The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

[Cl. VII.

CHAPTER VII.
THE ELLIPTIC PLANE TRIGONOMETRY.
~ 80. The following treatment of the Elliptic Trigonometry
is due to Gerard and Mansion. Gerard discussed the Hyperbolic Trigonometry on these lines.* Mansion showed that
the method discovered by Gerard was applicable also to the
Elliptic case.The notation to be employed has first to be explained.
Let OA and OA' be two lines meeting at O at right angles.
Let OL be a third line making an acute angle with OA and OA'.
Let P be any point upon the line OL, such that OP <.
A
M L
r
0 x M A
FIG. 95.
Let PM and PM' be the perpendiculars to OA and OA'.
We denote OM, MP, and OP by x, y, and r; and OM' and
M'P by y' and x'.
* Gerard, Sur la (Geometrie non euclidienne (Paris, 1892). Cf. also
Young, " On the Analytical Basis of Non-Euclidian Geometry," Amer.
Journ. of Math., vol. xxxiii. p. 249 (1911); and Coolidge, Non-Euclidean
Geometry, ch. iv. (Oxford, 1909).
t Mansion, Principes Fondamentaux de la Geomdtrie non euclidienne
de Riemann (Paris, 1895).