SCIENCE ABSOLUTE OP SPACE. 57 

 or putting -^-=k, \el, 



/ being- infinitesimal at the same time as k. 

 Therefore, for the limit, 1 = 1 and consequently 

 I=& 



The circle traced on the sphere-limit with 

 the arc r of the curve-limit for radius, has for 

 length 2-r. Therefore, 



OjK=2rr=2-* tan z=*i (Y Y" 1 ). 



In the rectilineal A where a, p designate the 

 angles opposite the sides a, b, we have ( 25) 



sin a:sin ,i=Qa:Qt>=-i(AA- 1 ): -i(B B" 1 ) 

 =sin (^^l) :sin (fc I). 



Thus in plane trigonometry as in spherical 

 trigonometry, the sines of the angles are to 

 each other as the sines of the opposite sides, 

 only that on the sphere the sides are reals, 

 and in the plane we must consider them as 

 imaginaries, just as if the plane were an 

 imaginary sphere. 



We may arrive at this proposition without a 

 preceding determination of the value of I. 



0* 



If we designate the constant by q, we 



tan z 



shall have, as before 



=*q (Y-Y- 1 ), 



