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From: Herbert Prinz (no email)
Date: Thu Oct 07 2004 - 16:53:52 EDT
George Huxtable wrote:
> Ptolemy writes down what is effectively a table of sines (actually, chords
> of a circle).
>
> At half-degree intervals of angle A Ptolemy computes the lengths of the
> chord of a circle which subtends angle A, effectively 2 sin (A/2), from A =
> 0 to 180 deg, to remarkably high accuracy, entirely from geometrical
> arguments. So this is effectively a table of sines, at quarter-degree
> intervals, between 0 and 90 degrees.
>
But take note of a subtlety. When the geometrical arguments lead to a dead end,
Ptolemy is prepared to drop the whole apparatus of pure geometry and resorts to
a numerical procedure. Starting out with 72d and 60d from the pentagon and
hexagon, he obtains the chords of angles 12d, 6d,...,..3/4d by continued
bisection. But how to get the chord of 1/2d? He knows that the trisection of the
angle is impossible with rule and dividers. With the help of some proportions,
he estimates an upper and lower limit and observes that they agree to the
required number of decimals. That's good enough for his purposes. He takes
astronomy out of the realm of philosophy and treats it like an engineer.
Trying hard to give the appearance that what we are discussing here has anything
to do with navigation, I shall mention in passing the name of Cotter. (I know,
George, that you maintain this list...)
In A History etc., p.16 he writes that "Ptolemy had the theorem
"sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
"This theorem is usually known as Ptolemean Theorem."
Now that you have seen the relevant chapter in the Almagest, would you agree
with this?
Herbert Prinz
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