APPENDIX I. 



REMARKS ON THE PRECEDING TREATISE, 

 BY BOLYAI PARKAS. 



[From Vol. II of Tentamen, pp. 380-383.] 



Finally it may be permitted to add something 

 appertaining to the author of the Appendix in 

 the first volume, who, however, may pardon me 

 if something I have not touched with his acute- 

 ness. 



The thing consists briefly in this: the form- 

 ulas of spherical trigonometry (demonstrated 

 in the said Appendix independently of Euclid's 

 Axiom XI) coincide with the formulas of plane 

 trigonometry, if (in a way provisionally speak- 

 ing) the sides of a spherical triangle are ac- 

 cepted as reals, but of a rectilineal triangle 

 as imaginaries; so that, as to trigonometric 

 formulas, the plane may be considered as an 

 imaginary sphere, if for real, that is accepted 

 in which sin rt. Z. \. 



Doubtless, of the Euclidean axiom has been 

 said in volume first enough and to spare: for 



