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I know that back in the Stone Age, people used tables on this thing called paper to look up values for functions like $\sin$ and $\ln$. But how did the guys who wrote the tables calculate those values? I know that they could have used Taylor series or the integral definition of $\ln$ with Riemann sums, but did they really do that for each number in the tables? Maybe they used a ruler and a unit circle for the trigonometric ratios?

Also, does anyone know what computers use to do the same?

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Don't you want "how" and not "what"? –  Qiaochu Yuan Dec 4 '12 at 3:55
    
Were there such tables in print before the advent of computers? I can't imagine someone calculating all these things on an abacus... I guess they used Taylor, Pade, Newton-methods for solving equations (e.g. $\sqrt{a}$ is the positive root of $x^2=a$). Also things like en.wikipedia.org/wiki/Approximations_of_%CF%80 , curve fitting methods (splines)... there are numerous ways (some have been around for centuries). –  Pantelis Sopasakis Dec 4 '12 at 4:06
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There is a big literature on how Napier (and Briggs) calculated a variant of $\ln$ (Napier) and $\log_{10}$ (Briggs). In the bad old days BC (before calculators), Science students had tables of logarithms, and slide rules. Engineering students used a belt holster for their slide rules. –  André Nicolas Dec 4 '12 at 8:29

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As far as I am aware, approximating polynomials have usually been the most popular method since they have been understood for performing these calculations, both in computers and in the computation of tables. In ancient times, most trigonometric tables were constructed by physical measurement of carefully constructed triangles- there are stories of Greek mathematicians drawing strings across fields so that they could get more accurate measures. Some other methods, such as nomography and linear interpolation between known control points have been used with various degrees of success. Also, for computation of roots and logarithms, for a long time we were limited to the same simple brute force algorithms we teach children in grade school- it was the best option that was clear at the time.

It is also important to note that computation of mathematical tables typically employed many people working full working days, for many months.

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Nowadays, computers use the same algorithms, right? They're just much, much faster. –  Neal Dec 4 '12 at 3:58
    
Well, for specific problems there are optimizations available. For example, we can use Newton's method for calculating roots (which, by the way, is an old method), and there are some nifty simplifications we can use for calculating trig functions. While the methods are essentially the same, they are better informed and tend to employ greater insight into a problem- as a result of our greater understanding or the problems. Just compare calculating e by integrating e^x from 0 to 1. Euler's method works, but RK4 is much better. The actual optimizations are usually quite problem specific. –  deftfyodor Dec 4 '12 at 4:07
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It is in my opinion not a good idea to propagate the non-story of actual physical measurement with strings. True, a story has been around forever to the effect Eyptions used the $3$-$4$-$5$ triangle to produce right angles, useful, I guess, in constructing pyramids. The story is based on a fragment that says they used stretched ropes in their work. No mention made of right angles, or $3$-$4$-$5$. all that is much later interpretation. Anyway, as any carpenter can tell you, there are much better methods to construct good right-angles (level, plumb line). For the methods that were used (more) –  André Nicolas Dec 4 '12 at 8:03
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for tables of chords (trig functions came later), see Ptolemy's Almagest, first section. By the way, Ptolemy's Theorem was used for the calculations. The methods are ingenious, sophisticated. "Exact" calculations in some cases, down to $3^\circ$. Bisection, equivalents of addition formulas, nicely thought out interpolation. The work was continued through the Middle Ages, in India, the Moslem world, and even Europe. A big story. High accuracy was needed because of the demands of astronomy (and astrology). –  André Nicolas Dec 4 '12 at 8:10

You can often find discussions of this in old trigonometry texts. For example, see Chapter IX: Construction of Trigonometric Tables (pp. 82-93) in Isaac Todhunter’s Plane Trigonometry (1882).

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