62 



SCIENCE. 



[N. S. Vol. IX. No. 211. 



604) Cayley discusses the question whether the 

 new definitions of distance depend upon that of 

 distance in the ordinary sense, since it is obvi- 

 ously unsatisfactory to use one conception of 

 distance in defining a more general conception 

 of distance. 



His earlier view was to regard coordinates 

 ' not as distances or ratios of distances, but as 

 an assumed fundamental notion, not requiring 

 or admitting of explanation.' Later he re- 

 garded them as ' mere numerical values, at- 

 tached arbitrarily to the point, in such wise 

 that for any given point the ratio x : y has a de" 

 terminate numerical value,' and inversely. 



But in 1871 Klein had explicitly recognized 

 this difficulty and indicated its solution. He 

 says : " The cross ratios (the sole fixed ele- 

 ments of projective geometry) naturally must 

 not here be defined, as ordinarily happens, as 

 ratios of sects, since this would assume the 

 knowledge of a measurement. In von Staudt's 

 Beitriigen zur Geometric der Lage (§ 27. n. 

 393), however, the necessary materials are given 

 for defining a cross ratio as a pure number. 

 Then from cross ratios we may pass to homo- 

 geneous point- and plane-coordinates, which, in- 

 deed, are nothing else than the relative values 

 of certain cross ratios, as von Staudt has like- 

 wise shown (Beitraege, § 29. n. 411)." 



This solution was not satisfactory to Cayley, 

 who did not think the difficulty removed by the 

 observations of von Staudt that the cross ratio 

 {A, B, P, Q) has "independently of any notion 

 of distance the fundamental properties of a 

 numerical magnitude, viz. : any two such ratios 

 have a sum and also a product, such sum and 

 product being each of them a like ratio of four 

 points determinable by purely descriptive con- 

 structions." 



Consider, for example, the product of the 

 ratios (A, S, P, Q) and {A/ B/ P/ Q'). We 

 can construct a point B such that (A/ B/ P/ 

 Q') = {A,B, Q, B). The product of (^, B, P, Q) 

 and {A, B, Q, B) is said to be {A, B, P, B). 

 This last definition of a product of two cross 

 ratios, Cayley remarks, is in effect equivalent 

 to the assumption of the relation dist. (PQ) 

 + dist. {QB) = dist. (PB). 



The original importance of this memoir to 

 Cayley lay entirely in its exhibiting metric as a 



branch of descriptive geometry. That this gen- 

 eralization of distance gave pangeometry was 

 first pointed out by Klein in 1871. 



Klein showed that if Cayley's Absolute be 

 real we get Bolyai's system ; if it be imaginary 

 we get either spheric or a new system called by 

 Klein single elliptic ; if the Absolute be an im- 

 aginary point pair we get parabolic geometry ; 

 and if, in particular, the point pair be the cir- 

 cular points we get ordinary Euclid. 



It is maintained by B. A. W. Russell, in his 

 powerful essay on the Foundations of Geometry 

 (Cambridge, 1897), "that the reduction of met- 

 rical to projective properties, even when, as in 

 hyperbolic geometry, the Absolute is real, is 

 only apparent, and has merely a technical 

 validity. ' ' 



Cayley first gave evidence of acquaintance 

 with non-Euclidean geometry in 1865 in the 

 paper in the Philosophical Magazine, above-men- 

 tioned. 



Though this is six years after the Sixth Me- 

 moir, and though another six was to elapse 

 before the two were connected, yet this is, I 

 think, the very first appreciation of Lobachev- 

 sky in any mathematical journal. 



Baltzer has received deserved honor for in 

 1866 calling the attention of Holiel to Lobachev- 

 sky's ' GeometrischeUntersuchungen,' and from 

 the spring thus opened actually flowed the flood 

 of ever-broadening °non-Euclidean research. 



But whether or not Cayley's path to these 

 gold-fields was ever followed by any one else, 

 still he had therein marked out a claim for 

 himself a whole year before the others. 



In 1872, after the connection with the Sixth 

 Memoir had been set up, Cayley takes up the 

 matter in his paper, in the Mathemaiische An- 

 nalen,' On the Non-Euclidean Geometry,' which 

 begins as follows: "The theory of the non- 

 Euclidean geometry, as developed in Dr. Klein's 

 paper ' Ueber die Nicht-Euclidische Geometrie,' 

 may be illustrated by showing how in such a 

 system we actually measure a distance and an 

 angle, and by establishing the trigonometry of 

 such a system." 



I confine myself to the ' hyperbolic ' case of 

 plane geometry : viz., the Absolute is here a real 

 conic, which for simplicity I take to be a circle ; 

 and I attend to the points within the circle. 



