1884.] Non-Euclidian Plane Geometry. 85 



the Lobatschewskian 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. 



6. The surface of the sphere is, like the plane, homogeneous, 

 isotropic, and palintropic. We may on the spherical surface construct 

 as above a right-angled triangle ABC, wherein the side c and the 

 angle B are arbitrary ; and (corresponding to the before-mentioned 

 formulae for the plane) we then have 



tan a = tan c cos B, sin b= sin c sin B, 

 whence also 



tan 6= tan c cos A, sina=sincsin A. 

 We deduce from these 



tan 2 a , sin 2 & - 



tan 2 c sin 3 c 



leading to cos 2 c=cos 2 acos 2 &; and then 



sin b -D sin a , A 



=cosctanB, =cosctanA, 



tan a tan b 



gi ving cos a cos 6 = cos'c tan A tan B ; 



that is tan A tan B= , which is>l. 



cos a cos b 



Hence A+B>a right angle, or in the right-angled trangle ACB, 

 the sum A + B + C of the angles is >two right angles. Whence 

 also in any triangle ABC whatever, dividing it into two right- 

 angled triangles by means of a perpendicular let fall from an angle 

 on the opposite side, we have the sum A-fB + C of the angles 

 >two right angles. And we obtain, moreover, 



a = tan" 1 (tan c cos B) + tan" 1 (tan b cos C) , 

 6= tan" 1 (tan a cos C) + tan -1 (tan c cos A), 

 c=tan~ 1 (tan b cos A) -f tan" 1 (tan a cos B), 



which lead to all the formulae of spherical trigonometry. 



7. Suppose the radius of the sphere to be 1/X : then a, 6, c being 

 the lengths of the sides, the lengths in spherical measure are Xa, \b, \c ; 

 and we must in the formulae instead of a, b, c write Xa, X&, Xc respec- 

 tively. In particular for the imaginary sphere x 2 +y' 2 + z*= l,we 

 have \=i, and we must instead of a, b, c write ai, bi, ci respectively. 

 The fundamental formulas for the right-angled triangle thus become 



tanh a=tanh c cos B, sinh fc=sinh c sin B, 



