827] 
ON THE NON-EUCLIDIAN PLANE GEOMETRY. 
221 
the space, that is, which we become acquainted with by experience, but which is the 
representation lying at the foundation of all physical experience. 
3. I propose in the present paper to develope further the geometry of the 
pseudosphere. In regard to the name, and the subject generally, I refer to two 
memoirs by Beltrami, “ Teoria fondamentale degli spazii di curvatura costante,” Annali 
di Matem., t. n. (1868—69), pp. 232—255, and “ Saggio di interpretazione della geometria 
non-Euclidea,” Battaglini, Giorn. di Matem., t. VI. (1868), pp. 284—312, both translated, 
Ann. de VÉcole Normale, t. vi. (1869) ; in the last of these, he speaks of surfaces of 
constant negative curvature as “ pseudospherical,” and in a later paper, “ Sulla superficie 
di rotazione che serve di tipo aile superficie pseudosferiche,” Battaglini, Giorn. di Matem., 
t. x. (1872), pp. 147—160, 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 VAnalyse à la Géométrie (1850), and the generating curve is there 
spoken of as “ bien connue des géomètres.” 
4. In ordinary plane geometry, take (fig. 1) a line Bx, and on it a point B 
from B, in any direction, draw the line BA ; take upon it a point A, and from 
Fig. l. 
A 
this point, at right angles to Bx, draw Ay, cutting 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 Bx, 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 = c cos B, b = c sin B. 
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 
b = c cos A, a = c sin .4, 
and consequently cos A = sin B, sin A = cos B; whence sin (A 4- B) = 1, cos (A + B) = 0, and 
thence A + B = a right angle, or A + B + G = two right angles.