How was the first log table put together?

Question

Henry Briggs compiled the first table of base-$10$ logarithms in 1617, with the help of John Napier. My question is: how did he calculate these logarithms? How were logarithms calculated back then?

I've found these pages to be fairly useful, but they don't seem to say much in the way of what I'm looking for. Any answers or useful references would be appreciated.

Well, you just hammer down a few logs and a plate, and you get a table. :-) — Asaf Karagila, Sep 05 '13 at 17:14
@HagenvonEitzen if he did it naturally, it would be a natural-log table :P — Ben Grossmann, Sep 05 '13 at 17:29
@exk an interesting proposition, given that Brook Taylor introduced Taylor series 100 years later. To be sure though, the Mercator series was first published in 1668, and isn't a terribly efficient means of calculation except for values sufficiently near $1$. — Ben Grossmann, Sep 05 '13 at 18:22
@Omnomnomnom - I didn't specify that it was an informed guess — exk, Sep 05 '13 at 18:27
@exk Sorry, I didn't mean to be particularly mean about it. I meant only to convey that I had considered the possibility and, unfortunately, it doesn't fit the bill. — Ben Grossmann, Sep 05 '13 at 18:28
At least he said "I would guess" in there, unlike a lot of other responses in math.stackexchange ... — GEdgar, Sep 05 '13 at 21:00
Richard Feynman gives an account of Briggs's work in Chapter 22, "Algebra" of "The Feynman Lectures on Physics, Volume 1". — awkward, Sep 05 '13 at 23:14
See http://paramanands.blogspot.com/2012/07/logarithms-using-square-roots.html — Paramanand Singh, Oct 27 '15 at 06:13
Feynman's Lectures are online so search Briggs in ch 22 (to complete awkward 's comment). — Raymond Manzoni, May 15 '19 at 09:01

score 8 · Accepted Answer · edited Apr 08 '25 at 21:16

At MAA you may find 'Logarithms: The early history of a familiar function' while Napier's logarithms are described with care in Roegel's article 'Napier's ideal construction of the logarithms' (rather less nice than the usual ones since using $10^7$ as reference!). A shorter description was given by Lexa in 'Remembering John Napier and His Logarithms' and should provide a good starting point. I'll add this paper by Clark & Montelle and this MT $2021$ lecture.

Napier's work itself appears in 'A Description of the Admirable Table of Logarithms' : Edward Wright's $1616$ translation of Napier's Latin book. A more recent translation was provided by Ian Bruce. A book from $1915$ named 'Napier tercentenary memorial volume' is proposed by archive.org.

Thank you for a very thorough answer. I'll be sure to look at those. — Ben Grossmann, Sep 06 '13 at 15:50

score 5 · Answer 2 · answered Mar 23 '15 at 08:10

5

You can find the details in e: The Story of a Number. The basic idea is that square roots are easy to calculate. If you want for example $\log_{10}2$ (the number such $2=10^{\log_{10}2}$): $$10^{0.25} = 10^{1/4} = 1.778...< 2 < 3.162... = 10^{1/2} = 10^{0.5},$$ i.e., $$0.25 < \log_{10}2 < 0.5$$ and multiplying/dividing for $10^{1/2^k}=\sqrt{\sqrt{\cdots\sqrt{10}}}$ you can get better approximations.

Also important: the successive square roots of 10 are calculated once and can be used many times.

answered Mar 23 '15 at 08:10

Martín-Blas Pérez Pinilla

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multiplying and dividing by the square and higher roots of 10 can get very ugly very fast when working to 40 digits. (Euler was a wiz at square roots.) It would be much easier if you could divide by 1.1111111, 1.01010101, 1.001001001, etc. Now the division is simple! Shift and subtract. You can do it on an abacus, though two might be handier. The table of logarithms is greatly reduced. I think that Napier's original log table was not a log table at all, but was a table of powers of 1.0000001... created by shift and add. – richard1941 Aug 26 '16 at 05:37

How was the first log table put together?

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