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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: J Cora
Date: 2009 Jan 26, 07:12 -0800

```greetings,

I looked at the ptolemy table a few years ago and tried to
work out an example for some values.  I have several
and am including one here.  Hopefully no glaring errors?
The file is postscript and was done by hand - text editor.
Windows users may need to install ghostscript and
linux user myself.

http://mirror.cs.wisc.edu/pub/mirrors/ghost/GPL/gs863/gs863w32.exe

http://mirror.cs.wisc.edu/pub/mirrors/ghost/ghostgum/gsv49w32.exe

On Sat, Jan 24, 2009 at 4:44 PM,   wrote:
>
> George, you wrote:
> "Unless I've made an error, which is always possible, it shows that over
1800 years ago, there already existed a method that was more precise than
what Frank is suggesting, though the knowledge became lost."
>
> Yes, the method I described was not intended to be Ptolemy's method (which
you already described) and it was not intended to be the best approach. I was
merely trying to demonstrate the basic approach to generating a trig table
without using series expansions.
>
> And you wrote:
> "To increase the precision, Frank might (as he suggested) have to reduce his
starting value below 1º, which would then have entailed even more iterations,
each of which would have to be carried out to high precision and without
arithmetical errors"
>
> Right. But it's not as much work as you might imagine. Let's start with the
sine of one minute of arc. At that small angle, for some level of accuracy,
we can assume that the sine is equal to the arc. So sin(1')=(pi/180)/60.
Assuming we're in the early modern era, near the beginning of the history of
modern navigation, c.1700, anyone who needs to knows pi to at least fifteen
digits, but let's go with pi=3.141593. Then use the sine sum formula
> sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b)
> and cos(a)=sqrt(1-sin(a)*sin(a))
> to take us from 1' to 5'. At that point we can jump to 10', then to 15' and
30', and from there to 1 degree. After that proceed as before, one degree,
and you have a very accurate table of sines and cosines. If each calculation
is done to nine digits accuracy, the final table is accurate to nearly seven
places. The largest errors are for sin/cos of 45 degrees which come out to
0.70710680 and 0.70710676 respectively where they should both be 0.70710678.
If we run the calculation up from 1 degree and down from 45, the results are
even better.
>
> You concluded:
> "Of course, Ptolemy had to perform similar tasks, without even the benefit of
> modern decimal arithmetic."
>
> Yes, but he had grad students... er...slaves! er... same thing. It's a huge
amount of work to do this by hand, no doubt. Thank the gods for the printing
press and its information storage descendants so we never have to do it
again.
>
> -FER
>
>
> >
>

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```

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