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If I'm understanding this correctly, before programmable computers were invented, the only way to do complex calculations was to use a slide rule.

But hold on — to construct a slide rule, you need to somehow compute the logarithms of all the numbers on the rule. But calculating logarithms is hard. Like, really hard!

So how on Earth did people make slide rules having logarithmic markings on them?

(Indeed, some slide rules have a bunch of additional scales for trigonometry and so on. Same question: how do you calculate where to put the tick marks?)

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  • $\begingroup$ By carefully measuring them? $\endgroup$ – Angina Seng May 1 '19 at 16:52
  • $\begingroup$ It should be slide in the title. And computing exact logarithms is hard but approximating (Taylor serier ;) ) is easy. $\endgroup$ – Mohammad Zuhair Khan May 1 '19 at 16:52
  • $\begingroup$ @MohammadZuhairKhan ...although perhaps not all that easy if you haven’t got a calculator, or even a slide rule?! $\endgroup$ – jst345 May 1 '19 at 17:31
  • $\begingroup$ But newotn's approximation/ taylor series are all things that I believe would work. Besides, rulers have a $\pm 1 mm$ uncertainty so I doubt we even need to be that accurate. $\endgroup$ – Mohammad Zuhair Khan May 1 '19 at 17:33
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    $\begingroup$ Early logarithm tables were computed by repeated multiplications of a constant (close to $1$). Not really hard, but tedious. $\endgroup$ – Yves Daoust May 1 '19 at 18:00
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There were extensive and tables of logarithms around the same time as the slide rule was invented. Have a look at these two Wikipedia articles:

In 1624 [Briggs's] Arithmetica Logarithmica, appeared in folio, a work containing the logarithms of thirty-thousand natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000). (https://en.wikipedia.org/wiki/History_of_logarithms#Early_tables)

and

The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. (https://en.wikipedia.org/wiki/Slide_rule#History)

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Keep in mind that you have just to compute once the logarithms, and then you may use them. Briggs had already computed the logarithm tables with enough precision at the beginning of the 17th century...

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