How did Jost Bürgi's logarithms work?

Question

According to what I have gathered from the internet Jost Bürgi came up with the idea of logarithms (as he called Progress Tabulen) after learning a correspondence between arithmetic and geometric sequences. We know this as the product rule for exponents: if $n \longleftrightarrow a^n$ and $m \longleftrightarrow a^m,$ then $$n+m \longleftrightarrow a^n\times a^m.$$

From what I understood, he first computed $r^n$ for $r=1.0001.$ First 15 rows of the table look as below, but should contain $\left\lceil{\log_{1.0001}(10)}\right\rceil =23028$ rows in total.

\begin{array}{|c|c|} \hline n & 1.0001^n \\ \hline 0 & 1.0000000000 \\ 1 & 1.0001000000 \\ 2 & 1.0002000100 \\ 3 & 1.0003000300 \\ 4 & 1.0004000600 \\ 5 & 1.0005001000 \\ 6 & 1.0006001500 \\ 7 & 1.0007002100 \\ 8 & 1.0008002801 \\ 9 & 1.0009003601 \\ 10 & 1.0010004501 \\ 11 & 1.0011005502 \\ 12 & 1.0012006602 \\ 13 & 1.0013007803 \\ 14 & 1.0014009104 \\ 15 & 1.0015010505 \\ \hline \end{array}

But there should be a few more steps to complete his construction. Because this table (with 23028 entries) only allows us to multiply numbers as long as their product does not exceed 10, or the sum of corresponding logarithms does not exceed 23028. Can somebody, who has studied this, summarize to me how he overcame this challenge?

Answer 1

Products larger than 10 can be handled by reducing the exponent of 1.0001 by multiples of $23027$, which removes factors of $10$ because $1.001^{23027}=9.99999780$ is extremely close to 10.

For example:

$3 \times6=1.0001^{10987} \times 1.0001^{17918}=1.0001^{28905}=1.0001^{23027} \times 1.0001^{5878}$.

$3 \times 6=1.0001^{23027} \times 1.0001^{5878}=10.00 \times 1.800=18$.

If you look at the original tables which are quite hard to read or the reproduction of the tables by Denis Roegel, you'll see that Bürgi's tables have the "red numbers" (the logarithms) running from 0 to 230270 in steps of 10 and the "black numbers" running from 100000000 (100 million) to 999999780 (approximately one billion.) The red numbers are scaled by a factor of 10 while the black numbers are scaled by a factor of 100 million.

Repeating our calculation of $3 \times 6$ using Bürgi's table, we see that the red number corresponding to 300 million is 109870 and the red number corresponding to 600 million is 179180. Adding these two red numbers gives a sum of 289050. Reducing this sum by 230270 leaves 58780. This red number of 58780 corresponds to the black number of 179997110, which is $1.8 \times 10^{8}$.