98 



NATURE 



[yunc 2, 1881 



Greek commerce and for the interchange of Greek thought, 

 and a briUiant period followed — one of the most memor- 

 able in the history of the world." Athens became the 

 centre of all intellectual movement. To her— the Athens 

 of Pericles — came Hippocrates of Chios, and "in this 

 city geometry was first published." 



Our author agrees with Hankel (against Proclus) as to 

 the important influence of the Eleatics (Parmenides and 

 Zeno), " not only on the development of geometry at that 

 time (circ. 450 B.C.), but further on its subsequent pro- 

 gress in respect of method." Clairaut, in his " Elements 

 of Geometry" (recently translated by Dr. Kaines, the 

 original te.xt is cited by Dr. AUman), notices this influence 

 in the case of Euclid. The paradoxes of Zeno led to the 

 banishment of the Infinite (which plays so important a 

 part in the modern treatment), " the infinitely small as 

 well as the infinitely great." What Hippocrates may 

 very fairly be supposed to have done in relation to the 

 squaring of the circle is, we think, well put. " SimpUcius 

 has preserved in his ' Comm. to Phys. Ausc' of Aristotle 

 a pretty full and partly literal extract from the ' History 

 of Geometry ' of Eudemus." It is to Bretschneider we 

 owe a careful revision and emendation of this fragment, 

 and our author has skilfully attempted to determine what 

 is Simplicius and what is simply Eudemus in this account. 

 It is curious that Bretschneider merely notices the " cir- 

 cumstantiality of the construction and the long-windedness 

 and the over-elaboration of the proofs," and Hankel 

 expresses surprise that this fragment, " 150 years older 

 than Euclid's Elements, already bears that character, 

 typically fixed by the latter, which is so peculiar to the 

 geometry of the Greeks." Had the present pamphlet 

 been confined to the elucidation of this single matter it 

 would have had a sufficient raison d'etre. 



The next geometer whose contributions to geometry 

 are determined and discussed is Democritus, more usually 

 regarded as a philosopher. At this stage, too, our author 

 takes stock, and shows that the progress made in this 

 (about) half-century interval since Pythagoras mainly 

 '' concerns the circle." 



We note the connection of the name of Hippocrates 

 with another of the famous problems of antiquity, viz. the 

 duplication of the cube : he seems to have been the first 

 to reduce this question to the finding of two mean pro- 

 portionals between two given straight lines, the greater 

 of which is double the less. Many interesting particulars 

 are given in connection with this problem. The general 

 problem is stated to have been first solved by " Archytas 

 of Tarentura, then by his pupil Eudoxus of Cnidus, and 

 thirdly by Menajchmus, a pupil of Eudoxus"; this last 

 used "the conic sections which he had discovered." A 

 third problem, the tri-section of an angle, also came to 

 the front about this time. Dr. Allman fully discusses this 

 also, and shows that it was one which was fairly within 

 the reach of a Pythagorean. Montucla however attributes 

 to Hippias of Elis, a contemporary of Socrates, the in- 

 vention of the quadratrix (the quadratrix of Dinostratus), 

 by means of which (in a quite different way from Sylvester's 

 Fan) an angle can be not only trisected, but divided into 

 any number of equal parts. Allman sides with Hankel 

 and shows the improbability of Hippias being the in- 

 ventor, but he is against him when he refers the method 

 of exhaustions to Hippocrates of Chios. It will have 



been seen that the great geometer of this period is Hippo- 

 crates, " who seems to have attracted notice as well by 

 the strangeness of his career as by his striking discovery 

 of the quadrature of the lune." The unfavourable state- 

 ments of Aristotle, Eudemus, Jamblichus, and Eutocius 

 are examined, and part of the summing up i.s, " We may 

 fairly assume that Hippocrates imperfectly understood 

 some of the matter to which he had listened ; and that, 

 later, when he published what he had learned, he did not 

 faithfully render what had been communicated to him." 



An examination of this pamphlet still further shows 

 that the writer, while carefully using the recent works of 

 Bretschneider, Hankel, Cantor, and others, has himself 

 gone over the original authorities and formed his own 

 opinions upon the difficult questions that turn up. It is, 

 in our opinion, a most valuable contribution to the subject, 

 and we shall be glad when the piecemeal work in " Her- 

 mathena" is done, and the book appears, as we believe it 

 is the writer's intention that it should appear, in proper 

 book form as one work. 



OUR BOOK SHELF 

 The Zoological Record for iZyg. Being Volume Sixteen 

 of the Record of Zoological Literature. Edited by 

 Edward Caldwell Rye, F.Z.S. (London : John Van 

 Voorst, for the Zoological Record Association, 1881.) 

 The editor's promise to the members of the Zoological 

 Record Association has been kept, and the Record for 

 1879 was published in the month of April in this year. 

 We gladly note in addition his confident expectation that 

 the Record for 1 880 will be published ere the present year 

 ends. This sixteenth volume contains nearly 700 pages 

 of well-condensed records of the literature of zoology of 

 1879. The lion's share of the hard work has fallen to 

 Mr. W. F. Kirby, who, with Mr. McLachlan, records the 

 literature of the Insecta. The Rev. O. P. Cambridge 

 gives the record of the Arachnida for 1878 and 1879. 

 The Vermes and Echinoderms are done by Prof. Jeffrey 

 Bell, and the Ccelenterata and Protozoa are elaborated 

 by A. G. Bourne, S. J. Hickson, and Stuart Ridley. 

 The works on the Mammals are recorded by W. A. 

 Forbes ; on the Birds by Howard Saunders ; on the 

 Reptiles and Fishes — alas ! that we should have to write 

 it— by the late gifted A. W. E. O'Shaughnessy. Prof. E. 

 von Martens still records the literature of the MoUusca 

 and MoUuscoidea, the only recorder still remaining as such 

 of that small group who came to the assistance of Dr. 

 Guntherin 1864. We miss from last year's list of recorders 

 Dr. C. Liitken, who served during his seven years well and 

 faithfully; in him knowledge and experience of the subject 

 he worked at were combined with much tact. The British 

 Association, the Royal Society, and the Zoological Society 

 of London have, as is now usual, handsomely assisted in 

 aid of the publication of this most useful volume. 



The most useful index to new genera and sub-genera 

 seems most carefully done. The list of new genera is for 

 the year almost 1000 ; so that evidently the zoological 

 kingdom is not as yet worked out. 



Wiltshire Rainfall, 1880. (Marlborough: C. Perkins 

 and Son, Times Office). 



The compilers of this carefully-printed and, for the class 

 of publications, luxuriously got-up annual merit our hearty 

 commendation for the general excellence of the v\ork thus 

 put before us. From its physical geography Wiltshire 

 forms a well-marked rainfall region, it being a little to the 

 north of the centre of the county that the two Avons 

 and several tributaries of the Thames take their rise. 

 From this plateau the country slopes northward to the 



