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

BEGINNINGS OF TRIGONOMETRY 253
forms ' '
liber aggregativus ' and liber cle principiis universalibus'.
Each of these expressions may well mean the work
of Apollonius which Marinus refers to as the General
'
Treatise ' (fj kccOoXov wpayfjiaTeia). There is no apparent
reason to doubt that the remark in question was really
contained in Menelaus's original work ; and, even if it is an
Arabian interpolation, it is not likely to have been made
without some definite authority. If then Apollonius was the
discoverer of the proposition, the fact affords some ground for
thinking that the beginnings of trigonometry go as far back,
at least, as Apollonius. Tannery 1 indeed suggested that not
only Apollonius but Archimedes before him may have compiled
a ' table of chords ', or at least shown the way to such
a compilation, Archimedes in the work of which we possess
only a fragment in the Measurement of a Circle^ and Apollonius
in the cdkvtoklov, where he gave an approximation to the value
of tt closer than that obtained by Archimedes; Tannery
compares the Indian Table of Sines in the Surya-Siddhdnta,
where the angles go by 24ths of a right angle (l/24th = 3° 45',
2/24ths=7° 30', &c), as possibly showing Greek influence.
This is, however, in the region of conjecture ; the first person
to make systematic use of trigonometry is, so far as we know,
Hipparchus.
Hipparchus, the greatest astronomer of antiquity, was
born at Nicaea in Bithynia. The period of his activity is
indicated by references in Ptolemy to observations made by
him the limits of which are from 161 B.C. to 126 B.C. Ptolemy
further says that from Hipparchus's time to the beginning of
the reign of Antoninus Pius (a.d. 138) was 265 years. 2 The
best and most important observations made by Hipparchus
were made at Rhodes, though an observation of the vernal
equinox at Alexandria on March 24, 146 B.C., recorded by him
may have been his own. His main contributions to theoretical
and practical astronomy can here only be indicated in the
briefest manner.
1
Tannery, Recherches sur Vhist. de Vastronomie ancienne, p. 64.
2 Ptolemy, Syntaxis, vii. 2 (vol. ii, p. 15).
254 TRIGONOMETRY
The work of Hipparchus.
Discovery of precession.
1. The greatest is perhaps his discovery of the precession
of the equinoxes. Hipparchus found that the bright star
Spica was, at the time of his observation of it, 6° distant
from the autumnal equinoctial point, whereas he deduced from
observations recorded by Timocharis that .Timocharis had
made the distance 8°. Consequently the motion had amounted
to 2° in the period between Timocharis's observations, made in
283 or 295 B.C., and 129/8 B.C., a period, that is, of 154 or
166 years; this gives about 46-8" or 43-4" a year, as compared
with the true value of 50-3757".
Calculation of mean lunar month.
2. The same discovery is presupposed in his work On the
length of the Year, in which, by comparing an observation
of the summer solstice by Aristarchus in 281/0 B.C. with his
own in 136/5 B.C., he found that after 145 years (the interval
between the two dates) the summer solstice occurred half
a day-and-night earlier than it should on the assumption of
exactly 365J days to the year; hence he concluded that the
tropical year contained about ^§o^n °f a day-and-night less
than 3 65\ days. This agrees very nearly with Censorinus's
statement that Hipparchus's cycle was 304 years, four times
the 76 years of Callippus, but with 111,035 days in it
instead of 111,036 (
= 27,759x4). Counting in the 304 years
12x304 + 112 (intercalary) months, or 3,760 months in all,
Hipparchus made the mean lunar month 29 days 12 hrs.
44 min. 2-| sec, which is less than a second out in comparison
with the present accepted figure of 29-53059 days!
3. Hipparchus attempted a new determination of the sun's
motion by means of exact equinoctial and solstitial observations;
he reckoned the eccentricity of the sun's course
and fixed the apogee at the point 5° 30' of Gemini. More
remarkable still was his investigation of the moon's
course. He determined the eccentricity and the inclination
of the orbit to the ecliptic, and by means of records of
observations of eclipses determined the moon's period with
extraordinary accuracy (as remarked above). We now learn