270 



NATURE 



[January 23, 1902 



confirmation of her views of the Christian cross being 

 originally related to the swastika. The Chinese, taking 

 this view, naturally called the Christians " Cross-wor- 

 shippers." In Japan and India Mrs. Nuttall has found 

 many parallels, and in Mesopotamia to this day the men of 

 Saba appear to worship the Pole-star. The religious 

 literature of ancient Babylonia and Assyria contains 

 many passages which prove that the Semites who em- 

 ployed the cuneiform character held many views in 

 common with the Mexicans ; while an elaborate 

 e.\amination of Egyptian works has provided Mrs. 

 Nuttall with a large number of proofs that in Me.xico, 

 China, and Egypt the views held in respect of certain 

 astronomical phenomena were identical. 



One of the most interesting sections of the book before 

 us is that which treats of civilisations in general, and which 

 contains Mrs. Nuttall's general summary and conclusions 

 about the meaning of the facts which she has so diligently 

 compiled. To detail these would greatly lengthen an 

 article which is already inordinately long, and the reader 

 will, no doubt, prefer to peruse these for himself; but we 

 may briefly point out that the central idea of the work is 

 that the swastika, which was first employed as a year 

 sign, became later the symbol of the Four Quarters, of 

 quadruplicate division, and of a stable central power, 

 whose rule extended in four directions and controlled the 

 entire Heaven. Human society was divided into four 

 groups, and territorial organisations were formed in four 

 parts. Early civilisations were founded on astronomical 

 principles, on which also rested the worship of the gods. 

 In the case of America, certain elements of culture are 

 assumed to be due to " Mediterranean seafarers " and to 

 transported refugees and would-be colonists ; the basis, 

 however, of both foreign and native civilisations was the 

 recognition of immutable laws governing the universe, 

 " attained, by both races, by long-continued observation 

 of Polaris and the 'northern' constellations." The use 

 of Mrs. Nuttall's volume is much facilitated by the excel- 

 lent index, which fills thirty-four pages of matter printed 

 in double columns in small type, and which merits great 

 praise. We could have wished that a bibliography had 

 been added and more references to the public literature 

 of early symbolism ; to say this is not to detract from 

 the commendation w-hich the book justly deserves, for a 

 classified list of authorities could be compiled from Mrs. 

 Nuttall's notes, and it would be useful to everybody 

 interested in the subject. 



GEOMETRY— NOT IN EUCLID'S ORDER. 

 Primer of Geometry, comprising the Subject- Matter of 

 Euclid L-IV., treated by the Methods of Pure 

 Geometry. By H. \V. Croome Smith, B.A. Pp. xvi -I- 

 100. (London: Macmillan and Co., Ltd., 1901.) 

 Price 2s. 

 T^IlIS little book is another attack on Euclid, and its 

 main object is to exhibit an elementary course of 

 geometry in a system of natural sequence — Euclid's 

 order and method being, of course, ignored. Although 

 in the preface the author adopts a severely logical style 

 and successfully maintains a strong case against our 

 conservative Euclidians, it seems to us that in one re- 

 spect he is in error. His work is divided into three 

 NO. 1682, VOL. 65] 



chapters, headed " Straight Lines and Rectilineal 

 Figures," "The Circle," and "Areas." In the first 

 chapter no mention of a circle occurs, and the author 

 taxes Euclid with an illogical mode of procedure in the 

 following words : — • 



" It is at least questionable logic to make use of the 

 circle in the early stages, and subsequently to use the 

 properties thus demonstrated of lines, angles, &c., in 

 demonstrating the properties of the circle. " 



Justice to Euclid compels us to maintain that this 

 charge is substantially unjust, because the only use made 

 by him of the circle in the early stages (Book I. of 

 Euclid) relies on the facts that the radius is a line of 

 constant length, and (in prop. xii. of Book I.) that if a 

 circle cuts a right line once, it will cut it again. No one 

 can quarrel justly with these assumptions, or can 

 seriously describe them as involving " properties " of the 

 circle which require antecedent demonstration. When 

 criticising Euclid, we must remember that geometry is 

 not wholly a system of pure or formal logic — it implies 

 the sense of sight, sensuous intuition in space. 



One disadvantage of ignoring the circle wholly in the 

 early stages, as is done by Mr. Croome Smith, is that 

 we get into serious difficulties with regard to the con- 

 ception and measurement of angles. He identifies an 

 angle with "change in direction" — "this change in 

 direction., which has nothing to do with the length of the 

 Ime, is what we mean by angular magnitude " (p. 6) ; 

 "the angle is measured by the amount of revolution of 

 a straight line when turned about the vertex in the plane 

 of the lines from the one to the other." True ; but how 

 are we to get a quantitative meaning of the word 

 "revolution" itself? It is, without the aid of the con- 

 ception of a divided circle, or of a system of superposition, 

 just as vague and undefined as Euclid's own term " in- 

 clination." It appears to us that Mr. Croome Smith 

 wrestles vainly with a definition of a right angle on p. 7. 

 He imagines a right line OA to revolve round O into the 

 position 01'., which is o.\ reversed, and he says, 

 "in the position oc, midway between o.\ and 01! in the 

 course of its revolution, the turning line makes with 0.\ 

 or oi: an angle which is half the preceding angle : such 

 an angle therefore is also an angle of lons/iint magnitude, 

 and is called a right angle." 



In this definition there is one little word — " midway " 

 — the precise meaning of which we should wish to know. 

 We fear that it is hopelessly vague without the notion of 

 a circular protractor, or something more than the author 

 is willing to give us. Hence we think that his definition 

 of an angle and his method of measuring angular mag- 

 nitude are not successful. 



Nevertheless, criticism of this kind must not condemn 

 a book which has several merits. A judicious teacher 

 will always be able to supplement imperfect definitions. 

 There is, perhaps, far too much straining after complete- 

 ness of definition and verbal exactness in writers on 

 geometry ; for some of the most simple notions in the 

 subject are things which cannot be defined with absolute 

 accuracy, and the writer as well as the teacher must take 

 it for granted that the pupil has already an adequate 

 notion of the thing described— f.^., a point, a right line, 

 a plane surface. 



Mr. Croome Smith rightly discards Euclid's limitation 



