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

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.
214 SUCCESSORS OF THE GREAT GEOMETERS
work in which we find the division of the ecliptic circle into
360 parts ' ' or degrees. The author says, after the preliminarypropositions,
'The circle of the zodiac having been divided into 360 equal
circumferences (arcs), let each of the latter be called a degree
in space (fiolpa tottiktj, 'local' or 'spatial part'). And similarly,
supposing that the time in which the zodiac circle
returns to any position it has left is divided into 360 equal
times, let each of these be called a degree in time (/ioipa
XpOVLKTj)'
From the word KaXeiaOco (' let it be called ') we may perhaps
infer that the terms were new in Greece. This brings us to
the question of the origin of the division (1) of the circle of
the zodiac, (2) of the circle in general, into 360 parts. On this
question innumerable suggestions have beerf made. With
reference to (1) it was suggested as long ago as 1788 (by Formaleoni)
that the division was meant to correspond to the
number of days in the year. Another suggestion is that it
would early be discovered that, in the case of any circle the
inscribed hexagon dividing the circumference into six parts
has each of its sides equal to the radius, and that this would
naturally lead to the circle being regularly divided into six
parts ; after this, the very ancient sexagesimal system would
naturally come into operation and each of the parts would be
divided into 60 subdivisions, giving 360 of these for the whole
circle. Again, there is an explanation which is not even
geometrical, namely that in the Babylonian numeral system,
which combined the use of 6 and 10 as bases, the numbers 6,
60, 360, 3600 were fundamental round numbers, and these
numbers were transferred from arithmetic to the heavens.
The obvious objection to the first of these explanations
(referring the 360 to the number of days in
the solar year) is
that the Babylonians were well acquainted, as far back as the
monuments go, with 365-2 as the number of days in the year.
A variant of the hexagon- theory is the suggestion that a
natural angle to be discovered, and to serve as a measure of
others, is the angle of an equilateral triangle, found by drawing
a star # like a six-spoked wheel without any circle. If
the base of a sundial was so divided into six angles, it would be