A history of Greek mathematics - Wilbourhall.org

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ZENODORUS. HYPSICLES 213
their surfaces equal to that of the sphere, Zenodorus confined
himself to proving (1) that the sphere is greater if the other
solid with surface equal to that of the sphere is a solid formed
by the revolution of a regular polygon about a diameter
bisecting it as in Archimedes, On the Sphere and Cylinder,
Book I, and (2) that the sphere is greater than any of
the regular solids having its surface equal to that of the
sphere.
Pappus's treatment of the subject is
more complete in that
he proves that the sphere is greater than the cone or cylinder
the surface of which is equal to that of the sphere, and further
that of the five regular solids which have the same surface
that which has more faces is the greater. 1
Hypsicles (second half of second century B.C.)
has already
been mentioned (vol. i, pp. 419-20) as the author of the continuation
of the Elements known as Book XIV. He is quoted
by Diophantus as having given a definition of a polygonal
number as follows
'
If there are as many numbers as we please beginning from
1 and increasing by the same common difference, then, when
the common difference is 1, the sum of all the numbers is
a triangular number; when 2, a square; when 3, a pentagonal
number [and so on]. And the number of angles is called
after the number which exceeds the common difference by 2,
and the side after the number of terms including 1.'
This definition amounts to saying that the nth. a-gonal number
(1 counting as the first) is \n {
2 + {n— 1) (a — 2) }. If, as is
probable, Hypsicles wrote a treatise on polygonal numbers, it
has not survived. On the other hand, the 'AvacpopiKos (Ascensiones)
known by his name has survived in Greek as well as in
Arabic, and has been edited with translation. 2 True, the
treatise (if it really be by Hypsicles, and not a clumsy effort
by a beginner working from an original by Hypsicles)
does no credit to its author; but it is in some respects
interesting, and in particular because it is the first Greek
1
Pappus, v, Props. 19, 38-56.
2
Manitius, Des Hypsikles Schrift Anaphorikos, Dresden, Lehmannsche
Buchdruckerei, 1888.