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

HYPSICLES 215
natural to divide each of the sixth parts into either 10 or 60
parts; the former division would account for the attested
division of the clay into 60 hours, while the latter division on
the sexagesimal system would give the 360 time-degrees (each
of 4 minutes) making up the day of 24 hours. The purely
arithmetical explanation is defective in that the series of
numbers for which the Babylonians had special names is not
60, 360, 3600 but 60 (Soss), 600 (Ner), and 3600 or 60 2 (Sar).
On the whole, after all that has been said, I know of no
better suggestion than that of Tannery. 1 It is certain that
both the division of the ecliptic into 360 degrees and that of
the wxOrjpepw into 360 time-degrees were adopted by the
Greeks from Babylon. Now the earliest division of the
ecliptic was into 12 parts, the signs, and the question is, how
were the signs subdivided ? Tannery observes that, according
to the cuneiform inscriptions, as well as the testimony of
Greek authors, the sign was divided into parts one of which
(dargatu) was double of the other (murra n), the former being
l/30th, the other (called stadium by Manilius) l/60th, of the
sign ;
the former division would give 360 parts, the latter 720
parts for the whole circle. The latter division was more
natural, in view of the long-established system of sexagesimal
fractions; it also had the advantage of corresponding tolerably
closely to the apparent diameter of the sun in comparison
with the circumference of the sun's apparent circle. But, on
the other hand, the double fraction, the l/30th, was contained
in the circle of the zodiac approximately the same number of
times as there are days in the year, and consequently corresponded
nearly to the distance described by the sun along the
zodiac in one day. It would seem that this advantage was
sufficient to turn the scale in favour of dividing each sign of
the zodiac into 30 parts, giving 360 parts for the whole
circle. While the Chaldaeans thus divided the ecliptic into
360 parts, it does not appear that they applied the same division
to the equator or any other circle. They measured angles
representing 2°, so that the complete
in general by ells, an ell
circle contained 180, not 360, parts, which they called ells.
The explanation may perhaps be that the Chaldaeans divided
1
Tannery, 'La coudee astronomique et les anciennes divisions du
cercle ' (Memo ires scientifiques, ii, pp. 256-68).
216 SUCCESSORS OF THE GREAT GEOMETERS
the diameter of. the circle into 60 ells in accordance with their
usual sexagesimal division, and then came to divide the circumference
into 180 such ells on the ground that the circumference
is roughly three times the diameter. The measurement
in ells and dactyli (of which there were 24 to the ell)
survives in Hipparchus (On the Phaenomena of Eudoxus and
Aratus), and some measurements in terms of the same units
are given by Ptolemy. It was Hipparchus who first divided
the circle in general into 360 parts or degrees, and the
introduction of this division coincides with his invention of
trigonometry.
The contents of Hypsicles's tract need not detain us long.
The problem is : If we know the ratio which the length of the
longest day bears to the length of the shortest day at any
given place, to find how many time-degrees it takes any given
sign to rise ; and, after this has been found, the author finds
what length of time it takes any given degree in any sign to
rise, i.e. the interval between the rising of one degree-point on
the ecliptic and that of the next following. It is explained
that the longest day is
the time during which one half of the
zodiac (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) rises,
and the shortest day the time during which the other half
(Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini) rises.
Now at Alexandria the longest day is to the shortest as 7
to 5; the longest therefore contains 210 'time-degrees', the
shortest 150. The two quadrants Cancer-Virgo and Libra-
Sagittarius take the same time to rise, namely 105 timedegrees,
and the two quadrants Capricornus-Pisces and Aries-
Gemini each take the same time, namely 75 time-degrees.
It is further assumed that the times taken by Virgo, Leo,
Cancer, Gemini, Taurus, Aries are in descending arithmetical
progression, while the times taken by Libra, Scorpio, Sagittarius,
Capricornus, Aquarius, Pisces continue the same descending
arithmetical series. The following lemmas are used
and proved :
I. If a 1?
a 2
...a n
,
a
n + J
> a
w+2
... a 2w is a descending arithmetical
progression of 2 n terms with 8 ( = a x
—