624 



NA TURE 



[October 24, 1901 



Mrs. Bryant, both as an expert logician and as the 

 daughter of a fellow of Trinity College, Dublin (Rev. 

 W. A. Willock), who had no belief in the appropriateness 

 of Euclid's book except to "grown-up, hard-headed, 

 thinking men," was sure to remove from the path of the 

 young pupil as much of the essential ditliculty of Euclid 

 as could be removed consistently with the retention of 

 the book as the basis of school instruction. 



To follow the subject in detail, we notice that the 

 editors have deliberately left out alternative proofs of the 

 "Asses' Bridge" on the ground that Euclid's proof is 

 found by experience to be more readily understood than 

 any of the alternative proofs — a statement which surely 

 cannot be well founded. What can be more simple than 

 the proof founded on the superposition of two identical 

 triangles? And, again, if we imagine the bisector of the 

 vertical angle to be drawn, we have the result as a direct 

 consequence of prop. iv. It is not to the point to object 

 that Euclid will not allow us to imagine this bisector 

 unless we can show how to draw it ; if the bisector were 

 drawn, the result would follow — -that is proof enough. 

 At the end of Book i. there is a large collection of 

 worked-out theorems and problems ; and we may speci- 

 ally notice the excellent exposition of the method of 

 analysis and synthesis in pp. 102-106, which will greatly 

 help the pupil who is learning this method of attacking 

 problems. Besides these worked questions, there is a 

 collection of 100 unworked exercises in illustration of 

 Book i. 



In Book ii. the fundamental propositions 12 and 13 are 

 proved as an extension of the proposition of Pythagoras 

 (47, Bk. i.) by the famous old windmill figure so familiar 

 to us all ; and, as the editors inform us, this proof is 

 found in Lardner's Euclid, but cannot be traced further 

 back. It is strange that the editors of our school Euclids 

 should have overlooked this most interesting and graphic 

 proof. Lardner's Euclid, now seldom seen, is— even com- 

 pared with the best modern editions — a work of great 

 usefulness and high merit. 



There is a note at the end of Book ii. (p. 14S; the sub- 

 stance of which is that pure geometry must be kept 

 severely apart from all arithmetical conceptions ; and this 

 is followed (p. 150) by a still more remarkable note 

 stating that " in all examinations" the use of -t- and -, 

 of the abbreviation AB^ for the square on AB, and of 

 the abbreviation AB . BC for the rectangle AB, BC, is 

 permitted in writing out all theorems and problems of 

 geometry, provided that these are not] given in Euclid's 

 text. 



Why such an extraordinary distinction and restriction 

 should exist is incomprehensible to us, and remains so 

 even after we have read the excuse put forward for 

 it by the present editors ; and after this excuse comes 

 the statement 



" the use of these symbols ought never to be allowed 

 at any time until it is clear that AB- and AB . BC 

 are used by the student simply as the shortest way of 

 writing the square on AB and the rectangle contained by 

 AB and BC, respectively." 



Thus the divorce of all arithmetical conception — and, 



indeed, all quantitative conception — from geometry is 



advocated ; and if the restriction were really carried out 



both by teachers and by examiners (which it is not), the 



NO. 1669, VOL. 64] 



teaching of the subject would be rendered much more , 

 slow and difficult than it is at present. ; 



Book iii. ends with a very large collection of worked- l 

 out questions followed by 100 e.xercises, a very good 

 feature being the association of each famous result with ' 

 the name of its discoverer ; and a similar remark may be ., 

 made with regard to Book iv. Book v. is omitted, only 

 the definitions required in Book vi. being given. 1 

 Euclid's test of proportion — i.e. of the equality of the ; 

 ratio A : B to the ratio C : D — is given and applied to six 

 special cases (p. 293) under the heading "Theory of 1 

 Proportion." This test is, of course, that C : D will be 

 the same as .^ : B if when otA = ;/B we have wC^wD ; 

 and we wonder whether any beginner in the world is 

 introduced to the notion of the equality of ratios by this 

 means. Probably without a single exception, every boy 

 is first told that 4 : 2 is the same as 6 : 3, because 2 is 

 contained in 4 just as often as 3 is contained in 6 ; and 

 even if the one quantity were not contained an integer 

 number of times in the other, he would be prepared to 

 admit and understand the equality of ratios if this 

 number was an endless decimal, provided it was the ' 

 same for the two compared ratios. Euclid's test must 

 infallibly be received by the beginner merely as the ipse 

 dixit of Euclid ; the beginner cannot understand its 

 validity apart from arithmetical notions ; and it seems 

 rather grotesque to find it formally employed to prove 

 such a trifle as " magnitudes which have the same ratio 

 to the same magnitude must be equal." Lardner has, as 

 usual, some excellent remarks on this criterion ; but his 

 exposition amounts to no justification that could possibly 

 convince the mind of a beginner. Hear also the opinion 

 of the Rev. W. .'\. Willock on the question ("Elementary 

 Geometry of the Right Line and Circle," p. ix.) : — 



"The criterion of proportion used is that of Elrington, 

 by .y«/5muUiples. This test is here adopted because it is 

 more readily understood by young students, and also more 

 conformable to the common notions of proportion. More- 

 over, it holds good, in all strictness, for commensurable 

 magnitudes ; and, as to the incommensurable, it holds 

 equally good if the equisubmultiples taken of the first 

 and third terms be infinitesimals. . . . The right con- 

 clusion as to the two tests is, probably, that both should 

 be given in a tfeatise on elementary geometry, each 

 having its own peculiar advantages." 



At the end of Book vi. follows what may be regarded 

 as a small encycloptcdia of important results and methods 

 — coaxal circles, harmonic ranges, poles and polars, 

 centres of similitude, inversion, maxima and minima, &c. 

 — an invaluable collection, excellently handled. 



Book xi. calls for no detailed remarks : its accompany- 

 ing illustrations are of the same high order of merit as 

 that which characterises all the special work of the 

 editors. 



OUR BOOK SHELF. 

 The Life-History of British Serpents and their Local 



Distribution in the British Isles. By Gerald R. 



Leighton, M.D. Pp. xvi -f 383. 8vo. Illustrated. 



(London : W. Blackwood and Sons, 1901.) Price Ss. 



net. 

 The idea of supplying the " field-naturalists of the British 

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