Januaey 13, 1899.] 



SCIENCE. 



61 



actually presented itself in the question of the 

 determination of the number of curves which 

 satisfy given conditions ; the conditions imply 

 relations between the coefficients in the equa- 

 tion of the curve ; and for the better under- 

 standing of these relations it was expedient to ■ 

 consider the coeflficients as the coordinates of a 

 point in a space of the proper dimensionality." 



For a dozen years after it was written the 

 Sixth Memoir on Quantics would not have been 

 enumerated in a Bibliography of non-Euclidean 

 geometry, for its author did not see that it gave 

 a generalization which was identifiable with 

 that initiated by Bolyai and Lobach^vski, though 

 afterwards, in his address to the British Asso- 

 ciation, in 1883, he attributes the fundamental 

 idea involved to Riemann, whose paper was 

 written in 1854. 



Says Cayley : "In regarding the physical 

 space of our experience as possibly non-Euclid- 

 ean, Riemann's idea seems to be that of modify- 

 ing the notion of distance, not that of treating 

 it as a locus in four-dimensional space." 



What the Sixth Memoir was meant to do was 

 to base a generalized theory of metrical geome- 

 try on a generalized definition of distance. 



As Cayley himself says: " * -» * the 

 theory in effect is that the metrical properties 

 of a figure are not the properties of the figure 

 considered per se apart from everything else, 

 but its properties when considered in connection 

 with another figure, viz., the conic termed the 

 absolute." 



The fundamental idea that a metrical property 

 could be looked at as a projective property of 

 an extended system had occurred in the French 

 school of geometers. Thus Laguerre (1853) so 

 expresses an angle. Cayley generalized this 

 French idea, expressing all metrical properties 

 as projective relations to a fundamental config- 

 uration. 



We may illustrate by tracing how Cayley 

 arrives at his projective definition of distance. 

 Two projective primal figures of the same kind 

 of elements and both on the same bearer are 

 called conjective. When in two conjective 

 primal figures one particular element has the 

 same mate to whichever figure it be regarded 

 as belonging, then every element has this 

 property. 



Two conjective figures, such that the elements 

 are mutually paired (coupled), form an involu- 

 tion. If two figures forming an involution have 

 self-correlated elements these are called the 

 double elements of the involution. 



An involution has at most two double ele- 

 ments, for were three self-correlated all would 

 be self-correlated. If an involution has two 

 double elements these separate harmonically 

 any two coupled elements. An involution is 

 completely determined by two couples. 



From all this it follows that two point-pairs A, 

 B and A^, i?, define an involution whose double 

 points D, Z>j are determined as that point-pair 

 which is harmonically related to the two given 

 point-pairs. 



Let the pair A, B ha fixed and called the 

 Absolute. Two new points A^, B^ are said (by 

 definition) to be equidistant from a double point 

 D defined as above. D is said to be the ' center ' 

 of the pair A^, i?,. Inversely, if A^ and D be 

 given, i?j is uniquely determined. 



Thus, starting from two points P and Pj, we 

 determine P.^ such that Pj is the center of P and 

 Pj, then determine P3 so that P^ is the center of 

 Pi and P3, etc. ; also in the opposite direction 

 we determine an analogous series of points 

 P — 1, P — 2, .... We have, therefore, a series 

 of points 



....,P— , P-i, P, P„P„P3 



at ' equal intervals of distance.' Taking the 

 points P, Pi to be indefinitely near to each other, 

 the entire line will be divided into a series of 

 equal infinitesimal elements. 



In determining an analytic expression for the 

 distance of two points Cayley introduced the 

 inverse cosine of a certain function of the coor- 

 dinates, but in the Note which he added in the 

 Collected Papers he recognizes the improve- 

 ment gained by adopting Klein's assumed defi- 

 nition for the distance of any two points P, Q : 



, AP.BQ 

 dist. (P® = clog^^^^^, 



where A, B are the two fixed points giving the 

 Absolute. 



This definition preserves the fundamental re- 

 lation 



dist. (P® + dist. {QR) = dist. (PP). 



In this note (Col. Math. Papers, Vol. 2, p. 



