A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

MENELAUS OF ALEXANDRIA 261
treatise about the hydrostatic balance, i.e. about the determination
of the specific gravity of homogeneous or mixed
bodies, in the course of which he mentions Archimedes and
Menelaus (among others) as authorities on the subject; hence
the treatise (3) must have been a book on hydrostatics discussing
such problems as that of the crown solved by Archimedes.
The alternative proof of Eucl. I. 25 quoted by
Proclus might have come either from the Elements of Geometry
or the Book on triangles. With regard to the geometry, the
'
liber trium fratrum ' (written by three sons of Musa b. Shakir
in the ninth century) says that it contained a solution of
duplication of the cube, which is none other than that of
Archytas. The solution of Archytas having employed the
intersection of a tore and a cylinder (with a cone as well),
there would, on the assumption that Menelaus reproduced the
solution, be a certain appropriateness in the suggestion of
Tannery 1 that the curve which Menelaus called the napd8o£os
ypa/jL/xi] was in reality the curve of double curvature, known
by the name of Viviani, which is the intersection of a sphere
with a cylinder touching it internally and having for its
diameter the radius of the sphere.
case of Eudoxus's hipiDopede, and it
the
This curve is a particular
has the property that the
portion left outside the curve of the surface of the hemisphere
on which it lies is equal to the square on the diameter of the
sphere ; the fact of the said area being squareable would
justify the application of the word napdSogos to the curve,
and the quadrature itself would not probably be beyond the
powers of the Greek mathematicians, as witness Pappus's
determination of the area cut off between a complete turn of
a certain spiral on a sphere and the great circle touching it at
the origin. 2
The Sphaerica of Menelaus.
This treatise in three Books is fortunately preserved in
the Arabic, and although the extant versions differ considerably
in form, the substance is beyond doubt genuine
the original translator was apparently Ishaq b. Hunain
(died A. D. 910). There have been two editions, (1) a Latin
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.
1
Tannery, Memoires scientifiqites^ ii, p. 17.
2
Pappus, iv, pp. 264-8.
math. Wissenschaften,Heft xiv. 1902).
2
Pappus, vi, p. 476. 16.