262 
TRIGONOMETRY 
translation by Maurolycus (Messina, 1558) and (2) Halleys 
edition (Oxford, 1758). The former is unserviceable because 
Maurolycus’s manuscript was very imperfect, and, besides 
trying to correct and restore the propositions, he added 
several of his own. Halley seems to have made a free 
translation of the Hebrew version of the work by Jacob b. 
Machir (about 1273), although he consulted Arabic manuscripts 
to some extent, following them, e.g., in dividing the work into 
three Books instead of two. But an earlier version direct 
from the Arabic is available in manuscripts of the thirteenth 
to fifteenth centuries at Paris and elsewhere; this version is 
without doubt that made by the famous translator Gherard 
of Cremona (1114-87). With the help of Halley’s edition, 
Gherard’s translation, and a Leyden manuscript (930) of 
the redaction of the work by Abü-Nasr-Mansür made in 
A.d. 1007-8, Björnbo has succeeded in presenting an adequate 
reproduction of the contents of the Sphaerica} 
Book I. 
In this Book for the first time we have the conception and 
definition of a spherical triangle. Menelaus does not trouble 
to give the usual definitions of points and circles related to 
the sphere, e.g. pole, great circle, small circle, but begins with 
that of a spherical triangle as ‘ the area included by arcs of 
great circles on the surface of a sphere ’, subject to the restric 
tion (Def. 2) that each of the sides or legs of the triangle is an 
arc less than a semicircle. The angles of the triangle are the 
angles contained by the arcs of great circles on the sphere 
(Def. 3), and one such angle is equal to or greater than another 
according as the planes containing the arcs forming the first 
angle are inclined at the same angle as, or a greater angle 
than, the planes of the arcs forming the other (Defs. 4, 5). 
The angle is a right angle if the planes of the arcs are at right 
angles (Def. 6). Pappus tells us that Menelaus in4iis Sphaerica 
calls the figure in question (the spherical triangle) a ‘ three- 
side ’ (TpLTrXevpoi') 1 2 ; the word triangle (rpiyoovov) was of course 
1 Björnbo, Studien über Menelaos' Sphärik (Abhandlungen zur Gesch. d. 
math. Wissenschaften, Heft xiv. 1902). 
2 Pappus, vi, p. 476.16.