1884.] Non-Euclidian Plane Geometry. 83 



generally, I refer to two memoirs by Beltrami, " Teoria fondamentale 

 degli Spazii di Curvatura Costante," Annali di Matem., t. ii. 

 (1808-69), pp. 232-255, and " Saggio di interpretazione della 

 Geometria non-Euclidea," Giornale di Matem., t. vi (1868), pp. 

 284-312, both, translated, Ann. de 1'Ecole Normale, t. vi (1869) ; in 

 the last of these he speaks of surfaces of constant negative curvature as 

 il pseudospherical," and in a later paper, " Sulla superficie di rotazione 

 che serve di tipo alle superficie pseudosferiche," Gior. di Matem., 

 t. x (1872), pp. 147-151, he treats of the particular surface which 

 I have called the pseudosphere. The surface is mentioned, Note iv 

 of Liouville's edition of Monge's " Application de 1' Analyse a la 

 Geometric " (1850), and the generating curve is there spoken of as 

 " bien connue des geometres." 



FIG. 1. 



4. In ordinary plane geometry, take (fig. 1) a line B.c, and on it a 

 point B ; from B, in any direction, draw the line BA ; take upon it 

 a point A, and from this point, at right angles to B, draw Ay, cut- 

 ting it at C. We have thus a triangle ACB, right-angled at C ; 

 and we may denote the other angles, and the lengths of the sides, 

 by A, B, c, a, b, respectively. In the construction of the figure the 

 length c and the angle B are arbitrary. 



The plane is a surface which is homogeneous, isotropic, and 

 palintropic. That is, whatever be the position of B, the direction of 

 Ito, and the sense in which the angle B is measured, we have the 

 same expressions for a, b as functions of c, B ; these expressions, of 

 course, are 



a=ccosB, &=csinB. 



But considering Ay as the initial line, and AB=c, as a line drawn 

 from A at an inclination thereto =A, we have in like manner 



6=e cos A, a=c sin A, 



and consequently cos A= sinB, sinA= cosB ; whence sin (A + B)=], 



'.. ''" o 2 



