January 13, 1899.] 



SCIENCE. 



63 



I use the simple letters, a, A, . . to denote 

 (linear or angular) distances measured in the 

 ordinary manner ; and the same letters with a 

 superscript stroke (i, ^, . . to denote the same 

 distances measured according to the theory. 

 The radius of the Absolute is for convenience 

 taken to be = 1 ; the distance of any point from 

 the center can, therefore, be represented as the 

 sine of an angle. 



The distance BC, or say a, of any two points 

 B,C is by definition as follows : 



BICJ 

 BJ^I 



:*] 



(where I, J" are the intersections of the line BC 

 with the circle). 



As for the trigonometry " the formulas are, in 

 fact, similar to those of spherical trigonometry 

 with only cosh a, sinh a, etc., instead of cos a, 

 sin a, etc." 



Cayley returned again to this matter in his 

 celebrated Presidential Address to the British 

 Association (1883), saying there: "It is well 

 known that Euclid's twelfth axiom, even in 

 Play fair's form of it, has been considered as 

 needing demonstration ; and that Lobatsch^v- 

 sky constructed a perfectly consistent theory, 

 whei'ein this axiom was assumed not to hold 

 good, or say a system of non-Euclidean plane 

 geometry. There is a like system of non- 

 Euclidean solid geometry." 



"But suppose the physical space of our ex- 

 perience to be thus only approximately Euclid- 

 ean space, what is the consequence which fol- 

 lows?" 



The very next year this ever-interesting sub- 

 ject recurs in the paper (May 27, 1884) ' On the 

 Non-Euclidean Plane Geometry.' "Thus the 

 geometry of the pseudo-sphere, using the ex- 

 pression straight line to denote a geodesic of 

 the surface, is the Lobatschevskian geometry; 

 or, rather, I would say this in regard to the 

 metrical geometry, or trigonometry, of the sur- 

 face; for in regard to the descriptive geometry 

 the statement requires some qualification * * * 

 this is not identical with the Lobatschevskian 

 geometry, but corresponds to it in a manner such 

 as that in which the geometry of the surface of 

 the circular cylinder corresponds to that of the 

 plane. I would remark that this realization of. 



the Lobatschevskian geometry sustains the 

 opinion that Euclid's twelfth axiom is un- 

 demoustrable." 



But here this necessarily brief notice must 

 abruptly stop. Cayley, in addition to his won- 

 drous originality, was assuredly the most learned 

 and erudite of mathematicians. Of him in his 

 science it might be said he knew everything, 

 and he was the very last man who ever will 

 know everything. His was a very gentle, 

 sweet character. Sylvester told me he never 

 saw him angry but once, and that was (both 

 were practicing law !) when a messenger broke 

 in on one of their interviews with a mass of 

 legal documents — new business for Cayley. In 

 an access of disgust, Cayley dashed the docu- 

 ments upon the floor. 



George Bruce Halsted. 



Austin, Texas. 



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