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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: First sine table (Ptolemy)
From: Lu Abel
Date: 2009 Jan 21, 22:17 -0800

```I, too, would love to hear how ancient tables were created.

Most of us who have taken a college-level math course or looked through
various handbooks have seen the infinite series for sines, cosines, and
other functions.   But two things I remember from having taken a
graduate-level course in numerical analysis, especially as might be used
on computers:

1.   Truncating an infinite series can produce unexpected results.
Most famous of these is the Gibbs Phenomenon, where the Fourier
transform for a square wave (an infinite series of sine waves which
should sum up to the square wave) in fact produces a square wave with
large "ears" if the series is truncated.   See
http://en.wikipedia.org/wiki/Gibbs_phenomenon

2.  Numerical algorithms for approximating functions through a finite
series often use coefficients different from those in the infinite
series to produce more accurate results.

James N Wilson wrote:
> Were infinite series used in the initial creation of tables? I find in my
> ancient college handbook listings of series for just about everything,
> including the trigonometric functions.
>
> Jim Wilson
> ____________________________________________________________

>
> >
>

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