Preface

mm
t
W
III
tHHWi'illHJiiiill
4
I
I
mm
m\
'.lii \
'ill' mmmw
■
Cihntmt ut
^ TV, -l.
(lullini^
^rci^tulr^ Int '^'^^^^^isley College Alumnae Assoc.
Jhx iHnnnriitm
Helen A. Shafer
c>
I
MATHEMATICAL RECREATIONS AND ESSAYS
^O-
. MACMILLAN A?fD CO.. Limited
XONDON • BOMBAY • CALCUTTA MELBCURNE
THE 3rACMILLAN C03IPANY
NEW TOKK • BOSTON • CHICAGO DALLAS • SAX FRANCISCO
THE MAC]\IILLAX CO. OF CANADA. Ltd.
TORONTO
MATHEMATICAL RECKEATIONS AND ESSAYS
BY
W.^W. EOUSE BALL,
FELLOW OF TRINITY COLLEGE, CAMBRIDGE.
SIXTH EBITIOy.
MACMILLAN AND CO., LIMITED ST MARTIN'S STREET, LONDON
1914
[All rights reserved]
Copyright
First Edition, February, 1892.
Second Edition, May, 1892.
Third Edition, 1896.
Fourth Edition, 1905.
Fifth Edition, 1911. Sixth Edition, 1914.
1
PREFACE.
rpHE earlier part of this book contains an account of certain -^ Mathematical Recreations : this is followed by some Essays on subjects most of which are directly concerned with historical mathematical problems. I hasten to add that the conclusions are of no practical use, and that most of the results are not new. If therefore the reader proceeds further he is at least fore- warned. At the same time I think I may say that many of the questions discussed are interesting, not a few are associated with the names of distinguished mathematicians, while hitherto several of the memoirs quoted have not been easily accessible to English readers. A great deal of additional matter has been inserted since the work was first issued in 1892.
The book is divided into two parts, but in both parts I have excluded questions which involve advanced mathematics.
The First Part now consists of ten chapters, in which are described various problems and amusements of the kind usually termed Mathematical Recreations, Several of the questions mentioned in the first five chapters 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 left them out: in the absence of such a work, I thought it better to insert them and trust to the judicious reader to omit them altogether or to skim them as he feels inclined. I may add that in discussing problems where the complete
VI PREFACE
solutions are long or intricate I have been generally content to indicate memoirs or books where the methods are set out at length, and to give a few illustrative examples. In several cases I have also stated problems which still await solution.
The Second Part now consists of eleven chapters, mostly dealing with Historical Questions. It is with some hesitation that I have included in this part chapters on String Figures, Astrology, and Ciphers, but I think they may be interesting to my readers, even though the subjects are only indirectly con- nected with Mathematics.
I have inserted detailed references, as far as I know them, 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.
I am indebted to my friend Mr G. N. Watson for his kindness in reading the proof-sheets of this (the fifth) edition, and for many helpful suggestions and comments.
w. w. rousp: ball.
Trinity College, Cambridge. October, 1911.
NOTE TO THE SIXTH EDITION (1914).
In this edition, besides trivial corrections and changes, a few additional recreations have be^n inserted, and Chapter XVIII has been re-written.
Vil
TABLE OF CONTENTS.