Cambridge, January 21, 1865. 
472 note on lobatschewsky’s imaginary geometry. [362 
which equations (if only we write therein \tt — a', ^7r—b', \ir — c' in place of a, b', c' 
respectively) are in fact the equations given under a less symmetrical form in the 
curious paper “ Géométrie Imaginaire ” by N. Lobatschewsky, Rector of the University 
of Kasan, Crelle, vol. xvn. (1837), pp. 295—320. The view taken of them by the 
author is hard to be understood. He mentions that in a paper published five 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 accordingly 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 + G is < 7r, and the angles a, b', c' being 
calculated from the sides by the formulae 
, 1 //I '1 
cos a = ., cos b = ——, cos c = . 
cos ai cos bi cos ci 
(I have, as mentioned above, replaced Lobatschewsky’s a', b', c' by their complements) : 
the relation between the angles A, B, G and the subsidiary quantities a', b', c' which 
replace the sides, is given by the formulæ 
1 
cos A + cos B cos C 
cos a' 
sin B sin G 
1 
cos B 4- cos C cos A 
cos b' 
sin G sin A 
1 
cos G + cos A cos B 
cos c 
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, G each < £77-, 
may be omitted) contains only the real quantities A, B, G, a', b', c'; and is a system 
correlative to the equations of ordinary Spherical Trigonometry. 
It is hardly necessary to remark that the equation 
1 
> = cos ai 
cos a 
is Jacobi’s imaginary transformation in the Theory of Elliptic Functions. See, as to 
this, my paper “On the Transcendent gd. u = \ log tan + \ui),” Phil. Mag. vol. xxiv. 
(1862), pp. 19—22, [320].