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    Re: Finding longitude in the 12th century
    From: Alexandre Eremenko
    Date: 2012 Sep 1, 08:49 -0400

    I can tell the story briefly.
    > Ah, but then where did those log trig tables come from?
    Tables of trigonometric functions were first developed by Greek
    astronomers in the Roman Empire. Probably the oldest set which survived
    to our time in in Ptolemy (II century AD). These are tables of chords,
    but they are equivalent to the tables of sines, because of the very simple
    relation between the sine and the chord.
    They were computed by hand, using certain relations from trigonometry.
    For example, geometry tells us that sin 30d=1/2, then there are theorems
    which from the sine of an angle give you the sine of 1/2 of the angle,
    and from sines and cosines of two angles give you sine of their sum.
    Computing these tables with many digits was an enormous labor, especially if
    you take into account that they had no our decimal system.
    (Just try to add or multiply large Roman numerals, and you will see:-)
    They used Greek numerals, which seem to be even less convenient.
    There was a question whether people in medieval time really interested
    in calculations. Not much. But some such people always existed: astronomers
    (and astrologers).
    Next great breakthrough was the introduction of decimal system and
    However the main idea of logarithms existed before that. The idea is
    that logarithms transform multiplication into addition.
    Addition is relatively easy, while multilication is hard.
    To see this, try to multiply b y hand two 6-digit numbers.
    So the main property of log is that log(xy)=log(x)+log(y).
    However the trig functions have similar properties.
    Just slightly more complicated. Indeed, for example
    Suppose you want to multiply A and B. Using the trig tables find
    x and y such that cos(x)=A, cos(y)=B.
    Compute x+y and x-y. Then use the table to find their cos.
    Then the formula above gives you the product.
    So you only do additions, subtractions, division by 2, and look to the
    tables. This method is called prostapheresis.
    So you can use trig tables instead of log tables, to multiply.
    Logarithms and the decimal system were introduced almost simultaneously,
    Decimal system was promoted by Simon Stevin in the end of XVI century,
    and logarithms by John Napier (with important contribution of
    Henri Briggs, who advised to combine logs with decimal system, thus
    introducing convenient decimal logs) and Joost Burgi in early XVII.
    (So Columbus had no logarithms and no decimal system).
    Logs were computed by the following
    formula, which is essentially the definition:
    log(x)=area under the hyperbola y=1/t from t=1 to t=x. This area can be
    computed by various means, for example, approximating the region
    under the curve by many rectangles.
    I think it was Kepler who said that invention of logarithms increased
    10 times the life span of astronomers:-) The invention happened during his
    life time, and I suppose it made possible his great discoveries
    which were based on enormous amount of calculations.
    Napier and Burgi comuted log tables with enormous number of digits.
    They also computed logs of sines. Since then everyone can solve spherical
    triangles quickly.

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