232 Prof. Cayley on Lobatschewsky's Imaginary Geometry. 



1 . cos C + cos A cos B 



= COS CI = 



cos d sin A sin B 



which equations (if only we write therein — — a\ =■ —V, ^ — c' 



A A A 



in place of a\ b\ d respectively) are in fact the equations given 

 under a less symmetrical form in the curious paper " Geometrie 

 Imaginaire " by N. Lobatschewsky, Rector of the University of 

 Kasan, Crelle, vol. xvii. (1837) pp. 295-320. The view taken 

 of them by the author is hard to be understood. He mentions 

 that in a paper published live years previously in a scientific 

 journal at Kasan, after developing a new theory of parallels, he 

 had endeavoured to prove that it is only experience which obliges 

 us to assume that in a rectilinear triangle the sum of the angles 

 is equal to two right angles, and that a geometry may exist, if 

 not in nature at least in analysis, on the hypothesis that the 

 sum of the angles is less than two right angles ; and he accord- 

 ingly attempts to establish such a geometry, viz. a, b, c being 

 the sides of a rectilinear triangle, wherein the sum of the angles 

 A + B + C is <7r, and the angles a r , b\ d being calculated from 

 the sides by the formulas 



cos a' = ., cos b l = 7-.' cos c = r 



cos ai cos bi cos ci - 



(I have, as mentioned above, replaced Lobatschewsky's a! } b ! , d 

 by their complements): the relation between the angles A, B, C 

 and the subsidiary quantities a!, b 1 , d which replace the sides, is 

 given by the formulas 



1 _ cos A+ cos B cos C, 

 cos a' "" sin B sin C 

 1 cos B + cos C cos A 



cos b' sin C sin A 



1 _ cos C + cos A cos B 

 cos d ~ sin A sin B 



I do not understand this; but it would be very interesting 

 to find a real geometrical interpretation of the last-mentioned 

 system of equations, which (if only A, B, C are positive real 

 quantities such that A + B + C<7r; for the condition, A, B, C 

 each -<j7r, may be omitted) contains only the real quantities 

 A, B, C, a', b', d ; and is a system correlative to the equations 

 of ordinary Spherical Trigonometry. 



It is hardly necessary to remark that the equation 



= cos ai 



