Wikipedia:Reference desk/Archives/Mathematics/2016 February 2

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February 2

Computation of trig function before computers

What exactly were the steps, or the algorithm to find out the values of the logarithm, sine or cosine tables? Back then There were like 1000 values for each table, but if you did not have the table, was it really impossible to do these calculations? 186.146.10.154 (talk) 12:18, 2 February 2016 (UTC) (posted by SemanticMantis (talk) 16:07, 2 February 2016 (UTC))SemanticMantis (talk) 16:15, 2 February 2016 (UTC)

Trigonometric_tables gives some info on how they were computed. Let us know if there's something in there you don't understand. SemanticMantis (talk) 16:49, 2 February 2016 (UTC)

The earliest method would just be to construct the triangles and measure the values directly, perhaps interpolating to find the in-between values. StuRat (talk) 17:03, 2 February 2016 (UTC)

See Taylor series. The method of computing the values in the log, sine, or cosine tables were essentially the same as the method that the computer uses behind the scenes. If you did not have the table and were not skilled in doing the laborious pencil-and-paper Taylor series, it was impossible to do the calculations accurately, although there were estimation techniques. If you did not have the table and were skilled in the laborious pencil-and-paper calculation, I suppose (but am guessing) that you could contract with a publisher to develop their version of the table. Robert McClenon (talk) 18:25, 2 February 2016 (UTC)

The following series can be evaluated by hand to as many terms as accuracy demands.

Natural logarithm

of z where 0 < z < 2,

$\ln(z)={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots$

Common logarithm

$\log _{10}(x)={\frac {\ln(x)}{\ln(10)}}$

Sine

${\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \end{aligned}}$

where x is in radians (1 radian = 180/pi)

Cosine

${\begin{aligned}\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \end{aligned}}$

AllBestFaith (talk) 20:38, 3 February 2016 (UTC)

The above series for ln(z) converges to slowly to be useful for numerical computation. Use

$\ln((z+1)(z-1)^{-1})=2(z^{-1}+3^{-1}z^{-3}+5^{-1}z^{-5}+\cdots )$

for |z|>1. (See Abramowitz and Stegun formula 4.1.28). Bo Jacoby (talk) 09:00, 4 February 2016 (UTC).

John Napier published tables of ln(x) in 1614. He could not know what equations would be published in 1964. Henry Briggs may have used a finite-difference method some years later. The judgement "converges to[sic] slowly" need not apply where the calculator devotes years to his task. AllBestFaith (talk) 13:32, 4 February 2016 (UTC)

We have this article: History_of_logarithms. Bo Jacoby (talk) 17:57, 4 February 2016 (UTC).