124 



NA TURE 



[June 9, 1892 



has left us little opportunity of adding to or correcting 

 what he has thus reproduced from his note-books. 



There are two sources to which every writer on the 

 subject of the earlier part must apply, viz. the " Probl^mes 

 plaisans et d^lectables," by C. G. Bachet, Sieur de 

 Meziriac, and Ozanam's " Rdcrdations mathematiques et 

 physiques." These Mr. Ball carefully discusses as to 

 editions and their respective merits. 



The work before us is divided into two parts : mathe- 

 matical recreations and mathematical problems and 

 speculations. The former consists of seven chapters. In 

 the first chapter are collected together numerous problems 

 with numbers, watches, and cards. Some of these last 

 are interesting to the mathematician, and have been dis- 

 cussed in the Messenger of Mathematics and the " Reprint 

 from \he Educational Times." The Middle Ages furnish 

 some curious questions, and an antique problem in deci- 

 mation is associated with the name of Josephus, but 

 these are well-known instances. Bachet's weights problem 

 calls for mention. It finds a place in the author's algebra ; 

 the omissions in Bachet's argument, Mr. Ball notes, have 

 been supplied by Major MacMahon (see Nature, vol. 

 xlii. pp. 1 13, 1 14). Mersenne's numbers have been treated 

 recently at some length by Mr. Ball in the Messenger of 

 Mathematics (vol. xxi. pp. 34-40) ; in this account it is 

 stated that 2<'i - i = 2 305 843 009 213 693 951 is the 

 biggest known prime. Fermat claims some space (cf. 

 Nature, vol. xviii. pp. 104, 344). Of his so-called last 

 theorem (no integral values of x, _y, z, can be found to 

 satisfy the equation x" -\-y" — s", if n is an integer 

 greater than 2) we read : — 



" This proposition has acquired extraordinary celebrity 

 from the fact that no general demonstration of it has 

 been given, but there is no reason to doubt that it is true." 



It is fitting that we should give Mr. Ball's grounds for 

 this belief. 



" Fermat was a mathematician of quite the 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 its conclusions have little practical import- 

 ance, and since his time it has been discussed by only a 

 few mathematicians, while even of them not many have 

 made it their chief study. This is the explanation of the 

 fact that it took more than a century before some of the 

 simpler results which Fermat had enunciated were 

 proved, and thus it is not surprising that a proof of the 

 theorem which he succeeded in establishing only towards 

 the close of his life should involve great difficulties." 



Proofs have been given in the cases of « = 3, 4, 5, 

 7, 14 (cf. pp. 28, 29). Many subjects of interest 

 take up the second chapter, as " Geometrical Fal- 

 lacies" (every triangle is equilateral, and the whole is 

 equal to a part : this latter we think we have seen in an 

 article by De Morgan) ; curious "Proofs by Dissection" 

 (cf. Messenger of Mathematics, vol. vi. p. 87), there is a 



printer's error (p. 35, 1. 9 up) of tan~^ — in place of 



40 



tan-^ — ; " Colouring Maps" (only four colours necessary 



to colour a map of a country, divided into districts, 

 in such a way that no two contiguous districts shall 

 be of the same colour), the literature of this problem 

 is brought fully up to date ; an account is given of the 

 NO. I 1 80, VOL. 46] 



results of Cayley's "Contour and Slope Lines," and 

 of Clerk Maxwell's " Hills and Dales." Then follow 

 " Statical Games of Position " (" Three in a Row " and 

 " Tesselation," both problems connected with the name 

 of Sylvester) ; " Dynamical Games of Position " ("■ Shunt- 

 ing," " Ferry-boat Problems," and numerous counter, 

 pawn, and solitaire problems), and a glance at " Para- 

 dromic Rings." 



Chapter iii. treats of " Some Mechanical Questions," as 

 "Perpetual Motion," the "Underhand Cut on a Tennis 

 Ball " {Messenger of Mathematics, vol. vii.), the " Boome- 

 rang," and the " Flight of Birds" (Nature, 1890-91). In 

 chapter iv. we have a miscellaneous lot, the fifteen puzzle, 

 Chinese rings, the fifteen school-girls problem, and such 

 card problems as Gergonne's pile problem, the mouse- 

 trap, and many others. Chapter v. discusses " Magic 

 Squares," and chapters vi. and vii. are devoted to " Uni- 

 cursal Problems." These are Euler's problem (more fully 

 discussed by Listing, " Topologie," and Tait), mazes, geo- 

 metrical trees, the Hamiltonian game, and the knight's 

 path on a chess-board. All these matters are treated 

 lucidly, and with sufficient detail for the ordinary reader, 

 and for others there is ample store of references. 

 There is no chance of catching Mr. Ball tripping in his 

 use of books, and his ready access to mathem.atical 

 journals can hardly be surpassed, so that we have not 

 come upon any new facts. We may mention, however, 

 in connection with the knight's path, that there is a short 

 article, accompanied by diagrams, on the subject in the 

 Leisure Hour (December 20, 1873), by H. Meyer, of the 

 Hannoverische Anzeiger. 



The second part, in its opening chapter, gives at some 

 length an account of the three classical proplems, viz. the 

 duplication of the cube, the trisection of an angle, and the 

 quadrature of the circle. Chapter ix., on astrology, has 

 many curious details relating to that science, and gives a 

 facsimile of Cardan's drawing of the horoscope of Edward 

 VI., with an abstract of Cardan's account. On the whole 

 matter of the chapter Mr. Ball writes : — 



" Though the practice of astrology was connected so 

 often with impudent quackery, yet one ought not to forget 

 that nearly every physician and man of science in 

 mediaeval Europe was an astrologer. These observers 

 did not consider that its rules were definitely established^ 

 and they laboriously collected much of the astronomical 

 evidence that was to crush their art. Thus, though there 

 never was a time when astrology was not practised by 

 knaves, there was a period of intellectual development 

 when it was accepted honestly as a difficult but real 

 science." 



De Morgan, it may be remembered, in the " Budget '* 

 (p. 278) says :— 



" If anything ever had a fair trial, it was astrology. The 

 idea itself is natural enough. A human being, set down on 

 this earth, without any tradition, would probably suspect 

 that the heavenly bodies had somethng to do with 

 the guidance of aflfairs." 



" Hyperspace," which occupies chapter x., has a full 

 bibliography (compiled by G. B. Halsted, American 

 Journal of Mathematics, \o\%. i. and ii.), forms the subject 

 of one of Mr. Hinton's interesting " Scientific Romances" 

 (cf. Nature, vol. xxxi. p. 431), and is connected with Dr. 

 Abbott's "Flatland" (Nature, vol. xxxi. p. 76). Mr 



