1. A Short Account of the History of Mathematics (1888), by W W Rouse Ball.
This book, most of which is a transcript of some lectures I delivered this year, gives a concise account of the history of mathematics. I have tried to make it as little technical as possible, and I hope that it will be intelligible to any one who is acquainted with the elements of mathematics. Partly to facilitate this, partly to gain space, I have generally made use of modern notation in quoting any results; the reader must therefore recollect that while the matter is the same as that of any writer to whom reference is made his proof is sometimes translated into a more convenient and familiar language.
The subject-matter of this book is a historical summary of the development of mathematics, illustrated by the lives and discoveries of those to whom the progress of the science is mainly due. It may serve as an introduction to more elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know. The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics. In the second edition, issued in 1893, I rearranged parts of it, and introduced a good deal of additional matter. The scheme of arrangement will be gathered from the table of contents at the end of this preface. Shortly it is as follows. The first chapter contains a brief statement of what is known concerning the mathematics of the Egyptians and Phoenicians; this is introductory to the history of mathematics under Greek influence. The subsequent history is divided into three periods: first, that under Greek influence, chapters ii to vii; second, that of the middle ages and renaissance, chapters viii to xiii; and lastly that of modern times, chapters xiv to xix. In discussing the mathematics of these periods I have confined myself to giving the leading events in the history, and frequently have passed in silence over men or works whose influence was comparatively unimportant. Doubtless an exaggerated view of the discoveries of those mathematicians who are mentioned may be caused by the non-allusion to minor writers who preceded and prepared the way for them, but in all historical sketches this is to some extent inevitable, and I have done my best to guard against it by interpolating remarks on the progress of the science at different times. Perhaps also I should here state that generally I have not referred to the results obtained by practical astronomers and physicists unless there was some mathematical interest in them. In quoting results I have commonly made use of modern notation; the reader must therefore recollect that, while the matter is the same as that of any writer to whom allusion is made, his proof is sometimes translated into a more convenient and familiar language. The greater part of my account is a compilation from existing histories or memoirs, as indeed must be necessarily the case where the works discussed are so numerous and cover so much ground. When authorities disagree I have generally stated only that view which seems to me to be the most probable; but if the question be one of importance, I believe that I have always indicated that there is a difference of opinion about it.
The following pages contain an account of the development of the study of mathematics in the university of Cambridge, and the means by which proficiency in that study was at various times tested. The general arrangement is as follows. The first seven chapters are devoted to an enumeration of the more eminent Cambridge mathematicians arranged chronologically. I have in general contented myself with mentioning the subject-matter of their more important works, indicating the methods of exposition which they adopted, but I have not attempted to give a detailed analysis of their writings. ... The following chapters deal with the manner in which at different times mathematics was taught, and the means by which proficiency in the study was tested.
This work on elementary algebra has been written at the request of the Syndics of the Cambridge University Press, and is intended to include those parts of the subject which most Schools and Examination Boards consider as covered by the adjective 'elementary'. The discussion, herein contained, of Permutations and Combinations, the Binomial Theorem, and the Exponential Theorem - subjects which are sometimes included in Elementary Algebra, and sometimes excluded from it - should be regarded as introductory to their treatment in larger textbooks. I have in general followed the order of arrangement and method of presenting the subject which are traditional in England. ... I am indebted to the kindness of the Secretaries of the Cambridge Local Examinations Syndicate and of the Oxford and Cambridge Schools Examination Board for permission to use the papers and questions which have been set in the examinations held under their authority. A large number of the examples inserted at the end of each chapter are, except for a few verbal alterations, derived from one or other of these sources.
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 accessible to English readers. The book is divided into two parts, but in both parts I have included questions which involve advanced mathematics. The first part consists of seven chapters, in which are included various 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 number and situation; the fifth chapter contains a concise account of magic squares; and the sixth and seventh chapters deal with some unicursal problems. Several of the questions mentioned in the first three 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 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 historical. 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.
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 Mathematical 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.
6.2. Review by: William John Greenstreet.
The Mathematical Gazette 3 (54) (1905), 256-257.
In 1892 Mr W W Rouse Ball published a volume entitled 'Mathematical Recreations and Problems of Past and Present Times' (Messrs. Macmillan). A fourth edition has now made its appearance with the 240 pages of the original increased to almost 400. There are three new chapters, dealing respectively with the History of the Mathematical Tripos at Cambridge, Mersenne's Numbers, Cryptography and Ciphers. Part II, which was headed "Mathematical Problems and Speculations," is now altered to "Miscellaneous Essays and Problems." .... We might naturally expect from the author of 'The History of the Study of Mathematics at Cambridge' a lucid and systematic account of the Tripos, and we are not disappointed. We see how the word has changed its meaning as few words have, "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 foolscap paper, from a paper to a list of names, and from a list of names to a system of examination." With the author, we regret to note the disappearance during the last ten years of the famous Tripos verses, which in their time caused a good deal of harmless and innocent amusement. "Mr Tripos" had been an unchartered libertine for three hundred and twenty years, off and on; he was allowed to say anything he liked in these verses "so long as it was not dull and was scandalous"; and, to quote the exhortation of the University officials, he always remembered of recent years while exercising his privilege of humour to be modest withal.
The American Mathematical Monthly 20 (2) (1913), 61.
A great deal of new matter has been added to this interesting book since it was first published in 1892. The fifth edition contains almost 250 pages more than the first and about 100 pages more than the fourth. The work on "Kirkman's School Girls Problem" has been enlarged and made into a separate chapter. There is a paragraph on the same problem as proposed independently by J Steiner in a somewhat more general form. There is a new chapter of 20 pages on "The Parallel Postulate," and one of 6 pages on the "Insolubility of the Algebraic Quintic." Those who amused themselves in their youth by making figures known as 'Cat's Cradles' by twisting on the hands a loop of string will be interested in the new chapter on "String Figures." The subject is more extensive than most people think. The chapter is 32 pages long and is not supposed to be a complete discussion.
National Mathematics Magazine 14 (6) (1940), 357-358.
For nearly half a century this delightful classic has furnished amusement to many who would explore mathematical realms for recreation, and it has stimulated others to attack the unsolved problems of earlier editions and to generalize the results of some of the known theorems. It is thus a real event to have at hand a new edition published 17 years after the tenth, which appeared in 1992. The revision has been extensive, although there has been no essential change in the character of the material included in the volume. The fifth chapter of the old edition on mechanical recreations, the eighth on bees and their cells, and the fifteenth on string figures have been omitted. The twelfth, devoted to miscellaneous problems (Chinese rings, problems connected with packs of cards, etc.), has been broken up and distributed among other chapters. The fourteenth, on cryptographs and ciphers, has been completely rewritten by Abraham Sinkov, a cryptanalyst in the U. S. War Department. The new material is found in arithmetical recreations, Chapter 2; geometrical recreations, Chapter 3; polyhedra, Chapter 5, which is entirely new; magic squares, Chapter 7; and map-colouring problems, Chapter 8. Chapter 5 is handsomely illustrated by two half-tone plates and numerous figures, which greatly illuminate the text.
8.2. Review by: F P White.
The Mathematical Gazette 23 (256) (1939), 422.
This is still Rouse Ball, but with a difference. A large part of the tenth edition is reprinted without change; this needs no description here. Three chapters have, however, been omitted entirely, the one on string figures, which Rouse Ball himself had very much cut down after the appearance of his separate book on the subject in 1920, the short chapter on bees and their cells, and the chapter on mechanical recreations. This last, which begins with Zeno's paradoxes and includes a mention of cut and spin, might well be expanded, as Rouse Ball himself suggested, into an independent work; it is about time that someone dealt authoritatively with the mystery of the " new ball" in cricket.
8.3. Review by: W D R.
The Mathematics Teacher 33 (1) (1940), 46.
This revised edition is for the most part the tenth edition reprinted, and most teachers are familiar enough with the old edition to make further comment here unnecessary. However, the revised edition has omitted three chapters of the old edition, namely, "String Figures," "Bees and Their Cells," and "Mechanical Recreations." Chapter II has been radically changed as has also Chapter VIII on "Map Colouring Problems." A great deal of Chapter III on "Geometric Recreations" and Chapter VII on "Magic Squares" is new; Chapter XII has been broken up and distributed among the first, third, fourth, and eleventh chapters. Finally, Chapter V on "Polyhedra" is new, and Chapter XIV on "Cryptographs and Ciphers" has been completely rewritten.
The High School Journal 26 (1/2) (1943), 44-45.
This is the last edition of an older book by Mr Ball by the same title. The original edition came out in 1892. The book has been so popular that eleven different editions were published. Besides these there were four different reprints of the tenth and two of the eleventh editions. This alone gives one some idea of the place this book has taken in our mathematics literature. The book consists of fourteen chapters. Chapters I and II deal with "Arithmetical Recreations," Chapters III and IV with "Geometrical Recreations," Chapter V with "Polyhedra," Chapter VI with "Chess-Board Recreations," Chapter VII with "Magic Squares," Chapter VIII with "Map-Colouring Problems," Chapter IX with "Unusual Problems," Chapter X with "Kirkman's School-girl Problem," Chapter XI with "Miscellaneous Problems," Chapter XII with "Three Classical Geometrical Problems," Chapter XIII with "Calculating Prodigies," while Chapter XIV deals with cryptography and cryptanalysis. In these chapters the reader finds excellent treatments of many of the interesting problems that have intrigued men's minds throughout the ages. It is amazing that such a variety of problems have presented themselves to mankind for solution and still more amazing that they have been successfully solved. The ingenious devices that have been invented to solve these problems are really more wonderful than the mechanical devices which have so richly blessed us. There is no doubt that this is the outstanding book in this field. It is really a classic. Any person who has leisure time and the urge to follow some of the interesting by-paths which some of our greatest intellects have blazed for us should treasure this book. The reading of it and the study of the problems here presented would keep one's mind out of mischief for many hours and would challenge its alertness and acumen.
The Mathematics Teacher 69 (1) (1976), 86.
It goes without saying this book is an established work, as the twelve editions of it since 1892 verify. This twelfth edition follows the eleventh, which appeared in 1939. The changes in this edition include new material on primes, recent work in factorization, consideration of some new unsolved problems, and the latest in formation on the map colouring problems. Some maze problems have been added, and problems which mathematical ingenuity can solve that the computer cannot. Naturally, if you have the 1939 edition you would have much of what is in this 1974 edition, but 35 years of new material still makes this a must to add to your mathematics library.
The Mathematics Teacher 81 (4) (1988), 325.
Rouse Ball and Coxeter's 'Mathematical Recreations and Essays' "contains descriptions of various problems of the kind usually termed Mathematical Recreations, and a few Essays on some analogous questions." Generally, the text, which has had "all matter which involved advanced mathematics" deleted, is clearly written, and information on both notation and problem origin is carefully referenced. Although not remarkable for the materials included because it is a republication of Rouse Ball's first edition (1892), the text is nonetheless a publication that in its revised form would be an excellent supplementary text for the high school student. Coxeter's revision contains fourteen chapters with a very useful table of contents. As a supplementary book, its sections on calculating prodigies, magic squares, and cryptographic systems alone present great motivational devices. I would highly recommend it as an addition to any educator's personal library. It is clearly written, informative, and very well referenced.
Newton's 'Principia' is the classic of English mathematical writing. Three editions were brought out during Newton's lifetime - the first in 1687; the second, edited by Cotes, in 1713; the third, edited by Pemberton, in 1726. I had at one time hoped to publish a critical edition of the work, with a prefatory account of its origin and history, notes showing the form in which it was printed originally and the changes introduced in 1713 and 1726, accompanied where desirable by an analytical commentary, together with various other propositions on the subject which are extant among Newton's papers though they were not incorporated in any of the editions that he issued. I am unlikely in the immediate future to find time to carry out this plan, but I think it possible that there may be a sufficient number of mathematicians interested in the subject to justify the publication of this essay on the history of Newton's work, though even when thus limited the following notes are on some points less full than I should have wished to make them had leisure permitted. ... This sketch of the origin and history of the 'Principia' falls naturally into six divisions. These deal (a) with Newton's investigations in 1666; (b) with his investigations in 1679; (c) with his investigation in 1684, of which the chief results are embodied in the 'De Motu' of 1685; (d) with the compilation and publication of the 'Principia', 1685-1687; (e) with the contents of the 'Principia'; and (f) with the subsequent history of the work and the preparation of the later editions, but this head I do not propose to discuss in any great detail.
... to any unacquainted with the leading facts here given, I hope even a sketch like this may prove not uninteresting; while to those (teachers or learners), who have no time to read larger works, but who hold, as I do, that a knowledge of the history of a science lends interest to its study and often increases its educational value, I trust that this primer may be of some use.
13.2. Review by: Edwin Mortimer Blake.
Science, New Series 5 (113) (1897), 352-353.
The object of the 'Primer,' as well set forth in its introduction, is "to give a popular account of the history of mathematics, including therein some notice of the lives and surroundings of those to whom its development is mainly due, as well as their discoveries. Such a sketch, written in non-technical language and confined to less than 140 pages, can contain nothing beyond a bare outline of the subject, and, of course, is not intended for those to whom it is familiar." It consists of the author's larger work reduced in size by the omission of all detailed and highly technical matter. In a few places the pruning process has been carried too far. ... The book affords to students in our high schools and colleges a means of gaining, with a small expenditure of time, a sufficiently complete history of the mathematical subjects they are studying, to give them a much greater appreciation of and interest for such subjects.
Several years ago the Editor of the 'College Monographs' conceived the idea of a series of volumes dealing separately with the Colleges of our two ancient Universities; since then many historical and illustrated works carrying out, in some degree, this idea have been published. But the Editor believes that there is still a need for a set of short, well-written, and illustrated handbooks of moderate price. The 'College Monographs', of which this volume is the first, have been planned to supply this need. They are written by members of the society with which they are concerned, and aim at giving (1) a concise description of the buildings, (2) a recital of the origin and history of the community, (3) an account of collegiate life, manners, and customs, both past and present, and (4) a record of distinguished sons ...
The Mathematical Gazette 5 (80) (1909), 113-114.
Mr W W Rouse Ball's admirable History is now translated into French and Italian, and he is to be congratulated on this substantial recognition of the undoubted merits of that book. We could, however, have wished that the two large volumes published by Messrs Hermann had more completely represented work done by Mr Rouse Ball himself in the field in which he has made for himself a name. It is not that the additions which bring his volume of 520 English pages to nearly 700 tall pages in the French edition are not full of interesting matter ... .
This volume contains papers on some questions of Cambridge history put together, mostly for undergraduate societies and magazines, at various times during the last twenty-five years. I have included a memoir, written for a London Society, on Newton's Principia, a work that profoundly affected the development of University studies in the eighteenth century, and a chapter on the History of the Mathematical Tripos, which at one time appeared in my Mathematical Recreations and Essays, since these are concerned with Cambridge subjects.
17.1. From the Preface.
I have taken advantage of a second edition to add a few more papers, and at the suggestion of friends have changed the title to 'Cambridge Notes, chiefly concerning Trinity College and the University'.
The making of String Figures is a game common among primitive people. Its study by men of science is a recent development, their researches have, however, already justified its description as a hobby, fascinating to most people and readily mastered. The following pages contain a lecture which I gave last spring at the Royal Institution, London, on these figures and their history; to it I have appended full directions for the construction of several easy typical designs, arranged roughly in order of difficulty, and, for those who wish to go further, lists of additional patterns and references. The only expense necessary to anyone who takes up the pastime is the acquisition of a piece of good string some seven feet long ...
18.2. Review by: The Editors.
The Mathematical Gazette 10 (151) (1921), 255-256.
There are few amusements so inexpensive as the making of string figures. With about seven feet of string and this little book, we are told by the author that "amusement will be forthcoming to while away many a vacant hour." Most of us are familiar from our nursery days with "cat's cradle," but many may be unaware that this and hundreds of similar devices are to be found amusing the inhabitants of many parts of the world, and that of late these games have become a subject of investigation on the part of the ethnologists. Mr C F Jayne and Mr A C Haddon have both published works on string figures, and that by the former contains nearly 900 diagrams. We cordially recommend to our readers Mr Rouse Ball's Lecture, which was delivered last spring at the Royal Institution, London.