122 



NATURE 



[December 12, 1895 



we learn that Alkhojandi, about looo a.d., discovered 

 that the equation .r^ -x- y^ — z^ does not admit of rational 

 solutions. 



A brief outline of the period 1200- 1500 brings this 

 excellent book to its conclusion. That it is the outcome 

 of independent and careful research is evident ; and the 

 reader's appreciation of this fact is rather enhanced than 

 otherwise by Dr. Zeuthen's graceful acknowledgment of 

 his obligations to his predecessors in the same field, more 

 particularly to Cantor and Tannery. 



Mr. Ball's "Primer" is a work of a very different scope. 

 Its object 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 of their discoveries." It is ex- 

 pressly said that it is not intended for those to whom the 

 subject is familiar. The plan adopted is to give a series 

 of brief biographies in chronological order, interspersed 

 with occasional paragraphs on particular periods. The 

 necessary element of " human interest " is supplied by a 

 number of anecdotes. Many of these are pertinent 

 enough ; others are certainly superfluous. Why should 

 half a page be devoted to the unhappy matrimonial 

 experiences of Kepler? or, again, what is the value of 

 the information that Descartes in Paris was " modestly 

 clad in green taffety?" 



Still, the book is entertaining, and, although very 

 sketchy, fulfils its purpose well enough until we come 

 to the last section, which treats of recent mathe- 

 matics. Here the difficulties of the subject, and the 

 narrow limits of his plan, have been too much for the 

 author. That this should be the case is not altogether 

 surprising ; but some of the faults of omission and com- 

 mission are too serious to be passed over. 



Thus in the paragraph on Cauchy, no mention is made 

 of his work on the theory of numbers, or of his great 

 memoir on waves. The statement that " the rule for 

 iindinj the principal values of integrals was enunciated 

 by him, and the calculus of residues was his invention," 

 is much as if one should say " Newton discovered the 

 binomial theorem and wrote the 'Principia.'" Worst 

 of all, the account concludes with the remark : "In 

 many of his memoirs the feverish haste with which they 

 were thrown off is too visible, and several are marred by 

 obscurity, repetition of old results, and blunders." Such 

 criticism of a great genius is in questionable taste, and is 

 apt to recoil on the person who makes it. It seems to 

 us rather obscure to say (p. 132) that " in this theory the 

 th eta-functions are independent of the form of their 

 space-boundaries"; and that Eisenstein "considered 

 the theorems relating to the possibility of representing a 

 number as a sum of squares, and showed that the general 

 theorem was limited to eight squares." Will not the 

 general reader infer from this that no number is the sum 

 of more than eight squares ? 



Then as to madvertent errors (it would be unkind to 

 call them blunders) : (i) it is not true that "the only 

 regular polygons which can be constructed by elementary 

 geometry are those of which the number of sides is 

 2'" (2" -|- i), where m and n are integers and 2" -j- i is a 

 prime " ; even Euclid could construct a regular quin- 

 decagon ; (2) the theory of ternary quadratic forms is 

 not due to Eisenstein ; (3) Eisenstein did not give a rule 

 NO. 1363, VOL. 53] 



for distinguishing whether a given series represents an 

 algebraical or a transcendental function (see Heine's 

 " Kugelfunctionen," 2nd edition, i, p. 50) ; (4) Abel did 

 not prove that it is impossible to solve a quintic equation 

 by means of radicals, but the quite different proposition 

 that a root of the general quintic cannot be expressed in 

 terms of its coefficients by means of radicals. 



P"inally, the omission of all notice of Galois is entirely 

 inexplicable. The pathetic story of his death appeals to 

 universal sympathy, and might even draw a tear from 

 the hardened general reader ; while the influence of his 

 work upon recent analysis is, perhaps, second only to 

 that of Riemann. 



Fault-finding is not pleasant, and is apt to bulk too 

 largely in a review. Mr. Ball's readers may not be im- 

 pressed by the fact that Cayley " introduced the so-called 

 'absolute,'" and they may be inclined to think that 

 " homaloidal hyper-space " is a somewhat technical ex- 

 pression ; but they will find plenty of amusement in the 

 " Primer," and a good deal of instructive reading ; while, 

 for reasons which are different, but each sufficient, the 

 occasional lapsus calami will do neither the instructed 

 nor the uninstructed reader any harm. G. B. M. 



THE SPIDERS OF BURMA. 

 Descriptive Catalogue of the Spiders of Burma, based 

 upon the Collection tnade by Eugene W. Dates, and 

 preserved in the British Museu7n. By Dr. T. Thorell 

 (London : printed by order of the Trustees, 1895.) 



DURING his residence in British Burma, in the capa- 

 city of civil engineer, Mr. E. W. Gates availed him- 

 self of the rare opportunities of travelling, afforded by his 

 official duties, to investigate certain portions of the fauna 

 of the country, choosing as objects of special study such 

 diverse groups as Scorpions, Whip-Scorpions, and 

 Spiders ; Centipedes and MiUipedes; and Birds. It was, 

 we believe, primarily his intention to work out all his 

 collections himself upon his return to England on fur- 

 lough. In fact, while still in the East he published, in 

 i\ie.fournal of the Asiatic Society of Bengal, descriptions 

 of his new species of Whip-Scorpions {Thclyphonus), and 

 shortly after his arrival, his paper upon the Indian and 

 Burmese species of Scorpions of the genus Isonietrus 

 appeared in the fournal of the Bombay N(itural History 

 Society. But further than this, his studies in the inverte- 

 brate portion of his material did not go ; and realising 

 the impossibility of grappling in the space of time at his 

 disposal with the vast number of species of spiders and 

 myriapods that he had procured, he generously pre- 

 sented these in their entirety to the Trustees of the 

 British Museum, and devoted his energies to the study 

 of the Birds of British India and Burma, of which he 

 had already acquired considerable knowledge. In the. 

 course of the next few years, the Centipedes and Milli- 

 pedes were determined and reported upon in a series of 

 memoirs that appeared in the Antiali del Museo Civico 

 di Genova. The spiders, however, were, at Mr. Gates's 

 request, submitted for examination to Dr. T. Thorell, 

 who had already made himself an authority upon 

 Burmese Arachnida, in connection with the study 

 of the material of this group amassed under the 

 auspices of the Marchese G. Doria by that practised 



