208 



NATURE 



[August 17, ign 



form of apparatus are spoken about, but that they do 

 not appear to have been verified by competent 

 authority. 



The reviewer is of opinion that further experiments 

 are desirable, and that these should be directed mainly 

 towards ascertaining whether or not the movements 

 of the "diviner's" rod are caused by any influence 

 outside himself. The experiments are difficult to carry 

 out, because it is clearly fair that the conditions should 

 be those acceptable to the "diviner"; these vary 

 greatly, few "diviners" being- entirely in agreement 

 when asked to describe clearly the extent of their 

 powers. J. Wertheimer. 



DIOPHANTINE ANALYSIS. 

 Diophantus of Alexandria: a Study in the History 

 of Greek Algebra. By Sir T. L. Heath, K.C.B. 

 Second edition, with a Supplement containing an 

 Account of Fermat's Theorems and Problems con- 

 nected with Diophantine Analysis and some Solu- 

 tions of Diophantine Problems by Euler. Pp. 

 vii + 387. (London : Cambridge University Press, 

 1910.) Price 12s. 6d. net. 



THIS is far from being a mere reprint of the first 

 edition ; in fact, it is in great part a new work, 

 which, in conjunction with Tannery's critical edition 

 of the " Arithmetica," makes Diophantus at last 

 accessible to the ordinary reader. 



The introduction, besides giving a historical 

 account of Diophantus, the MSS. of his works, and 

 the writers who have dealt with them, contains most 

 interesting and valuable sections on Diophantus's 

 notation and methods of solution. As to the first, we 

 are astounded, as in the case of Archimedes, at the 

 ease with which enormous numbers are computed, 

 in spite of the cumbrous Greek notation. An in- 

 stance in point is the famous cattle problem (attri- 

 buted to Archimedes), which is briefly discussed on 

 pp. 12 1-4. Its solution involves the Pellian equation 

 / 2 -4729494H :: =i, and according to Sir T. Heath's 

 calculations, the value of one of the unknowns of 

 the problem would be a number containing 206,545 

 digits. Of the methods of Diophantus not much can 

 be said, because he uses so many ingenious devices 

 to suit different problems ; but we may note his dex- 

 terity in choosing his unknown quantity, and his 

 curious plan of "working back" by a sort of rule 

 of false position. A good example of the latter is 

 v. 29 (p. 224): "To find three squares such that the 

 sum of their squares is a square," where it will be 

 seen that an insufficient assumption is corrected and 

 modified in a sort of tentative way until a solution 

 is found. 



It would be unprofitable to go into any detail here 

 on the nature of Diophantine problems in general; 

 to appreciate them it is necessary to read Diophantus, 

 Fermat, and Euler. By a very happy inspiration, 

 the present volume has been made to include all the 

 notes of Fermat upon Diophantus, and extracts from 

 his correspondence with Fr^nicle and others; besides 

 ilii-; we have solutions of seventeen Diophantine 

 problems by Euler, which are models for those who 

 O. 2l8l. VOL. 87 1 



feel inclined to work in this fascinating field. There 

 can be little doubt that there are still numbers of 

 arithmetical problems to be solved by Diophantine 

 methods, and Fermat's method of reduction 

 (descente) for proving the impossibility of certain 

 indeterminate equations awaits rediscovery and de- 

 velopment. Moreover, the theory of algebraic forms 

 and symmetric functions ought surely to lead to new 

 arithmetical applications of a Diophantine type. To 

 give an example of the sort of thing we mean : Let 

 x, y, z be three variables ; we have identically 



tx(y- zf=(y-z)(z-x)(x-y) (x +y+z). 



Now put x, y, s = ! 3 , i)', f 3 , and suppose that 

 ( 3 + >l 3 + C 3 = m(i(- t then the previous identity leads 

 at once to 



AS+B' + C^/.vABC 

 with 



A, B, C = tW-C),rtC -f 3 ), Cif-V*)- 



This is a partial sample of what Fermat would call a 

 descente ; of course, it is now well known as the 

 theory of residuation of points on cubic cuA'es, but 

 it is interesting to see how it results from an ele- 

 mentary algebraic identity, and there are still arith- 

 metical problems in this connection which do not 

 appear to have been solved. 



It may interest those who are unacquainted with 

 the subject to give one typical Diophantine problem 

 and its solution. The problem is "To find two posi- 

 tive integers such that their sum is a square, and 

 the sum of their squares a biquadrate." One solu- 

 tion is (4565486027761, 1061652293520), and it has 

 been proved by Lagrange that, as Fermat "confi- 

 dently asserted," this is the simplest solution. More 

 exactly : the same problem may be put in the form, 

 " Find a right-angled triangle such that the hypo- 

 tenuse and the sum of the sides are both squares," 

 and Fermat's assertion was that the above solution 

 gave the smallest of such triangles. 



In conclusion, it may be remarked that there is a 

 crux in the Greek text which does not seem to have 

 been finally disposed of. After putting the problem, 

 " To find two numbers such that their sum and 

 product are given numbers," Diophantus adds the 

 condition, " the square of half the sum must exceed 

 the product by a square number." «m S<r tovtq 

 Tr\a<rfj.aTiK6v. It would be possible to translate 

 this, "This is artificial" (as opposed to "natural"), 

 but there does not seem to be any point in this. On 

 the other hand, to translate "This can be seen from 

 a model " would give good sense, because we should 

 only have to replace a diagram in Euclid by a corre- 

 sponding arrangement of counters; unfortunately, 

 this seems to read more into the text than is legiti- 

 mate. Neither of these alternatives is proposed in 

 the note on p. 140; the editor prefers, on the whole, 

 Xylander's effictum aliunde, which is not far in 

 sense from "artificial," in the context. The same 

 phrase occurs in two other places, and in each case 

 we can give a quasi-geometrical arrangement of 

 counters to show that the condition is necessary; so 

 far, this is in favour of the second alternative sug- 

 gested above. G. B. M. 



