Project Gutenberg’s Mathematical Recreations and Essays, by W. W. Rouse Ball This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with 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. MATHEMATICAL RECREATIONS AND ESSAYS BY W.W. ROUSE BALL Fellow and Tutor of Trinity College, Cambridge. FOURTH EDITION MACMILLAN AND CO., Limited NEW YORK: THE MACMILLAN COMPANY 1905 [All rights reserved.] Produced by Joshua Hutchinson, David Starner, David Wilson and the Online Distributed Proofreading Team at http://www.pgdp.net Transcriber's notes Most of the open questions discussed by the author were settled during the twentieth century. The author's footnotes are labelled using printer's marks*; footnotes showing where corrections to the text have been made are labelled numerically1. Minor typographical corrections are documented in the LATEX source. This document is designed for two-sided printing. Consequently, the many hyperlinked cross-references are not visually distinguished. The document can be recompiled for more comfortable on-screen viewing: see comments in source LATEX code. 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 amusements 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 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. 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 TABLE OF CONTENTS. vi PART I. \Les hommes ne sont jamais plus ing´enieux que dans l'invention des jeux; l'esprit s'y trouve `a son aise. Apr`esles jeux qui d´ependent unique- ment des nombres viennent les jeux o`uentre la situ- ation. Apr`esles jeux o`un'entrent que le nombre et la situation viendraient les jeux o`uentre le mou- vement. Enfin il serait a souhaiter qu'on e^utun cours entier des jeux, trait´esmath´ematiquement." (Leibnitz: letter to De Montmort, July 29, 1715.) 1 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 works on mathematical recreations include several such problems, which are obvious to any one acquainted with the elements of algebra, but which to many who are ignorant of that subject possess the same kind of charm that some mathematicians find in the more recondite proposi- tions of higher arithmetic. I shall devote the bulk of this chapter to these elementary problems, but I append a few remarks on one or two questions in the theory of numbers. Before entering on the subject of the chapter, I may add that a large proportion of the elementary questions mentioned here and in the following two chapters are taken from one of two sources. The first of these is the classical Probl`emesplaisans et d´electables, by C.G. Ba- chet, sieur de M´eziriac,of which the first edition was published in 1612 and the second in 1624: it is to the edition of 1624 that the references hereafter given apply. Several of Bachet's problems are taken from the writings of Alcuin, Pacioli di Burgo, Tartaglia, or Cardan, and possi- bly some of them are of oriental origin, but I have made no attempt to add such references. The other source to which I alluded above is Ozanam's R´ecr´eationsmath´ematiqueset 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, van Etten, and Oughtred: this part is excellent, but the same cannot be said of the additions due to Ozanam. In the Biographie Universelle al- lusion is made to subsequent editions issued in 1720, 1735, 1741, 1778, 2 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 any knowledge. 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 in any of the books current in England on mathematical amusements as many as a dozen puzzles which are not contained in one of these four volumes. Fresh editions were issued in 1741, 1750 (the second volume of which bears the date 1749), 1770, and 1790. The edition of 1750 is said to have been corrected by Montucla on condition that his name should not be associated with it; but the edition of 1790 is the earliest one in which reference is made to these corrections, though the editor is referred to only as Monsieur M***. Montucla expunged most of what was actually incorrect in the older editions, and added several historical notes, but unfortunately his scruples prevented him from striking out the accounts of numerous trivial experiments and truisms which over- load the work. An English translation of the original edition appeared in 1708, and I believe ran through four editions, the last of them being published in Dublin in 1790. Montucla's revision of 1790 was translated by C. Hutton, and editions of this were issued in 1803, in 1814, and (in one 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 a large part of most compilations of mathematical amusements. They are given here mainly for their historical|not for their arithmetical| interest; and perhaps a mathematician may well omit them, and pass at once to the latter part of this chapter. These questions are of the nature of tricks or puzzles and I follow the usual course and present them in that form. I may note however that most of them are not worth proposing, even as tricks, unless either the modus operandi is disguised or the result arrived at is different from that expected; but, as I am not writing on conjuring, I refrain from alluding to the means of disguising the operations indicated, and give merely a bare enumeration of the steps essential to the success of the method used, though I may recall the fundamental rule that no trick, however good, will bear immediate repetition, and that, if it is 4 ARITHMETICAL RECREATIONS.