August 6, 1896] 



NA TURE 



3'5 



analytical geometry. In fact, a considerable part of 

 Apollonius's treatise is coordinate geometry pure and 

 simple ; but it is expressed throughout in a strictly 

 geometrical form. This is not without its advantages, 

 both theoretical and practical: it avoids the thorny question 

 of the continuity of numerical quantity, and it compels 

 the reader to realise the meaning of every step that is 

 taken. It is not unlikely that a well-trained Greek 

 mathematician could follow the geometrical demonstra- 

 tions as easily as the modern analyst can assimilate the 

 corresponding algebraical proofs ; it is anything but 

 easy for one who has been brought up on the system now 

 current to familiarise himself with the methods and points 

 of view which prevailed in the age of the Ptolemies. 



Still the labour is well worth undertaking, and Mr. 

 Heath's edition will do much to lighten it. It may be 

 well to state at once that the book will not relieve the 

 serious student of the duty of consulting the original 

 text. The editor, after performing the laborious task of 

 literally translating the whole treatise, decided, very 

 wisely, we think, not to publish it in that form. Instead 

 of this, he has recast it into a form similar to that 

 employed in most text-books on geometrical conies ; he 

 has occasionally condensed several propositions into one, 

 made some slight rearrangements of order, and omitted, 

 or merely given an abstract of, a certain number of pro- 

 positions which are either of slight importance, or 

 indirect proofs of converses by the usual reductio ad 

 absurdum. 



The result is that the English reader has before him 

 the substance of .'Vpollonius's great work, in a notation 

 with which he is himself familiar, while at the same time 

 he may, with a slight effort, read it back into the geo- 

 metrical form of the original. In this sense it deserves to 

 be called an edition, and is not a mere caricature tricked 

 out with modern " improvements." Apart from the nota- 

 tion, the book really gives a trustworthy presentation of 

 the contents and method of the original ; the amount of 

 alteration which the actual text has undergone may be 

 estimated by the literal transcripts of Book III. Prop. 54, 

 and Book II. Prop. 50 (one case), which will be found on 

 pp. Ixxxix-xciv of the Introduction. Some may object 

 that the condensation is excessive ; but we are inclined 

 to think that this is not the case, when we consider the 

 object which the editor had in view, namely to provide 

 an edition " so entirely remodelled by the aid of accepted 

 modern notation as to be thoroughly readable by any 

 competent mathematician." 



In this praiseworthy aim Mr. Heath has certainly suc- 

 ceeded, and it may be hoped that the " Conies " will now 

 attract the attention which it undoubtedly deserves. The 

 more the treatise is examined, the more evident become 

 its power and comprehensiveness. ApoUonius begins by 

 considering any plane section of a circular cone, not 

 necessarily right, and at once obtains a result equivalent 

 to the equations of the parabola, ellipse and hyperbola 

 in the forms 



f- = px, r = px± ti- 



referred to a diameter and the tangent at one end of it ; 

 / being the parameter, and </ the corresponding diameter. 

 It is to be observed that, in the first instance, .A.pollonius 

 speaks of l/ie diameter, namely the particular one 

 NO. 1397, VOL. 54] 



associated with the axial triangle of the cone. He then 

 goes on to prove the existence of a conjugate diameter, 

 and shows that any chord through the centre is bisected 

 there : then, after a discussion of tangents, comes a very 

 remarkable section, in which the transition is made from 

 the original diameter and the tangent at one end of it to 

 any other diameter and corresponding tangent. Every 

 one is more or less aware of the fact that ApoUonius 

 practically solved the problem of drawing normals to a 

 conic from any point in its plane ; it is perhaps hardly 

 so well known that he was acquainted with many of the 

 focal properties of central conies, with the auxiliary 

 circle, and with the harmonic properties of poles and 

 polars. Oddly enough, the focus-directrix property of a 

 conic does not appear, and was apparently unknown to 

 .i^pollonius ; the directrix is never used or mentioned, 

 and the foci of a central conic are obtained by a con- 

 struction equivalent to AS.SA' = CB-. For this reason, 

 no doubt, the focus of a parabola is not used or men- 

 tioned. But, with this one exception, almost all the 

 principal theorems of ordinary geometrical conies are to 

 be found in this treatise, composed more than twenty-one 

 centuries ago. 



The te.xt of Mr. Heath's edition is preceded by a very 

 valuable introduction, in which will be found an excellent 

 account of the earlier history of conic sections among 

 the Greeks, followed by an instructive essay on the 

 characteristics and methods of ApoUonius. This, with 

 the appendix on the terminology of Greek geometry, 

 will be of great service to those who may feel attracted 

 towards research in the history of mathematics ; a 

 subject not interesting to many, but fascinating to the 

 few who combine the instinct of an antiquarian with 

 the necessary linguistic knowledge and mathematical 

 ability. 



One or two suggestions may perhaps be made in 

 anticipation of another edition. A glossary of Greek 

 technical terms, or at any rate an index of them, with 

 references to the pages of the introduction where they 

 are explained, would be a useful addition. The figures, 

 on the whole, are clear, but some of them might be more 

 accurately drawn ; and in some of the longer and more 

 difficult propositions it is very inconvenient to have to 

 turn back to look at a figure on a previous page. 



G. B. M. 



THE HARE, FROi^f THE FIELD TO THE 

 TABLE. 

 Fur and Feather Series. — The Hare. Edited by A. E. T. 

 Watson. i2mo. Pp. 263. Illustrated. (London : Long- 

 mans, Green, and Co., 1896.) 

 FROM the first it was evident that the beautifully 

 illustrated volumes of the " Fur and Feather Series'' 

 would appeal more to the sportsman and the bon-vivant 

 than to the naturalist. That this is the case with the 

 present issue may be inferred from the fact that out of 

 a total of 263 pages, only a paltry sixty-two are devoted 

 to what the author calls the natural history of the hare. 

 As a matter of fact, it is impossible to apply the term 

 "natural history" to the subject of more than the first 

 forty-eight pages ; the third chapter in Mr. Macpherson's 



