Motivation for Napier's Logarithms

In the wikipedia article on logarithms, I am clueless about the approach and motivation for the following computations done by Napier (and the mysterious appearance of Euler's number) in this section. Please help me understand:

By repeated subtractions Napier calculated $10^7(1 − 10^{−7})^L$ for L ranging from 1 to 100. The result for $L=100$ is approximately $0.99999 = 1 − 10^{-5}$. Napier then calculated the products of these numbers with $10^7(1 − 10^{−5})^L$ for L from 1 to 50, and did similarly with $0.9995 ≈ (1 − 10^{−5})^{20}$ and $0.99 ≈ 0.995^{20}$. These computations, which occupied 20 years, allowed him to give, for any number ''N'' from 5 to 10 million, the number ''L'' that solves the equation

$$N=10^7 {(1-10^{-7})}^L$$

In modern notation, the relation to natural logarithms is:

$$L = \log_{(1-10^{-7})} \!\left( \frac{N}{10^7} \right) \approx 10^7 \log_{ \frac{1}{e}} \!\left( \frac{N}{10^7} \right) = -10^7 \log_e \!\left( \frac{N}{10^7} \right)$$

where the very close approximation corresponds to the observation that

$${(1-10^{-7})}^{10^7} \approx \frac{1}{e}. $$

I understand the computations, but I don't know what's going on.

Solution 1:

A good reference is John Napier and the invention of logarithms, 1614, by E. W. Hobson (1914). You can also read the original source works by Napier: Mirifici logarithmorum canonis descriptio (English: A description of the admirable table of logarithmes, 1616), and Mirifici logarithmorum canonis constructio (English: The construction of the wonderful canon of logarithms, 1889).

Here is a summary.

We now think of the logarithm function as the inverse of exponentation, but Napier was working at a time when even the notion of exponentiation was not a common one. (This was all also before calculus, computation with infinite series, or coordinate geometry.) Instead, his crucial insight was a certain definition of "logarithms" which satisfied the following

Proposition: If a set of numbers is in geometric progression, then their logarithms are in arithmetic progression.

So, roughly speaking, Napier constructed sets of numbers in geometric progression, and found their logarithms using linear interpolation. For the geometric progressions, he chose common ratios like $(1-\frac1{10^7})$ (others he used were $(1-\frac1{10^5})$, $(1-\frac1{2000})$ and $(1-\frac1{100})$) because it's easy to multiply a number by it: subtract from the number the result of shifting it $7$ decimal places to the right. For instance, to seven decimal places, $9999998.0000001(1-\frac1{10^7})$ is

$$\begin{align} 9999998&.0000001 \\ -\phantom{000000}0&.9999998 \\ -----&----- \\ = 9999997&.0000003\end{align}$$

This is what Wikipedia means by "by repeated subtraction".

Specifically, his logarithm was defined as follows.

Imagine a line $TS$ of length $R$ (= $10^7$), along which a point $P$ moves from $T$ to $S$ such that its velocity is proportional to its distance from $S$. Meanwhile, another point $Q$ on a different line, starting at $T_1$ when $P$ is at $T$, moves at uniform velocity (the velocity of $P$ when at $T$).
DEFINITION: If when the point $P$ is at $P_1$ the point $Q$ is at $Q_1$, then the logarithm of the length $P_1S$ is defined to be the length $T_1Q_1$.
So $l(R)=0$ and $l(x) \to \infty$ as $x \to 0^+$. Also $l(x)<0$ for $x>R$.

Napier showed, with an essentially sound argument, that when $Q_1, Q_2, \dots$ are covered after equal times, i.e., the lengths $T_1Q_1, T_1Q_2, \dots$ are in arithmetic progression, then the lengths $P_1S, P_2S, \dots$ are in geometric progression. This is the proposition mentioned above.

In modern notation, the Napier logarithm $l(x)$ he defined is (and what he calculated is an approximation to) $10^7 \log_{ \frac{1}{e}} \left(\frac{x}{10^7} \right) = -10^7\ln x + 10^7\ln (10^7)$. We can see this as follows: if $x$ is the length $PS$, and $y = l(x)$ is the length $T_1Q$, then $\frac{dx}{dt} = - \frac{Vx}{R}$ where $V$ is the initial velocity when $P$ is at $T$, and $\frac{dy}{dt} = V$, so $\frac{dy}{dx} = - \frac{R}{x}$, and solving this differential equation (with the initial condition $l(R)=0$) gives $y = -R\ln x + R\ln R$. (Involved here is the observation that $\frac{d}{dx} \ln x = \frac1x$, which seems due to Newton a few decades later. See the timeline at the bottom of this page.)

The first few values are more or less equal to $\log_{(1-10^{-7})} \left( \frac{x}{10^7} \right)$. (To see this approximately, like Napier: pick some small time unit, and suppose that after $1$ time unit $y=k$ and $x = Rr$ for some $r<1$. (If the time unit is small, the distance $R(1-r) \approx k$.) Then after $t$ time units $y = kt$ and $x = Rr^t$. So $t = \log_r(x/R)$ and $y = k\log_r(x/R)$ which for $r = (1-1/R)$ and $k \approx R(1-r) = 1$ gives $y = \log_{1-1/R}(x/R)$.) Later parts of his table are approximations in different ways, depending on the corresponding geometric progressions.

The modern definition of logarithms so that $\log 1 = 0$ and $\log 10 = 1$ (well, he was thinking of $\log 10 = 10^{10}$, but this just means computing logarithms to ten digits) occurred to Napier only after he had begun working on his original plan, so in his publications he only proposes an outline of how this table could be constructed. It was Henry Briggs who published the first table of base-10 logarithms.

To be precise, Napier's table gave the "logarithms" of sines of angles from $0^\circ$ to $90^\circ$. The then definition of $\mathop{Sine} \theta$, dating all the way back from Aryabhata in the 5th century, was (for some fixed radius $R$) the length of the half-chord that subtends angle $\theta$ in a circle of radius $R$. In modern notation, $\mathop{Sine} \theta = R\sin \theta$. Napier chose $R = 10^7$. (So $\mathop{Sine} 0^\circ = 0$ and $\mathop{Sine} 90^\circ = 10^7$.) His table gave the "logarithms" of the Sines of equidistant angles, so although it gave logarithms of numbers from $0$ to $10^7$, these numbers were not equally spaced. Here is part of a page from his tables (see more here):

The last row says (in modern notation, and omitting digits after the decimal point) that $10^7\sin(9^\circ 15') = 1607426$, that $l(1607426) = 18279507$ (actually it should have been $18279511$), and reading from the right, it gives the Sine of the complement angle $80^\circ 45'$ (or equivalently the Cosine of that angle), the logarithm of that Sine, and the difference between the two logarithms (useful essentially because $\log\sin\theta -\log\cos\theta=\log\tan\theta$, etc).

About the choice of $10^7$ (concisely explained in Henry's answer), note that at the time decimal notation was also not standard. So instead of putting digits after the decimal point and expecting the readers to understand, he had to multiply by a large power of 10 so that enough digits are in the integral part (before where the decimal point would be). Hobson's book says

His choice was made with a view to making the logarithms of the sines of angles between $0^\circ$ and $90^\circ$, i.e., of numbers between $0$ and $10^7$, positive and so as to contain a considerable integral part.

And Napier's Constructio says

In these progressions we require accuracy and ease in working. Accuracy is obtained by taking large numbers for a basis; but large numbers are most easily made from small by adding cyphers [zeroes]. Thus instead of 100000, which the less experienced make the greatest sine, the more learned put 10000000, …

And so on. It's a bit humorous really, from our perspective, but the Constructio is illuminating for historians who want to know how the first logarithm tables were constructed.

Solution 2:

Napier chose $7$ to balance precision against effort: $6$ would be quicker but less accurate; $8$ would be more accurate but take longer to construct the tables. He multiplied by $10^7$ so it appeared he was working with integer logarithms.

For example, if he wanted to multiply $0.78$ by $0.56$, he would multiply each by $10^7$ and look up $7800000$ in the table of $N$ and find a value of $L$ of about $2484613$, and similarly look up $5600000$ to find something like $5798185$.

Adding these two together he would get $8282798$, and looking that up in the table for $L$ he would find a value for $N$ of about $4368000$. Dividing by $10^7$ gives the result of $0.4368$.