Subject: Re: [HM] Radian Measure
From: Bob Stein (bstein@csusb.edu)
Date: Wed Feb 09 2000 - 13:13:31 EST
Check the work of Roger Cotes. He didn't use the word "radian", but in
effect he did have that measure. I quote from the book on Cotes by R. Gowing
Bickley:
In his preface to Logometria, Part 2, Cotes introduces the concept of
the measure of an angle. There, he considered that enough had been said
about the measures of ratios but a word was needed about angles. The
circular arc, intercepted between the legs of an angle and having its centre
at the point of the angle, was the natural measure; but this would vary
according to the size of the circle, therefore a modulus was needed. . . .
Robert Smith expands the idea in his Editor's Notes . Following closely the
form of the argument in Cotes' Proposition 1 in Logometria, Part 1, where
the modular ratio is shown to be 2.71828, . . . (i.e. the ratio whose
measure is always equal to the modulus) Smith arrives at the idea of a
modular angle, i.e., an angle whose measure is always equal to the radius,
and shows it to be 57.295 degrees. This is probably the first published
calculation of one radian in degrees. Smith first derives the series for
arc sin m, where m is the measure of an angle, uses Newton's method of
reversion of series to find a series for m, and then puts m equal to the
chosen modulus M. In this way he finds the series for the sine of the angle
whose measure is equal to the modulus-the modular angle. It is analogous,
step by step, to Cotes' determination of the modular ratio in Logometria,
Proposition 1. Smith says he found it in a small paper of Cotes';
unfortunately, this paper has not survived.
Bob Stein
>
> In the book by Cajori on notation and symbols there is a very short
> reference to when radians showed up. Could someone help with a better
> reference or with a brief but more extended history.
>
> Many thanks.
>
> Robert Mena, from Long Beach
>
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