I 20 



NA TURE 



Wcc. 9, 1880 



through the researches of Riemann and Helmholtz, 

 although Grassmann had ah-eady published, in 1844, his 

 classical but long-neglected ' Ausdehnungslehre.' '' In this 

 connexion we can merely refer to the admirable lecture 

 by Clifford, " The Postulates of the Science of Space." 

 There is a good statement of Euclid's assumptions, but 

 we shall refer only to that which is made in I. 4, thus 

 enunciated by De Morgan : "Any figure maybe removed 

 from place to place without alteration of form, and a 

 plane figure may be turned round on the plane." This is 

 employed by Prof Henrici, as it has been by many others, 

 to prove I. 5, with this difference, that he does it after 

 Mr. Dodgson has made Euclid say there is "too much 

 of the Irish Bull about it, and that it reminds one too 

 vividly of the man who walked down his own throat, to 

 deserve a place in a strictly philosophical treatise." But 

 the difterence between these two writers is a radical one, 

 and is not confined to the above solitary instance. The 

 treatment of Book I. (the remarks on axiom xii. in con- 

 nection with I. 28, 29 are valuable) calls for no special 

 comment. In Book II. we have the propositions dis- 

 cussed symbolically and proved by the aid of laws inves- 

 tigated by Sir W. Rowan Hamilton and Grassmann : 

 laws familiar to more advanced students, but which are 

 here put in a manner within the grasp, we think, of 

 junior students. The book is one, however, to which this 

 class never take very kindly, and requires patience and 

 illustration on the part of the teacher. We can, from the 

 outline here given, guess how Prof. Henrici will treat 

 this part of geometry in his forthcoming second volume. 

 The remarks upon the Fourth Book conclude with a 

 *'few theorems not given by Euchd," but they are readily 

 derived from (if not explicitly stated in) Euclid's con- 

 structions. Of Book V. there is a careful sketch, and our 

 author shows " Why the usual algebraical treatment of 

 proportion is not really sound." (Here we may refer also 

 to Mr. A. J. Ellis's "Euclid's Conception of Ratio and 

 Proportion " in his "Algebra identified with Geometry," 

 and in a simpler form in a lecture at the College of 

 Preceptors.) Books VI., XL, XII. need not delay us. 

 We come now to the Projective Geometrj-, which we 

 should much like to see reproduced in pamphlet form 

 for use in colleges or schools. We notice Prof. Henrici 

 states, " In Euclid's Elements almost all] propositions 

 refer to the magnitude of lines, angles, areas, or 

 volumes, and therefore to measurement." This, too, 

 is our own view, and we presume it is what Mr. Wilson 

 intended when he says : " Every theorem may be shown 

 to be a means of indirectly measuring some magnitude" ; 

 whether it be so or not, at any rate Mr. Dodgson 

 cannot impugn the Professor's more guarded statement. 

 Those properties of figures which do not alter by 

 projection are projective properties : there is a slight 

 omission in the illustrations given, an exception should, 

 we think, have been made in the case when the plane of 

 projection is perpendicular to the plane upon which 

 the quadrilateral, or circle, or other figure is projected. 

 The points of difterence between the two sciences are 

 well put. "In Euclid each proposition stands by itself; 

 its connection with others is never indicated ; the leading 

 ideas contained in its proof are not stated ; general 

 principles do not exist. In the modern methods, on the 

 other hand, the greatest importance is attached to the 



leading thoughts which pervade the whole ; and general 

 principles, which bring whole groups of theorems under 

 one aspect, are given rather than separate propositions. 

 The whole tendency is to generalisation." Euclid, it is 

 open to remark, throughout his work, avoids the infinite, 

 whereas the modern geometry, like a good Samaritan, 

 takes the most tender care of it. The systems adopted 

 by Prof. Henrici are principally the methods of projection 

 and correspondence— as handled by Von Staudt in the 

 "Geometrie der Lage," and by Grassman in his above- 

 cited work. We should like to analyse this sketch in 

 detail, but we must forbear. For curves of two dimensions 

 it is quite too delicious for us to mar it by such scant and 

 imperfect treatment as we could here give it, and we must 

 content ourselves with giving the heads of the several 

 sub-sections. After the statement of definitions and preh- 

 minary explanations, we have segments of a line, projec- 

 tion and cross-ratios (Cliftbrd's name for the anharmonic 

 ratios of Chaslcs), correspondence, curves and cones of 

 second order or second class, pole and polar, diameters 

 and axes of conies, involution, involution determined by 

 a conic on a line — foci, pencil of conies. The con- 

 clusion of the essay on the conies is that we arrive 

 at the definitions from which our English text-books 

 usually start. So the mode of treatment will be 

 seen to be novel to the majority of English students. 

 The concluding sections (six columns) on ruled quadric 

 surfaces, but more especially on twisted cubics, seem to 

 us to bear on their faces tokens of having been somewhat 

 hurriedly written, so are not quite up to the high standard 

 of the previous work. At the close Prof. Henrici refers 

 his readers to Reye's "Geometrie der Lage" for "a 

 more exhaustive treatment of the subject." " Scarcely 

 any use has been made of algebra, and it would have 

 been even possible to avoid this little, as is done by 

 Reye." Prof. Clerk Maxwell, in a note to us, com- 

 mended, in his own quaint way,, this work of Reye. A. 

 short list of references is appended. 



We could have wished that the " Analytical Geometry" 

 had also been intrusted to Prof. Henrici, more especially 

 that we might have seen how he would have connected 

 the two together, and also that we might have had the 

 subject discussed from a Continental point of view. We 

 have sufficiently comprehensive and good treatises already 

 by English writers, some of which are adorned with much 

 of Prof. Cayley's work, and we feel, too, that had our 

 author had carte blanche for space, he would have done 

 his work well ; whereas in attempting to pack much 

 matter into a small space we think he has assumed much 

 which is not familiar to some, and yet at the same time 

 which is elementary to others who are advanced students. 

 Nor does the article, to our mind, thoroughly serve for 

 purposes of reference, though, no doubt, it goes some way 

 to this end. The secret may be that " Pure Geometry " 

 is more limited in its range, has, on one side, to do with 

 a book known to almost all, and, on the other side, even 

 does not reach, for the generality, beyond the conic sec- 

 tions ; " Analytical Geometry," on the other hand, has to 

 do with everything that relates to curves and surfaces, of 

 whatever sort they may be. Prof. Cayley takes the line 

 of analytical geometry "as a method," and confines him- 

 self, in his twenty-four and a half columns, to the con- 

 sideration of the applications of Cartesian co-ordinates 



