314 



NA TURE 



[February i, 1900 



has been great improvement, but it must be confessed 

 that there is still need of increased carefulness. Thus, if 

 we take (quite at random) p. 792, we have a paper by 

 BUtschli stated to extend from p. 291 to p. 593 of the 

 Arch. Entiuickmcch., which is incredible ; Haacke's 

 text-book called a Grunddiss ; Hertwig's " Streitfragen" 

 wrongly spelt ; Hickson's paper on the medusae of 

 Millepora cited where it seems irrelevant — trivial mis- 

 takes all of them, but too many for one page, and it is so 

 elsewhere. All the same, this third volume of" L'Annee 

 Biologique " is a fine piece of work, and every biologist 

 will wish it the success it deserves. J. A. T. 



AN ARITHMETICAL MISCELLANY. 

 Exercices d' Arithmetique. Par J. Fitz-Patrick et G. 

 Chevrel. Deuxi&me edition. Pp. xiv -I- 680. (Paris : 

 A. Hermann, 1900.) 



THIS second edition of a very entertaining book 

 differs from the first by the inclusion of more than 

 500 new and unsolved examples, and a supplement on 

 commercial arithmetic, which, no doubt, will be found 

 very useful by the French schoolmaster, but is so in- 

 congruous with the rest of the work as to recall Horace's 

 well-known parable of the mermaid and its analogues in 

 literature and art. 



Apart from this concession to the practical, the 

 authors, largely imbued with the spirit of Edouard Lucas, 

 have provided their readers with a varied store of illus- 

 trations of Diophantine arithmetic and of numerous 

 fundamental propositions in the theory of numbers. 

 Their solutions are very clear and simple (though they 

 might, with advantage, have made more use of the 

 notation of congruences), and they will undoubtedly 

 succeed in promoting a more general and intelligent 

 interest in the theory of arithmetic. 



Many of the examples are of a very elementary 

 character ; but there are some which deserve the atten- 

 tion of expert mathematicians. For instance (p. 366), 

 we have Lucas's determination of all the prime factors of 

 (a^260 _ ^i2oo^/(^ _ ^j^ where a, b are the roots of x'^ = x-2 ; 

 the last five of these primes being 



125541359, 25215201901, 34449677641, 153790567559, 

 733268745721. 



This result is said to have been verified by M. Le 

 Lasseur. Again (p. 158), the Rev. Father J, Pervou- 

 chine, of Perm, has found that 2'^^^ + i, comprising 

 2525223 digits, is divisible by 167772161 ( = 5.2^'^-!- i), 

 which is prime. Here are mysteries which we must 

 leave to Lieut.-Colonel Cunningham and Mr. Bickmore 

 to unravel. 



An agreeable element of humour is supplied by 

 Question 399, on the interpretation of Art. 757 of the 

 Civil Code ; that ambiguous drafting is not wholly un- 

 known on the other side of the Channel is a surprise 

 which is not without its consolations. 



It would be tedious to detail even the more con- 

 spicuous features of this handsome volume ; enough to 

 say that every student of arithmetic will find in it some- 

 thing to arouse his interest and extend his knowledge. 

 If he is a novice, the study of this book will help him to 

 appreciate the works of at least the earlier masters, such 

 NO. 1579, VOL. 61] 



as Euler and Lagrange ; if he is a veteran, he will find 

 recreation in turning over its pages in his leisure 

 moments. 



There is one reflection which a perusal of the work can 

 hardly fail to suggest. The province of arithmetic is so 

 definite that one would expect its methods to be marked 

 by a general uniformity. But this is far from being the 

 case ; and there is, in particular, an unmistakable con- 

 trast between Diophantine arithmetic and the severe, but 

 noble science founded by Lagrange, Gauss and Kummer, 

 which we may distinguish as the analytical theory of 

 numbers. Their points of contact in such things as the 

 elementary theory of congruences and of residues only 

 serve, at present, to accentuate their divergences ; it may 

 almost be affirmed that they appeal to different classes of 

 mind. To use a metaphor, we may say that one is the 

 primitive gold-mining of the individual prospector, the 

 other the systematic working of a quartz reef with the 

 help of modern machinery. Just now the analytica 

 method holds the field ; there are several reasons for 

 this — the development of the theory of algebraic integers, 

 the influence of function-theory, the general " arithmetic- 

 ising " of analysis ; but a reaction is almost certain to 

 come. It must be remembered that all the available evi- 

 dence seems to show that Fermat's methods were essen- 

 tially Diophantine ; and there is very good reason to 

 believe that he was in possession of some peculiar 

 analysis, the secret of which died with him and still 

 awaits rediscovery. Whether this is so or not, there 

 can be no doubt that the cultivation of Diophantine 

 methods deserves more attention than it receives. The 

 risk of failure is great ; but the chance of finding a 

 treasure island exists, and ought to appeal to that spirit 

 of adventure which dwells in every mathematician who is 

 worthy of the name. G. B. M. 



MISSIONARY ANTHROPOLOGY. 

 In Dwarf Land and Cannibal Country. A Record of 

 Travel and Discovery in Central Africa: By A. B. 

 Lloyd. With an introduction by the President of the 

 Church Missionary Society. Pp. xxiv 4-385. (London : 

 T. Fisher Unwin, 1899.) 



FURTHER information regarding the dwarfs of the 

 north-eastern part of the Congo Basin is one of the 

 main desiderata in African anthropology. We therefore 

 turned to this volume hoping, from its title and size, for 

 detailed measurements of these dwarfs, convincing evi- 

 dence as to whether they belong to several tribes or are 

 all clans of one tribe, and for further light on their 

 beliefs and folklore. But we are disappointed, for the 

 book adds practically nothing to our knowledge of this 

 group of dwarfs, and the title is misleading. The book 

 narrates the story of Mr. Lloyd's missionary labours and 

 adventures from July 14, 1894, to the end of 1898 ; most 

 of these three and a half years were spent in the Uganda 

 Protectorate, and the author's acquaintance with the 

 Congo dwarfs was obtained between October 6 and 1 5, 

 1898. The account of his experiences with this people 

 occur only within some seventeen pages, whereas 368 

 are devoted to " Out of Dwarf Land." 



The bulk of the book is occupied with an account of 



