TITLE: How was the first log table put together?
QUESTION [16 upvotes]: 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.

REPLY [3 votes]: 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.