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    First sine table (Ptolemy)
    From: George Huxtable
    Date: 2009 Jan 21, 17:34 -0000

    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{at}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{at}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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