240 



NATURE :'0 



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[_yuly lo, 1879 



EUCLID AND HIS MODERN RIVALS 

 Euclid and His Modern Rivals. By Charles L. Dodgson, 



M.A. (London : Macmillan, 1879.) 

 Elementary Geometry. Books i. — iv., containing the 

 subjects of Euclid's first six books; following the 

 syllabus of geometry prepared by the Geometrical 

 Association. By J. M. Wilson, M.A. Fourth edition. 

 (London : Macmillan, 1878.) 



BY a curious chance these two works reached our 

 hands nearly on the same day, and as Mr. Dodgson 

 devotes a great portion of his space (62 pp.) to the con- 

 sideration of Mr. Wilson's Geometries, we have thought it 

 well to notice the two authors at the same time. As 

 however it is patent from the fact of Mr. Wilson's work 

 having reached a fourth edition, that his method is not 

 imknown to, and, may we add, not unappreciated by, a 

 large section of mathematical teachers, we shall at once 

 pass on to a consideration of Mr. Dodgson' s book, only 

 noticing Mr. Wilson's book in connection with the criti- 

 cisms put forward in "Euclid and His Modern Rivals." 



A few words by way of introduction. Mr. Dodgson 

 has been a teacher of geometry at Oxford, we believe, for 

 nearly five-and-twenty years, and during that time has 

 had frequent occasion to examine candidates in that sub- 

 ject. For a great part of the above-stated period things 

 went pretty smoothly, and King Euclid held undisputed 

 sway in the " Schools ; " but eleven years ago a troubler 

 of the geometrical Israel came upon the scene, and read 

 a paper before the Mathematical Society, entitled " Euclid 

 as a Text-Book of Elementary Geometry." The agitation 

 thus commenced acquired strength, and at length, in 

 consequence of a correspondence carried on in these 

 columns, the Geometrical Association was formed. A 

 prime mover in this matter was that Mr. Wilson who 

 wrote the paper, and subsequently brought out the 

 geometry cited. Mr. Dodgson is one of the gentlemen 

 opposed to this change, and the moving cause of the 

 present lUad is the "vindication of Euclid'smasterpiece." 

 Another consequence of the agitation is that many have 

 tried their prentice hands on the production of new geo- 

 metries — " rivals," our author calls them — " forty-five were 

 left in my rooms to-day." Can we wonder then, that, his 

 soul being stirred within him, he should overhaul a selec- 

 tion of them to see what blots he could " spot " in them ? 

 He might well have taken for his motto one once familiar 

 to us — 



" If there's a hole in a' your coat?, 



I rede ye tent it ; 

 A chiel's amang ye takin' notes, 



An' faith he'll prent it ! " 



Our author's criticism takes a peculiar form, but we shall 

 not blame him for this, for he has afforded us much 

 amusement, and we quite hold with the Horatian hne he 

 cites in extenuation of his mode of procedure : "Riden- 

 tem dicere verum quid vetat .? " We believe he has made 

 a good many hits, but at times his wit, we think, has led 

 him too far. We shall not, however, here give any 

 account of his plot — we prefer to refer our readers to the 

 work itself — but confine our notice to the remarks upon 

 Mr. Wilson's books, and upon Mr. Morell's " Euclid 

 Simplified." ' 



Mr. Dodgson devotes forty-eight pages to Mr. Wilson's 



A work we ourselves had occasion very strongly to condemn — see 

 Nature, vol. xiii. p. loz. 



" Elementary Geometry " (second edition, 1 869). We can 

 hardly see why so much space should be devoted to a 

 work which seems tacitly to have been withdrawn by the 

 author, or, at any rate, to have been considered inferior 

 to the work under review. Is it that the "scene" was 

 written some time since, and was considered to be too 

 good to be sacrificed ? Happily it is not our business to 

 defend Mr. Wilson' s views on " direction ; " he is perfectly 

 competent to defend his own views, and no doubt, should 

 he see fit, will do so at the right time. 



"Minos"— who argues for Mr. Dodgson— himself 

 seems to think that his remarks will now and again be 

 considered hypercritical. Take the following : — 



Niematid {i\it general representative of the "rivals," 

 quoting from the "Elementary Geometry"). Two 

 straight lines that meet one another form an angle at the 

 point where they meet (p. 5). Min. Do you mean that 

 they form it "at the point," and nowhere else? Nie. I 

 suppose so. Min. I fear you allow your angle no magni- 

 tude, if you limit its existence to so small a locality ! 

 Nie. Well, we (/i?«'/ mean "nowhere else." Min. {medi- 

 tatively). You mean at the point— and somewhtre else.? 

 Where else, if you please ? Nie. We mean— we don't 

 quite know why we put in the words at all. Let us say 

 " Two straight lines that meet one another form an angle." 

 Min. Very well. It hardly tells us what an angle is, and, 

 so far, it is inferior to Euclid's definition; but it may 

 pass. Again (p. 73), A'ie. reads, P. 5, Ax. 5, "Angles 

 are equal when they could be placed on one another so 

 that their vertices would coincide in position, and their 

 arms in direction." Min. " Placed on one another ! " Did 

 you ever see the child's game, where a pile of four hands 

 is made on the table, and each player tries to have a 

 hand at the top of the pile .' A^zV. I know the game. Min. 

 Well, did you ever see both players succeed at once ? A'ie. 

 No. Min. Whenever that feat is achieved you may then 

 expect to be able to place two angles " on one another !" 

 You have hardly, I think, grasped the physical fact that, 

 when one of two things is on the other, the second is 

 underneath the first. But perhaps I am hypercritical. 



What the text means is, of course, that B, C, D 

 could be placed upon A ox A upon B, C, D, so as 

 to coincide. A still more striking instance is p. 160. 

 Mr. Wilson adopts the syllabus-definition, "When one 

 straight line stands upon another straight line and makes 

 the adjacent angles equal, each of the angles is called a 

 right angle"— a. definition, by the way, remarkably like 

 Euclid's. Minos says, "allow me to present you with a 

 figure, as I see the Syllabus does not supply one^ 



A Here AB 'stands upon' B C, and makes the 



B adjacent angles equal. How do you like these 



C ' right angles ?' " 

 This is a hit, of course, indeed a double hit, the one 

 farcical in its illustration, the other sober enough, for the 

 Syllabus considers that two angles (a major, and a minor, 

 conjugate) are formed by two straight lines drawn from a 

 point. Mr. Dodgson is very amusing upon the "straight'' 

 angle, and, no doubt, would be equally so upon the equi- 

 valent " flat " angle. A good phrase is still a desideratum, 

 but De Morgan long ago pointed out that "the angle 

 made by a straight line with its continuation is a definite 

 angular magnitude," and considered its half to be the 

 best definition of a right angle. 



