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Hewitt Schlereth had asked about the construction of early mathematical 
tables, such as log tables.

I wonder if the attached table will interest others, as it impressed me.

It's a translation of part of a table from Ptolemy's "Almagest", which he 
wrote a few years after 150 AD

Ptolemy didn't quite have our concept of sines, so it shows what he called a 
"table of chords", which, with an adjustment or two, is exactly the same as 
a table of sines of angles, to remarkably high accuracy. This table he used 
as a tool, in many places later in the Almagest, just as we would use sines 
of angles today.

If a chord, across a circle of radius r, subtends at its centre an arc A, 
then the length of the chord is 2 r sin (A/2) (just sketch it to convince 
yourself). That's what Ptolemy has tabulated, for increments in A of 
1/2-degree, over a range of 180 degrees, which is the maximum arc possible 
for a a chord. And he has taken r to be equal to 60 (arbitrary) units.

To compare with a modern table of sines, then, we have to halve the arc A, 
shown in the left column. So that part of the table shown in the attachment, 
showing, in the left column, arc A in steps of 1/2 from 0 to 32, actually 
represents angles A/2 fom 0 to 16º in quarter-degrees. The rest of the 
table, in several further pages not shown, extends that angle A/2 to 90º, 
which is shown as a maximum in the left column of the table as an arc A of 
180.

What about the chords of those angles? Ptolemy had no way of dividing them 
into decimal parts, but he was familiar with sexadecimal divisions, which 
had been handed down from the Babylonians. So, after his units, in the left 
column of his chords, comes another column which is 60ths, and yet another 
column which is 3600ths of a unit. So to get to his exact chord-lengths, you 
have to take those fractions of the numbers given, and add them up.

Finally, remember that these were chords of a circle of a circle of radius 
60, diameter 120, so the maximum chord was 120 units. So to turn them into 
our familiar sines, we have to divide the given chord lengths by 120. It 
turns out that Ptolemy produced what we would now call a table of sines, to 
breathtakingly high precision.

It's simplest to see by trying an example from the part-table in the 
attachment. Choose an arc at random, say 26 1/2 degrees, which corresponds 
to an actual angle of half that, 13 1/4 º. Now take Ptolemy/s chord, of 27 / 
30 /14 units. Turn that into decimal units, as 27 + 30/60 + 14 /3600, or 
27.503889 chord-units in decimal. Now divide by the diameter of 120 and we 
get 0.2291991. Compare that with sin 13.25º, for which a calculator gives 
0.2292000. a difference just a bit over 1 part in a million! And that's not 
a fluke; if you make the same comparison for other arcs in the table, you 
will find corresponding precision, though I expect there may be a blunder ot 
two to be unearthed, if you examined every entry.

The disbelieving should try a different angle, and check it out. I have 
attached only one-sixth of the complete table; if anyone would like the 
rest, just ask, or look at the website referred to below.

I haven't mentioned the third column, which is nothing more than our 
familiar table of differences, for linear interpolation. He shows the 
difference in the chord that corresponds to a change of one 60th of a unit 
in each arc; that is, to a change in angle of 1/120 of a degree. These 
differences are shown, in the fourth column, down to one 216000th of a chord 
unit; a precision that's irrelevant, as I see it.

How on Earth did Ptolemy do it? Remarkably, he used pure Euclidean geometry. 
From first principles, he derived the lengths of sides of regulat polygons 
inscribed within a circle. And then went on to show how chords of sums and 
differences of two angles could be derived from those, and built up his 
table in that way. He gives several pages of explanation, with diagrams.

All this understanding soon became lost to Western culture when scientific 
knowledge succumbed to religious orthodoxy, and only survived in Arab / 
Persian cultures, through which it was returned to Europe over 1000 years 
later.

My text comes from chapters 10 and 11 of book 1 of the Almagest, translated 
by Taliaferro, which comes with works of Coperrnicus and Kepler in "Great 
books of the Western World", 16, published by Encyclopaedia Brittanica, 1952 
edition, pages 14-24. Fortunately, that translation of those very pages, 
with the full table, has been digitised and can be read at-
www.cs.xu.edu/math/math147/02f/ptolemy/ptolemytext.html

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.







 contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 

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