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how were trig tables calculated, prior to electronics? Prior to calculus?

Is there an algorithm, like with roots?

Googling online, I see talk about either using Taylor series or some tricks with half or sum/difference. But I kind of wonder if you just had to fill out a table of trig values from scratch, say to the 0.1 degree, how they did it. Did they faff around with half angles and the like. Or just crank Taylor? Or was there some other algorithm? Like...I can't find a good explanation for how they did it in practice. Must have been someone who did it first. And heck...did Taylor series even exist back then?

P.s Thinking about it, I can see how you basically just need to do sine up to 45. Can use definitions and/or Pythagorean identity to give the other functions. And then there's a sort of symmetry at 45 (with sine of 50 equal to cos of 40). But still...you'd be looking at having to construct values for 45*10=450 sines...from some sort of algorithm. Plus the work to extend it to the other functions arithmetically (including root algorithm for the cosines).

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  • $\begingroup$ See The trigonometric functions $\endgroup$ Commented Jan 28 at 15:21
  • $\begingroup$ It is not just mathematical relations that can be exploited. It is possible to find specific values roughly by construction, mechanical or drawing or whatnot. $\endgroup$ Commented Jan 28 at 18:13
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    $\begingroup$ Not my area of expertise, so just a pointer to literature: Menso Folkerts, Dieter Launert, and Andreas Thom, "Jost Bürgi's method for calculating sines." Historia Mathematica, Vol. 43, No. 2, May 2016, pp. 133-147 (DOI). "In the first part of the article the standard method is explained which was rooted in Greek antiquity with Ptolemy's computation of chords and which was used in the Arabic-Islamic tradition and in the Western European Middle Ages for calculating chords as well as sines. The main part of the article deals with Bürgi's way." $\endgroup$ Commented Jan 28 at 18:18
  • $\begingroup$ Thanks all. The Burgi paper is particularly interesting. Seems clear from it that it was not trivial to get decimal tables through Taylor's time or so. Not as simple or brute force mechanical as the square root algorithm. $\endgroup$ Commented Jan 28 at 20:45
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    $\begingroup$ Copied from my 4 Dec 2012 Mathematics SE answer: 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). $\endgroup$ Commented Jan 28 at 21:37

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The table of logarithms has been computed for the first time by Napier. The general idea was the following: you take a number $a$ very close to $1$ and compute powers $a^n$. Then $n=\log a^n/\log a$. Of course $\log xy=\log x+\log y$ is used to simplify the calculation.

For technical detail you may consult the book of Napier himself (an English translation is available online) and/or this volume of comments to it:

Napier Tercentenary memorial volume, edited by Cargill Gilston Knott, publ. by Royal Soc. Edinburgh, 1915 (available online).

Napier computed his logarithms (and logarithms of sines) with 7 significant digits, so most subsequent publications could simply copy from his tables, usually rounding. I don't have to mention that Taylor series and Newton-Raphson iteration formula were invented almost a century after Napier.

Tables of trigonometric functions were computed by Ptolemy. He did not have sines or cosines, using instead the chords $$\mathrm{chd}\, x=2\sin(x/2).$$ His method was roughly speaking the following: he used extensively the addition law, especially the double angle formula. He first computed the chord of 1 degree with high accuracy, and then used addition formula and interpolation to compute the rest. Besides Ptolemy's own book, there are several modern commentaries which explain all detail:

O. Neugebauer, A history of ancient mathematical astronomy, in 3 vols. Springer, 1975.

O. Pedersen, A Survey of the Almagest: With Annotation and New Commentary by Alexander Jones, Springer, 2011.

(Both books and Almagest itself are available online)

Let me add the following story. In 1811, a a Russian spy mission was captured by the Japanese, and they spent 3 years in captivity. The Japanese tried to extract from them as much useful information as they could. A captain's mate computed for them complete tables of sines and logarithms (from scratch), which were unknown in Japan at that time.

I wonder, how many modern shipmates can perform this feat:-)

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  • $\begingroup$ It is sad that you were so close to mentioning another important relevant factoid of history: For a short time just before logarithms, prosthaphæresis was the main method to do what logarithms did—converting between multiplication and addition. Copernicus's book referenced it all the time, and Napier was also using it before he invented logarithms. $\endgroup$ Commented 2 days ago
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    $\begingroup$ In connection with the "Russian spy mission" you mention, have a look here with the story of a japanese cartographer sentenced to death in the early 19th century for having shared with europeans his knowledge about maps of Japan. $\endgroup$ Commented 2 days ago
  • $\begingroup$ @Jean Marie Becker: Yes, indeed the official goal; of this mission was cartography of the coast of Japan. The official Japanese cause of detention was that another party of Russians landed, burned several villages and killed people several years ago. $\endgroup$ Commented yesterday

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