A history of Greek mathematics - Wilbourhall.org

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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
1
Tannery, Memoires scientifiqites^ ii, p. 17.
2
Pappus, iv, pp. 264-8.