5s SCIENCE ABSOLUTE OF 



whence we deduce the same proportion as 

 above, taking for / 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 IK- 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 -j must therefore be zero, and 

 consequently i= oo. 



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 // (0) = sin B tan //(/&amp;gt;). 



a being tlu- hvpothenusr of a right-angled tri 

 angle, /&amp;gt; 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~ l ):(P P~ ] ). 



Now. putting f () r brevity, u(k)=k\ we 

 have tan J/ : tan la = (cot a tan a) : (cot / 

 -tan/ )^(A_A- ) :(P-P- )=,1 : sin B. 



