**Subject: **Re: [HM] Radian Measure

**From: **Glen Van Brummelen (*gvanbrum@bennington.edu*)

**Date: **Thu Mar 09 2000 - 10:58:18 EST

**Next message:**Ralph A. Raimi: "Re: [HM] Mathematics and Time"**Previous message:**Barron, Alfred [PRI]: "Re: [HM] Mathematics and Time"**In reply to:**Dinesh Maheshwari: "Re: [HM] Radian Measure"**Reply:**Glen Van Brummelen: "Re: [HM] Radian Measure"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

Hello all,

Regarding the radius of Hipparchus' circle for his chord table, Dinesh

wrote:

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

gvanbrum@bennington.edu

Ph. (802)-440-4467

Fax (802)-440-4461

Home (802)-440-8142

**Next message:**Ralph A. Raimi: "Re: [HM] Mathematics and Time"**Previous message:**Barron, Alfred [PRI]: "Re: [HM] Mathematics and Time"**In reply to:**Dinesh Maheshwari: "Re: [HM] Radian Measure"**Reply:**Glen Van Brummelen: "Re: [HM] Radian Measure"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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