May 6, 1920] 



NATURE 



289 



on " is a point. The editor keeps the time- 

 honoured rendering- " evenly " — we should prefer 

 "equably " or "indifferently"; but this is of little 

 importance, because, whatever term is used, it 

 will have to be carefully explained. In any case, 

 Simson's definition is a gross misrepresentation of 

 Euclid, who evidently, however obscurely, states 

 a property of a straight line in relation to all the 

 points upon it. 



Owing- to the great popularity of the Elements, 

 it is unlikely that we shall ever find out exactly 

 the contributions made to geometry by Euclid's 

 predecessors. Naturally, we should like to know 

 what were the attainments of the Pythagorean 

 school, and how far they were arithmetical or 

 geometrical respectively. A still greater satis- 

 faction would be to know how far the Greek 

 theory of proportion was developed by Eudoxus. 

 As presented in the Elements, it may fairly be 

 called the crowning triumph of Greek mathe- 

 matics ; it is so near absolute perfection that no 

 recent analysis can amend it, except perhaps by 

 explicitly introducing what is known as the axiom 

 of Archimedes. The question is, How far, if in 

 any way, did Euclid improve upon Eudoxus 's 

 exposition ? 



We do not know whether Sir T. Heath intends 

 to publish all the other books of the Elements. 

 Those who are interested in Greek mathematical 

 thought would be grateful for a similar edition 

 of Euclid's arithmetical books, especially the 

 tenth, which, as De Morgan pointed out long ago 

 (in Smith's Dictionary of Classical Biography), 

 contains an exhaustive discussion of a particular 

 family of irrationals. Another boon would be an 

 edition like this of some of the books of Apol- 

 lonius's "Conies," especially those which virtu- 

 ally give the equations of conies referred to a 

 principal diameter and the tangent at a vertex. 



There is one point on which we venture, with 

 all deference, to disagree with the editor. On 

 p. 175 and elsewhere he translates ttoXXw fiu^mv 

 by "much greater." The Greek idiom is peculiar, 

 but unless we are to make it absurd we must 

 render iroAX<S by "all the more," or some such 

 phrase. 



One remark in conclusion. Forty or fifty years 

 ago, when a blind idolatry of Simson's "Euclid" 

 was still the vogue, Euclid's fifth book was never 

 read, and its theorems were assumed on the basis 

 of Todhunter's "Algebra." In other words, the 

 greatest achievement of Greek mathematics was 

 absolutely ' ignored. It is a great mistake to 

 assume that all who are in favour of modern 

 methods of teaching are wholly out of sympathy 

 with the classic exponents of their subject. On 

 NO. 2636, VOL. 105] 



the contrary, those of them who are sane psycho- 

 logists will always bear in mind that the progress 

 of the individual is, in a way, a condensed image 

 of the progress of the race, and they will be the 

 last to ignore the historical development of their 

 subject, whether it be mathematics, or philosophy, 

 or chemistry, or anything else. G. B. M. 



The Earliest Flint Implements. 



Pre-PalaeoUthic Man. By J. Reid Moir. Pp. 

 67 + 29 plates. (Ipswich: W. E. Harrison; 

 London : Simpkin, Marshall, Hamilton, Kent, 

 and Co., Ltd., n.d.) Price 75. 6d. 



IN this little volume Mr. Reid Moir treats of 

 the various forms of flaked flints found in 

 deposits older than those in which ordinary 

 palaeolithic implements occur. He also describes 

 the experiments in flint-fracture which have con- 

 vinced him that the specimens in question are 

 examples of human workmanship. Most of the 

 matter has already appeared in various papers 

 by the author, and many of the illustrations, are 

 from these papers, but the whole is a useful 

 summary which the general reader will be glad 

 to have in so convenient a form. 



Among flints like the so-called eoliths, which 

 are very little shaped by chipping, it must natur- 

 ally be difficult to decide which have been flaked 

 by man for his own use, but Mr. Reid Moir con- 

 siders that they can be distinguished by the 

 shape and appearance of the flake-scars. Accord- 

 ing to his experiments, the scar left by fortuitous 

 percussion is comparatively wide and truncated, 

 and often marked by concentric lines, while that 

 made by human flaking is longer than it is wide, 

 tapering at the far end, and not marked by con- 

 centric lines. If this criterion be trustworthy, it is 

 evident that man's earliest handiwork can be 

 recognised, for when he first began to use stone 

 he must have selected pieces which were already 

 of the needed shape, and he merely trimmed 

 certain edges for greater effectiveness. 



Man's first efforts to shape a real implement 

 are supposed to have resulted in the rostro- 

 carinate type, and this by further chipping gradu- 

 ally passed into the familiar palaeolith. Mr. Reid 

 Moir describes the process of change, as further 

 discussed in his memoir in the Philosophical 

 Transactions which was noticed in Nature for 

 April I, p. 146. He also, as in the memoir just 

 mentioned, expresses the opinion that the Acheu- 

 lean and the Mousterian forms of palaeoliths have 

 been derived from the rostro-cari nates in some- 

 what different ways. The speculation is interest- 



