Subject: Re: [HM] Radian Measure
From: Kim Plofker (Kim_Plofker@Brown.edu)
Date: Mon Mar 06 2000 - 19:45:32 EST
James Landau said:
> It can be argued (but I would consider such an argument to be half-facetious)
> that Archimedes used radian measure when he made his famous approximation to
> pi by inscribing and circumscribing a circle with a 96-sided polygon. By
> computing the length of one side of the two polygons, Archimedes was
> approximating the length of an arc of radian measure 2 pi / 96.
>
> More seriously, there is the formula
>
> arctan x = x - x^3/3 + x^5/5 - X^7/7 + ...
>
> Most texts say this formula was discovered by Gregory in 1671 and Leibniz in
> 1674, but the original discoverer was Madhava in India circa 1400. Since
> this formula gives x in radians, can we not argue seriously that Madhava,
> Gregory, and Leibniz all worked with radian measure?
Hmmm. Madhava was actually solving for the angle in minutes, owing
to the nature of his trigonometric setup. He says (or Sankara, his
student's son's student's student, implies he said):
"The product of the given Sine and the Radius divided by the Cosine
is the first result. Make the square of the Sine the multiplier and
the square of the Cosine the divider; then a series of results is
to be computed continuously from the first result and so forth. Divide
these in order by the odd numbers 1, 3, and so on. Subtract the sum
of the even results from the sum of the odd results; that [remainder]
is the arc."
So his formula for an angle k would work out to something like this,
in symbolic notation:
k = (Sin k)(R)/Cos k - (Sin^3 k)(R)/(3)(Cos^3 k) + (Sin^5 k)(R)/(5)(Cos^5 k)
...
where Sin and Cos are the usual Indian trigonometric functions equivalent
to R times our sin and cos, and R is the Indian trigonometric radius not
equal to 1. Most often, R=3438, approximately the number of minutes in a
circle divided by 2 pi; Madhava is using a much more exact 3437.something.
So his answer comes out in minutes, the normal unit of measure for angles.
If one's going to consider the use of R \approx C/(2\pi), and trigonometric
functions normalized to that R, to be "functionally equivalent" to the
concept of radians, then its origin has to be pushed back quite a bit
farther: Indian trigonometry uses R = 3438 at least since the time of
Aryabhata (it's a handy value for small-angle approximations, for one
thing).
With best wishes,
Kim Plofker
Department of the History of Mathematics
Brown University
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