book is at all dry. On the contrary the biographical sketches frequently

contain amusing anecdotes, and many of the theorems mentioned are very clearly explained so as to bring them within the grasp of those who are only acquainted with elementary mathematics. — Nature.
Le style de M. Ball est clair et dlegant, de nombreux aper9us rendent facile de suivre le fil de son exposition et de frequentes citations permettent £l celui qui le desire d'approfondir les recherches que I'auteur n'a pu qu'- effleurer. . . . Get ouvrage pourra devenir tr^s utile comme manuel d'histuire des mathdmatiques pour les etudiants, et il ne sera pas ddplace dans lea biblioth^ques des savants. — Bibliotheca Mathematica,
The author modestly describes his work as a compilation, but it is thoroughly well digested, a due proportion is observed between the various parts, and when occasion demands he does not hesitate to give an independ- ent judgment on a disputed ])oint. His verdicts in such instances appear to us to be generally sound and reasonable. ... To many readers who have not the courage or the opportunity to tackle the ponderous volumes of Montucla or the (mostly) ponderous treatises of German writers on special periods, it may be somewhat of a surprise to find what a wealth of human interest attaches to the history of so "dry" a subject as mathematics. We are brought into contact with many remarkable men, some of whom have 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 least one thorough blackguard (Cardan) ; and Mr. Ball's pages abound with quaint and amusing touches characteristic of the authors under con- sideration, or of the times in which they \\ve6..— Manchester Guardian.
There can be no doubt that the author has done his work in a very excel- lent way. . . . There is no one interested in almost any part of mathematical science who will not welcome such an exposition as the present, at once popu- larly written and exact, embracing the entire subject. . . . Mr. Ball's work is destined to become a standard one on the subject. — The Glasgow Herald.
A most interesting book, not only for those who are mathematicians, but for the much larger circle of those who care to trace the course of general scientific progress. It is written in such a way that those who have only an elementary acquaintance with the subject can find on almost every page something of general interest. — The Oxford Magazine.
La lecture en est singulierement attachante et instructive. — Bulletin des sciences mathematiquea.
495 A PRIMER OF THE
HISTORY OF MATHEMATICS
By W. W. rouse BALL. [Fourth Edition, 19 U. Fp. iv + 149. l^nce 2s. net.]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This book contains a sketch in popular language of the history of mathematics ; it includes notices of the lives and surround- ings of those to whom the development of the subject is mainly due as well as of their discoveries.
This Primer is written in the agreeable style with which the author has made us acquainted in his previous essays ; and we are sure that all readers of it will be ready to say that Mr. Ball has succeeded in the hope he has formed, that "it may not be uninteresting" even to those who are unac- quainted with the leading facts. It is just the book to give an intelligent young student, and should allure him on to the perusal of Mr. Ball's "Short Account." The present work is not a mere rdchauffd of that, though naturally most of what is here given will be found in equivalent form in the larger work. . . . The choice of material appears to us to be such as should lend interest to the study of mathematics and increase its educational value, which has been the author's aim. The book goes well into the pocket, and la excellently printed. — The Academy.
"We have here a new instance of Mr. Rouse Ball's skill in giving in a small space an intelligible account of a large subject. In 137 pages we have a sketch of the progress of mathematics from the earliest records up to the middle of this century, and yet it is interesting to read and by no means a mere catalogue. — 2'fie Manchester Guardian.
It is not often that a reviewer of mathematical works can confess that he has read one of them through from cover to cover without abatement of interest or fatigue. But that is true of Mr. Rouse Ball's wonderfully enter- taining little "History of Mathematics," which we heartily recommend to even the quite rudimentary mathematician. The cai)able mathematical master will not fail to find a dozen interesting facts therein to season his teaching. — The Saturday Review.
A fascinating little volume, which should be in the hands of all who do not possesis the more elaborate History of Matheiiiatics by the same author. — The Mathematical Gazette.
This excellent sketch should be in the hands of every student, whether he is studying mathematics or no. In moat cases there is an unfortunate lack
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496
of knowledge upon this subject, and we welcome anything that will help to supply the deficiency. The primer is written in a concise, lucid and easy manner, and gives the reader a general idea of the progress of mathematics that is both interesting and 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 quarter of the space. ... Of course, what he now gives is a bare outline of the subject, but it is ample for all except the most advanced proficients. There is no question that, as the 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 for any mathematical student who has made some way with the calculus than a careful perusal of this little book. — The Educational Times,
The author has done good service to mathematicians by engaging in work in this special field. . . , The Primer gives, in a brief compass, the history of the advance of this branch of science when under Greek influence, during the Middle Ages, and at the Renaissance, and then goes on to deal with the introduction of modern analysis and its recent developments. It refers to the life and work of the leaders of mathematical thought, adds a new and enlarged value to well-known problems by treating of their inception and history, and lights up with a warm and personal interest a science which Bome of its detractors have dared to call dull and cold. — The Educational Review.
It is not too much to say that this little work should be in the possession of every mathematical teacher. . . . The Primer gives in a small compass the leading events in the development of mathematics. ... At the same time, it is no dry chronicle of facts and theorems. The biographical sketches of the great workers, if short, are pithy, and often amusing. Well-known propo- sitions will attain a new interest for the pui^il as 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" should read and study . . . , and those who wish to learn how to think will find advantage in 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 a highly readable and instructive chapter in the history of education. The condensation has been skilfully done, the reader's interest being sus- tained by the introduction of a good deal of far from tedious detail. — Tlie Glasgow Herald.
Mr. W. W. Rouse Ball is well known as the author of a very clever history of mathematics, besides useful works on kindred subjects. His latest pro- duction is A Primer of the History of Mathematics, a book of one hundred and forty pages, giving in non-technical language a full, concise, and readable narrative of the development of the science from the days of the Ionian Greeks until the jiresent time. Anyone with a leaning towards algebraic or geometrical studies will be intensely interested in this account of progress from primitive usages, step by step, to our present elaborate systems. The lives of the men who by their research and discovery helped along the good work are described briefly, but graphically. . . . The Primer should become a standard text-book. — The Literary World.
This is a capital little sketch of a subject on which Mr. Ball is an acknow- ledged authority, and of which too little is generally known. Mr. Ball, moreover, writes easily and well, and has the art of saying wliat he has to say in an interesting style. — Tlie School Guardian.
497 MATHEMATICAL
RECREATIONS AND ESSAYS
By W. W. rouse BALL. [Siwth Edition, 1914. Pp. xvi + 492. Price 8s. Gd. net]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This work is divided into two parts ; the first is on mathematical recreations and puzzles, the second includes some miscellaneous essays and an account of various problems of historical interest. In both parts questions which involve advanced mathematics are excluded.
The mathematical recreations include numerous elementary questions and paradoxes, as well as problems such as the proposi- tion that to colour a map not more than four colours are necessary, the explanation of the effect of a cut on a tennis ball, the fifteen puzzle, the eight queens problem, the fifteen school-girls, the con- struction of magic squares, the theory and history of mazes, and the knight's path on a chess-board.
The second part commences with sketches of the history of the Mathematical Tripos at Cambridge, and of some half-dozen cele- brated problems in mathematics. These are followed by essays on String Figures, Astrology, and Ciphers. The last three chapters are devoted to an account of certain hypotheses as to the nature of Space and Mass, and the means of measuring Time.
Mr. Ball has attained a position in the front rank of writers on suhjects connected with the history of mathematics, and tliis brochure will add another to his successes in this field. Li it he has collected a mass of information bearing upon matters 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 left few ears for other gleaners. The nature of the work is comjiletely indicated to the mathematical student by its title. Does lie want to revive his acquaintance with the Probltmes Flatsaris et Dclectahles of Bachet, or the lUcriations Mathcmatiqtiea et Physi'iues of
498
Ozanam? Let him take Mr. Ball for his companion, and he will have 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 'the fifteen puzzle,' 'Chinese rings,' 'the fifteen school-girls problem ' et id genus orane. Sufficient space is devoted to accounts of magic squares and uuicursal problems (such as mazes, the knight's path, and geo- metrical trees). These, and many other problems of equal interest, come under the head of 'Recreations.' The problems and speculations include an account of the Three Classical Problems ; there is also a brief sketch of Astrology ; and interesting outlines of the present state of our knowledge of hyper-space and of the constitution of matter. This enumeration badly indicates the matter handled, but it sufficiently states what the reader may expect to find. Moreover for the use of readers who may wish to pursue the several heads further, Mr. Ball gives detailed references to the sources from whence he has derived his information. These Mathematical Recreatiovs we can commend as suited for mathematicians and equally for others who wish to while away an occasional hour. — The Academy,
The idea of writing some such account as that before us must have been present to Mr. Ball's mind when he was collecting the material which he has so skilfully worked up into his History of Mathematics, We think this because . . - many bits of ore which would not suit the earlier work find a fitting niche in this. Howsoever the case may be, we are sure that non- mathematical, as well as mathematical, readers will derive amusement, and, we venture to think, profit withal, from a perusal of it. The author has gone very exhaustively over the ground, and has left us little opportunity of add- ing to or correcting what he has thus reproduced from his note-books. The work before us is divided into two parts : mathematical recreations and mathematical problems and speculations. All these matters are treated lucidly, and with sufficient detail for the ordinary reader, and for others there is ample store of references. . , . Our analysis shows how great an extent of ground is covered, and the account is fully pervaded by the attractive charm i>Ir. Ball knows so well how to 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 of mathematics. There is a fascination about this volume which results from a happy combination of puzzle and paradox. There is both milk for babes and strong meat for grown men. ... A great deal of the information is hardly accessible in any English books ; and Mr. Ball would deserve the gratitude of mathematicians for having merely collected the facts. But he has presented them with such lucidity and vivacity of style that there is not a dull page in the book ; and he has added minute and full bibliographical references which greatly enhance the value of his work. — The Cambridge Review.
Mathematicians with a turn for the paradoxes and puzzles connected with number, space, and time, in which their science abounds, will delight in Mathematical Recreations and Problems of Past and Present Times. — The
Times.
Mathematicians have their recreations ; and Mr. Ball sets forth the humours of mathematics in a book of deepest interest to the clerical reader, and of no little attractiveness to the layman. The notes attest an enormous amount of research. — The National Observer.
Mr. Ball, to whom we are already indebted for two excellent Histories of Mathematics, has just produced a book which will be thoroughly appreciated by those who enjoy the settiijg of the wits to work. . . . He has collected a vast amount of information 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 the assortment. — I'he Observatory.
400
Mr. Rouse Ball has the true gift of story-telling, and he writes so pleasantly that though we enjoy the fulness of his knowledge we aro tempted to forget the considerable amount of labour involved in the preparation of his book. He gives us the history and the matliematics of many problems . . . and where the limits of his work prevent him from dealing fully with the points raised, like a true worker he gives us ample references to original memoirs. . . . The