NATURE 



241 



EUCLID AND HIS EDITORS. 

 Euclids Elements of Geometry. Edited for the Syndics 

 of the Press by H. M. Taylor, M.A. Books I.-VI., 

 XI., XII. Pp. xxii + 658. (Cambridge : at the Uni- 

 versity Press, 1895.) 

 THE appearance of a school edition of " Euclid's 

 Elements," published under the auspices of the 

 Cambridge University Press, provokes reflections upon 

 the strange position so long maintained in this country by 

 Robert Simson's authorised version (so to speak) of the 

 work of the Alexandrian geometer. For more than a 

 hundred years the Simsonian text enjoyed an unchallenged 

 supremacy ; and not so very long ago any proposal to 

 amend it, or to teach elementary geometry by means of 

 some other book, was regarded as something very like 

 profanity. 



It is only in accordance with the nature of things that 

 this professed veneration for Euclid should coexist with a 

 profound ignorance of the real Elements., and of the 

 other extant works of their author. To this day the 

 reputation of Euclid is not unlike that of the wizard 

 Virgil in the Middle Ages. Most educated Englishmen 

 are quite unaware of the existence of those books of the 

 Elements which are not read in schools ; there is even a 

 legend current in some quarters that they were destroyed 

 in MS. by Euclid's wife I For the only good critical 

 edition of Euclid's works (Heiberg's, in the Teubner 

 Series) we must go to Germany ; and it would be interest- 

 ing to know how many English mathematicians are 

 acquainted with the original text, and how many English 

 scholars have had the curiosity to find out whether 

 "parallel<7piped" (rhyming with "biped," by the way) is 

 vouched for by the Greek. To crown the absurdity, it is 

 just that book of the Elements which is of greatest per- 

 manent value that, by common consent, is never read. 



After all, the study of the real Euclid may very fitly 

 remain the privilege of the minority ; the really urgent 

 need is that geometry should be taught rationally and 

 effectively in our schools. Happily, we have outlived the 

 glacial period which still prevailed in the early part of 

 this century ; and all good authorities are practically 

 agreed upon the main lines of reform. The weight of 

 opinion among experienced teachers seems still to be in 

 favour of retaining at least the framework of the Elements, 

 and of following, generally, Euclid's sequence of pro- 

 positions. In favour of this course there is something to 

 be said, independently of the base consideration of 

 examination requirements. That there is a certain advan- 

 tage in a recognised order of propositions will probably 

 be admitted by most of those who have wandered in the 

 chaos of geometrical conies ; and with regard to Euclid's 

 methods of proof, it is doubtful whether the various 

 alternatives which have been suggested are really easier 

 for a schoolboy to learn and understand. A beginner is 

 very apt to appeal to his powers of intuition in a quite 

 illegitimate way ; and in trying to reproduce a proof 

 which depends upon superposition or symmetry he often 

 loses himself in a haze of words, and fails to give a sound 

 demonstration. Proofs of this kind may very well be in- 

 NO. 1368, VOL. 53] 



eluded, of course ; but should not, we think, take the 

 place of the more formal ones in the text. 



Mr. Taylor's book is one of several which, while harm- 

 lessly masquerading as " editions " of Euclid, are really 

 excellent treatises on elementary geometry, based upon 

 the lines which Euclid has laid down. Signs of the 

 wholesome reform that has taken place, meet the eye on 

 every page. The antiquated terms, the clumsy repe- 

 titions, the tiresome rigmaroles of Simson's text are done 

 away with ; notes and explanations are given where 

 necessary ; additional propositions are introduced ; and 

 there is an abundance of exercises, carefully graduated, 

 and properly distributed throughout the book, instead of 

 being hidden away at the end of it. One of the extra- 

 ordinary superstitions of former days was that nobody 

 could do a geometrical deduction unless he had previously 

 learnt three books of Euclid by heart ; it is to be hoped 

 that this ridiculous theory is at length abandoned. Some 

 boys, of course, can never do a " rider," even the easiest ; 

 but those who have any capacity can be started success- 

 fully after learning the first five propositions, or even 

 before. 



School Euclids may be roughly divided into two 

 classes, according as symbols of abbreviation are used or 

 avoided. While such a notation as AB'^ for " the square 

 on AB" is decidedly objectionable, the use of symbols of 

 mere abbreviation is a matter of taste. Personally we 

 detest them, and rejoice that they do not appear in Mr. 

 Taylor's treatise. Brevity has been secured in the proper 

 way, by a careful choice of words, and not by a host of 

 contractions and ugly symbols for " circle," " parallelo- 

 gram," and so on. 



There are several attractive features in Mr. Taylor's 

 book to which attention may be drawn. The selection of 

 additional propositions is very good, and it is needless to 

 say that the proofs given are very elegant. There is a. 

 most interesting collection of proofs of Pythagoras's 

 theorem (i. 47) ; Ptolemy's theorem is proved by means 

 of Book iii.; and Gergonne's construction for the circles 

 touching three given circles, is to be found on p. 458. 

 There are also sections dealing with poles and polars, 

 coaxal circles, projective rows and pencils, Pascal's 

 and Brianchon's theorems for the circle, centres of 

 similitude, and inversion (including an account of Peau- 

 cellier's cell) ; besides this, there is a long supplement to 

 Book xi., which discusses, inter alia., properties of tetra- 

 hedra and parallelepipeds, spherical geometry, the regular 

 solids, and the elements of perspective. In conclusion, 

 we have the determination of the surface and volume of 

 a sphere. It will be seen from this mere list how liberally 

 Mr. Taylor has interpreted his editorial function, and how 

 many important theories he has contrived to touch upon ; 

 at the same time, the book is anything but " stodgy," and 

 cannot fail to interest and stimulate an intelligent reader. 

 There is a good inde.x, and here and there brief historical 

 notes are given. 



It is instructive to observe how Mr. Taylor has dealt 

 with Euclid's text. He explains in the preface that he 

 began by translating the first book ; he ended by " giving 

 up all idea of simple translation, and retaining merely 

 the substance of the work, following closely Euclid's 

 sequence of propositions in Books i. and ii., at all events." 

 To see what this means, let us take propositions 1-26 



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