How did people calculate numerical values of transcendental and trigonometric functions?

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?

• 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
• 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
• You appear to have missed a second, and equally important question: How were the tables checked? – Kapil Dec 19 '18 at 12:42

• 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
• 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