Subject: [HM] Chord Table Possibly by Hipparchus of Rhodes
From: Barnabas Hughes (hcedu037@csun.edu)
Date: Sun Jan 02 2000 - 13:56:22 EST
Dear Colleagues,
Relying on Theon of Alexandria the historian Thomas L. Heath [1] noted that
Hipparchus of Rhodes composed 12 books of chords. More recently G.L. Toomer
[2] suggested that 12 sections on chord tables probably describes what
Hipparchus constructed. In fact, following upon a remark by Otto
Neugebauer [3], his mentor and colleague at Brown University, he recreated a
plausible table of 24 values, beginning with 7.5 degrees and proceeding in
that increment to 180 degrees, using (what is now considered certain) a
radius of 3438 minutes for the base circle, the Pythagorean Theorem, the
constructible chords of 60 and 90 degrees, and a half-chord formula. The
last tool, which Toomer considered "... the sole non-trivial element in
the computation of his [Hipparchus'] chord table" (p. 14), prompted the
following investigation.
I began to wonder how such an accomplished geometer, as Hipparchus certainly
was, could have overlooked using the chord of 72 degrees (from the Euclidean
construction of the pentagon) to construct a chord table with more entries
than the one described above. Suppose he had composed such a table, what
would it look like? Beginning on Hipparchus' base circle of radius = 3438'
and with Euclid's method for constructing the chord of a pentagon, I
computed the chord of 72 degrees to be 4041.6'. Thereafter, using the
half-chord formula, the Pythagorean Theorem, complementary angles, and
supplementary angles, I constructed a table of 34 entries from 4.5 degrees
to 85.5 degrees, with 6 vacancies each lying 4.5 degrees from its nearest
neighbor. Interestingly enough, five of the six vacancies belonged to
common multiples of 4.5 and 7.5, the latter derived from 90 degrees, the
sixth vacancy filled with the chord of 180 degrees equal to twice the
radius. On the face of it, there appears no reason why Hipparchus might not
have composed such a table almost as easily as I did (almost, because I used
a TI-83 to do the computing). Hence, I am led to speculate that the 7.5
degree table of chords was an earlier effort of Hipparchus (it would be so
obvious to begin with the chords of 90 degrees and 60 degrees to see where
these values might lead) and that a larger table from the chord of 72
degrees might have been a more mature accomplishment, although it seems not
to have survived except in my wonderment.
Has anything like this appeared in the literature? Comments will be welcomed.
Resources: [1] T.L. Heath (1931), A Manual of Greek Mathematics (Oxford:
Clarendon Press), p. 398. [2] G.J. Toomer, "The Chord Table of Hipparchus
and the Early History of Greek Trigonometry," in Centaurus (1974) 18:6-28.
[3] Otto Neugebauer, "On Some Aspects of Early Greek Astronomy," in
Proceedings of the American Philosophical Society (1972) 116(3):243-251.
Barney Hughes
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