APPENDIX II. 



SOME POINTS IN JOHN BOLYAl s APPENDIX 



COMPARED WITH LOBACHEVSKI, 



BY WOLFGANG BOLYAI. 



[From Kurzer Grundrtss, p. 82.] 



Lobachevski and the author of the Appendix 

 each consider two points A, B, of the sphere- 

 limit, and the corresponding axes 

 ray AM, ray BN (23). 



They demonstrate that, if , /?, 

 r designate the arcs of the circle 

 limit AB, CD, HL, separated by 

 segments of the axis AC = 1, AH 

 =x, we have 



L- f ] * 

 r~ UJ 



Lobachevski represents the value of - by 



0~ x , e having some value &amp;gt;1, dependent on the 

 unit for length that we have chosen, and able 

 to be supposed equal to the Naperian base. 



The author of the Appendix is led directly 

 to introduce the base of natural logarithms. 



