58 SCIENCE ABSOLUTE OF SPACE. 



whence we deduce the same proportion as 

 above, taking for i the distance for which the 

 ratio I is equal to e. 



If axiom XI is not true, there exists a de- 

 terminate i, which must be substituted in the 

 formulas. 



If, on the contrary, this axiom is true, we 

 must make in the formulas i= oo. Because, in 



this case, the quantity -=Y is always =1, the 



sphere-limit being a plane, and the axes being 

 parallel in Euclid's sense. 



The exponent \ must therefore be zero, and 

 consequently i= GO. 



It is easy to see that Bolyai's formulas of 

 plane trigonometry are in accord with those of 

 Lobachevski. 



Take for example the formula of 37, 



tan // (#)=sin B tan //(/), 



a being the hypothenuse of a right-angled tri- 

 angle, p one side of the right angle, and B the 

 angle opposite to this side. 



Bolyai's formula of 31, I, gives 



1 : sin B=(A-A- 1 ):(P-P- ] ). 



Now, putting for brevity, i/7 (Jc)=k' , we 

 have tan 2p ' : tan 2a ' (cot a ' tan a ' } : (cot p ' 

 / = A-A- 1 P-P- 1 ^! : sin B. 



