Re: [HM] Radian Measure

Subject: Re: [HM] Radian Measure
From: Glen Van Brummelen (
Date: Thu Mar 09 2000 - 10:58:18 EST

Hello all,

Regarding the radius of Hipparchus' circle for his chord table, Dinesh
> May I request Kim to briefly describe as to how G.J. Toomer et al
> (as I comprehend form the post) inferred that "Hipparchus too used
> trigonometric radius R = 3438".
> I do not have access to Toomers's article on Hipparchus and
> would appreciate a brief explanation of his logic behind the
> aforementioned inference.

Noel Swerdlow reviewed Toomer's article in Math Reviews, and his
review raised doubts about his Toomer's ingenious attempt to
reconstruct Hipparchus' base circle radius. I hope I'm not violating
any netiquette by letting Swerdlow speak for himself:

"One form of early Indian sine table is computed at $3{\textstyle\frac
3{4}}\sp \circ$
intervals with a unit radius $(=\sin 90\sp \circ)$ of $3438'$ $(\approx
360\sp \circ·60/(2\pi))$.
The author believes that the Indian table was adapted from an earlier chord
table by Hipparchus
at $7{\textstyle\frac 1{2}}\sp \circ$ intervals with the same unit radius.
That Hipparchus used
a chord table at $7{\textstyle\frac 1{2}}\sp \circ$ intervals, that is,
one-half a "step"
(bathmos), is very likely, but that its radius was $3438'$ rather than, say,
$3600'$ $(=1,0,0)$
is not so certain. The author attempts an ingenious, although round-about,
demonstration that,
had it worked, would have provided all but conclusive proof. Unfortunately,
it doesn't quite work.

"In Almagest IV, 11 Ptolemy reports that Hipparchus used two sets of three
lunar eclipses to find,
for an eccentric lunar model, that $R/e=3144/327{\textstyle\frac 2{3}}$,
and, for an epicyclic
model, that $R/r=3122{\textstyle\frac 1{2}}/247{\textstyle\frac 1{2}}$. The
author recomputes
$e$ and $r$ from the eclipses using a $3438'$ chord table to see whether
these admittedly curious
ratios follow. His results, taking into consideration a possible error by
Hipparchus, are close,
although in the reviewer's opinion not close enough to prove that Hipparchus
used such a table.
In any case, since publishing the article, the author has found that the
interval between the
eclipses of March 19/20 and Sept. 12, 199 B.C., which had been read as
$176\sp {\text d}{\textstyle\frac 1{3}}\sp {\text h}$, should be
$176\sp {\text d}1{\textstyle\frac 1{3}}\sp {\text h}$, and this change
vitiates the computation of
$R/r$. Thus, the unit radius of Hipparchus' chord table remains doubtful.

"Despite the unsuccessful result, this article is still a valuable study of
the early stages of Greek
trigonometry, and in particular, of the computation of a chord table from
only the half-angle and
supplementary angle (i.e. Pythagorean) theorems, without the addition and
subtraction theorems used
by Ptolemy in Almagest I, 10."

Cheers, Glen Van Brummelen


Glen Van Brummelen
Bennington College
Bennington, VT, USA 05201
Ph. (802)-440-4467
Fax (802)-440-4461
Home (802)-440-8142

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