MENELAUS OF'ALEXANDRIA 
261 
treatise about the hydrostatic balance, i.e. about the deter 
mination 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 dis 
cussing such problems as that of the crown solved by Archi 
medes. 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 5 (written by three sons of Musa b. Shakir 
in the ninth century) says that it contained a solution of the 
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 irapdSogos 
ypapyri was in reality the curve of double curvature, known 
by the name of Yiviani, which is the intersection of a sphere 
with a cylinder touching it internally and having for its 
diameter the radius of the sphere. This curve js a particular 
case of Eudoxus’s hippopede, and it 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 irapdSogo? 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 treaiüe in three Books is fortunately preserved in 
the Arabic, and although the extant versions differ con 
siderably 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 
1 Tannery, Mémoires scientifiques, ii, p. 17. 2 Pappus, iv, pp. 264-8.