A history of Greek mathematics - Wilbourhall.org

262 TRIGONOMETRY
translation by Maurolycus (Messina, 1558) and (2) Halley's
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 Abu-Nasr-Mansur made in
A.D. 1007-8, Bjornbo has succeeded in presenting an adequate
reproduction of the contents of the Sphaerica. 1
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 ',
restriction
(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 in his Sphaerica
calls the figure in question (the spherical triangle) a threeside
' (rpnrXefpo^) 2 ;
' the word triangle (Tpiyatvov) was of course
i
1
Bjornbo, Studien uber Menelaos' Spharik (Abhandlungen zur Gesch. d.
math. Wissenschaften,Heft xiv. 1902).
2
Pappus, vi, p. 476. 16.