# Is there a way of retrieving the number $e$ from the tables of Napier?

I am looking for a way to relate the numbers in the tables of Napier Mirifici Logarithmorum Canonis Descriptio with the number $$e$$. For example, as Napier used the number $$\left(1-\frac{1}{10^7}\right)$$ to calculate the logarithms. His successive calculations might have generated a number close to $$\left(1-\frac{1}{10^7}\right)^{10^7}$$. Therefore, the tables might have contained a number close to $$3,678,794$$, which is equal to the integer part of $$10^7\times\left(1-\frac{1}{10^7}\right)^{10^7}$$, as $$\left(1-\frac{1}{10^7}\right)^{10^7}$$ gets close to $$\frac{1}{e}$$ and Napier multiplies the result by $$10^7$$.

However, we cannot use this reasoning directly. Is there a way of relating a number in the table with the number $$e$$? Maybe taking the $$\arcsin$$ of a number in the table or doing another kind of calculation?

My goal is to make the findings of Napier closer to a modern reader and to clarify the evolution of the ideas that brought the definition of the number $$e$$.

The entries of the tables (as seen, for example, here by following the link to the "spreadsheet tables") give an angle $$\theta$$ (in degrees and minutes), then $$\sin\theta$$, then $$-\ln\sin\theta$$.
If $$-\ln\sin\theta = 1$$, then $$\sin\theta = \frac1e$$, and $$\theta = \arcsin \frac1e$$, which is between $$21^\circ 35'$$ and $$21^\circ 36'$$ (closer to the former). Even without anachronistically using a calculator to compute that, we can find this entry by looking through the $$-\ln\sin\theta$$ column for a value close to $$1$$. And we find (on page 44 of the PDF) a table with the following fragment:
$$\begin{array}{cccc} \text{Degree} \\ 21 \\ \\ \text{Minute} & \text{Sine} & \text{Logarithm} & \cdots \\ \vdots & \vdots & \vdots \\ 35 & 3678541 & 10000690 & \cdots \\ 36 & 3681246 & 9993339 & \cdots \\ \vdots & \vdots & \vdots \end{array}$$
This tells us that the value of $$\sin\theta$$ which gives $$-\ln\sin\theta = 1$$ (the value which we eventually identify as $$\frac1e$$) is between $$0.3678541$$ and $$0.3681246$$.