A history of Greek mathematics - Wilbourhall.org

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
—