Geometrie zu Grunde liegen, written in 1854, GottingerAbhandlungen, 1866–7, vol. xiii, pp. 131–152 (translated in Nature, May 1 and 8,1873, vol. viii, pp. 14–17, 36–37); H.L.F. von Helmholtz, Gottinger Nachrichten,June 3, 1868, pp. 193–221; and E. Beltrami, Saggio di Interpretazione della Ge-ometria non-Euclidea, Naples, 1868, and the Annali di Matematica, series 2,vol. ii, pp. 232–255: see an article by von Helmholtz in the Academy, Feb. 12,1870, vol. i, pp. 128–131. Within the last twenty-five years the theory has beentreated by several mathematicians.
A bibliography of hyper-space, compiled by G.B. Halsted, appeared in theAmerican Journal of Mathematics, vol. i (1878), pp. 261–276, 384–385; andvol. ii (1879), pp. 65–70.
277
• Project • Gutenberg • #26839 •
278 HYPER-SPACE. [CH. XII
account for the origin of our conceptions of them, for these theorieslead to no practical result and rest on assertions which are incapableof definite proof—a foundation which does not commend itself to ascientific student. Space, time, and matter cannot be defined; but themeans of measuring them and the investigation of their properties fallwithin the domain of mathematics.
I devote this chapter to considerations connected with space, leav-ing the subjects of time and mass to the following two chapters.
I shall confine my remarks on the properties of space to two spec-ulations which recently have attracted considerable attention. Theseare (i) the possibility of the existence of space of more than three di-mensions, and (ii) the possibility of kinds of geometry, especially of twodimensions, other than those which are treated in the usual text-books.These problems are related. The term hyper-space was used originallyof space of more than three dimensions, but now it is often employed todenote also any Non-Euclidean space. I attach the wider meaning to it,and it is in that sense that this chapter is on the subject of hyper-space.
In regard to the first of these questions, the conception of a worldof more than three dimensions is facilitated by the fact that there is nodifficulty in imagining a world confined to only two dimensions—whichwe may take for simplicity to be a plane, though equally well it might bea spherical or other surface. We may picture the inhabitants of flatlandas moving either on the surface of a plane or between two parallel andadjacent planes. They could move in any direction along the plane,but they could not move perpendicularly to it, and would have noconsciousness that such a motion was possible. We may suppose themto have no thickness, in which case they would be mere geometricalabstractions: or, preferably, we may think of them as having a smallbut uniform thickness, in which case they would be realities.
Several writers have amused themselves by expounding and illus-trating the conditions of life in such a world. To take a very simpleinstance, in flatland—or any even dimensional space—a knot is impos-sible, a simple alteration which alone would make some difference inthe experience of the inhabitants as compared with our own.
If an inhabitant of flatland was able to move in three dimensions, hewould be credited with supernatural powers by those who were unableso to move; for he could appear or disappear at will, could (so far asthey could tell) create matter or destroy it, and would be free from so
• Project • Gutenberg • #26839 •
CH. XII] HYPER-SPACE. 279
many constraints to which the other inhabitants were subject that hisactions would be inexplicable by them.
We may go one step lower, and conceive of a world of onedimension—like a long tube—in which the inhabitants could moveonly forwards and backwards. In such a universe there would be linesof varying lengths, but there could be no geometrical figures. To thosewho are familiar with space of higher dimensions, life in line-land wouldseem somewhat dull. It is commonly said that an inhabitant couldknow only two other individuals; namely, his neighbours, one on eachside. If the tube in which he lived was itself of only one dimension,this is true; but we can conceive an arrangement of tubes in two orthree dimensions, where an occupant would be conscious of motion inonly one dimension, and yet which would permit of more variety in thenumber of his acquaintances and conditions of existence.
Our conscious life is in three dimensions, and naturally the ideaoccurs whether there may not be a fourth dimension. No inhabitant offlatland could realize what life in three dimensions would mean, though,if he evolved an analytical geometry applicable to the world in which helived, he might be able to extend it so as to obtain results true of thatworld in three dimensions which would be to him unknown and incon-ceivable. Similarly we cannot realize what life in four dimensions is like,though we can use analytical geometry to obtain results true of thatworld, or even of worlds of higher dimensions. Moreover the analogyof our position to the inhabitants of flatland enables us to form someidea of how inhabitants of space of four dimensions would regard us.
Just as the inhabitants of flatland might be conceived as beingeither mere geometrical abstractions, or real and of a uniform thicknessin the third dimension, so, if there is a fourth dimension, we may beregarded either as having no thickness in that dimension, in which eventwe are mere (geometrical) abstractions—as indeed idealist philosophershave asserted to be the case—or as having a uniform thickness in thatdimension, in which event we are living in four dimensions although weare not conscious of it. In the latter case it is reasonable to supposethat the thickness in the fourth dimension of bodies in our world issmall and possibly constant; it has been conjectured also that it iscomparable with the other dimensions of the molecules of matter, andif so it is possible that the constitution of matter and its fundamentalproperties may supply experimental data which will give a physical
• Project • Gutenberg • #26839 •
280 HYPER-SPACE. [CH. XII
basis for proving or disproving the existence of this fourth dimension.
If we could look down on the inhabitants of flatland we could seetheir anatomy and what was happening inside them. Similarly an in-habitant of four-dimensional space could see inside us.
An inhabitant of flatland could get out of a room, such as a rectan-gle, only through some opening, but, if for a moment he could step intothree dimensions, he could reappear on the other side of any bound-aries placed to retain him. Similarly, if we came across persons whocould move out of a closed prison-cell without going through any of theopenings in it, there might be some reason for thinking that they did itby passing first in the direction of the fourth dimension and then backagain into our space. This however is unknown.
Again, if a finite solid was passed slowly through flatland, the in-habitants would be conscious only of that part of it which was in theirplane. Thus they would see the shape of the object gradually changeand ultimately vanish. In the same way, if a body of four dimensionswas passed through our space, we should be conscious of it only as asolid body, namely, the section of the body by our space, whose formand appearance gradually changed and perhaps ultimately vanished.It has been suggested that the birth, growth, life, and death of ani-mals may be explained thus as the passage of finite four-dimensionalbodies through our three-dimensional space. I believe that this ideais due to Mr Hinton.
The same argument is applicable to all material bodies. The im-penetrability and inertia of matter are necessary consequences; the con-servation of energy follows, provided that the velocity with which thebodies move in the fourth dimension is properly chosen: but the inde-structibility of matter rests on the assumption that the body does notpass completely through our space. I omit the details connected withchange of density as the size of the section by our space varies.
We cannot prove the existence of space of four dimensions, but itis interesting to enquire whether it is probable that such space actu-ally exists. To discuss this, first let us consider how an inhabitant offlatland might find arguments to support the view that space of threedimensions existed, and then let us see whether analogous argumentsapply to our world. I commence with considerations based on geometryand then proceed to those founded on physics.
Inhabitants of flatland would find that they could have two tri-
• Project • Gutenberg • #26839 •
CH. XII] HYPER-SPACE. 281
angles of which the elements were equal, element to element, and yetwhich could not be superposed. We know that the explanation of thisfact is that, in order to superpose them, one of the triangles would haveto be turned over so that its undersurface came on to the upper side,but of course such a movement would be to them inconceivable. Possi-bly however they might have suspected it by noticing that inhabitantsof one-dimensional space might experience a similar difficulty in com-paring the equality of two lines, ABC and CB′A′, each defined by a setof three points. We may suppose that the lines are equal and such thatcorresponding points in them could be superposed by rotation roundC—a movement inconceivable to the inhabitants—but an inhabitantof such a world in moving along from A to A′ would not arrive at thecorresponding points in the two lines in the same relative order, andthus might hesitate to believe that they were equal. Hence inhabitantsof flatland might infer by analogy that by turning one of the trianglesover through three-dimensional space they could make them coincide.
We have a somewhat similar difficulty in our geometry. We can con-struct triangles in three dimensions—such as two spherical triangles—whose elements are equal respectively one to the other, but which can-not be superposed. Similarly we may have two spirals whose elementsare equal respectively, one having a right-handed twist and the othera left-handed twist, but it is impossible to make one fill exactly thesame parts of space as the other does. Again, we may conceive of twosolids, such as a right hand and a left hand, which are exactly simi-lar and equal but of which one cannot be made to occupy exactly thesame position in space as the other does. Those are difficulties similarto those which would be experienced by the inhabitants of flatland incomparing triangles; and it may be conjectured that in the same wayas such difficulties in the geometry of an inhabitant in space of onedimension are explicable by temporarily moving the figure into spaceof two dimensions by means of a rotation round a point, and as suchdifficulties in the geometry of flatland are explicable by temporarilymoving the figure into space of three dimensions by means of a rota-tion round a line, so such difficulties in our geometry would disappearif we could temporarily move our figures into space of four dimensionsby means of a rotation round a plane—a movement which of courseis inconceivable to us.
Next we may enquire whether the hypothesis of our existence in
• Project • Gutenberg • #26839 •
282 HYPER-SPACE. [CH. XII
a space of four dimensions affords an explanation of any difficulties orapparent inconsistencies in our physical science*. The current concep-tion of the luminiferous ether, the explanation of gravity, and the factthat there are only a finite number of kinds of matter, all the atomsof each kind being similar, present such difficulties and inconsistencies.To see whether the hypothesis of a four-dimensional space gives anyaid to their elucidation, we shall do best to consider first the analogousproblems in two dimensions.
We live on a solid body, which is nearly spherical, and which movesround the sun under an attraction directed to it. To realize a corre-sponding life in flatland we must suppose that the inhabitants live onthe rim of a (planetary) disc which rotates round another (solar) discunder an attraction directed towards it. We may suppose that theplanetary world thus formed rests on a smooth plane, or other surfaceof constant curvature; but the pressure on this plane and even its ex-istence would be unknown to the inhabitants, though they would beconscious of their attraction to the centre of the disc on which theylived. Of course they would be also aware of the bodies, solid, liquid,or gaseous, which were on its rim, or on such points of its interior asthey could reach.
Every particle of matter in such a world would rest on this planemedium. Hence, if any particle was set vibrating, it would give upa part of its motion to the supporting plane. The vibrations thuscaused in the plane would spread out in all directions, and the planewould communicate vibrations to any other particles resting on it. Thusany form of energy caused by vibrations, such as light, radiant heat,electricity, and possibly attraction, could be transmitted from one pointto another without the presence of any intervening medium which theinhabitants could detect.
If the particles were supported on a uniform elastic plane film, theintensity of the disturbance at any other point would vary inverselyas the distance of the point from the source of disturbance; if on auniform elastic solid medium, it would vary inversely as the squareof that distance. But, if the supporting medium was vibrating, then,wherever a particle rested on it, some of the energy in the plane would
* See a note by myself in the Messenger of Mathematics, Cambridge, 1891,vol. xxi, pp. 20–24, from which the above argument is extracted. The ques-tion has been treated by Mr Hinton on similar lines.
• Project • Gutenberg • #26839 •
CH. XII] HYPER-SPACE. 283
be given up to that particle, and thus the vibrations of the interveningmedium would be hindered when it was associated with matter.
If the inhabitants of this two-dimensional world were sufficientlyintelligent to reason about the manner in which energy was transmittedthey would be landed in a difficulty. Possibly they might be unable toexplain gravitation between two particles—and therefore between thesolar disc and their disc—except by supposing vibrations in a rigidmedium between the two particles or discs. Again, they might be ableto detect that radiant light and heat, such as the solar light and heat,were transmitted by vibrations transverse to the direction from whichthey came, though they could realize only such vibrations as were intheir plane, and they might determine experimentally that in order totransmit such vibrations a medium of great rigidity (which we may callether) was necessary. Yet in both the above cases they would have alsodistinct evidence that there was no medium capable of resisting motionin the space around them, or between their disc and the solar disc. Theexplanation of these conflicting results lies in the fact that their universewas supported by a plane, of which they were necessarily unconscious,and that this rigid elastic plane was the ether which transmitted thevibrations.
Now suppose that the bodies in our universe have a uniform thick-ness in the fourth dimension, and that in that direction our universerests on a hom*ogeneous elastic body whose thickness in that directionis small and constant. The transmission of force and radiant energy,without the intervention of an intervening medium, may be explainedby the vibrations of the supporting space, even though the vibrationsare not themselves in the fourth dimension. Also we should find, as infact we do, that the vibrations of the luminiferous ether are hinderedwhen it is associated with matter. I have assumed that the thicknessof the supporting space is small and uniform, because then the inten-sity of the energy transmitted from a source to any point would varyinversely as the square of the distance, as is the case; whereas if thesupporting space was a body of four dimensions, the law would be thatof the inverse cube of the distance.
The application of this hypothesis to the third difficulty mentionedabove—namely, to show why there are in our universe only a finitenumber of kinds of atoms, all the atoms of each kind having in common
• Project • Gutenberg • #26839 •
284 HYPER-SPACE. [CH. XII
a number of sharply defined properties—will be given later*.
Thus the assumption of the existence of a four-dimensional hom*o-geneous elastic body on which our three-dimensional universe rests,affords an explanation of some difficulties in our physical science.
It may be thought that it is hopeless to try to realize a figure infour dimensions. Nevertheless attempts have been made to see whatthe sections of such a figure would look like.
If the boundary of a solid is ϕ(x, y, z) = 0, we can obtain some ideaof its form by taking a series of plane sections by planes parallel to z = 0,and mentally superposing them. In four dimensions the boundary ofa body would be ϕ(x, y, z, ω) = 0, and attempts have been made torealize the form of such a body by making models of a series of solidsin three dimensions formed by sections parallel to ω = 0. Again, we canrepresent a solid in perspective by taking sections by three co-ordinateplanes. In the case of a four-dimensional body the section by each ofthe four co-ordinate solids will be a solid, and attempts have been madeby drawing these to get an idea of the form of the body. Of course afour-dimensional body will be bounded by solids.
The possible forms of regular bodies in four dimensions, analogousto polyhedrons in space of three dimensions, have been discussed byMr Stringham†.
I now turn to the second of the two problems mentioned at thebeginning of the chapter: namely, the possibility of there being kindsof geometry other than those which are treated in the usual elementarytext-books. This subject is so technical that in a book of this natureI can do little more than give a sketch of the argument on which theidea is based.
The Euclidean system of geometry, with which alone most peopleare acquainted, rests on a number of independent axioms and postu-lates. Those which are necessary for Euclid’s geometry have, withinrecent years, been investigated and scheduled. They include not onlythose explicitly given by him, but some others which he unconsciouslyused. If these are varied, or other axioms are assumed, we get a differ-ent series of propositions, and any consistent body of such propositions
* See below, p. 318 (3).† American Journal of Mathematics, 1880, vol. iii, pp. 1–14.
• Project • Gutenberg • #26839 •
CH. XII] HYPER-SPACE. 285
constitutes a system of geometry. Hence there is no limit to the numberof possible Non-Euclidean geometries that can be constructed.
Among Euclid’s axioms and postulates is one on parallel lines,which is usually stated in the form that if a straight line meets twostraight lines, so as to make the two interior angles on the same sideof it taken together less than two right angles, then these straight linesbeing continually produced will at length meet upon that side on whichare the angles which are less than two right angles. Expressed in thisform the axiom is far from obvious, and from early times numerousattempts have been made to prove it*. All such attempts failed, andit is now known that the axiom cannot be deduced from the other ax-ioms assumed by Euclid. It can be replaced by other statements aboutparallel lines, such as that the distance between two parallel lines is al-ways the same, but such alternative statements, though perhaps primafacie more axiomatic, are not to be preferred to Euclid’s form, since hisstatement brings out prominently a characteristic feature of the spacewith which he is concerned.
The earliest conception of a body of Non-Euclidean geometry wasdue to the discovery, made independently by Saccheri, Lobatschewsky,and John Bolyai, that a consistent system of geometry of two dimen-sions can be produced on the assumption that the axiom on parallels isnot true, and that through a point a number of straight (that is, geode-tic) lines can be drawn parallel to a given straight line. The resultinggeometry is called hyperbolic.
Riemann later distinguished between boundlessness of space andits infinity, and showed that another consistent system of geometry oftwo dimensions can be constructed in which all straight lines are of a fi-nite length, so that a particle moving along a straight line will return toits original position. This leads to a geometry of two dimensions, calledelliptic geometry , analogous to the hyperbolic geometry, but character-ized by the fact that through a point no straight line can be drawnwhich, if produced far enough, will not meet any other given straightline. This can be compared with the geometry of figures drawn on thesurface of a sphere.
Thus according as no straight line, or only one straight line, or apencil of straight lines can be drawn through a point parallel to a given
* Some of the more interesting and plausible attempts have been collected byJ. Richard in his Philosophie de Mathematiques, Paris, 1903.
• Project • Gutenberg • #26839 •
286 HYPER-SPACE. [CH. XII
straight line, we have three systems of geometry of two dimensionsknown respectively as elliptic, parabolic or homaloidal or Euclidean,and hyperbolic.
In the parabolic and hyperbolic systems straight lines are infinitelylong. In the elliptic they are finite. In the hyperbolic system there areno similar figures of unequal size; the area of a triangle can be deducedfrom the sum of its angles, which is always less than two right angles;and there is a finite maximum to the area of a triangle. In the ellipticsystem all straight lines are of the same finite length; any two linesintersect; and the sum of the angles of a triangle is greater than tworight angles. In the elliptic system it is possible to get from one pointto a point on the other side of a plane without passing through theplane, namely, by going the other way round the straight line joiningthe two points; thus a watch-dial moving face upwards continuouslyforward in a plane in a straight line in the direction from the mark vito the mark xii will finally appear to a stationary observer with itsface downwards; and if originally the mark iii was to the right of theobserver it will finally be on his left hand.
In spite of these and other peculiarities of hyperbolic and ellipticalgeometries, it is impossible to prove by observation that one of themis not true of the space in which we live. For in measurements in eachof these geometries we must have a unit of distance; and if we live ina space whose properties are those of either of these geometries, andsuch that the greatest distances with which we are acquainted (ex. gr.the distances of the fixed stars) are immensely smaller than any unit,natural to the system, then it may be impossible for our observations todetect the discrepancies between the three geometries. It might indeedbe possible by observations of the parallaxes of stars to prove that theparabolic system and either the hyperbolic or elliptic system were false,but never can it be proved by measurements that Euclidean geometryis true. Similar difficulties might arise in connection with excessivelyminute quantities. In short, though the results of Euclidean geometryare more exact than present experiments can verify for finite things,such as those with which we have to deal, yet for much larger thingsor much smaller things or for parts of space at present inaccessible tous they may not be true.
If however we go a step further and ask what is meant by sayingthat a geometry is true or false, I can only quote the remark of Poincare,
• Project • Gutenberg • #26839 •
CH. XII] HYPER-SPACE. 287
that the selection of a geometry is really a matter of convenience, andthat that geometry is the best which enables us to state the physicallaws in the simplest form. This opinion has been strongly controverted,but at any rate it expresses one view of the question.
The above refers only to hyper-space of two dimensions. Natu-rally there arises the question whether there are different kinds of Non-Euclidean space of three or more dimensions. Riemann showed thatthere are three kinds of Non-Euclidean space of three dimensions hav-ing properties analogous to the three kinds of Non-Euclidean space oftwo dimensions already discussed. These are differentiated by the testwhether at every point no geodetical surface, or one geodetical surface,or a fasciculus of geodetical surfaces can be drawn parallel to a givensurface: a geodetical surface being defined as such that every geodeticline joining two points on it lies wholly on the surface. It may be addedthat each of the three systems of geometry of two dimensions describedabove may be deduced as properties of a surface in each of these threekinds of Non-Euclidean space of three dimensions.
It is evident that the properties of Non-Euclidean space of threedimensions are deducible only by the aid of mathematics, and cannotbe illustrated materially, for in order to realize or construct surfacesin Non-Euclidean space of two dimensions we think of or use modelsin space of three dimensions; similarly the only way in which we couldconstruct models illustrating Non-Euclidean space of three dimensionswould be by utilizing space of four dimensions.
We may proceed yet further and conceive of Non-Euclidean ge-ometries of more than three dimensions, but this remains, as yet, anunworked field.
Returning to the former question of Non-Euclidean geometries, Iwish again to emphasize the fact that, if the axioms enunciated byEuclid are replaced by others, it is possible to construct other con-sistent systems of geometry. Some of these are interesting, but thosewhich have been mentioned above have a special importance, from thesomewhat sensational fact that they lead to no results necessarily in-consistent with the properties, as far as we can observe them, of thespace in which we live; we are not at present acquainted with any othersystems which are consistent with our experience.
• Project • Gutenberg • #26839 •
CHAPTER XIII.
TIME AND ITS MEASUREMENT.
The problems connected with time are totally different in characterfrom those concerning space which I discussed in the last chapter. Ithere stated that the life of people living in space of one dimensionwould be uninteresting, and that probably they would find it impossibleto realize life in space of higher dimensions. In questions connected withtime we find ourselves in a somewhat similar position. Mentally, wecan realize a past and a future—thus going backwards and forwards—actually we go only forwards. Hence time is analogous to space of onedimension. Were our time of two dimensions, the conditions of ourlife would be infinitely varied, but we can form no conception of whatsuch a phrase means, and I do not think that any attempts have beenmade to work it out.
I shall concern myself here mainly with questions concerning themeasurement of time, and shall treat them rather from a historicalthan from a philosophical point of view.
In order to measure anything we must have an unalterable unit ofthe same kind, and we must be able to determine how often that unit iscontained in the quantity to be measured. Hence only those things canbe measured which are capable of addition to things of the same kind.
Thus to measure a length we may take a foot-rule, and by applyingit to the given length as often as is necessary, we shall find how manyfeet the length contains. But in comparing lengths we assume as theresult of experience that the length of the foot-rule is constant, or ratherthat any alteration in it can be determined; and, if this assumptionwas denied, we could not prove it, though, if numerous repetitions of
288
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 289
the experiment under varying conditions always gave the same result,probably we should feel no doubt as to the correctness of our method.
It is evident that the measurement of time is a more difficult matter.We cannot keep a unit by us in the same way as we can keep a foot-rule; nor can we repeat the measurement over and over again, for timeonce passed is gone for ever. Hence we cannot appeal directly to oursensations to justify our measurement. Thus, if we say that a certainduration is four hours, it is only by a process of reasoning that we canshow that each of the hours is of the same duration.
The establishment of a scientific unit for measuring durations hasbeen a long and slow affair. The process seems to have been as follows.Originally man observed that certain natural phenomena recurred afterthe interval of a day, say from sunrise to sunrise. Experience—forexample, the amount of work that could be done in it—showed thatthe length of every day was about the same, and, assuming that thiswas accurately so, man had a unit by which he could measure durations.The present subdivision of a day into hours, minutes, and seconds isartificial, and apparently is derived from the Babylonians.
Similarly a month and a year are natural units of time though it isnot easy to determine precisely their beginnings and endings.
So long as men were concerned merely with durations which wereexact multiples of these units or which needed only a rough estimate,this did very well; but as soon as they tried to compare the differentunits or to estimate durations measured by part of a unit they founddifficulties. In particular it cannot have been long before it was noticedthat the duration of the same day differed in different places, and thateven at the same place different days differed in duration at differenttimes of the year, and thus that the duration of a day was not aninvariable unit.
The question then arises as to whether we can find a fixed unit bywhich a duration can be measured, and whether we have any assurancethat the seconds and minutes used to-day for that purpose are all ofequal duration. To answer this we must see how a mathematicianwould define a unit of time. Probably he would say that experienceleads us to believe that, if a rigid body is set moving in a straightline without any external force acting on it, it will go on moving inthat line; and those times are taken to be equal in which it passes overequal spaces: similarly, if it is set rotating about a principal axis passing
• Project • Gutenberg • #26839 •
290 TIME AND ITS MEASUREMENT. [CH. XIII
through its centre of mass, those times are taken to be equal in whichit turns through equal angles. Our experiences are consistent withthis statement, and that is as high an authority as a mathematicianhopes to get.
The spaces and the angles can be measured, and thus durationscan be compared. Now the earth may be taken roughly as a rigid bodyrotating about a principal axis passing through its centre of mass, andsubject to no external forces affecting its rotation: hence the time ittakes to turn through four right angles, i.e. through 360◦, is alwaysthe same; this is called a sidereal day: the time to turn through onetwenty-fourth part of 360◦, i.e. through 15◦, is an hour: the time to turnthrough one-sixtieth part of 15◦, i.e. through 15′, is a minute: and so on.
If, by the progress of astronomical research, we find that there areexternal forces affecting the rotation of the earth, mathematics wouldhave to be invoked to find what the time of rotation would be if thoseforces ceased to act, and this would give us a correction to be appliedto the unit chosen. In the same way we may say that although anincrease of temperature affects the length of a foot-rule, yet its changeof length can be determined, and thus applied as a correction to thefoot-rule when it is used as the unit of length. As a matter of factthere is reason to think that the earth takes about one sixty-sixth of asecond longer to turn through four right angles now than it did 2500years ago, and thus the duration of a second is just a trifle longer to-day than was the case when the Romans were laying the foundationsof the power of their city.
The sidereal day can be determined only by refined astronomicalobservations and is not a unit suitable for ordinary purposes. Therelations of civil life depend mainly on the sun, and he is our naturaltime keeper. The true solar day is the time occupied by the earth inmaking one revolution on its axis relative to the sun; it is true noonwhen the sun is on the meridian. Owing to the motion of the sunrelative to the earth, the true solar day is about four minutes longerthan a sidereal day.
The true solar day is not however always of the same duration.This is inconvenient if we measure time by clocks (as now for nearlytwo centuries has been usual in Western Europe) and not by sun-dials,and therefore we take the average duration of the true solar day asthe measure of a day: this is called the mean solar day. Moreover
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 291
to define the noon of a mean solar day we suppose a point to moveuniformly round the ecliptic coinciding with the sun at each apse, andfurther we suppose a fictitious sun, called the mean sun, to move in thecelestial equator so that its distance from the first point of Aries is thesame as that of this point: it is mean noon when this mean sun is onthe meridian. The mean solar day is divided into hours, minutes, andseconds; and these are the usual units of time in civil life.
The time indicated by our clocks and watches is mean solar time;that marked on ordinary sun-dials is true solar time. The differencebetween them is the equation of time: this may amount at some periodsof the year to a little more than a quarter of an hour. In England wetake the Greenwich meridian as our origin for longitudes, and insteadof local mean solar time we take Greenwich mean solar time as thecivil standard.
Of course mean time is a comparatively recent invention. TheFrench were the last civilized nation to abandon the use of true time:this was in 1816.
Formerly there was no common agreement as to when the day be-gan. In parts of ancient Greece and in Japan the interval from sunriseto sunset was divided into 12 hours, and that from sunset to sunriseinto 12 hours. The Jews, Chinese, Athenians, and, for a long time, theItalians, divided their day into 24 hours, beginning at the hour of sun-set, which of course varies every day: this method is said to be still usedin certain villages near Naples, except that the day begins half-an-hourafter sunset—the clocks being re-set once a week. Similarly the Baby-lonians, Assyrians, Persians, and until recently the modern Greeks andthe inhabitants of the Balearic Islands counted the twenty-four hours ofthe day from sunrise. Until the middle of last century, the inhabitantsof Basle reckoned the twenty-four hours from our 11.0 p.m. The ancientEgyptians and Ptolemy counted the twenty-four hours from noon: thisis the practice of modern astronomers. In Western Europe the day istaken to begin at midnight—as was first suggested by Hipparchus—andis divided into two equal periods of twelve hours each.
The week of seven days is an artificial unit of time. It had its originin the East, and was introduced into the West by the Roman emper-ors, and, except during the French Revolution, has been subsequentlyin general use among civilized races. The names of the days are de-rived from the seven astrological planets, arranged, as was customary,
• Project • Gutenberg • #26839 •
292 TIME AND ITS MEASUREMENT. [CH. XIII
in the order of their apparent times of rotation round the earth, namely,Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon. Thetwenty-four hours of the day were dedicated successively to these plan-ets: and the day was consecrated to the planet of the first hour.
Thus if the first hour was dedicated to Saturn, the second wouldbe dedicated to Jupiter, and so on; but the day would be Saturn’s day.The twenty-fourth hour of Saturn’s day would be dedicated to Mars,thus the first hour of the next day would belong to the Sun; and theday would be Sun’s day. Similarly the next day would be Moon’s day;the next, Mars’s day; the next, Mercury’s day: the next, Jupiter’s day;and the next, Venus’s day.
The astronomical month is a natural unit of time depending on themotion of the moon, and containing about 291
2days. The months of the
calendar have been evolved gradually as convenient divisions of time,and their history is given in numerous astronomies. In the originalJulian arrangement the months in a leap year contained alternately31 and 30 days, while in other years February had 29 days. This wasaltered by Augustus in order that his month should not be inferior toone named after his uncle.
The solar tropical year is another natural unit of time. Accordingto a recent determination, it contains 365.242216 days, that is, 365d.
5h. 48m. 47s..4624.The Egyptians knew that it contained between 365 and 366 days,
but the Romans did not profit by this information, for Numa is saidto have reckoned 355 days as constituting a year—extra months beingoccasionally intercalated, so that the seasons might recur at about thesame period of the year.
In 46 b.c. Julius Caesar decreed that thenceforth the year shouldcontain 365 days, except that in every fourth or leap year one additionalday should be introduced. He ordered this rule to come into force onJanuary 1, 45 b.c. The change was made on the advice of Sosigenesof Alexandria.
It must be remembered that the year 1 a.d. follows immediately1 b.c., that is, there is no year 0, and thus 45 b.c. would be a leapyear. All historical dates are given now as if the Julian calendar wasreckoned backwards as well as forwards from that year*. As a matter offact, owing to a mistake in the original decree, the Romans, during the
* Herschel, Astronomy, London, 11th ed. 1871, arts. 916–919.
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 293
first 36 years after 45 b.c., intercalated the extra day every third year,thus producing an error of 3 days. This was remedied by Augustus, whodirected that no intercalation of an extra day should be made in any ofthe twelve years a.u.c. 746 to 757 inclusive, but that the intercalationshould be again made in the year a.u.c. 761 (that is, 8 a.d.) andevery succeeding fourth year.
The Julian calendar made the year, on an average, contain 365.25days. The actual value is, very approximately, 365.242216 days. Hencethe Julian year is too long by about 111
4minutes: this produces an
error of nearly one day in 128 years. If the extra day in every thirty-second leap year had been omitted—as was suggested by some unknownPersian astronomer—the error would have been less than one day in100,000 years. It may be added that Sosigenes was aware that his rulemade the year slightly too long.
The error in the Julian calendar of rather more than eleven min-utes a year gradually accumulated, until in the sixteenth century theseasons arrived some ten days earlier than they should have done. In1582 Gregory XIII corrected this by omitting ten days from that year,which therefore contained only 355 days. At the same time he decreedthat thenceforth every year which was a multiple of a century shouldbe or not be a leap year according as the multiple was or was notdivisible by four.
The fundamental idea of the reform was due to Lilius, who diedbefore it was carried into effect. The work of framing the new calendarwas entrusted to Clavius, who explained the principles and necessaryrules in a prolix but accurate work* of over 700 folio pages. The planadopted was due to a suggestion of Pitatus made in 1552 or perhaps1537: the alternative and more accurate proposal of Stoffler, made in1518, to omit one day in every 134 years being rejected by Lilius andClavius for reasons which are not known.
Clavius believed the year to contain 365.2425432 days, but heframed his calendar so that a year, on the average, contained 365.2425days, which he thought to be wrong by one day in 3323 years: in realityit is a trifle more accurate than this, the error amounting to one dayin about 3600 years.
The change was unpopular, but Riccioli† tells us that, as those
* Romani Calendarii a Greg. XIII, restituti Explicatio, Rome, 1603.† Chronologia Reformata, Bonn, 1669, vol. ii, p. 206.
• Project • Gutenberg • #26839 •
294 TIME AND ITS MEASUREMENT. [CH. XIII
miracles which take place on fixed dates—ex. gr. the liquefaction ofthe blood of S. Januarius—occurred according to the new calendar,the papal decree was presumed to have a divine sanction—Deo ipsohuic correctioni Gregorianae subscribente—and was accepted as a nec-essary evil.
In England a bill to carry out the same reform was introduced in1584, but was withdrawn after being read a second time; and the changewas not finally effected till 1752, when eleven days were omitted fromthat year. In Roman Catholic countries the new style was adoptedin 1582. In Scotland the change was made in 1600. In the GermanLutheran States it was made in 1700. In England, as I have said above,it was introduced in 1752; and in Ireland it was made in 1782. It iswell known that the Greek Church still adheres to the Julian calendar.
The Mohammedan year contains 12 lunar months, or 35413
days,and thus has no connection with the seasons.
The Gregorian change in the calendar was introduced in order tokeep Easter at the right time of year. The date of Easter depends onthat of the vernal equinox, and as the Julian calendar made the year ofan average length of 365.25 days instead of 365.242216 days, the vernalequinox came earlier and earlier in the year, and in 1582 had regrededto within about ten days of February.
The rule for determining Easter is as follows*. In 325 the NiceneCouncil decreed that the Roman practice should be followed; and after463 (or perhaps, 530) the Roman practice required that Easter-dayshould be the first Sunday after the full moon which occurs on or nextfollowing the vernal equinox—full moon being assumed to occur onthe fourteenth day from the day of the preceding new moon (thoughas a matter of fact it occurs on an average after an interval of rathermore than 143
4days), and the vernal equinox being assumed to fall on
March 21 (though as a matter of fact it sometimes falls on March 22).This rule and these assumptions were retained by Gregory on the
ground that it was inexpedient to alter a rule with which so manytraditions were associated; but, in order to save disputes as to theexact instant of the occurrence of the new moon, a mean sun and amean moon defined by Clavius were used in applying the rule. Oneconsequence of using this mean sun and mean moon and giving an
* De Morgan, Companion to the Almanac, London, 1845, pp. 1–36; Ibid., 1846,pp. 1–10.
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 295
artificial definition of full moon is that it may happen, as it did in 1818and 1845, that the actual full moon occurs on Easter Sunday. In theBritish Act, 24 Geo. II. cap. 23, the explanatory clause which definesfull moon is omitted, but practically full moon has been interpretedto mean the Roman ecclesiastical full moon; hence the Anglican andRoman rules are the same. Until 1774 the German Lutheran Statesemployed the actual sun and moon. Had full moon been taken to meanthe fifteenth day of the moon, as is the case in the civil calendar, thenthe rule might be given in the form that Easter-day is the Sunday onor next after the calendar full moon which occurs next after March 21.
Assuming that the Gregorian calendar and tradition are used, therestill remains one point in this definition of Easter which might lead todifferent nations keeping the feast at different times. This arises fromthe fact that local time is introduced. For instance the difference oflocal time between Rome and London is about 50 minutes. Thus theinstant of the first full moon next after the vernal equinox might occurin Rome on a Sunday morning, say at 12.30 a.m., while in England itwould still be Saturday evening, 11.40 p.m., in which case our Easterwould be one week earlier than at Rome. Clavius foresaw the difficulty,and the Roman Communion all over the world keep Easter on that dayof the month which is determined by the use of the rule at Rome. Butpresumably the British Parliament intended time to be determined bythe Greenwich meridian, and if so the Anglican and Roman dates forEaster might differ by a week; whether such a case has ever arisen orbeen discussed I do not know, and I leave to ecclesiastics to say howit should be settled.
The usual method of calculating the date on which Easter-day fallsin any particular year is involved, and possibly the following simplerule* may be unknown to some of my readers.
Let m and n be numbers as defined below. (i) Divide the numberof the year by 4, 7, 19; and let the remainders be a, b, c, respectively.(ii) Divide 19c + m by 30, and let d be the remainder. (iii) Divide2a + 4b + 6d + n by 7, and let e be the remainder. (iv) Then theEaster full moon occurs d days after March 21; and Easter-day is the(22 + d + e)th of March or the (d + e− 9)th day of April, except thatif the calculation gives d = 29 and e = 6 (as happens in 1981) then
* It is due to Gauss, and his proof is given in Zach’s Monatliche Correspondenz,August, 1800, vol. ii, pp. 221–230.
• Project • Gutenberg • #26839 •
296 TIME AND ITS MEASUREMENT. [CH. XIII
Easter-day is on April 19 and not on April 26, and if the calculationgives d = 28, e = 6, and also c > 10 (as happens in 1954) then Easter-day is on April 18 and not on April 25, that is, in these two casesEaster falls one week earlier than the date given by the rule. Thesetwo exceptional cases cannot occur in the Julian calendar, and in theGregorian calendar they occur only very rarely. It remains to state thevalues of m and n for the particular period. In the Julian calendar wehave m = 15, n = 6. In the Gregorian calendar we have, from 1582to 1699 inclusive, m = 22, n = 2; from 1700 to 1799, m = 23, n = 3;from 1800 to 1899, m = 23, n = 4; from 1900 to 2099, m = 24, n = 5;from 2100 to 2199, m = 24, n = 6; from 2200 to 2299, m = 25, n = 0;from 2300 to 2399, m = 26, n = 1; and from 2400 to 2499, m = 25,n = 1. Thus for the year 1908 we have m = 24, n = 5; hence a = 0,b = 4, c = 8; d = 26; and e = 2: therefore Easter Sunday will be onthe 19th of April. After the year 4200 the form of the rule will haveto be slightly modified.
The dominical letter and the golden number of the ecclesiasticalcalendar can be at once determined from the values of b and c. Theepact, that is, the moon’s age at the beginning of the year, can bealso easily calculated from the above data in any particular case; thegeneral formula was given by Delambre, but its value is required sorarely by any but professional astronomers and almanack-makers thatit is unnecessary to quote it here.
We can evade the necessity of having to recollect the values of mand n by noticing that, if N is the given year, and if {N/x} denotesthe integral part of the quotient when N is divided by x, then m is theremainder when 15 + ξ is divided by 30, and n is the remainder when6 + η is divided by 7: where, in the Julian calendar, ξ = 0, and η = 0;and, in the Gregorian calendar, ξ = {N/100} − {N/400} − {N/300},and η = {N/100} − {N/400} − 2.
If we use these values of m and n, and if we put for a, b, c, theirvalues, namely, a = N −4{N/4}, b = N −7{N/7}, c = N −19{N/19},the rule given on the previous page takes the following form. “Divide19N −{N/19}+ 15 + ξ by 30, and let the remainder be d. Next divide6(N + d + 1) − {N/4} + η by 7, and let the remainder be e. ThenEaster full moon is on the dth day after March 21, and Easter-day ison the (22 + d + e)th of March or the (d + e − 9)th of April as thecase may be; except that if the calculation gives d = 29, and e = 6,
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 297
or if it gives d = 28, e = 6, and c > 10, then Easter-day is on the(d + e − 16)th of April.”
Thus, if N = 1899, we divide 19(1899)−99+15+(18−4−6) by 30,which gives d = 5, and then we proceed to divide 6(1899+5+1)− 474+(18−4−2) by 7, which gives e = 6: therefore Easter-day is on April 2.
The above rules cover all the cases worked out with so much labourby Clavius and others*.
I may add here a rule, quoted by Zeller, for determining the dayof the week corresponding to any given date. Suppose that the pthday of the qth month of the year N anno domini is the rth day of theweek, reckoned from the preceding Saturday. Then r is the remainderwhen p + 2q + {3(q + 1)/5}+ N + {N/4}− η is divided by 7; providedJanuary and February are reckoned respectively as the 13th and 14thmonths of the preceding year.
For instance, Columbus first landed in the New World on Oct. 12,1492. Here p = 12, q = 10, N = 1492, η = 0. If we divide 12 + 20 + 6+ 1492 + 373 by 7 we get r = 6; hence it was on a Friday. Again,Charles I was executed on Jan. 30, 1649. Here p = 30, q = 13,N = 1648, η = 0, and we find r = 3; hence it was on a Tuesday.As another example, the battle of Waterloo was fought on June 18,1815. Here p = 18, q = 6, N = 1815, η = 12, and we find r = 1;hence it took place on a Sunday.
I proceed now to give a short account of some of the means ofmeasuring time which were formerly in use.
Of devices for measuring time, the earliest of which we have anypositive knowledge are the styles or gnomons erected in Egypt andAsia Minor. These were sticks placed vertically in a horizontal pieceof ground, and surrounded by three concentric circles, such that everytwo hours the end of the shadow of the stick passed from one circle toanother. Some of these have been found at Pompeii and Tusculum.
The sun-dial is not very different in principle. It consists of a rodor style fixed on a plate or dial; usually, but not necessarily, the style isplaced so as to be parallel to the axis of the earth. The shadow of the
* Most of the above-mentioned facts about the calendar are taken from Delambre’sAstronomie, Paris, 1814, vol. iii, chap. xxxviii; and his Histoire de l’astronomiemoderne, Paris, 1821, vol. i, chap. i: see also A. De Morgan, The Book ofAlmanacs, London, 1851; S. Butcher, The Ecclesiastical Calendar, Dublin, 1877;and C. Zeller, Acta Mathematica, Stockholm, 1887, vol. ix, pp. 131–136: on thechronological details see J.L. Ideler, Lehrbuch der Chronologie, Berlin, 1831.
• Project • Gutenberg • #26839 •
298 TIME AND ITS MEASUREMENT. [CH. XIII
style cast on the plate by the sun falls on lines engraved there whichare marked with the corresponding hours.
The earliest sun-dial, of which I have read, is that made by Berosusin 540 b.c. One was erected by Meton at Athens in 433 b.c. Thefirst sun-dial at Rome was constructed by Papirius Cursor in 306 b.c.Portable sun-dials, with a compass fixed in the face, have been longcommon in the East as well as in Europe. Other portable instrumentsof a similar kind were in use in medieval Europe, notably the sun-rings,hereafter described, and the sun-cylinders*.
I believe it is not generally known that a sun-dial can be so con-structed that the shadow will, for a short time near sunrise and sunset,move backwards on the dial†. This was discovered by Nonez. The ex-planation is as follows. Every day the sun appears to describe a circleround the pole, and the line joining the point of the style to the sundescribes a right cone whose axis points to the pole. The section ofthis cone by the dial is the curve described by the extremity of theshadow, and is a conic. In our latitude the sun is above the horizonfor only part of the twenty-four hours, and therefore the extremity ofthe shadow of the style describes only a part of this conic. Let QQ′
be the arc described by the extremity of the shadow of the style fromsunrise at Q to sunset at Q′, and let S be the point of the style andF the foot of the style, i.e. the point where the style meets the planeof the dial. Suppose that the dial is placed so that the tangents drawnfrom F to the conic QQ′ are real, and that P and P ′, the points ofcontact of these tangents, lie on the arc QQ′. If these two conditionsare fulfilled, then the shadow will regrede through the angle QFP asits extremity moves from Q to P , it will advance through the anglePFP ′ as its extremity moves from P to P ′, and it will regrede throughthe angle P ′FQ′ as its extremity moves from P ′ to Q′.
If the sun’s apparent diurnal path crosses the horizon—as alwayshappens in temperate and tropical latitudes—and if the plane of thedial is horizontal, the arc QQ′ will consist of the whole of one branchof a hyperbola, and the above conditions will be satisfied if F is withinthe space bounded by this branch of the hyperbola and its asymptotes.
* Thus Chaucer in the Shipman’s Tale, “by my chilindre it is prime of day,” andLydgate in the Siege of Thebes, “by my chilyndre I gan anon to see. . . that itdrew to nine.”
† Ozanam, 1803 edition, vol. iii, p. 321; 1840 edition, p. 529.
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 299
As a particular case, in a place of latitude 12◦ N. on a day when thesun is in the northern tropic (of Cancer) the shadow on a dial whoseface is horizontal and style vertical will move backwards for about twohours between sunrise and noon.
If, in the case of a given sun-dial placed in a certain position, theconditions are not satisfied, it will be possible to satisfy them by tiltingthe sun-dial through an angle properly chosen. This was the rational-istic explanation, offered by the French encyclopaedists, of the miraclerecorded in connection with Isaiah and Hezekiah*. Suppose, for in-stance, that the style is perpendicular to the face of the dial. Drawthe celestial sphere. Suppose that the sun rises at M and culminatesat N , and let L be a point between M and N on the sun’s diurnalpath. Draw a great circle to touch the sun’s diurnal path MLN at L,let this great circle cut the celestial meridian in A and A′, and of thearcs AL, A′L suppose that AL is the less and therefore is less thana quadrant. If the style is pointed to A, then, while the sun is ap-proaching L, the shadow will regrede, and after the sun passes L theshadow will advance. Thus if the dial is placed so that a style whichis normal to it cuts the meridian midway between the equator and thetropic, then between sunrise and noon on the longest day the shadowwill move backwards through an angle
sin−1(cos ω sec 12ω)− cot−1{sin ω cos(l − 1
2ω)(cos2 l − sin2 ω)−
12} ,
where l is the latitude of the place and ω is the obliquity of the ecliptic.The above remarks refer to the sun-dials in ordinary use. In 1892
General Oliver brought out in London a dial with a solid style, thesection of the style being a certain curve whose form was determinedempirically by the value of the equation of time as compared with thesun’s declination†. The shadow of the style on the dial gives the localmean time, though of course in order to set the dial correctly at anyplace the latitude of the place must be known: the dial may be alsoset so as to give the mean time at any other locality whose longituderelative to the place of observation is known.
The sun-ring or ring-dial is another instrument for measuring solartime‡. One of the simplest type is figured in the accompanying diagram.
* 2 Kings, chap. xx, vv. 9–11.† An account of this sun-dial with a diagram was given in Knowledge, July 1,
1892, pp. 133, 134.‡ See Ozanam, 1803 edition, vol. iii, p. 317; 1840 edition, p. 526.
• Project • Gutenberg • #26839 •
300 TIME AND ITS MEASUREMENT. [CH. XIII
The sun-ring consists of a thin brass band, about a quarter of an inchwide, bent into the shape of a circle, which slides between two fixedcircular rims—the radii of the circles being about one inch. At onepoint of the band there is a hole; and when the ring is suspended froma fixed point attached to the rims so that it hangs in a vertical planecontaining the sun, the light from the sun shines through this hole andmakes a bright speck on the opposite inner or concave surface of thering. On this surface the hours are marked, and, if the ring is properlyadjusted, the spot of light will fall on the hour which indicates thesolar time. The adjustment for the time of year is made as follows.The rims between which the band can slide are marked on their outeror convex side with the names of the months, and the band containingthe hole must be moved between the rims until the hole is opposite tothat month for which the ring is being used.
For determining times near noon the instrument is reliable, but forother hours in the day it is accurate only if the time of year is properlychosen, usually near one of the equinoxes. This defect may be correctedby marking the hours on a curved brass band affixed to the concavesurface of the rims. I possess two specimens of rings of this kind. These
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 301
rings were distributed widely. Of my two specimens, one was bought inthe Austrian Tyrol and the other in London. Astrolabes and sea-ringscan be used as sun-rings.
Clepsydras or water-clocks, and hour-glasses or sand-clocks, affordother means of measuring time. The time occupied by a given amountof some liquid or sand in running through a given orifice under the sameconditions is always the same, and by noting the level of the liquidwhich has run through the orifice, or which remains to run through it,a measure of time can be obtained.
The burning of graduated candles gives another way of measuringtime, and we have accounts of those used by Alfred the Great for thepurpose. Incense sticks were used by the Chinese in a similar way.
Modern clocks and watches* comprise a train of wheels turned bya weight, spring, or other motive power, and regulated by a pendulum,balance, fly-wheel, or other moving body whose motion is periodic andtime of vibration constant. The direction of rotation of the hands of aclock was selected originally so as to make the hands move in the samedirection as the shadow on a sun-dial whose face is horizontal—the dialbeing situated in our hemisphere.
The invention of clocks with wheels is attributed by tradition toPacificus of Verona, circ. 850, and also to Gerbert, who is said to havemade one at Magdeburg in 996: but there is reason to believe that thesewere sun-clocks. The earliest wheel-clock of which we have historicalevidence was one sent by the Sultan of Egypt in 1232 to the EmperorFrederick II, though there seems to be no doubt that they had beenmade in Italy at least fifty years earlier.
The oldest clock in England of which we know anything was oneerected in 1288 in or near Westminster Hall out of a fine imposed ona corrupt Lord Chief Justice. The bells, and possibly the clock, werestaked by Henry VIII on a throw of dice and lost, but the site wasmarked by a sun-dial, destroyed about sixty years ago, and bearing theinscription Discite justiciam moniti. In 1292 a clock was erected inCanterbury Cathedral at a cost of £30. One erected at GlastonburyAbbey in 1325 is at present in the Kensington Museum and is in regularaction. Another made in 1326 for St Alban’s Abbey showed the astro-nomical phenomena, and seems to have been one of the earliest clocksthat did so. One put up at Dover in 1348 is still in good working order.
* See Clock and Watch Making by Lord Grimthorpe, 7th edition, London, 1883.
• Project • Gutenberg • #26839 •
302 TIME AND ITS MEASUREMENT. [CH. XIII
The clocks at Peterborough and Exeter were of about the same date,and portions of them remain in situ. Most of these early clocks wereregulated by horizontal balances: pendulums being then unknown. Ofthe elaborate clocks of a later date, that at Strasburg made by Dasy-podius in 1571, and that at Lyons constructed by Lippeus in 1598, areespecially famous: the former was restored in 1842, though in a mannerwhich destroyed most of the ancient works.
In 1370, Vick constructed a clock for Charles V with a weight asmotive power and a vibrating escapement—a great improvement onthe rough time-keepers of an earlier date.
The earliest clock regulated by a pendulum seems to have beenmade in 1621 by a clockmaker named Harris, of Covent Garden, Lon-don, but the theory of such clocks is due to Huygens*. Galileo had dis-covered previously the isochronism of a pendulum, but did not applyit to the regulation of the motion of clocks. Hooke made such clocks,and possibly discovered independently this use of the pendulum: heinvented or re-invented the anchor pallet.
A watch may be defined as a clock which will go in any position.Watches, though of a somewhat clumsy design, were made at Nurem-berg by P. Hele early in the sixteenth century—the motive power beinga ribbon of steel, wound round a spindle, and connected at one endwith a train of wheels which it turned as it unwound—and possiblya few similar time-pieces had been made in the previous century. Bythe end of the sixteenth century they were not uncommon. At thistime they were usually made in the form of fanciful ornaments suchas skulls, or as large pendants, but about 1620 the flattened oval formwas introduced, rendering them more convenient to carry in a pocketor about the person. In the seventeenth century their construction wasgreatly improved, notably by the introduction of the spring balance byHuygens in 1674, and independently by Hooke in 1675—both mathe-maticians having discovered that small vibrations of a coiled spring, ofwhich one end is fixed, are practically isochronous. The fusee had beenused by R. Zech of Prague in 1525, but was re-invented by Hooke.
Clocks and watches are usually moved and regulated in the mannerindicated above. Other motive powers and other devices for regulatingthe motion may be met with occasionally. Of these I may mention aclock in the form of a cylinder, usually attached to another weight as
* Horologium Oscillatorium, Paris, 1673.
• Project • Gutenberg • #26839 •
CH. XIII] TIME AND ITS MEASUREMENT. 303
in Atwood’s machine, which rolls down an inclined plane so slowly thatit takes twelve hours to roll down, and the highest point of the facealways marks the proper hour*.
A water-clock made on a somewhat similar plan is described byOzanam† as one of the sights of Paris at the beginning of the lastcentury. It was formed of a hollow cylinder divided into various com-partments each containing some mercury, so arranged that the cylinderdescended with uniform velocity between two vertical pillars on whichthe hours were marked at equidistant intervals.
Other ingenious ways of concealing the motive power have beendescribed in the columns of La Nature‡. Of such mysterious timepiecesthe following are not uncommon examples, and probably are known tomost readers of this book. One kind of clock consists of a glass dialsuspended by two thin wires; the hands however are of metal, and theworks are concealed in them or in the pivot. Another kind is madeof two sheets of glass in a frame containing a spring which gives tothe hinder sheet a very slight oscillatory motion–imperceptible excepton the closest scrutiny–and each oscillation moves the hands throughthe requisite angles. Some so-called perpetual motion timepieces weredescribed above on pages 77–78. Lastly, I have seen in France a clockthe hands of which were concealed at the back of the dial, and carriedsmall magnets; pieces of steel in the shape of insects were placed on thedial, and, following the magnets, served to indicate the time.
The position of the sun relative to the points of the compass de-termines the solar time. Conversely, if we take the time given by awatch as being the solar time—and it will differ from it by only a fewminutes at the most—and we observe the position of the sun, we canfind the points of the compass§. To do this it is sufficient to point thehour-hand to the sun, and then the direction which bisects the anglebetween the hour and the figure xii will point due south. For instance,if it is four o’clock in the afternoon, it is sufficient to point the hour-hand (which is then at the figure iiii) to the sun, and the figure iion the watch will indicate the direction of south. Again, if it is eight
* Ozanam, 1803 edition, vol. ii, p. 39; 1840 edition, p. 212; or La Nature, Jan. 23,1892, pp. 123, 124.
† Ozanam, 1803 edition, vol. ii, p. 68; 1840 edition, p. 225.‡ See especially the volumes issued in 1874, 1877, and 1878.§ The rule is given by W.H. Richards, Military Topography, London, 1883, p. 31,
though it is not stated quite correctly. I do not know who first enunciated it.
• Project • Gutenberg • #26839 •
304 TIME AND ITS MEASUREMENT. [CH. XIII
o’clock in the morning, we must point the hour-hand (which is thenat the figure viii) to the sun, and the figure x on the watch gives thesouth point of the compass.
Between the hours of six in the morning and six in the evening theangle between the hour and xii which must be bisected is less than180◦, but at other times the angle to be bisected is greater than 180◦;or perhaps it is simpler to say that at other times the rule gives thenorth point and not the south point.
The reason is as follows. At noon the sun is due south, and it makesone complete circuit round the points of the compass in 24 hours. Thehour-hand of a watch also makes one complete circuit in 12 hours.Hence, if the watch is held in the plane of the ecliptic with its faceupwards, and the figure xii on the dial is pointed to the south, boththe hour-hand and the sun will be in that direction at noon. Both moveround in the same direction, but the angular velocity of the hour-handis twice as great as that of the sun. Hence the rule. The greatest errordue to the neglect of the equation of time is less than 2◦. Of course inpractice most people, instead of holding the face of the watch in theecliptic, would hold it horizontal, and in our latitude no serious errorwould be thus introduced.
In the southern hemisphere where at noon the sun is due norththe rule requires modification. In such places the hour-hand of a watch(held face upwards in the plane of the ecliptic) and the sun move inopposite directions. Hence, if the watch is held so that the figure xiipoints to the sun, then the direction which bisects the angle betweenthe hour of the day and the figure xii will point due north.
• Project • Gutenberg • #26839 •
CHAPTER XIV.
MATTER AND ETHER THEORIES.
Matter, like space and time, cannot be defined, but either thestatement that matter is whatever occupies space or the statementthat it is anything which can be seen, touched, or weighed, suggests itsmore important characteristics to anyone already familiar with it.
The means of measuring matter and some of its properties aretreated in most text-books on mechanics, and I do not propose to dis-cuss them. I confine the chapter to an account of some of the hypothesesby physicists as to the ultimate constitution of matter, but I excludemetaphysical conjectures, which from their nature are mere assertionsincapable of proof and are not subject to mathematical analysis. Thequestion is intimately associated with the explanation of the phenom-ena of attraction, light, chemistry, electricity, and other branches ofphysics.
I commence with a list of some of the more plausible of the hy-potheses formerly proposed which accounted for the obvious propertiesof matter, and shall then discuss how far they explain or are consis-tent with other facts*. The interest of the list is largely historical, forwithin the last few years new views as to the constitution of matter
* I have based my account mainly on Recent Advances in Physical Science, byP.G. Tait, Edinburgh, 1876 (chaps, xii, xiii); and on the article Atom by J. ClerkMaxwell in the Encyclopaedia Britannica or his Collected Works, vol. ii, pp. 445–484: see also W.M. Hicks’s address, Report of the British Association (Ipswichmeeting), 1895, vol. lxv, pp. 595–606. For the more recent speculations seeJ.J. Thomson, Electricity and Matter, Westminster, 1904, and J. Larmor, Aetherand Matter, Cambridge, 1900.
305
• Project • Gutenberg • #26839 •
306 MATTER AND ETHER THEORIES. [CH. XIV
have been propounded, but the details of these more recent hypothesesare so complicated and technical that only professional mathematicianscan understand them. Accordingly I allude to them only briefly.
I. Hypothesis of Continuous Matter. It may be sup-posed that matter is hom*ogeneous and continuous, in which case thereis no limit to the infinite divisibility of bodies. This view was heldby Descartes*.
This conjecture is consistent with the facts deducible by untrainedobservation, but there are many other phenomena for which it doesnot account; moreover there seems to be no way of reconciling such astructure of matter either with the facts of chemical changes or with theresults of spectrum analysis. At any rate the theory must be regardedas extremely improbable.
II. Atomic Theories. If matter is not continuous we mustsuppose that every body is composed of aggregates of molecules. If so,it seems probable that each such molecule is built up by the associationof two or more atoms, that the number of kinds of atoms is finite, andthat the atoms of any particular kind are alike. As to the nature of theatoms the following hypotheses have been made.
(i) Popular Atomic Hypothesis. The popular view is that everyatom of any particular kind is a minute indivisible article possessingdefinite qualities, everlasting in its form and properties, and infinitelyhard.
This statement is plausible, but the difficulties to which it leadsappear to be insuperable. In fact we have reason to think that the atomswhich form a molecule are composite systems in incessant vibration ata rate characteristic of the molecule, and it is most probable that theyare elastic.
Newton seems to have hazarded a conjecture of this kind whenhe suggested† that the difficulties, connected with the fact that thevelocity of sound was one-ninth greater than that required by theory,might be overcome if the particles of air were little rigid spheres whosedistance from one another under normal conditions was nine times the
* Descartes, Principia, vol. ii, pp. 18, 23.† Newton, Principia, bk. ii, prop. 50.
• Project • Gutenberg • #26839 •
CH. XIV] MATTER AND ETHER THEORIES. 307
diameter of any one of them. This was ingenious, but obviously theview is untenable, because, if such a structure of air existed, the aircould not be compressed beyond a certain limit, namely, about 1/1021stpart of its original volume, which has been often exceeded. The trueexplanation of the difficulty noticed by Newton was given by Laplace.
(ii) Boscovich’s Hypothesis. In 1759 Boscovich suggested* thatthe facts might be explained by supposing that an atom was an in-finitely small indivisible mass which was a centre of force—the law offorce being attractive for sensible distances, alternately attractive andrepulsive for minute distances, and repulsive for infinitely small dis-tances. In this theory all action between bodies is action at a distance.
He explained the apparent extension of bodies by saying that twoparts are consecutive (or similarly that two bodies are in contact) whenthe nearest pair of atoms in them are so close to one another that therepulsion at any point between them is sufficiently great to prevent anyother atom coming between them. It is essential to the theory that theatom shall have a mass but shall not have dimensions.
This hypothesis is not inconsistent with any known facts, but it hasbeen described, perhaps not unjustly, as a mere mathematical fiction,and certainly it is opposed to the apparent indications of our senses.At any rate it is artificial, though it may be a prejudice to regard thatas an argument against its adoption. To some extent this view wasaccepted by Faraday.
Lord Kelvin, better known as Sir William Thomson, has shown†
that, if we assume the existence of gravitation, then each of the abovehypotheses will account for cohesion.
(iii) Hypothesis of an Elastic Solid Ether. Some physicists havetried to explain the known phenomena by properties of the mediumthrough which our impressions are derived. By postulating that allspace is filled with a medium possessed of many of the characteristicsof an elastic solid, it has been shown by Fresnel, Green, Cauchy, Neu-mann, MacCullagh, and others that a large number of the propertiesof light and electricity may be explained. In spite of the difficulties towhich this hypothesis necessarily leads, and of its inherent improbabil-ity, it has been discussed by Stokes, Lame, Boussinesq, Sarrau, Lorenz,
* Philosophiae naturalis Theoria redacta ad unicam Legem Virium, Vienna, 1759.† Proceedings of the Royal Society of Edinburgh, April 21, 1862, vol. iv, pp. 604–
606.
• Project • Gutenberg • #26839 •
308 MATTER AND ETHER THEORIES. [CH. XIV
Lord Rayleigh, and Kirchhoff.This hypothesis has been modified and rendered somewhat more
plausible by von Helmholtz, Lommel, Ketteler*, and Voigt, who basedtheir researches on the assumption of a mutual reaction between theether and the material molecules located in it: on this view the prob-lems connected with refraction and dispersion have been simplified. Fi-nally, Sir William Thomson, in his Baltimore Lectures, 1885, suggesteda mechanical analogue to represent the relations between matter andthis ether, by which a possible constitution of the ether can be realized.He also suggested later a form of labile ether , from whose propertiesmost of the more familiar physical phenomena can be deduced, pro-vided the arrangement can be considered stable; a labile ether is anelastic solid, and its properties in two dimensions may be comparedwith those of a soap-bubble film, in three dimensions.
It is, however, difficult to criticise any of these hypotheses as atheory of the constitution of matter until the arrangement of the atomsor their nature is more definitely expressed.
III. Dynamical Theories. In recent years the suggestionhas been made that the so-called atoms may be forms of motion(ex. gr. permanent eddies) in one elementary material known as theether; on this view all the atoms are constituted of the same mat-ter, but the physical conditions are different for the different kindsof atoms. It has been said that there is an initial difficulty in anysuch hypothesis, since the all-pervading elementary fluid must possessinertia, so that to explain matter we assume the existence of a fluidpossessing one of the chief characteristics of matter. This is true asfar as it goes, but it is not more unreasonable than to attribute all thefundamental properties of matter to the atoms themselves, as is doneby many writers. The next paragraph contains a statement of one ofthe earliest attempts to formulate a dynamical atomic hypothesis.
(i) The Vortex Ring Hypothesis. This hypothesis assumes thateach atom is a vortex ring in an incompressible frictionless hom*oge-neous fluid.
Vortex rings—though, since friction is brought into play, of an im-perfect character—can be produced in air by many smokers. Better
* Theoretische Optik, Braunschweig, 1885.
• Project • Gutenberg • #26839 •
CH. XIV] MATTER AND ETHER THEORIES. 309
specimens can be formed by taking a cardboard box in one side ofwhich a circular hole is cut, filling it with smoke, and hitting the op-posite side sharply. The tendency of the particles forming a ring tomaintain their annular connection may be illustrated by placing sucha box on one side of a room in a direct line with the flame of a lightedcandle on the other side. If properly aimed, the ring will travel acrossthe room and put out the flame. If the box is filled only with air, sothat the ring is not visible, the experiment is more effective.
In 1858 von Helmholtz* showed that a closed vortex filament in aperfect fluid is indestructible and retains certain characteristics alwaysunaltered. In 1867 Sir William Thomson propounded† the idea thatmatter consists of vortex rings in a fluid which fills space. If the fluidis perfect we could neither create new vortex rings nor destroy thosealready created, and thus the permanence of the atoms is explained.Moreover the atoms would be flexible, compressible, and in incessantvibration at a definite fundamental rate. This rate is very rapid, andSir William Thomson gave the number of vibrations per second of asodium ring as probably being greater than 1014.
By a development of this hypothesis Prof. J.J. Thomson‡ showed,some years ago, that chemical combination may be explained. He sup-posed that a molecule of a compound is formed by the linking togetherof vortex filaments representing atoms of different elements: this ar-rangement may be compared with that of helices on an anchor ring.For stability not more than six filaments may be combined together,and their strengths must be equal. Another way of explaining chemi-cal combination on the vortex atom hypothesis has been suggested byW.M. Hicks. It is known§ that a spherical mass of fluid, whose interiorpossesses vortex motion, can move through liquid like a rigid sphere,and he has shown that one of these spherical vortices can swallow upanother, thus forming a compound element.
(ii) The Vortex Sponge Hypothesis. Any vortex atom hypothe-sis labours under the difficulty of requiring that the density of the fluid
* Crelle’s Journal, 1858, vol. lv, pp. 25–55; translated by Tait in the PhilosophicalMagazine, June, 1867, supplement, series 4, vol. xxxiii, pp. 485–512.
† Proceedings of the Royal Society of Edinburgh, Feb. 18, 1867, vol. vi, pp. 94–105.‡ A Treatise on the Motion of Vortex Rings, Cambridge, 1883.§ See a memoir by M.J.M. Hill in the Philosophical Transactions of the Royal
Society, London, 1894, part i, pp. 213–246.
• Project • Gutenberg • #26839 •
310 MATTER AND ETHER THEORIES. [CH. XIV
ether shall be comparable with that of ordinary matter. In order toobviate this and at the same time to enable it to transmit transver-sal radiations Sir William Thomson suggested what has been termed,not perhaps very happily, the vortex sponge hypothesis*: this restson the assumption that laminar motion can be propagated through aturbulently moving inviscid liquid. The mathematical difficulties con-nected with such motion have prevented an adequate discussion of thishypothesis, and I therefore confine myself to merely mentioning it.
These hypotheses, of vortex motion in a fluid, account for the in-destructibility of matter and for many of its properties. But in order toexplain statical electrical attraction it would seem necessary to supposethat the ether is elastic; in other words, that an electric field must be afield of strain. If so, complete fluidity in the ether would be impossible,and hence the above theories are now regarded as untenable.
(iii) The Ether-Squirts Hypothesis. Prof. Karl Pearson† has sug-gested another dynamical theory in which an atom is conceived as apoint at which ether is pouring into our space from space of four di-mensions.
If an observer lived in two dimensional space filled with ether andconfined by two parallel and adjacent surfaces, and if through a holein one of these surfaces fresh ether were squirted into this space, thevariations of pressure thereby produced might give the impression of ahard impenetrable body. Similarly an ether-squirt from space of fourdimensions into our space might give us the impression of matter.
It seems necessary on this hypothesis to suppose that there arealso ether-sinks, or atoms of negative mass; but as ether-squirts andether-sinks would repel one another we may suppose that the latterhave moved out of the universe known to our senses.
By defining the mass of an atom as the mean rate at which etheris squirting into our space at that point, we can deduce the Newtonianlaw of gravitation, and by assuming certain periodic variations in therate of squirting we can deduce some of the phenomena of cohesion, ofchemical action, and of electromagnetism and light. But of course thehypothesis rests on the assumption of the existence of a world beyondour senses.
* Philosophical Magazine, London, October, 1887, series 5, vol. xxiv, pp. 342–353.† American Journal of Mathematics, 1891, vol. xiii, pp. 309–362.
• Project • Gutenberg • #26839 •
CH. XIV] MATTER AND ETHER THEORIES. 311
(iv) The Electron Hypothesis. MacCullagh, in 1837 and 1839,proposed to account for optical phenomena on the assumption of anelastic ether possessing elasticity of the type required to enable it to re-sist rotation. This suggestion has been recently modified and extendedby Dr J. Larmor*, and, as now enunciated, it accounts for many of theelectrical and magnetic (as well as the optical) properties of matter.
The hypothesis is however very artificial. The assumed ether isa rotationally elastic incompressible fluid. In this fluid Larmor intro-duces monad electric elements or electrons, which are nuclei of radialrotational strain. He supposes that these electrons constitute the basisof matter. He further supposes that an electrical current consists ofa procession of these electrons, and that a magnetic particle is one inwhich these entities are revolving in minute orbits. Dynamical consid-erations applied to such a system lead to an explanation of nearly allthe more obvious phenomena. By further postulating that the orbitalmotion of electrons in the atom constitute it a fluid vortex it is possi-ble to apply the hydrodynamical pulsatory theory of Bjerknes or Hicksand obtain an explanation of gravitation.
Thus on this view mass is explained as an electrical manifestation.Electricity in its turn is explained by the existence of electrons, that is,of nuclei of strain in the ether, which are supposed to be in incessant andrapid motion. Whilst, to render this possible, properties are attributedto the ether which are apparently inconsistent with our experience ofthe space it fills. Put thus, the hypothesis seems very artificial. Perhapsthe utmost we can say for it is that, from some points of view, it may,so far as analysis goes, be an approximation to the true theory; inany case much work will have to be done before it can be consideredestablished even as a working hypothesis.
Most of the above was written in 1891. Since then investigationson radio-activity have opened up new avenues of conjecture which tendto strengthen the electron theory as a working hypothesis. More thanthirty years ago Clerk Maxwell had shown that light and electricity wereclosely connected phenomena. It was then believed that both were dueto waves in the hypothetical ether, but it was supposed that the phe-nomena of matter on the one side and of light and electricity on theother were sharply distinguished one from the other. The differences,
* Philosophical Transactions of the Royal Society, London, 1894, pp. 719–822;1895, pp. 695–743.
• Project • Gutenberg • #26839 •
312 MATTER AND ETHER THEORIES. [CH. XIV
however, between matter and light tend to disappear as investigationsproceed. In 1895 Rontgen established the existence of rays which couldproduce light, which had the same velocity as light, which were notaffected by a magnet, and which could traverse wood and certain otheropaque substances like glass. A year later Becquerel showed that ura-nium was constantly emitting rays which, though not affecting the eyeas light, were capable of producing an image on a photographic plate.Like Rontgen rays they can go through thin sheets of metal; like heatrays they burn the skin; like electricity they generate ozone from oxy-gen. Passed into the air they enable it to conduct the electric current.Their speed has been measured and found to be rather more than halfthat of light and electricity. It was soon found that thorium possessed asimilar property, but in 1903 Prof. Curie showed that radium possessedradio-activity to an extent previously unsuspected in any body, and infact the rays were so powerful as to make the substance directly visible.Further experiments showed that numerous bodies are radio-active, butthe effects are so much more marked in radium that it is convenient touse that substance for most experimental purposes.
Radium gives off no less than three kinds of rays besides a radio-active emanation. In these discharges there appears to be a gradualchange from what had been supposed to be an elementary form ofmatter to another. This leads to the belief that of the known forms ofmatter some, perhaps even all, are not absolutely stable. On the otherhand, it may be that only radio-active bodies are unstable, and that intheir disintegration we are watching the final stage in the evolution ofstable and constant forms of matter. It may, however, in any case turnout that some, or perhaps all, of the so-called elements may be capableof resolution into different combinations of electrons or electricity.
At an earlier date J.J. Thomson had concluded that the glow, seenwhen an electric current passes through a high vacuum tube, is due toa rush of minute particles across the tube. He calculated their weight,their velocity, and the charge of electricity transported by or repre-sented by them, and found these to be constant. They were deflectedlike Becquerel rays. All space seems to contain them, and electricity,if not identical with them, is at least carried by them. This suggestedthat these minute particles might be electrons. If so, they might thusgive the ultimate explanation of electricity as well as matter, and theatom of the chemist would be not an irreducible unit of matter, but a
• Project • Gutenberg • #26839 •
CH. XIV] MATTER AND ETHER THEORIES. 313
system comprising numerous such minute particles. These conclusionsare consistent with those subsequently deduced from experiments withradium. In 1904 the hypothesis was carried one stage further. In thatyear J.J. Thomson investigated the conditions of stability of certainsystems of revolving particles; and on the hypothesis that an atom ofmatter consists of a number of particles carrying negative charges ofelectricity revolving in orbits within a sphere of positive electrificationhe deduced many of the properties of the different chemical atoms cor-responding to different possible stable systems of this kind. His schemeled to results agreeing closely with the results of Mendeleeff’s periodichypothesis. An interesting consequence of this view is that Franklin’sdescription of electricity as subtle particles pervading all bodies, mayturn out to be substantially correct. It is also remarkable that cor-puscles somewhat analogous to those whose existence was suggested inNewton’s corpuscular theory of light should be now supposed to existin cathode and Becquerel rays.
(v) The Bubble Hypothesis*. The difficulty of conceiving the mo-tion of matter through a solid elastic medium has been met in anotherway, namely, by suggesting that what we call matter is a deficiency ofthe ether, and that this region of deficiency can move through the etherin a manner somewhat analogous to that in which a bubble can move ina liquid. To express this in technical language we may suppose the etherto consist of an arrangement of minute uniform spherical grains piledtogether so closely that they cannot change their neighbours, althoughthey can move relatively one to another. Places where the number ofgrains is less or greater than the number necessary to render the pilingnormal, move through the medium, as a wave moves through water,though the grains do not move with them. Places where the ether is inexcess of the normal amount would repel one another and move awayout of our ken, but places where it is below the normal amount wouldattract each other according to the law of gravity, and constitute par-ticles of matter which would be indestructible. It is alleged that thetheory accounts for the known phenomena of gravity, electricity, andlight, provided the size of its grains is properly chosen. Reynolds hascalculated that for this purpose their diameter should be rather morethan 5×10−18 centimetres, and that the pressure in the medium wouldbe about 104 tons per square centimetre. This theory is in itself more
* O. Reynolds, Submechanics of the Universe, Cambridge, 1903.
• Project • Gutenberg • #26839 •
314 MATTER AND ETHER THEORIES. [CH. XIV
plausible than the electron hypothesis, but its consequences have notyet been fully worked out.
Returning from these novel hypotheses to the classical theories ofmatter, we may now proceed a step further. Before a hypothesis onthe structure of matter can be ranked as a scientific theory we mayreasonably expect it to afford some explanation of three facts. Theseare (a) the Newtonian law of attraction; (b) the fact that there areonly a finite number of ultimate kinds of matter—such as oxygen, iron,etc.—which can be arranged in a series such that the properties of thesuccessive members are connected by a regular law; and (c) the mainresults of spectrum analysis.
In regard to the first point (a), we can say only that none of theabove theories are inconsistent with the known laws of attraction; andas far as the ether-squirts, the electron, and the bubble hypothesesare concerned, they have been elaborated into a form from which thegravitational law of attraction can be deduced. But we may still saythat as to the cause of gravity—or indeed of force—we know nothing.
Newton, in his Letters to Bentley, while declaring his ignorance ofthe cause of gravity, refused to admit the possibility of force acting ata finite distance through a vacuum. “You sometimes speak of gravity,”said he*, “as essential and inherent to matter: pray do not ascribethat notion to me, for the cause of gravity is what I do not pretendto know.” And in another place he wrote†, “’Tis inconceivable, thatinanimate brute matter should (without the mediation of somethingelse which is not material) operate upon and affect other matter withoutmutual contact; as it must if gravitation in the sense of Epicurus, beessential and inherent in it. . . That gravity should be innate, inherent,and essential to matter, so that one body may act upon another at adistance thro’ a vacuum, without the mediation of anything else, byand through which their action and force may be conveyed from one toanother, is to me so great an absurdity, that I believe no man who has inphilosophical matters a competent faculty of thinking can ever fall intoit. Gravity must be caused by an agent acting constantly according to
* Letter dated Jan. 17, 1693. I quote from the original, which is in the Libraryof Trinity College, Cambridge; it is printed in the Letters to Bentley, London,1756, p. 20.
† Letter dated Feb. 25, 1693; ibid., pp. 25, 26.
• Project • Gutenberg • #26839 •
CH. XIV] THE CAUSE OF GRAVITY 315
certain laws, but whether this agent be material or immaterial, I haveleft to the consideration of my readers.”
I have already alluded to conjectural explanations of gravity de-pendent on the ether-squirts, the electron, and the bubble hypotheses.Of other conjectures as to the cause of gravity, three, which do notinvolve the idea of force acting at a distance, may be here mentioned:
(1) The first of these conjectures was propounded by Newton inthe Queries at the end of his Opticks, where he suggested as a possibleexplanation the existence of a stress in the ether surrounding a particleof matter*.
This has been elaborated on a statical basis by Maxwell, whoshowed† that the stress would have to be at least 3000 times greaterthan that which the strongest steel would support. Sir William Thom-son (Lord Kelvin) has suggested‡ a dynamical way of producing thestress by supposing that space is filled with an incompressible fluid,constantly being annihilated by each atom of matter at a rate pro-portional to its mass, a constant supply being kept up at an infinitedistance. It is true that this avoids Maxwell’s difficulty, but we have noright to introduce such sinks and sources of fluid unless we have othergrounds for believing in their existence. The conclusion is that New-ton’s conjecture is very improbable unless we adopt the ether-squirtstheory: on that hypothesis it is a plausible explanation.
I should add that Maclaurin implies§ that though the above expla-nation was Newton’s early opinion, yet his final view was that he couldnot devise any tenable hypothesis about the cause of gravitation.
(2) In 1782 Le Sage of Geneva suggested‖ that gravity was causedby the bombardment of streams of ultramundane corpuscles. Thesecorpuscles are supposed to come in all directions from space and to beso small that inter-collisions are rare.
* Quoted by S.P. Rigaud in his Essay on the Principia, Oxford, 1838, appendix,pp. 68–70. On other guesses by Newton see Rigaud, text, pp. 61–62, and refer-ences there given.
† Article Attraction, in Encyclopaedia Britannica, or Collected Works, vol. ii,p. 489.
‡ Proceedings of the Royal Society of Edinburgh, Feb. 7, 1870, vol. vii, pp. 60–63.§ An Account of Sir Isaac Newton’s Philosophical Discoveries, London, 1748,
p. 111.‖ Memoires de l’Academie des Sciences for 1782, Berlin, 1784, pp. 404–432: see
also the first two books of his Traite de Physique, Geneva, 1818.
• Project • Gutenberg • #26839 •
316 MATTER AND ETHER THEORIES. [CH. XIV
A body by itself in space would receive on an average as many blowson one side as on another, and therefore would have no tendency tomove. But, if there are two bodies, each will screen the other from someof the bombarding corpuscles. Thus each body will receive more blowson the side remote from the other body than on the side turned towardsit. Hence the two bodies will be impelled each towards the other.
In order to make this force between two particles vary directly asthe product of their masses and inversely as the square of the distancebetween them, Le Sage showed that it was sufficient to suppose thatthe mass of a body was proportional to the area of a section at rightangles to the direction in which it was attracted. This requires thatthe constitution of a body shall be molecular, and that the distancesbetween consecutive molecules shall be very large compared with thesizes of the molecules. On the vortex hypothesis we may suppose thatthe ultramundane corpuscles are vortex rings.
This is ingenious, and it is possible that if the corpuscles were per-fectly elastic the theory might be tenable*. But the results of Maxwell’snumerical calculation show, first, that the particles must be imperfectlyelastic; second, that merely to produce the effect of the attraction of theearth on a mass of one pound would require that Le Sage’s corpusclesshould expend energy at the rate of at least billions† of foot-pounds persecond; and third, that it is probable that the effect of such a bombard-ment would be to raise the temperature of all bodies beyond a pointconsistent with our experience. Finally, it seems probable that thedistance between consecutive molecules would have to be considerablygreater than is compatible with the results given below.
Tait summed up the objections to these two hypotheses by say-ing‡, “One common defect of these attempts is. . . that they all demandsome prime mover, working beyond the limits of the visible universeor inside each atom: creating or annihilating matter, giving additionalspeed to spent corpuscles, or in some other way supplying the exhaus-tion suffered in the production of gravitation. Another defect is thatthey all make gravitation a mere difference-effect, as it were; therebyimplying the presence of stores of energy absolutely gigantic in com-
* See a paper by Sir William Thomson (Lord Kelvin) in the Proceedings of theRoyal Society of Edinburgh, Dec. 18, 1871, vol. vii, pp. 577–589.
† I use billion with the English (and not the French) meaning, that is, a billion= 1012.
‡ Properties of Matter, London, 1885, art. 164.
• Project • Gutenberg • #26839 •
CH. XIV] THE CAUSE OF GRAVITY 317
parison with anything hitherto observed, or even suspected to exist, inthe universe; and therefore demanding the most delicate adjustments,not merely to maintain the conservation of energy which we observe,but to prevent the whole solar and stellar systems from being instan-taneously scattered in fragments through space. In fact, the cause ofgravitation remains undiscovered.”
(3) There is another conjecture on the cause of gravity which Imay mention*. It is possible that the attraction of one particle onanother might be explained if both of them rested on a hom*ogeneouselastic body capable of transmitting energy. This is the case if ourthree-dimensional universe rests in the direction of a fourth dimensionon a four-dimensional hom*ogeneous elastic body (which we may call theether) whose thickness in the fourth dimension is small and constant.
The results of spectrum analysis lead us to suppose that everymolecule of matter in our universe is in constant vibration. On theabove hypothesis these vibrations would cause a disturbance in thesupporting space, i.e. in the ether. This disturbance would spread outuniformly in all directions; the intensity diminishing as the square ofthe distance from the centre of vibration, but the rate of vibration re-maining unaltered. The transmission of light and radiant heat may beexplained by such vibrations transversal to the direction of propaga-tion. It is possible that gravity may be caused by vibrations in thesupporting space which are wholly longitudinal or are compounded ofvibrations which are partly longitudinal and partly transversal in anyof the three directions at right angles to the direction of propagation.If we define the mass of a molecule as proportional to the intensityof these vibrations caused by it, then at any other point in space theintensity of the vibration there would vary as the mass of the moleculeand inversely as the square of the distance from the molecule; hence,if we may assume that such vibrations of the medium spreading outfrom any centre would draw to that centre a particle of unit mass atany other point with a force proportional to the intensity of the vibra-tion there, then the Newtonian law of attraction would follow. Thisconjecture is consistent either with Boscovich’s hypothesis or with thevortex theory. It would be interesting if the results of a branch of puremathematics so abstract as the theory of hyperspace should be found
* See an article by myself in the Messenger of Mathematics, Cambridge, 1891,vol. xxi, pp. 20–24.
• Project • Gutenberg • #26839 •
318 MATTER AND ETHER THEORIES. [CH. XIV
to be closely connected with one of the most fundamental problemsof material science.
I should sum up the effect of this discussion on gravity on therelative probabilities of the hypotheses as to the constitution of matterenumerated above, by saying that it does not enable us to discriminatebetween them.
The fact that the number of kinds of matter (chemical elements)is finite and the consequences of spectrum analysis are closely relatedand may be treated together. The results of spectrum analysis showthat every molecule of any species of matter, such as hydrogen, vibrateswith (so far as we can tell) exactly equal sets of periods of vibration.This then is one of the characteristics of the particular kind of matter,and it is probable that any explanation of why the molecules of eachkind have a definite set of periods of vibration will account also for thefact that the number of kinds of matter is finite.
Various attempts to explain why the molecules of matter are capa-ble only of certain definite periods of vibration have been made, and itmay be interesting if I give them briefly.
(1) To begin with, I may note the conjecture that it depends onproperties of time. This, however, is impossible, for the continuityof certain spectra proves that in these cases there is nothing whichprevents the period of vibration from taking any one of millions ofdifferent values: thus no explanation dependent on the nature of timeis permissible.
(2) It has been suggested that there may have been a sortingagency, and only selected specimens of the infinite number of speciesformed originally have got into our universe. The objection to this isthat no explanation is offered as to what has become of the excludedmolecules.
(3) The finite number of species might be explained by supposinga physical connection to exist between all the molecules in the universe,just as two clocks whose rates are nearly the same tend to go at thesame rate if their cases are connected.
Maxwell’s objection to this is that we have no other reason forsupposing that such a connection exists, but if we are living in a space offour dimensions as suggested above in chapter xii, this connection doesexist, for all the molecules rest on one and the same body. This body is
• Project • Gutenberg • #26839 •
CH. XIV] FINITE NUMBER OF KINDS OF MATTER 319
capable of transmitting vibrations, hence, no matter how the moleculeswere set vibrating originally, they would fall into certain groups, andall the members of each group would vibrate at the same rate. Itwas the possibility of obtaining thus a physical connection between thevarious particles in our universe that first suggested to me the idea ofa supporting medium in a fourth dimension.
(4) If we accept Boscovich’s hypothesis or that of an elastic solidether, and if we may lay it down as axiomatic that the mass of everysub-atom is the same, we may conceive that the number of ways ofcombining the sub-atoms into a permanent system is limited, and thatthe period of vibration depends on the form in which the sub-atomsare combined into an atom. This view is not inconsistent with anyknown facts. I may add that it is probable that the chemical atom isthe essential vibrating system, for the sodium spectrum, to take oneinstance, is the same as that of all its compounds.
(5) In the same way we may suppose that the vortex rings areformed so that they can have only a definite number of stable formsproduced by interlinking or kinking.
(6) Similarly we may modify the popular hypothesis by treat-ing the atoms as indivisible aggregates of sub-atoms which are in allrespects equal and similar, and can be combined in only a limited num-ber of forms which are permanent. But most of the old difficulties con-nected with the atoms arise again in connection with the sub-atoms.
(7) I am not aware that Maxwell discussed any other hypothesesin connection with this point, but it has been suggested recently that,if the various forms of matter were evolved originally out of some oneprimitive material, then there may have been periodic disturbances inthis matter when the atoms were being formed, such that they wereproduced only at some definite phase in the period*.
Thus, if the disturbance is represented by the swinging of a pendu-lum in a resisting medium, it might be supposed that the atoms wereformed at the points of maximum amplitude, and we should expect thatthe atoms successively thrown off would form a series having the prop-erties of its successive members connected by a regular periodic law.This conjecture, when worked out in some detail, led to the conclusionthat some elements which ought to have appeared in the series were
* See Crookes on Mendeleeff’s periodic law, Nature, Sept. 2, 1886, vol. xxxiv,pp. 423–432.
• Project • Gutenberg • #26839 •
320 MATTER AND ETHER THEORIES. [CH. XIV
missing, but it was possible to predict their properties and to suggestthe substances with which they were most likely to be found in com-bination. Guided by these theoretical conclusions a careful chemicalanalysis revealed the fact that such elements did exist.
That this hypothesis has led to new discoveries is something in itsfavour, but I do not wish to be understood to say that it is a theorywhich leads to results that have been verified subsequently. I shouldsay rather that we have obtained an analogy which is sufficiently likethe truth to suggest new discoveries. Such analogies are often the pre-cursors of laws, so that it is not unreasonable to hope that ere long ourknowledge of this border-land of chemistry and physics may be moredefinite, and thus that molecular physics may be brought within thedomain of mathematics. It is however very remarkable that J.J. Thom-son’s conclusions on the stability of the orbital systems he devisedshould agree so closely with Mendeleeff’s periodic law.
On the whole Maxwell thought that the phenomena point to acommon origin of all molecules of the same kind, that this was anevent not belonging to that order of nature under which we live, butmust have originated when or before the existing order was established,and that so long as the present order exists it is immutable.
This is equivalent to saying that we have arrived at a point be-yond which our limited experience does not enable us to carry theexplanation.
That we should be able to form an approximate idea of the sizeof the molecules of matter is a testimony to the extraordinary advancewhich mathematical physics has made recently.
Sir William Thomson, now Lord Kelvin, whose ingenuity seems toknow no limits—has suggested* four distinct methods of attacking theproblem. They lead to results which are not very different.
The first of these rests on an assertion of Cauchy that the phenom-ena of prismatic colours show that the distance between consecutivemolecules of matter is comparable with the wave-lengths of light. Tak-ing the most unfavourable case this would seem to indicate that in atransparent hom*ogeneous solid or liquid medium there are not more
* See Nature, March 31, 1870, vol. i, pp. 531–553; and Tait’s Recent Advances,pp. 303–318. The fourth method had been proposed by Loschmidt in 1863.
• Project • Gutenberg • #26839 •
CH. XIV] THE SIZE OF MOLECULES OF MATTER. 321
than 64 × 1024 molecules in a cubic inch, that is, that the distancebetween consecutive molecules is greater than 1/(4× 108)th of an inch.
The second method is founded on the amount of work required todraw out a film of liquid, such as a soap-bubble, to a given thickness.This can be calculated from experiments in a capillary tube, and itis found that, if a soap-bubble could be drawn out to a thickness of1/108th of an inch there would be but a few molecules in its thickness.This method is not quantitative.
Thirdly, Sir William Thomson proved that the contact phenomenaof electricity require that in an alloy of brass the distance betweentwo molecules, one of zinc and one of copper, shall be greater than1/(7×108)th of an inch; hence the number of molecules in a cubic inchof zinc or copper is not greater than 35 × 1025.
Lastly, the kinetic theory of gases leads to the conclusion thatcertain phenomena of temperature and viscosity depend, inter alia,on inter-molecular collisions, and so on the sizes and velocities of themolecules, while the average velocity with which the molecules moveincreases with the temperature. This leads to the conclusion that thedistance between two consecutive molecules of a gas at normal pres-sure and temperature is greater than 1/(6 × 106)th of an inch, and isless than 1/107th of an inch; while the actual size of the molecule is atrifle greater than 1/(3 × 1020)th of a cubic inch; and the number ofmolecules in a cubic inch is about 3 × 1020.
Thus it would seem that a cubic inch of gas at ordinary pressure andtemperature contains about 3 × 1020 molecules, all similar and equal,and each molecule has a volume of about 1/(3×1025)th of a cubic inch;while a cubic inch of the simplest solid or liquid contains rather lessthan 1027 molecules, and perhaps each molecule has a volume of about1/(3× 1026)th of a cubic inch. For instance, if a pea or a drop of waterwhose radius is 1/16th inch was magnified to the size of the earth, thenthere would be about thirty molecules in every cubic foot of it, andprobably the size of a molecule would be about the same as that of afives-ball. The average size of the minute drops of water in a very lightcloud can be calculated from the coloured rings produced when the sunor moon shines through it. The radius of a drop is about 1/30000th ofan inch. Such a drop therefore would contain about 2 × 1013 separatemolecules. In gases and vapours, the number of atoms required to makeup one of these molecules can be estimated, but in liquids the number
• Project • Gutenberg • #26839 •
322 MATTER AND ETHER THEORIES. [CH. XIV
is not as yet known.Loschmidt asserted that a cube whose side is 1/4000th of a mil-
limetre is the smallest object which can be made visible at the presenttime. Such a cube of oxygen or nitrogen would contain from 60 to 100millions of molecules of the gas. Also on an average about 50 elemen-tary molecules of the so-called elements are required to constitute onemolecule of organic matter. At least half of every living organism con-sists of water, and we may for the moment suppose that the remainderconsists of organic matter. Hence the smallest living being which is vis-ible under the microscope contains from 30 to 50 millions of elementarymolecules which are combined in the form of water, and from 30 to 50millions of elementary molecules which are combined so as to make notmore than one million organic molecules.
Hence a very simple organism might be built up out of as few as amillion similar organic molecules. Maxwell did not consider that thiswas sufficient to justify the current conclusions of physiologists, and saidthat they must not suppose that structural details of infinitely small di-mensions can furnish by themselves an explanation of the variety knownto exist in the properties and functions of the most minute organisms;but physiologists have replied that whether their conjectures be rightor wrong Maxwell’s argument is vitiated by his non-consideration ofdifferences due to the physical (as opposed to the chemical) structureof the organism and the consequent motions of the component parts.
• Project • Gutenberg • #26839 •
INDEX
Abbot, W., 177Abbott, E.A., 277Achilles and the Tortoise, 67Acts or Disputations, chap. vii
Agrippa, Cornelius, 122Ahrens, 97, 103, 123Airy, Sir Geo., 87, 194Aix, Labyrinth at, 152Albohazen on Astrology, 238Alcuin, 2, 55Alfred the Great, 301Alkarisimi on π, 217Alkborough, Labyrinth at, 152Anallagmatic Squares, 51Anaxagoras, 215Anderson, T., 272Angular Motion, 69Anstice, 103, 105–106Antipho, 215Apollonius, 206, 208, 216Archimedes, 72, 212, 215, 219Archytas on Delian Problem, 207Argyle, 258Arithmetic, Higher, 29–34Arithmetical Fallacies, 20–24Arithmetical Puzzles, 4–26— Recreations, chap. i
Arya Bhata on π, 216Asenby, Labyrinth at, 152Astrological Planets, 122, 240, 291
Astrology, chap. xAtomic Theories, chap. xivAtoms, Size of, 320Attraction, Law of, 314–318Augustine on Astrology, 244Augustus, 269, 292, 293Ayrton on Magic Mirrors, 87
Babbage, C., 192Bachet’s Problemes, 2–20, 27–29,
55, 113, 115, 122, 125, 132Bacon, Francis, 269Bailey, J.E., 251Ball, 171, 173, 205, 208, 224, 282Bardesan on Fate, 237Barrow, I., 172Baudhayana on π, 216Becquerel Rays, 312, 313Bedwell, T., 172Beltrami on Space, 277Benham on Spectrum Top, 87Bentley, Newton to, 314Bentley, R., 176Bernoulli, John, 22Berosus, 298Berri, de, 267Bertrand, J.L.F., 21Bertrand, L. (of Geneva), 159Besant on Hauksbee’s Law, 82Bhaskara on π, 217Bickmore, C.E., 230, 231, 236
323
• Project • Gutenberg • #26839 •
324 INDEX.
Biering on Delian Problem, 205Billingsley, H., 172Bills on Kirkman’s Problem, 109Binary Powers, Fermat on, 31–32Birch, J.G., 233Birds, Flight of, 85–86Bjerknes, 311Blackburn, J., 185Blundeville, T., 172Board, Mathematical, 198Boat-racing with a rope, 81–82Bolyai, J., 285Bonnycastle, J., 189Bordered Magic Squares, 135–136Boscovich on Matter, 307, 317, 319Boughton Green, Labyrinth at, 152Bouniakowski, V., on shuffling, 110Bourget, M.J., on shuffling, 110Bourlet, 18, 19Boussinesq on Ether, 307Brackets, in Tripos, 182, 188, 197,
198Brahmagupta on π, 217Breton on Mosaics, 152Brewster, Sir David, 249Briggs, H., 172Bristed, C.A., 192Bromton, 151Brouncker on π, 220Brown, J. (Saint), 190Bryan on Bird Flight, 86Bryso, 215Bubble Theory of Matter, 313Buckley, W., 172Burnside, Kirkman’s Problem, 103Butcher on the Calendar, 297
Caesar, Julius, 244, 269, 292
Calendar, the Civil, 292–294— the Ecclesiastical, 294–297— the Gregorian, 294–296— the Julian, 292–293Calendars, University, 186Cambridge, Mathematics, chap.
vii
— Studies at, chap. vii
Cantor on π, 215Cardan, 2, 93, 95, 237, 247–249, 275Cards, Problems with, 14–15, 26,
55, 109–120Carpmael, Kirkman’s Problem, 103Cartwright, W., 271Cauchy, 48, 307, 320Cayley, 44, 46, 103, 119, 154, 220Cellini, 249Cells of a Chess-board, 158Centrifugal Force, 71–72Ceulen, van, on π, 218, 219Challis MSS, 183Charles I, 269–271, 272, 297Charles V of Germany, 302Chartres, Labyrinth at, 152Chartres, R., 21, 41, 223Chasles on Trisection of Angle, 211Chaucer on the Sun-cylinder, 298Cheke, Sir John, 247Chess-board, Games on, 60–64,
103, 158–169— knights’ moves on, 63, 158–169— problems, 25, 60–64, 103,
158–169Chilcombe, Labyrinth at, 152Chinese on π, 217Chinese rings, 93–97Ciccolini on Chess, 164Cicero on Astrology, 244
• Project • Gutenberg • #26839 •
INDEX. 325
Ciphers, chap. xi
— Definition of, 252— Four types of, 259–267— Historical, 269–276— Requisites for good, 268–269Circle, Quadrature of, 212–223Cissoid, the, 206, 208, 213Clairaut on Trisection of Angle, 211Clarke, S., 172Classical Tripos, 196Claus, 91Clausen on π, 221Clavius on Calendar, 293, 294, 297Clepsydras, 301Clerk Maxwell, see MaxwellClerke, G., 172Clifford, 70Clocks, 77, 301–303Cnossus, Coins of, 151Coat and Waistcoat Trick, 64Coccoz, 37, 137Code-Book Ciphers, 274Cole, F.N., 225, 227, 230, 232, 234Colebrooke on Indian Algebra, 217Collini on Chess, 164Collins, Letter from J. Gregory, 220Colour-cube Problem, 51–52Colouring Maps, 44–46Columbus, 297Columbus’s Egg Puzzle, 74Comberton, Labyrinth at, 152Compasses, Watches as, 303–304Competition, in Tripos, 188, 192Composite Magic Squares, 134Conchoid, the, 206, 210, 213Cones moving uphill, 75Conradus, D.A., 261Continuity of Matter, 306Contour-lines, 47Cotes, R., 172
Counters, Games with, 48–50,58–63
Craig, J., 172Crassus, 244Cretan Labyrinth, 151, 152Cricket-Ball, Spin on, 85Crookes on Mendeleeff’s Laws, 319Cross-fours, 51Cryptographs, Definition of, 251— Three types of, 253–259Cryptography, chap. xi
Cube, Duplication of, 205–209Cubes, Coloured, 51–52Cubes, Skeleton, 26Cudworth on Sharp, 220Cumberland, R., 176–177Cumulative Vote, 26Cunningham, A.J.C., 32, 225, 228,
229, 230, 231, 234, 235Cureton on Syriac Astrology, 237Curie on radio-activity, 312Curiosa Physica, 86–87Cursor, 298Cusa on π, 217Cusps, Astrological, 238Cut on a Tennis-ball, 83–85Cutting Cards, Problems on, 15Cylinders, Sun-, 298
Dacres, A., 172Daedalus, Labyrinth of, 151D’Alembert, 21, 24Darwin, G.H., 42Dase on π, 221Dasypodius, 302Day, Definition of, 289— Commencement of, 291— Sidereal and Solar, 290Days of Week from date, 297Days of Week, Names of, 291–292Dealtry, W., 189, 190, 191
• Project • Gutenberg • #26839 •
326 INDEX.
De Berri, 267Decimation, 19–20Dee, J., 172De Fonteney on Ferry Problem, 55De Fouquieres, 49De Haan on π, 215, 218De Lagny on π, 221De la Hire on Magic Squares, 122,
123, 126–127, 128, 132–134De la Loubere on Magic Squares,
124–125, 137Delambre on Calendar, 296, 297Delannoy, 55, 60De la Pryme, 175Delian Problem, 205–209De Longchamps, G., 235De Moivre, on Knight’s Move, 159De Montmort, 1, 159De Morgan, A., 44, 67, 170, 173,
189, 190, 212, 214, 215,223, 246, 294
Denary scale of notation, 10, 11De Parville on Tower of Hanoı, 92De Polignac on Knight’s Move, 167De Rohan, 272De St Laurent, 110Descartes, 209, 211, 220, 306Des Ourmes on Magic Squares, 122Diabolic Squares, 136–137Dials, Sun-, 297–299Dickson, L.E., 118Diego Palomino, 19Digby, Lord, 271Digges, T., 172Diocles on Delian Problem, 208Diodorus on Lake Moeris, 150Dircks, H., 272Dirichlet on Fermat’s Theorem, 33Dissection, Proofs by, 42–44Dixon, A.C., 46, 103Dodecahedron Game, 155–158
Dominical Letter, 296Dominoes, 48, 141D’Ons-en-bray, Magic Squares, 122Doubly Magic Squares, 137Douglas, S., 192, 194Drach on Magic Squares, 123Drayton, 151Duplication of Cube, 205–209Durations, see TimeDurer, A., 122Dynamical Games, 52–64
Earnshaw, S., 87, 197Easter, Date of, 294–297Edward VI, 240, 247–249, 260— Horoscope of, 248Eight Queens Problem, 97–103Eisenlohr on Ahmes, 215Eisenstein, 32Electrons, 311–313Elliptic Geometry, 285–286Enestrom on π, 214Epicurus on Gravitation, 314Equilibrium, Puzzles on, 72–74Eratosthenes, 205Escott, E.B., 236Ether Theories, 307–310Ether-Squirts, 310Etten, van, 2, 10Euclid, 30, 36, 215Euclid’s Axioms &c., 285— Parallel Postulate, 285Euclidean Geometry, 284–287Euclidean Space, 284–287Euclid i. 32, 42Euclid i. 47, 42Euler, 29, 30, 32, 48, 122, 123, 136,
140, 159–163, 214, 220,221, 225, 227, 228
Euler’s Unicursal Problem,143–149
• Project • Gutenberg • #26839 •
INDEX. 327
Examination, Printed, 184, 196Exploration Problems, 26
Fallacies, Arithmetical, 20–24— Geometrical, 35–41— Mechanical, 67–69, 75–78Faraday on Matter, 307Fauquembergue, E., 230Fenn, J., 177Fermat on Binary Powers, 31–32Fermat, P., 29, 31–34, 123, 225,
226, 227, 228, 235Fermat’s Last Theorem, 32–34Ferry-boat Problems, 55–57Fifteen Puzzle, 88–91Fifteen School-girls, 103–109Figulus on Astrology, 244Firmicus on Astrology, 238Fitzpatrick, J., 59Flamsteed on Astrology, 246Flamsteed, J., 172Flat-land, 278–283Fluid Motion, 82–86Fluxions, 180, 183, 187, 189, 190,
192, 193Fonteney on Ferry Problem, 55Force, Definition of, 70Foster, S., 172Fouquieres on Ancient Games, 49Four-Colour Theorem, 44–46Fox on π, 223Frankenstein on Magic Pencils, 138Franklin, B., 313Frederick II of Germany, 301Frenicle, Magic Squares, 122, 135Frere, J., 178Fresnel on Ether, 307Frolow on Magic Squares, 136Frost, A.H., 103, 104–105, 123, 136
Galileo on Pendulum, 302Galton, 21Games, Dynamical, 52–64— Statical, 48–52— with Counters, 58–63Gases, Theory of, 321Gauss, 29, 34, 231, 277, 295Geodesic Problems, 57–58Geography, Physical, 46–48Geometrical Fallacies, 35–41Geometrical Recreations,
chap. ii
Geometry, Non-Euclidean,chap. xii
George I of England, 175Gerbert, 216, 301Gergonne’s Problem, 115–118Gill, Kirkman’s Problem, 106–107Glaisher, J.W.L., 98, 99, 171, 215Glamorgan, Earl of, 272Gnomons, 297Goldbach’s Theorem, 31Golden Number, 296Gooch, W., 184Gravity, Hypotheses on, 314–318Green on Ether, 307Greenwich, Labyrinth at, 153Gregorian Calendar, 293, 294Gregory XIII, 293–294Gregory, Jas., 213, 219Gregory of St Vincent, 209Gregory’s Series, 220Grienberger on π, 219Grille, The, 256Grimthorpe on Clocks, 301Gros on Chinese Rings, 95, 96Guarini’s Problem, 63Gun, Report of, 87Gunning, H., 184, 185Gunther, S., 97, 99, 123Guthrie on colouring maps, 44
• Project • Gutenberg • #26839 •
328 INDEX.
Haan, de, on π, 215, 218Halley on π, 221Halsted on Hyper-space, 277Hamilton, Archbishop, 247Hamilton, Sir Wm., 155–158Hamiltonian Game, 155–158Hampton Court, Maze at, 149, 153Hanoı, Tower of, 91–93Harris on pendulum clock, 302Harvey, J., 172Harvey, R., 172Hauksbee’s Law, 82–85Hayward, J., 260Heawood on Colouring Maps, 46Hegesippus on Decimation, 19Hele, P., 302Helmholtz, 78, 277, 309Helmholz, 308Henrietta Maria, 270Henry on Unicursal Problems, 143Henry VIII of England, 248, 301Hermary, 158Hero of Alexandria on π, 206, 216Herodotus on Lake Moeris, 150Herschel, Sir John, 192, 292Hezekiah, 299Hicks on Matter, 305, 309, 311Hiero of Syracuse, 72Higher Arithmetic, 29–34Hill, C., 273Hill, M.J.M., 309Hill, T., 172Hills and Dales, 46–48Hinton on Space, 277, 280, 282Hipparchus on hours of day, 291Hippias, 215Hippocrates of Chios, 206, 215Hodson, W., 183Holditch on Magic Squares, 123Homaloidal Geometry, 286Honorary Optimes, 174, 176, 196
Hood, T., 172Hooke on Timepieces, 302Horary Astrology, 238Hornbuckle, T.W., 190, 191Horner on Magic Squares, 123— Rules to cast, 239Horoscopes, chap. x— Example of, 248— Rules to cast, 238— Rules to read, 240–243Horrox, J., 172Hour-glasses, 301Hours, definition of, 290, 291Houses, Astrological, 238Huddling, 172, 173Hudson, C.T., on cards, 118Hudson, W.H.H., on cards, 111Hustler, J.D., 190Hutton, C., 3, 221Huygens, 209, 212, 213, 219, 220,
302Hyper-magic Squares, 136–137Hyper-space, chap. xiiHyperbolic Geometry, 285–286
Icosian Game, 155–158Ideler on the Calendar, 297Inertia, 70, 71Inwards on the Cretan Maze, 151Isaiah, 299
Jacob, E., 190, 191, 192Jacobi, 231, 235Jaenisch, 159, 164, 167, 168James II of England, 258Japanese Magic Mirrors, 87Jebb, J., 177, 179–181Johnson on Fifteen Puzzle, 88Jones on π, 221Jones on π, 214Josephus on Decimation, 19Julian Calendar, 292–293
• Project • Gutenberg • #26839 •
INDEX. 329
Julian’s Bowers, 152Julius Caesar, 244, 269Junior Optimes, 174, 181Jurin, J., 172
Kelvin, 307, 308, 309, 315, 316, 320Kempe on Colouring Maps, 45Ketteler on Ether, 308Kinetic Theory of Gases, 321Kirchhoff on Ether, 308Kirkman, T.P., 103, 108Kirkman’s Problem, 103–109Klein, 204Knight’s Path Problem,
158–169Knyghton, 151Konigsberg Problem, 143–149Kummer on Fermat’s Theorem, 33
Labile Ether, 308Labosne on Magic Squares, 132Labyrinths, 149–154Lacroix, 212Lagny on π, 221Lagrange, 192, 228Lagrange’s Theorem, 31La Hire, 122, 123, 126–127, 128,
132–134Laisant, C.A., 11, 236La Loubere, 124–125Lambert on π, 212Lambert on π, 212Lame, 33, 307Landry, 225, 227, 229, 230, 232Langley on Bird Flight, 86Laplace, 192Laplace on velocity of sound, 307Laquiere on Knight’s Path, 167Larmor on Electrons, 305, 311Latruaumont, 272Laughton, R., 172Lawrence, F.W., 236
Lax, W., 184, 185Lea on Kirkman’s Problem, 109Leap-year, 292–293Lebesgue on Fermat’s Theorem, 33Legendre, 33, 160, 168, 192, 212,
228, 231Leibnitz on Games, 1Lejeune Dirichlet on Fermat, 33Le Lasseur, 225, 227, 228, 229, 230Leonardo of Pisa on π, 217Le Sage on Gravity, 315, 316Leslie, J., 206, 210Leurechon, 2Lilius on the Calendar, 293Lilly on Astrology, 246Linde on Knight’s Path, 158Lindemann on π, 212Line-land, 279Lines of Slope, 47Lippeus, 302Listing’s Topologie, 65, 145Liveing on the Spectrum Top, 87Lobatschewsky, N.I., 277, 285Locke, J., 182, 184Lommel on Ether, 308London and Wise, 153Lorentz on Ether, 307Loschmidt on Molecules, 320, 322Loubere, de la, 124–125Louis XI of France, 245Louis XIV of France, 124Loyd, S., 17Lucas di Burgo, 2Lucas, E., 50, 55, 60, 61, 63, 91, 95,
103, 107, 143, 155, 229, 230Lucca, Labyrinth at, 152Lydgate on the Sun-cylinder, 298
• Project • Gutenberg • #26839 •
330 INDEX.
MacCullagh on Ether, 307Machin’s series for π, 221, 222Maclaurin on Newton, 315MacMahon, 28–29, 51Magic Bottles, 73Magic Mirrors, 87Magic Pencils, 137–140Magic Squares, chap. v
Magic Square Puzzles, 140–142Magnus on Hauksbee’s Law, 83Map Colour Theorem, 44–46Marie Antoinette, 273Mathematics, Cambridge, chap.
vii
Mathews, G.B., 234Matter, Constitution of, chap. xiv
— Hypotheses on, 306–314— Kinds of, limited, 318–320— Size of Molecules, 320–322Maxim on Bird Flight, 86Maxwell’s Demon, 87Maxwell, J. Clerk, 46, 87, 305, 311,
315, 316, 318, 319, 322Mazes, 149–154Mean Time, 290, 291Mechanical Recreations, chap.
iii
Medieval Problems, 16–20Menaechmus, 207Mendeleeff, 313, 319, 320Mersenne on Primes, 29Mersenne’s Numbers, chap. ix,
30, 224, 225Mesolabum, 207Metius on π, 218Meton, 298Meziriac, see BachetMilner, I., 182, 190Minding on Knight’s Path, 168Minos, 149, 205Minotaur, 151
Minutes, def. of, 290, 291Mirrors, Magic, 87Models, 78Moderators, chap. vii
Mohammed’s sign-manual, 148Moivre, A. De, 159Molecules, Size of, 320–322Money, Question on, 8–9Monge on Shuffling Cards, 109–111Months, 292Montmort, De, 1, 159Montucla, 3, 50, 72, 73, 122, 123,
132, 133, 159, 212, 213Moon, R., 123, 167, 168Morgan, A. De, see De MorganMorland, S., 172Morley on Cardan, 247Mosaic Pavements, 50, 152Moschopulus, 122, 126Motion in Fluids, 82–86Motion, Laws of, 66, 70–75— Paradoxes on, 67–69— Perpetual, 75–78Mousetrap, Game of, 119–120Muller (Regiomontanus), 217Mullinger, J.B., 202Mydorge, 2
Nasik Squares, 136–137Natal Astrology, 238Nauck, 97Neumann on Ether, 307Newton, 75, 172, 189, 190, 191, 209,
211, 213, 306, 313, 314, 315Newtonian Laws of Motion, 66–75Nicene Council on Easter, 294Niceron, 275Nicomedes, 206Nigidius on Astrology, 244Non-Euclidean Geometry,
284–287
• Project • Gutenberg • #26839 •
INDEX. 331
Nonez on Sun-dials, 298Notation, Denary scale of, 10Noughts and Crosses, 48Numa on the Year, 292Numbers, Perfect, 225— Puzzles with, 4–20— Theory of, 29–34
Oliver on Sun-dials, 299Ons-en-bray on Magic Squares, 122Oppert on π, 215Optimes, chap. vii, 174, 177, 196Oram on Eight Queens, 101Oughtred, W., 172Oughtred’s Recreations, 2, 10, 13,
16, 18, 73, 74Ourmes on Magic Squares, 122Ovid, 150Ozanam’s Recreations, 2, 10, 16,
20, 43, 50, 55, 64, 72, 73,74, 75, 77, 79, 93, 122, 123,132, 140, 159, 298, 299, 303
Ozanam, A.F., on Labyrinths, 152
π, 212–223, see table of contentsPacificus on Clocks, 301Pacioli di Burgo, 2Pairs of Cards Trick, 113–115Paley, W., 178, 184, 186, 190, 195Palomino, 19Pappus, 206, 210, 211Parabolic Geometry, 286Paradromic Rings, 64–65Parallels, Theory of, 285Parmentier on Knight’s Path, 159Parry on Sound, 87Parville on Tower of Hanoı, 92Pawns, Games with, 58–63Paynell, N., 172Pearson on Ether-Squirts, 310Pein on Ten Queens, 101Pencils, Magic, 137–140
Pepys, S., 271–272Perfect Numbers, 30, 225Permutation Problems, 26Perpetual Motion, 75–78Perrin, 11Perry on Magic Mirrors, 87Peterson on maps, 46Philo, 206Philoponus on Delian Problem, 205Physical Geography, 46–48Pierce on Kirkman’s Problem, 103Pile Problems, 115–119Pirie on π, 220Pitatus on the Calendar, 293Pittenger, 71Plana, G.A.A., 225, 227, 229, 232Planets (astrological), 122, 240, 291— Signification of, 240–242Plato on Delian Problem, 205, 206,
207Pliny, 150, 244Poe, E.A., 261, 264, 275Poignard, Magic Squares, 122, 123,
127Poincare, H., 286Poitiers, Labyrinth at, 152Polignac on Knight’s Path, 167Poll Examinations, 196Poll-Men, 174Pollock, F., 189–191Pompey, 244Porta, G., 275Power, Kirkman’s Problem, 103,
105Pratt on Knight’s Path, 164Pretender, The Young, 258Primes, 29Probabilities and π, 222Probabilities, Fallacies in, 24, 42Problem Papers, 183, 184, 186Ptolemy, 216, 237, 238, 291
• Project • Gutenberg • #26839 •
332 INDEX.
Purbach on π, 217Puzzles, Arithmetical, 4–29— Geometrical, 48–65— Mechanical, 67–75Pythagorean Symbol, 148
Quadrature of Circle, 212–223Queens Problem, Eight, 97–103Queens, Problems with, 97–103
Racquet-ball, Cut on, 83–85Railway Puzzles (shunting), 53–54Ramification, 154, 155Raphael on Astrology, chap. xRavenna, Labyrinth at, 152Rayleigh, 84, 86, 308Record, R., 172Regiomontanus on π, 217Reimer on Delian Problem, 205Reiss, 26, 63Relative Motion, 69Reneu, W., 175Renton, 42Reynolds, O., 313Rhind Papyrus, 215Riccioli on the Calendar, 293Rich, J., 271Richard, J., 69, 285Richards on use of compass, 303Richter on π, 222Riemann, G.F.B., 277, 285, 287Rigaud, S.P., 315Ring-Dial, 299–301Rockliff Marshes, Labyrinth at, 152Rodet on Arya-Bhata, 216Rodwell on Hyper-space, 277Roget, P.M., 163, 164–167Romanus on π, 218Rome, Labyrinth at, 152Rontgen Rays, 312Rooke, L., 172Rosamund’s Bower, 151
Rosen on Arab values of π, 217Rothschild, F., 245Routes on a Chess-board, 25, 158Row, Counters in a, 48–50, 58–61Rudio on π, 212Russell, B.A.W., 68Rutherford on π, 221
Saccheri, J., 277, 285Saffron Walden, Labyrinth at, 152Sailing, Theory of, 79–82Sand-clocks, 301Sarrau on Ether, 307Saunderson, N., 172Sauveur, Magic Squares, 122, 123,
136Scale of Notation, Denary, 10— Puzzles dependent on, 10–12School-girls, Fifteen, 103–109Schubert on π, 212Schumacher, 221Scott, Sir Walter, 245Scytale, The, 258Seconds, def. of, 290, 291Secret Communications, chap.
xi
Seelhoff, 225, 227, 230, 232Selander on π, 214Senate-House Examination,
chap. vii
Seneca on Astrology, 244Senior Optimes, 174, 177, 178Seventy-seven Puzzle, 44Shanks on π, 221Sharp on π, 220Shelton, T., 271, 272Sherwin’s Tables, 220Shuffling Cards, 109–111Shunting Problems, 53–54Sidereal Time, 290Simpson’s Euclid, 189
• Project • Gutenberg • #26839 •
INDEX. 333
Sixty-five Puzzle, 42Skeleton Cubes, 26Smith, A., on π, 223— Hen., on Numbers, 33— R., 172, 178— R.C., see RaphaelSnell on π, 219, 220Solar Time, 290Sosigenes on Calendar, 292, 293Sound, Problem in, 87— Velocity of, 306Southey on Astrology, 249Southwark, Labyrinth at, 153Space, Properties of, chap. xii
Spectrum Analysis, 306Spectrum Top, 87Spin on Cricket-ball, 85Spirits, Raising, 249Sporus on Delian Problem, 208Sprague on Eleven Queens, 101Squaring the Circle, 212–223Stability of Equilibrium, 72–74Statical Games, 48–52Steen on the Mousetrap, 119St Laurent on cards, 110Stoffler on the Calendar, 293Stokes on Ether, 307St Omer, Labyrinth at, 152Story on the Fifteen Puzzle, 88Strabo on Lake Moeris, 150Stringham on Hyper-space, 284Sturm, A., 205St Vincent, Gregory of, 209Styles, 297Suetonius, 269Sun-cylinders, 298Sun-dials, 297–299Sun-rings, 299–301Sun, the Mean, 291Svastika, 152Swift, 67
Sylvester, 49, 51, 108, 109
Tacitus on Astrology, 244Tait, 46, 59, 60, 145, 148, 305, 309,
316, 320Tanner on Shuffling Cards, 110Tarry, 57, 140, 150Tartaglia, 2, 16, 20, 27, 55Tate, 270Tavel, G.F., 191Taylor, B., 172Taylor, Ch., on Trisection, 211Tennis-ball, Cut on, 83–85Tesselation, 50–51Theory of Numbers, chap. ix
Thibaut on Baudhayana, 216Thompson on Magic Squares, 123Thomson, J.J., 305, 309, 312, 313,
320Thomson, Sir Wm., see KelvinThrasyllus on Astrology, 245Three-in-a-row, 48–50Three-pile Problem, 115–119Three-Things Problem, 18–19Tiberius on Astrology, 245Time, chap. xiii
— Equation of, 291— Measurement of, 288–291— Units of, 288–292Tissandier, 70Todhunter, J., 199Tonstall, C., 172Tower of Hanoı, 91–93Trastevere, Labyrinth at, 152Trees, Geometrical, 154–155Treize, Game of, 120Tricks with Numbers, 4–25Tripos, Mathematical, chap. vii
Tripos, Origin of term, 201–203Trisection of Angle, 210–212Tritheim, J., 275
• Project • Gutenberg • #26839 •
334 INDEX. [CH. XIV
Trollope on Mazes, 151Troy-towns, 152Turton, W.H., 39, 44, 102
Uhlemann on Astrology, 237Unicursal Problems, chap. vi
Van Ceulen on π, 218, 219Vandermonde, 63, 159, 163Van Etten, 2, 10Vase Problem, 16Vega on π, 221Vick on Clocks, 302Vieta, 209, 218Vince, S., 189, 190, 191Violle, Magic Squares, 122, 136Virgil, 150Voigt on Ether, 308Volpicelli on Knight’s Path, 158Von Helmholz, 78, 277, 308, 309Vortex rings, 308, 309— Spheres, 309— Sponges, 309, 310Voting, Question on, 26
Walecki, 107Walker, G.T., 21Wallis, J., 93, 95, 172, 220, 270Ward, S., 172Waring, E., 177, 185, 187, 190, 191Warnsdorff, Knight’s Path, 164Watch Problem, 13Watches, 77, 291, 301, 302— as Compasses, 303–304Water-clocks, 301, 303Waterloo, Battle of, 297Watersheds and Watercourses, 48Watson, R., 178, 186Waves, Superposition of, 87
Weber-Wellstein, 230Week, Days of, from date, 297Week, Names of Days, 291–292Weights Problem, the, 27–29Western on Binary Powers, 32Wheatstone on Ciphers, 269, 271Whewell, W., 171, 190, 191, 192,
193, 237, 245Whist, Number of Hands at, 26Whiston, W., 172Wiedemann on Lake Moeris, 150Wilkins on Ciphers, 253, 254, 258,
266, 275William III of England, 153Willis on Hauksbee’s law, 82Wilson on Ptolemy, 238Wilson’s Theorem, 191Wing, Labyrinth at, 152Wood, J., 185, 189, 190Woodhouse, R., 191, 192, 193Woolhouse, Kirkman’s Problem,
103Worcester, Marquis of, 272Wordsworth, C., 171, 177, 184, 186,
203Work, 72–75Wranglers, chap. vii, 174Wright, E., 172Wright, J.M.F., 194
Year, Civil, 292–294Year, Mohammedan, 294
Zach on π, 221Zech, R., 302Zeller, 297Zeno on Motion, 67–68Zodiac, Signs in Astrology, 239, 242
CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.
• Project • Gutenberg • #26839 •
335
A SHORT ACCOUNT OF THE
HISTORY OF MATHEMATICS
By W.W. ROUSE BALL.
[Third Edition. Pp. xxiv + 527. Price 10s. net.]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This book gives an account of the lives and discoveries of those mathe-maticians to whom the development of the subject is mainly due. Theuse of technicalities has been avoided and the work is intelligible to anyone acquainted with the elements of mathematics.
The author commences with an account of the origin and progressof Greek mathematics, from which the Alexandrian, the Indian, andthe Arab schools may be said to have arisen. Next the mathematicsof medieval Europe and the renaissance are described. The latter partof the book is devoted to the history of modern mathematics (begin-ning with the invention of analytical geometry and the infinitesimalcalculus), the account of which is brought down to the present time.
This excellent summary of the history of mathematics supplies a want whichhas long been felt in this country. The extremely difficult question, how far sucha work should be technical, has been solved with great tact. . . . The work containsmany valuable hints, and is thoroughly readable. The biographies, which includethose of most of the men who played important parts in the development of culture,are full and general enough to interest the ordinary reader as well as the specialist.Its value to the latter is much increased by the numerous references to authorities,a good table of contents, and a full and accurate index.—The Saturday Review.
Mr. Ball’s book should meet with a hearty welcome, for though we possessother histories of special branches of mathematics, this is the first serious attemptthat has been made in the English language to give a systematic account of theorigin and development of the science as a whole. It is written too in an attractive
• Project • Gutenberg • #26839 •
336
style. Technicalities are not too numerous or obtrusive, and the work is interspersedwith biographical sketches and anecdotes likely to interest the general reader. Thusthe tyro and the advanced mathematician alike may read it with pleasure andprofit.—The Athenæum.
A wealth of authorities, often far from accordant with each other, renders awork such as this extremely formidable; and students of mathematics have reasonto be grateful for the vast amount of information which has been condensed intothis short account. . . . In a survey of so wide extent it is of course impossible to giveanything but a bare sketch of the various lines of research, and this circ*mstancetends to render a narrative scrappy. It says much for Mr. Ball’s descriptive skill thathis history reads more like a continuous story than a series of merely consecutivesummaries.—The Academy.
We can heartily recommend to our mathematical readers, and to others also,Mr. Ball’s History of Mathematics. The history of what might be supposed a drysubject is told in the pleasantest and most readable style, and at the same timethere is evidence of the most careful research.—The Observatory.
All the salient points of mathematical history are given, and many of theresults of recent antiquarian research; but it must not be imagined that the bookis at all dry. On the contrary the biographical sketches frequently contain amusinganecdotes, and many of the theorems mentioned are very clearly explained so asto bring them within the grasp of those who are only acquainted with elementarymathematics.—Nature.
Le style de M. Ball est clair et elegant, de nombreux apercus rendent facile desuivre le fil de son exposition et de frequentes citations permettent a celui qui ledesire d’approfondir les recherches que l’auteur n’a pu qu’effleurer. . . . Cet ouvragepourra devenir tres utile comme manuel d’histoire des mathematiques pour lesetudiants, et il ne sera pas deplace dans les bibliotheques des savants.—BibliothecaMathematica.
The author modestly describes his work as a compilation, but it is thoroughlywell digested, a due proportion is observed between the various parts, and whenoccasion demands he does not hesitate to give an independent judgment on a dis-puted point. His verdicts in such instances appear to us to be generally sound andreasonable. . . . To many readers who have not the courage or the opportunity totackle the ponderous volumes of Montucla or the (mostly) ponderous treatises ofGerman writers on special periods, it may be somewhat of a surprise to find whata wealth of human interest attaches to the history of so “dry” a subject as math-ematics. We are brought into contact with many remarkable men, some of whomhave played a great part in other fields, as the names of Gerbert, Wren, Leibnitz,Descartes, Pascal, D’Alembert, Carnot, among others may testify, and with at leastone thorough blackguard (Cardan); and Mr. Ball’s pages abound with quaint andamusing touches characteristic of the authors under consideration, or of the timesin which they lived.—Manchester Guardian.
There can be no doubt that the author has done his work in a very excellentway. . . . There is no one interested in almost any part of mathematical science who
• Project • Gutenberg • #26839 •
337
will not welcome such an exposition as the present, at once popularly written andexact, embracing the entire subject. . . . Mr. Ball’s work is destined to become astandard one on the subject.—The Glasgow Herald.
A most interesting book, not only for those who are mathematicians, but forthe much larger circle of those who care to trace the course of general scientificprogress. It is written in such a way that those who have only an elementaryacquaintance with the subject can find on almost every page something of generalinterest.—The Oxford Magazine.
• Project • Gutenberg • #26839 •
338
A PRIMER OF THE
HISTORY OF MATHEMATICS
By W.W. ROUSE BALL.
[Second Edition. Pp. iv + 148. Price 2s. net.]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This book contains a sketch in popular language of the history of math-ematics; it includes some notice of the lives and surroundings of thoseto whom the development of the subject is mainly due as well as oftheir discoveries.
This Primer is written in the agreeable style with which the author has madeus acquainted in his previous essays; and we are sure that all readers of it will beready to say that Mr. Ball has succeeded in the hope he has formed, that “it maynot be uninteresting” even to those who are unacquainted with the leading facts. Itis just the book to give an intelligent young student, and should allure him on to theperusal of Mr. Ball’s “Short Account.” The present work is not a mere rechauffe ofthat, though naturally most of what is here given will be found in equivalent formin the larger work. . . . The choice of material appears to us to be such as shouldlend interest to the study of mathematics and increase its educational value, whichhas been the author’s aim. The book goes well into the pocket, and is excellentlyprinted.—The Academy.
We have here a new instance of Mr. Rouse Ball’s skill in giving in a smallspace an intelligible account of a large subject. In 137 pages we have a sketchof the progress of mathematics from the earliest records up to the middle of thiscentury, and yet it is interesting to read and by no means a mere catalogue.—TheManchester Guardian.
It is not often that a reviewer of mathematical works can confess that hehas read one of them through from cover to cover without abatement of interest orfatigue. But that is true of Mr. Rouse Ball’s wonderfully entertaining little “Historyof Mathematics,” which we heartily recommend to even the quite rudimentarymathematician. The capable mathematical master will not fail to find a dozeninteresting facts therein to season his teaching.—The Saturday Review.
• Project • Gutenberg • #26839 •
339
A fascinating little volume, which should be in the hands of all who do not pos-sess the more elaborate History of Mathematics by the same author.—The Math-ematical Gazette.
This excellent sketch should be in the hands of every student, whether he isstudying mathematics or no. In most cases there is an unfortunate lack of knowl-edge upon this subject, and we welcome anything that will help to supply thedeficiency. The primer is written in a concise, lucid and easy manner, and givesthe reader a general idea of the progress of mathematics that is both interestingand instructive.—The Cambridge Review.
Mr. Ball has not been deterred by the existence and success of his larger“History of Mathematics” from publishing a simple compendium in about a quarterof the space. . . . Of course, what he now gives is a bare outline of the subject, but itis ample for all except the most advanced proficients. There is no question that, asthe author says, a knowledge of the history of a science lends interest to its study,and often increases its educational value. We can imagine no better cathartic forany mathematical student who has made some way with the calculus than a carefulperusal of this little book.—The Educational Times.
The author has done good service to mathematicians by engaging in workin this special field. . . . The Primer gives, in a brief compass, the history of theadvance of this branch of science when under Greek influence, during the MiddleAges, and at the Renaissance, and then goes on to deal with the introduction ofmodern analysis and its recent developments. It refers to the life and work of theleaders of mathematical thought, adds a new and enlarged value to well-knownproblems by treating of their inception and history, and lights up with a warm andpersonal interest a science which some of its detractors have dared to call dull andcold.—The Educational Review.
It is not too much to say that this little work should be in the possession of everymathematical teacher. . . . The Primer gives in a small compass the leading events inthe development of mathematics. . . . At the same time, it is no dry chronicle of factsand theorems. The biographical sketches of the great workers, if short, are pithy,and often amusing. Well-known propositions will attain a new interest for the pupilas he traces their history long before the time of Euclid.—The Journal of Education.
This is a work which all who apprehend the value of “mathematics” shouldread and study. . . , and those who wish to learn how to think will find advantagein reading it.—The English Mechanic.
The subject, so far as our own language is concerned, is almost Mr. Ball’s own,and those who have no leisure to read his former work will find in this Primer ahighly-readable and instructive chapter in the history of education. The condensa-tion has been skilfully done, the reader’s interest being sustained by the introductionof a good deal of far from tedious detail.—The Glasgow Herald.
Mr. W.W. Rouse Ball is well known as the author of a very clever history ofmathematics, besides useful works on kindred subjects. His latest production is APrimer of the History of Mathematics, a book of one hundred and forty pages, givingin non-technical language a full, concise, and readable narrative of the development
• Project • Gutenberg • #26839 •
340
of the science from the days of the Ionian Greeks until the present time. Anyonewith a leaning towards algebraic or geometrical studies will be intensely interested inthis account of progress from primitive usages, step by step, to our present elaboratesystems. The lives of the men who by their research and discovery helped alongthe good work are described briefly, but graphically. . . . The Primer should becomea standard text-book.—The Literary World.
This is a capital little sketch of a subject on which Mr. Ball is an acknowledgedauthority, and of which too little is generally known. Mr. Ball, moreover, writeseasily and well, and has the art of saying what he has to say in an interestingstyle.—The School Guardian.
• Project • Gutenberg • #26839 •
341
MATHEMATICAL
RECREATIONS AND ESSAYS
By W.W. ROUSE BALL.
[Fourth Edition. Pp. xvi + 402. Price 7s. net.]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This work is divided into two parts; the first is on mathematical recre-ations and puzzles, the second includes some miscellaneous essays andan account of some problems of historical interest. In both parts ques-tions which involve advanced mathematics are excluded.
The mathematical recreations include numerous elementary ques-tions and paradoxes, as well as problems such as the proposition that tocolour a map not more than four colours are necessary, the explanationof the effect of a cut on a tennis ball, the fifteen puzzle, the eight queensproblem, the fifteen school-girls, the construction of magic squares, thetheory and history of mazes, and the knight’s path on a chess-board.
The second part commences with sketches of the history of theMathematical Tripos at Cambridge, of the three famous classical prob-lems in geometry (namely, the duplication of the cube, the trisection ofan angle, and the quadrature of the circle) and of Mersenne’s Numbers.These are followed by essays on Astrology and Ciphers. The last threechapters are devoted to an account of the hypotheses as to the natureof Space and Mass, and the means of measuring Time.
Mr. Ball has already attained a position in the front rank of writers on subjectsconnected with the history of mathematics, and this brochure will add another tohis successes in this field. In it he has collected a mass of information bearing uponmatters of more general interest, written in a style which is eminently readable,and at the same time exact. He has done his work so thoroughly that he has leftfew ears for other gleaners. The nature of the work is completely indicated to themathematical student by its title. Does he want to revive his acquaintance with theProblemes Plaisans et Delectables of Bachet, or the Recreations Mathematiques etPhysiques of Ozanam? Let him take Mr. Ball for his companion, and he will have
• Project • Gutenberg • #26839 •
342
the cream of these works put before him with a wealth of illustration quite delightful.Or, coming to more recent times, he will have full and accurate discussion of ‘thefifteen puzzle,’ ‘Chinese rings,’ ‘the fifteen school-girls problem’ et id genus omne.Sufficient space is devoted to accounts of magic squares and unicursal problems(such as mazes, the knight’s path, and geometrical trees). These, and many otherproblems of equal interest, come under the head of ‘Recreations.’ The problemsand speculations include an account of the Three Classical Problems; there is alsoa brief sketch of Astrology; and interesting outlines of the present state of ourknowledge of hyper-space and of the constitution of matter. This enumerationbadly indicates the matter handled, but it sufficiently states what the reader mayexpect to find. Moreover for the use of readers who may wish to pursue the severalheads further, Mr. Ball gives detailed references to the sources from whence hehas derived his information. These Mathematical Recreations we can commendas suited for mathematicians and equally for others who wish to while away anoccasional hour.—The Academy.
The idea of writing some such account as that before us must have been presentto Mr. Ball’s mind when he was collecting the material which he has so skilfullyworked up into his History of Mathematics. We think this because . . . many bits ofore which would not suit the earlier work find a fitting niche in this. Howsoever thecase may be, we are sure that non-mathematical, as well as mathematical, readerswill derive amusem*nt, and, we venture to think, profit withal, from a perusal ofit. The author has gone very exhaustively over the ground, and has left us littleopportunity of adding to or correcting what he has thus reproduced from his note-books. The work before us is divided into two parts: mathematical recreations andmathematical problems and speculations. All these matters are treated lucidly, andwith sufficient detail for the ordinary reader, and for others there is ample store ofreferences. . . . Our analysis shows how great an extent of ground is covered, andthe account is fully pervaded by the attractive charm Mr. Ball knows so well howto infuse into what many persons would look upon as a dry subject.—Nature.
A fit sequel to its author’s valuable and interesting works on the history ofmathematics. There is a fascination about this volume which results from a happycombination of puzzle and paradox. There is both milk for babes and strong meatfor grown men. . . . A great deal of the information is hardly accessible in anyEnglish books; and Mr. Ball would deserve the gratitude of mathematicians forhaving merely collected the facts. But he has presented them with such lucidityand vivacity of style that there is not a dull page in the book; and he has addedminute and full bibliographical references which greatly enhance the value of hiswork.—The Cambridge Review.
Mathematicians with a turn for the paradoxes and puzzles connected with num-ber, space, and time, in which their science abounds, will delight in MathematicalRecreations and Problems of Past and Present Times.—The Times.
Mathematicians have their recreations; and Mr. Ball sets forth the humoursof mathematics in a book of deepest interest to the clerical reader, and of no littleattractiveness to the layman. The notes attest an enormous amount of research.—The National Observer.
• Project • Gutenberg • #26839 •
343
Mr. Ball, to whom we are already indebted for two excellent Histories of Math-ematics, has just produced a book which will be thoroughly appreciated by thosewho enjoy the setting of the wits to work. . . . He has collected a vast amount ofinformation about mathematical quips, tricks, cranks, and puzzles—old and new;and it will be strange if even the most learned do not find something fresh in theassortment.—The Observatory.
Mr. Rouse Ball has the true gift of story-telling, and he writes so pleasantlythat though we enjoy the fulness of his knowledge we are tempted to forget theconsiderable amount of labour involved in the preparation of his book. He givesus the history and the mathematics of many problems . . . and where the limits ofhis work prevent him from dealing fully with the points raised, like a true workerhe gives us ample references to original memoirs. . . . The book is warmly to berecommended, and should find a place on the shelves of every one interested inmathematics and on those of every public library.—The Manchester Guardian.
A work which will interest all who delight in mathematics and mental exercisesgenerally. The student will often take it up, as it contains many problems whichpuzzle even clever people.—The English Mechanic and World of Science.
This is a book which the general reader should find as interesting as the mathe-matician. At all events, an intelligent enjoyment of its contents presupposes no moreknowledge of mathematics than is now-a-days possessed by almost everybody.—TheAthenæum.
An exceedingly interesting work which, while appealing more directly to thosewho are somewhat mathematically inclined, it is at the same time calculated tointerest the general reader. . . . Mr. Ball writes in a highly interesting manner on afascinating subject, the result being a work which is in every respect excellent.—The Mechanical World.
E um livro muito interessante, consagrado a recreios mathematicos, algunsdos quaes sao muito bellos, e a problemas interessantes da mesma sciencia, quenao exige para ser lido grandes conhecimentos mathematicos e que tem em graoelevado a qualidade de instruir, deleitando ao mesmo tempo.—Journal de scienciasmathematicas, Coimbra.
The work is a very judicious and suggestive compilation, not meant mainlyfor mathematicians, yet made doubly valuable to them by copious references. Thestyle in the main is so compact and clear that what is central in a long argument orprocess is admirably presented in a few words. One great merit of this, or any otherreally good book on such a subject, is its suggestiveness; and in running throughits pages, one is pretty sure to think of additional problems on the same generallines.—Bulletin of the New York Mathematical Society.
A book which deserves to be widely known by those who are fond of solvingpuzzles . . . and will be found to contain an admirable classified collection of inge-nious questions capable of mathematical analysis. As the author is himself a skilfulmathematician, and is careful to add an analysis of most of the propositions, it mayeasily be believed that there is food for study as well as amusem*nt in his pages. . . .Is in every way worthy of praise.—The School Guardian.
• Project • Gutenberg • #26839 •
344
Once more the author of a Short History of Mathematics and a History ofthe Study of Mathematics at Cambridge gives evidence of the width of his readingand of his skill in compilation. From the elementary arithmetical puzzles whichwere known in the sixteenth and seventeenth centuries to those modern ones themathematical discussion of which has taxed the energies of the ablest investigator,very few questions have been left unrepresented. The sources of the author’s in-formation are indicated with great fulness. . . . The book is a welcome addition toEnglish mathematical literature.—The Oxford Magazine.
• Project • Gutenberg • #26839 •
345
A HISTORY OF THE STUDY OF
MATHEMATICS ATCAMBRIDGE
By W.W. ROUSE BALL.
[Pp. xvi + 264. Price 6s.]
THE UNIVERSITY PRESS, CAMBRIDGE.
This work contains an account of the development of the study ofmathematics in the university of Cambridge from the twelfth centuryto the middle of the nineteenth century, and a description of the meansby which proficiency in that study was tested at various times.
The first part of the book is devoted to a brief account of the moreeminent of the Cambridge mathematicians, the subject matter of theirworks, and their methods of exposition. The second part treats of themanner in which mathematics was taught, and of the exercises andexaminations required of students in past times. A sketch is given ofthe origin and history of the Mathematical Tripos; this includes thesubstance of the earlier parts of the author’s work on that subject,Cambridge, 1880. To explain the relation of mathematics to otherdepartments of study an outline of the general history of the universityand the organization of education therein is added.
The present volume is very pleasant reading, and though much of it necessarilyappeals only to mathematicians, there are parts—e.g. the chapters on Newton, onthe growth of the tripos, and on the history of the university—which are full of inter-est for a general reader. . . . The book is well written, the style is crisp and clear, andthere is a humorous appreciation of some of the curious old regulations which havebeen superseded by time and change of custom. Though it seems light, it must rep-resent an extensive study and investigation on the part of the author, the essentialresults of which are skilfully given. We can most thoroughly commend Mr. Ball’svolume to all readers who are interested in mathematics or in the growth and theposition of the Cambridge school of mathematicians.—The Manchester Guardian.
• Project • Gutenberg • #26839 •
346
Voici un livre dont la lecture inspire tout d’abord le regret que des travauxanalogues n’aient pas ete faits pour toutes les Ecoles celebres, et avec autant de soinet de clarte. . . . Toutes les parties du livre nous out vivement interesse.—Bulletindes sciences mathematiques.
A book of pleasant and useful reading for both historians and mathematicians.Mr. Ball’s previous researches into this kind of history have already establishedhis reputation, and the book is worthy of the reputation of its author. It is morethan a detailed account of the rise and progress of mathematics, for it involvesa very exact history of the University of Cambridge from its foundation.—TheEducational Times.
Mr. Ball is far from confining his narrative to the particular science of whichhe is himself an acknowledged master, and his account of the study of mathematicsbecomes a series of biographical portraits of eminent professors and a record notonly of the intellectual life of the elite but of the manners, habits, and discussionsof the great body of Cambridge men from the sixteenth century to our own. . . . Hehas shown how the University has justified its liberal reputation, and how amplyprepared it was for the larger freedom which it now enjoys.—The Daily News.
Mr. Ball has not only given us a detailed account of the rise and progress ofthe science with which the name of Cambridge is generally associated but has alsowritten a brief but reliable and interesting history of the university itself from itsfoundation down to recent times. . . . The book is pleasant reading alike for themathematician and the student of history.—St. James’s Gazette.
A very handy and valuable book containing, as it does, a vast deal of interestinginformation which could not without inconceivable trouble be found elsewhere. . . .It is very far from forming merely a mathematical biographical dictionary, thegrowth of mathematical science being skilfully traced in connection with the suc-cessive names. There are probably very few people who will be able thoroughly toappreciate the author’s laborious researches in all sorts of memoirs and transactionsof learned societies in order to unearth the material which he has so agreeably con-densed. . . . Along with this there is much new matter which, while of great interestto mathematicians, and more especially to men brought up at Cambridge, will befound to throw a good deal of new and important light on the history of educationin general.—The Glasgow Herald.
Exceedingly interesting to all who care for mathematics. . . . After giving anaccount of the chief Cambridge Mathematicians and their works in chronologicalorder, Mr. Rouse Ball goes on to deal with the history of tuition and examinationsin the University . . . and recounts the steps by which the word “tripos” changed itsmeaning “from a thing of wood to a man, from a man to a speech, from a speechto two sets of verses, from verses to a sheet of coarse foolscap paper, from a paperto a list of names, and from a list of names to a system of examination.”—Neverdid word undergo so many alterations.—The Literary World.
In giving an account of the development of the study of mathematics in theUniversity of Cambridge, and the means by which mathematical proficiency wastested in successive generations, Mr. Ball has taken the novel plan of devoting the
• Project • Gutenberg • #26839 •
347
first half of his book to . . . the more eminent Cambridge mathematicians, and ofreserving to the second part an account of how at various times the subject wastaught, and how the result of its study was tested. . . . Very interesting informationis given about the work of the students during the different periods, with specimensof problem-papers as far back as 1802. The book is very enjoyable, and gives acapital and accurate digest of many excellent authorities which are not within thereach of the ordinary reader.—The Scots Observer.
• Project • Gutenberg • #26839 •
348
AN ESSAY ON
THE GENESIS, CONTENTS, AND HISTORYOF
NEWTON’S “PRINCIPIA”
By W.W. ROUSE BALL.
[Pp. x + 175. Price 6s. net.]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This work contains an account of the successive discoveries of Newtonon gravitation, the methods he used, and the history of his researches.
It commences with a review of the extant authorities dealing withthe subject. In the next two chapters the investigations made in 1666and 1679 are discussed, some of the documents dealing therewith be-ing here printed for the first time. The fourth chapter is devoted tothe investigations made in 1684: these are illustrated by Newton’s pro-fessorial lectures (of which the original manuscript is extant) of thatautumn, and are summed up in the almost unknown memoir of Febru-ary, 1685, which is here reproduced from Newton’s holograph copy. Inthe two following chapters the details of the preparation from 1685 to1687 of the Principia are described, and an analysis of the work isgiven. The seventh chapter comprises an account of the researches ofNewton on gravitation subsequent to the publication of the first editionof the Principia, and a sketch of the history of that work.
In the last chapter, the extant letters of 1678–1679 between Hookeand Newton, and of those of 1686–1687 between Halley and Newton, arereprinted, and there are also notes on the extant correspondence con-cerning the production of the second and third editions of the Principia.
For the essay which we have before us, Mr Ball should receive the thanks ofall those to whom the name of Newton recalls the memory of a great man. The
• Project • Gutenberg • #26839 •
349
Principia, besides being a lasting monument of Newton’s life, is also to-day theclassic of our mathematical writings, and will be so for some time to come. . . . Thevalue of the present work is also enhanced by the fact that, besides containing afew as yet unpublished letters, there are collected in its pages quotations from alldocuments, thus forming a complete summary of everything that is known on thesubject. . . . The author is so well known a writer on anything connected with thehistory of mathematics, that we need make no mention of the thoroughness of theessay, while it would be superfluous for us to add that from beginning to end it ispleasantly written and delightful to read. Those well acquainted with the Principiawill find much that will interest them, while those not so fully enlightened willlearn much by reading through the account of the origin and history of Newton’sgreatest work.—Nature.
An Essay on Newton’s Principia will suggest to many something solely mathe-matical, and therefore wholly uninteresting. No inference could be more erroneous.The book certainly deals largely in scientific technicalities which will interest expertsonly; but it also contains much historical information which might attract manywho, from laziness or inability, would be very willing to take all its mathematics forgranted. Mr. Ball carefully examines the evidence bearing on the development ofNewton’s great discovery, and supplies the reader with abundant quotations fromcontemporary authorities. Not the least interesting portion of the book is the ap-pendix, or rather appendices, containing copies of the original documents (mostlyletters) to which Mr. Ball refers in his historical criticisms. Several of these bearupon the irritating and unfounded claims of Hooke.—The Athenæum.
La savante monographie de M. Ball est redigee avec beaucoup de soin, et aplusieurs egards elle peut servir de modele pour des ecrits de la meme nature.—Bibliotheca Mathematica.
Newton’s Principia has world-wide fame as a classic of mathematical science.But those who know thoroughly the contents and the history of the book are aselect company. It was at one time the purpose of Mr. Ball to prepare a newcritical edition of the work, accompanied by a prefatory history and notes, andby an analytical commentary. Mathematicians will regret to hear that there is noprospect in the immediate future of seeing this important book carried to completionby so competent a hand. They will at the same time welcome Mr. Ball’s Essay onthe Principia for the elucidations which it gives of the process by which Newton’sgreat work originated and took form, and also as an earnest of the completedplan.—The Scotsman.
In this essay Mr. Ball presents us with an account highly interesting to math-ematicians and natural philosophers of the origin and history of that remark-able product of a great genius Philosophiae Naturalis Principia Mathematica, ‘TheMathematical Principles of Natural Philosophy,’ better known by the short termPrincipia. . . . Mr. Ball’s essay is one of extreme interest to students of physical sci-ence, and it is sure to be widely read and greatly appreciated.—The Glasgow Herald.
To his well-known and scholarly treatises on the History of Mathematics
• Project • Gutenberg • #26839 •
350
Mr. W.W. Rouse Ball has added An Essay on Newton’s Principia. Newton’s Prin-cipia, as Mr. Ball justly observes, is the classic of English mathematical writings;and this sound, luminous, and laborious essay ought to be the classical accountof the Principia. The essay is the outcome of a critical edition of Newton’s greatwork, which Mr. Ball tells us that he once contemplated. It is much to be hopedthat he will carry out his intention, for no English mathematician is likely to do thework better or in a more reverent spirit. . . . It is unnecessary to say that Mr. Ballhas a complete knowledge of his subject. He writes with an ease and clearnessthat are rare.—The Scottish Leader.
Le volume de M. Rouse Ball renferme tout ce que l’on peut desirer savoirsur l’histoire des Principes; c’est d’ailleurs l’œuvre d’un esprit clair, judicieux, etmethodique.—Bulletin des Sciences Mathematiques.
Mr. Ball has put into small space a very great deal of interesting matter, andhis book ought to meet with a wide circulation among lovers of Newton and thePrincipia.—The Academy.
Admirers of Mr. W.W. Rouse Ball’s Short Account of the History of Mathe-matics will be glad to receive a detailed study of the history of the Principia fromthe same hand. This book, like its predecessors, gives a very lucid account of itssubject. We find in it an account of Newton’s investigations in his earlier years,which are to some extent collected in the tract de Motu (the germ of the Principia)the text of which Mr. Rouse Ball gives us in full. In a later chapter there is a fullanalysis of the Principia itself, and after that an account of the preparation of thesecond and third editions. Probably the part of the book which will be found mostinteresting by the general reader is the account of the correspondence of Newtonwith Hooke, and with Halley, about the contents or the publication of the Prin-cipia. This correspondence is given in full, so far as it is recoverable. Hooke doesnot appear to advantage in it. He accuses Newton of stealing his ideas. His vain andenvious disposition made his own merits appear great in his eyes, and be-dwarfedthe work of others, so that he seems to have believed that Newton’s great perfor-mance was a mere expanding and editing of the ideas of Mr. Hooke—ideas whichwere meritorious, but after all mere guesses at truth. This, at all events, is the mostcharitable view we can take of his conduct. Halley, on the contrary, appears as aman to whom we ought to feel most grateful. It almost seems as though Newton’sphysical insight and extraordinary mathematical powers might have been largelywasted, as was Pascal’s rare genius, if it had not been for Halley’s single-heartedand self-forgetful efforts to get from his friend’s genius all he could for the enlight-enment of men. It was probably at his suggestion that the writing of the Principiawas undertaken. When the work was presented to the Royal Society, they under-took its publication, but, being without the necessary funds, the expense fell uponHalley. When Newton, stung by Hooke’s accusations, wished to withdraw a partof the work, Halley’s tact was required to avert the catastrophe. All the drudgery,worry, and expense fell to his share, and was accepted with the most generous goodnature. It will be seen that both the technical student and the general reader mayfind much to interest him in Mr. Rouse Ball’s book.—The Manchester Guardian.
Une histoire tres bien faite de la genese du livre immortel de Newton. . . .
• Project • Gutenberg • #26839 •
351
Le livre de M. Ball est une monographie precieuse sur un point important del’histoire des mathematiques. Il contribuera a accroıtre, si c’est possible, lagloire de Newton, en revelant a beaucoup de lecteurs, avec quelle merveilleuserapidite l’illustre geometre anglais a eleve a la science ce monument immortel,les Principia.—Mathesis.
• Project • Gutenberg • #26839 •
352
NOTES ON THE HISTORY OF
TRINITY COLLEGE,CAMBRIDGE
By W.W. ROUSE BALL.
[Pp. xiv + 183. Price 2s. 6d. net.]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This booklet gives a popular account of the History of Trinity Col-lege, Cambridge, and so far as the author knows, it is as yet (1905)the only work published on the subject. It was written mainly for theuse of his pupils, and contains such information and gossip about theCollege and life there in past times as he believed would be interestingto most undergraduates and members of the House.
This modest and unpretending little volume seems to us to do more for itssubject than many of the more formal volumes . . . treating of the separate col-leges of the English universities. . . . In nine short, extremely readable, and trulyinforming chapters it gives the reader a very vivid account at once of the origin anddevelopment of the University of Cambridge, of the rise and gradual supremacy ofthe colleges, of King’s Hall as founded by Edward II, of the suppression of King’sHall by Henry VIII on December 17, 1546, the foundation of Trinity College byroyal charter on December 19, and the subsequent fortunes of the premier collegeof Cambridge. The subject is in a way treated under the successive heads of thecollege, but this is quite subordinate to the handling and characterisation of thesubject under four great periods—namely, that during the Middle Ages, that dur-ing the Renaissance, that under the Elizabethan statutes, and that during the lasthalf-century. The colleges arose from the determination of the University to pre-vent students who were very young from seeking lodging, whether under the wing ofone or other of the religious orders—a circ*mstance which shows this University tohave been an essentially lay corporation. Early in the sixteenth century the collegehad absorbed all the members of the University, and henceforth the University waslittle more than the degree-granting body to students who lived and moved andhad their educational being under the colleges. . . . The University finally took theform of an aggregate of separate and independent corporations, with a federal con-stitution analogous in a rough sort of way to that of the United States of America,
• Project • Gutenberg • #26839 •
353
and different from similar corporations at Paris by the fact that these latter werealways subject to University supervision. . . . There is a good account of the effortnow going on to re-assert the University at the expense of the colleges. No onewho begins Mr. Ball’s book will lay it down till he has read it from beginning toend.—The Glasgow Herald.
It is a sign of the times, and a very satisfactory one, when . . . a tutor . . . takesthe trouble to make the history of his college known to his pupils. Considering thelack of good books about the Universities, we may thank Mr. Ball that he has beengood enough to print for a larger circle. Though he modestly calls his book only“Notes,” yet it is eminently readable, and there is plenty of information, as well asabundance of good stories, in its pages.—The Oxford Magazine.
Mr. Ball has put not only the pupils for whom he compiled these notes, but thelarge world of Trinity men, under a great obligation by this compendious but lucidand interesting history of the society to whose service he is devoted. The value ofhis contribution to our knowledge is increased by the extreme simplicity with whichhe tells his story, and the very suggestive details which, without much comment,he has selected, with admirable discernment, out of the wealth of materials athis disposal. His initial account of the development of the University is brief butextremely clear, presenting us with facts rather than theories, but establishing,with much distinctness, the essential difference between the hostels, out of whichthe more modern colleges grew, and that monastic life which poorer students wereoften tempted to join.—The Guardian.
An interesting and valuable book. . . . It is described by its author as “littlemore than an orderly transcript” of what, as a Fellow and Tutor of the College,he has been accustomed to tell his pupils. But while it does not pretend either tothe form or to the exhaustiveness of a set history, it is scholarly enough to rank asan authority, and far more interesting and readable than most academic historiesare. It gives an instructive sketch of the development of the University and of theparticular history of Trinity, noting its rise and policy in the earlier centuries ofits existence, until, under the misrule of Bentley, it came into a state of disorderwhich nearly resulted in its dissolution. The subsequent rise of the College and itsposition in what Mr. Ball calls the Victorian renaissance, are drawn in lines no lesssuggestive; and the book, as a whole, cannot fail to be welcome to every one whois closely interested in the progress of the College.—The Scotsman.
Mr. Ball has succeeded very well in giving in this little volume just what anintelligent undergraduate ought and probably often does desire to know about thebuildings and the history of his College. . . . The debt of the “royal and religiousfoundation” to Henry VIII is explained with fulness, and there is much interest-ing matter as to the manner of life and the expenses of students in the sixteenthcentury.—The Manchester Guardian.
• Project • Gutenberg • #26839 •
LICENSING. 355
End of the Project Gutenberg EBook of Mathematical Recreations and Essays,by
W. W. Rouse Ball
*** END OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS ***
***** This file should be named 26839-pdf.pdf or 26839-pdf.zip *****This and all associated files of various formats will be found in:
http://www.gutenberg.org/2/6/8/3/26839/
Produced by Joshua Hutchinson, David Starner, David Wilsonand the Online Distributed Proofreading Team athttp://www.pgdp.net
Updated editions will replace the previous one--the old editionswill be renamed.
Creating the works from public domain print editions means that noone owns a United States copyright in these works, so the Foundation(and you!) can copy and distribute it in the United States withoutpermission and without paying copyright royalties. Special rules,set forth in the General Terms of Use part of this license, apply tocopying and distributing Project Gutenberg-tm electronic works toprotect the PROJECT GUTENBERG-tm concept and trademark. ProjectGutenberg is a registered trademark, and may not be used if youcharge for the eBooks, unless you receive specific permission. If youdo not charge anything for copies of this eBook, complying with therules is very easy. You may use this eBook for nearly any purposesuch as creation of derivative works, reports, performances andresearch. They may be modified and printed and given away--you may dopractically ANYTHING with public domain eBooks. Redistribution issubject to the trademark license, especially commercialredistribution.
*** START: FULL LICENSE ***
THE FULL PROJECT GUTENBERG LICENSEPLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
To protect the Project Gutenberg-tm mission of promoting the freedistribution of electronic works, by using or distributing this work(or any other work associated in any way with the phrase "ProjectGutenberg"), you agree to comply with all the terms of the Full ProjectGutenberg-tm License (available with this file or online athttp://gutenberg.org/license).
Section 1. General Terms of Use and Redistributing Project Gutenberg-tmelectronic works
1.A. By reading or using any part of this Project Gutenberg-tmelectronic work, you indicate that you have read, understand, agree toand accept all the terms of this license and intellectual property
• Project • Gutenberg • #26839 •
356 LICENSING.
(trademark/copyright) agreement. If you do not agree to abide by allthe terms of this agreement, you must cease using and return or destroyall copies of Project Gutenberg-tm electronic works in your possession.If you paid a fee for obtaining a copy of or access to a ProjectGutenberg-tm electronic work and you do not agree to be bound by theterms of this agreement, you may obtain a refund from the person orentity to whom you paid the fee as set forth in paragraph 1.E.8.
1.B. "Project Gutenberg" is a registered trademark. It may only beused on or associated in any way with an electronic work by people whoagree to be bound by the terms of this agreement. There are a fewthings that you can do with most Project Gutenberg-tm electronic workseven without complying with the full terms of this agreement. Seeparagraph 1.C below. There are a lot of things you can do with ProjectGutenberg-tm electronic works if you follow the terms of this agreementand help preserve free future access to Project Gutenberg-tm electronicworks. See paragraph 1.E below.
1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation"or PGLAF), owns a compilation copyright in the collection of ProjectGutenberg-tm electronic works. Nearly all the individual works in thecollection are in the public domain in the United States. If anindividual work is in the public domain in the United States and you arelocated in the United States, we do not claim a right to prevent you fromcopying, distributing, performing, displaying or creating derivativeworks based on the work as long as all references to Project Gutenbergare removed. Of course, we hope that you will support the ProjectGutenberg-tm mission of promoting free access to electronic works byfreely sharing Project Gutenberg-tm works in compliance with the terms ofthis agreement for keeping the Project Gutenberg-tm name associated withthe work. You can easily comply with the terms of this agreement bykeeping this work in the same format with its attached full ProjectGutenberg-tm License when you share it without charge with others.
1.D. The copyright laws of the place where you are located also governwhat you can do with this work. Copyright laws in most countries are ina constant state of change. If you are outside the United States, checkthe laws of your country in addition to the terms of this agreementbefore downloading, copying, displaying, performing, distributing orcreating derivative works based on this work or any other ProjectGutenberg-tm work. The Foundation makes no representations concerningthe copyright status of any work in any country outside the UnitedStates.
1.E. Unless you have removed all references to Project Gutenberg:
1.E.1. The following sentence, with active links to, or other immediateaccess to, the full Project Gutenberg-tm License must appear prominentlywhenever any copy of a Project Gutenberg-tm work (any work on which thephrase "Project Gutenberg" appears, or with which the phrase "ProjectGutenberg" is associated) is accessed, displayed, performed, viewed,copied or distributed:
This eBook is for the use of anyone anywhere at no cost and withalmost no restrictions whatsoever. You may copy it, give it away orre-use it under the terms of the Project Gutenberg License includedwith this eBook or online at www.gutenberg.org
• Project • Gutenberg • #26839 •
LICENSING. 357
1.E.2. If an individual Project Gutenberg-tm electronic work is derivedfrom the public domain (does not contain a notice indicating that it isposted with permission of the copyright holder), the work can be copiedand distributed to anyone in the United States without paying any feesor charges. If you are redistributing or providing access to a workwith the phrase "Project Gutenberg" associated with or appearing on thework, you must comply either with the requirements of paragraphs 1.E.1through 1.E.7 or obtain permission for the use of the work and theProject Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or1.E.9.
1.E.3. If an individual Project Gutenberg-tm electronic work is postedwith the permission of the copyright holder, your use and distributionmust comply with both paragraphs 1.E.1 through 1.E.7 and any additionalterms imposed by the copyright holder. Additional terms will be linkedto the Project Gutenberg-tm License for all works posted with thepermission of the copyright holder found at the beginning of this work.
1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tmLicense terms from this work, or any files containing a part of thiswork or any other work associated with Project Gutenberg-tm.
1.E.5. Do not copy, display, perform, distribute or redistribute thiselectronic work, or any part of this electronic work, withoutprominently displaying the sentence set forth in paragraph 1.E.1 withactive links or immediate access to the full terms of the ProjectGutenberg-tm License.
1.E.6. You may convert to and distribute this work in any binary,compressed, marked up, nonproprietary or proprietary form, including anyword processing or hypertext form. However, if you provide access to ordistribute copies of a Project Gutenberg-tm work in a format other than"Plain Vanilla ASCII" or other format used in the official versionposted on the official Project Gutenberg-tm web site (www.gutenberg.org),you must, at no additional cost, fee or expense to the user, provide acopy, a means of exporting a copy, or a means of obtaining a copy uponrequest, of the work in its original "Plain Vanilla ASCII" or otherform. Any alternate format must include the full Project Gutenberg-tmLicense as specified in paragraph 1.E.1.
1.E.7. Do not charge a fee for access to, viewing, displaying,performing, copying or distributing any Project Gutenberg-tm worksunless you comply with paragraph 1.E.8 or 1.E.9.
1.E.8. You may charge a reasonable fee for copies of or providingaccess to or distributing Project Gutenberg-tm electronic works providedthat
- You pay a royalty fee of 20% of the gross profits you derive fromthe use of Project Gutenberg-tm works calculated using the methodyou already use to calculate your applicable taxes. The fee isowed to the owner of the Project Gutenberg-tm trademark, but hehas agreed to donate royalties under this paragraph to theProject Gutenberg Literary Archive Foundation. Royalty paymentsmust be paid within 60 days following each date on which youprepare (or are legally required to prepare) your periodic taxreturns. Royalty payments should be clearly marked as such andsent to the Project Gutenberg Literary Archive Foundation at the
• Project • Gutenberg • #26839 •
358 LICENSING.
address specified in Section 4, "Information about donations tothe Project Gutenberg Literary Archive Foundation."
- You provide a full refund of any money paid by a user who notifiesyou in writing (or by e-mail) within 30 days of receipt that s/hedoes not agree to the terms of the full Project Gutenberg-tmLicense. You must require such a user to return ordestroy all copies of the works possessed in a physical mediumand discontinue all use of and all access to other copies ofProject Gutenberg-tm works.
- You provide, in accordance with paragraph 1.F.3, a full refund of anymoney paid for a work or a replacement copy, if a defect in theelectronic work is discovered and reported to you within 90 daysof receipt of the work.
- You comply with all other terms of this agreement for freedistribution of Project Gutenberg-tm works.
1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tmelectronic work or group of works on different terms than are setforth in this agreement, you must obtain permission in writing fromboth the Project Gutenberg Literary Archive Foundation and MichaelHart, the owner of the Project Gutenberg-tm trademark. Contact theFoundation as set forth in Section 3 below.
1.F.
1.F.1. Project Gutenberg volunteers and employees expend considerableeffort to identify, do copyright research on, transcribe and proofreadpublic domain works in creating the Project Gutenberg-tmcollection. Despite these efforts, Project Gutenberg-tm electronicworks, and the medium on which they may be stored, may contain"Defects," such as, but not limited to, incomplete, inaccurate orcorrupt data, transcription errors, a copyright or other intellectualproperty infringement, a defective or damaged disk or other medium, acomputer virus, or computer codes that damage or cannot be read byyour equipment.
1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Rightof Replacement or Refund" described in paragraph 1.F.3, the ProjectGutenberg Literary Archive Foundation, the owner of the ProjectGutenberg-tm trademark, and any other party distributing a ProjectGutenberg-tm electronic work under this agreement, disclaim allliability to you for damages, costs and expenses, including legalfees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICTLIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSEPROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THETRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BELIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE ORINCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCHDAMAGE.
1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover adefect in this electronic work within 90 days of receiving it, you canreceive a refund of the money (if any) you paid for it by sending awritten explanation to the person you received the work from. If youreceived the work on a physical medium, you must return the medium with
• Project • Gutenberg • #26839 •
LICENSING. 359
your written explanation. The person or entity that provided you withthe defective work may elect to provide a replacement copy in lieu of arefund. If you received the work electronically, the person or entityproviding it to you may choose to give you a second opportunity toreceive the work electronically in lieu of a refund. If the second copyis also defective, you may demand a refund in writing without furtheropportunities to fix the problem.
1.F.4. Except for the limited right of replacement or refund set forthin paragraph 1.F.3, this work is provided to you ’AS-IS’ WITH NO OTHERWARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TOWARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.
1.F.5. Some states do not allow disclaimers of certain impliedwarranties or the exclusion or limitation of certain types of damages.If any disclaimer or limitation set forth in this agreement violates thelaw of the state applicable to this agreement, the agreement shall beinterpreted to make the maximum disclaimer or limitation permitted bythe applicable state law. The invalidity or unenforceability of anyprovision of this agreement shall not void the remaining provisions.
1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, thetrademark owner, any agent or employee of the Foundation, anyoneproviding copies of Project Gutenberg-tm electronic works in accordancewith this agreement, and any volunteers associated with the production,promotion and distribution of Project Gutenberg-tm electronic works,harmless from all liability, costs and expenses, including legal fees,that arise directly or indirectly from any of the following which you door cause to occur: (a) distribution of this or any Project Gutenberg-tmwork, (b) alteration, modification, or additions or deletions to anyProject Gutenberg-tm work, and (c) any Defect you cause.
Section 2. Information about the Mission of Project Gutenberg-tm
Project Gutenberg-tm is synonymous with the free distribution ofelectronic works in formats readable by the widest variety of computersincluding obsolete, old, middle-aged and new computers. It existsbecause of the efforts of hundreds of volunteers and donations frompeople in all walks of life.
Volunteers and financial support to provide volunteers with theassistance they need, is critical to reaching Project Gutenberg-tm’sgoals and ensuring that the Project Gutenberg-tm collection willremain freely available for generations to come. In 2001, the ProjectGutenberg Literary Archive Foundation was created to provide a secureand permanent future for Project Gutenberg-tm and future generations.To learn more about the Project Gutenberg Literary Archive Foundationand how your efforts and donations can help, see Sections 3 and 4and the Foundation web page at http://www.pglaf.org.
Section 3. Information about the Project Gutenberg Literary ArchiveFoundation
The Project Gutenberg Literary Archive Foundation is a non profit501(c)(3) educational corporation organized under the laws of thestate of Mississippi and granted tax exempt status by the Internal
• Project • Gutenberg • #26839 •
360 LICENSING.
Revenue Service. The Foundation’s EIN or federal tax identificationnumber is 64-6221541. Its 501(c)(3) letter is posted athttp://pglaf.org/fundraising. Contributions to the Project GutenbergLiterary Archive Foundation are tax deductible to the full extentpermitted by U.S. federal laws and your state’s laws.
The Foundation’s principal office is located at 4557 Melan Dr. S.Fairbanks, AK, 99712., but its volunteers and employees are scatteredthroughout numerous locations. Its business office is located at809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, [emailprotected]. Email contact links and up to date contactinformation can be found at the Foundation’s web site and officialpage at http://pglaf.org
For additional contact information:Dr. Gregory B. NewbyChief Executive and [emailprotected]
Section 4. Information about Donations to the Project GutenbergLiterary Archive Foundation
Project Gutenberg-tm depends upon and cannot survive without widespread public support and donations to carry out its mission ofincreasing the number of public domain and licensed works that can befreely distributed in machine readable form accessible by the widestarray of equipment including outdated equipment. Many small donations($1 to $5,000) are particularly important to maintaining tax exemptstatus with the IRS.
The Foundation is committed to complying with the laws regulatingcharities and charitable donations in all 50 states of the UnitedStates. Compliance requirements are not uniform and it takes aconsiderable effort, much paperwork and many fees to meet and keep upwith these requirements. We do not solicit donations in locationswhere we have not received written confirmation of compliance. ToSEND DONATIONS or determine the status of compliance for anyparticular state visit http://pglaf.org
While we cannot and do not solicit contributions from states where wehave not met the solicitation requirements, we know of no prohibitionagainst accepting unsolicited donations from donors in such states whoapproach us with offers to donate.
International donations are gratefully accepted, but we cannot makeany statements concerning tax treatment of donations received fromoutside the United States. U.S. laws alone swamp our small staff.
Please check the Project Gutenberg Web pages for current donationmethods and addresses. Donations are accepted in a number of otherways including checks, online payments and credit card donations.To donate, please visit: http://pglaf.org/donate
Section 5. General Information About Project Gutenberg-tm electronicworks.
• Project • Gutenberg • #26839 •
LICENSING. 361
Professor Michael S. Hart is the originator of the Project Gutenberg-tmconcept of a library of electronic works that could be freely sharedwith anyone. For thirty years, he produced and distributed ProjectGutenberg-tm eBooks with only a loose network of volunteer support.
Project Gutenberg-tm eBooks are often created from several printededitions, all of which are confirmed as Public Domain in the U.S.unless a copyright notice is included. Thus, we do not necessarilykeep eBooks in compliance with any particular paper edition.
Most people start at our Web site which has the main PG search facility:
http://www.gutenberg.org
This Web site includes information about Project Gutenberg-tm,including how to make donations to the Project Gutenberg LiteraryArchive Foundation, how to help produce our new eBooks, and how tosubscribe to our email newsletter to hear about new eBooks.
• Project • Gutenberg • #26839 •
