Napier’s Discovery of Logs

John Napier, the inventor of the first Log tables.

John Napier, the inventor of the first Log tables.

Logarithms were discovered by the Scottish mathematician John Napier in the early 1600’s. Napier was looking for ways to speed up arithmetic, and also developed what are now called Napier’s bones towards the same purpose. But we’ll focus on logs in this post.

Napier’s method of deriving logs is a little confusing. Napier came at the problem by considering the motion of two particles, one moving with constant velocity, and another whose motion slowed as it approached its destination. This was all in the days before calculus, so Napier had to explain how to relate quite complicated motions using only static or geometrical arguments. This is probably where a lot of the historical confusion comes from. Here, I’m just going to ‘cheat’ by using calculus to explain the basic ideas.

Napier’s two particles are linked by their motion in time. One moves to the right with constant speed $u$, travelling a distance $d=ut$ over time. Napier called this “arithmetic” motion. The second particle starts at one end of an interval of length $A$, and `falls’ back towards the other end $O$, but with a speed $v$ which is proportional to its current distance from $O$. Napier called this “geometric” motion.

Geometrical Motion on line OA

Let’s ‘cheat’ and use calculus to try and understand this second motion. Let $s$ be the current distance of the particle from $O$. Then the proportional velocity is given by $v=-\alpha s$, for some constant $\alpha$. And since the particle starts at $s=A$, we just need to solve the differential equation
\[ v=\frac{ds}{dt}= -\alpha s ; \qquad s(0)=A\]
which has solution
\[s=Ae^{-\alpha t}, \qquad \text{or} \qquad s=A\beta^{-t}\]

Plot of s(t), a decreasing exponential curve.
So the first thing we can see is that the particle never reaches $O$. It ‘falls’ to the left forever. Secondly, we can see that the particle reduces its current distance to $O$ by a fixed proportion $\beta$ with each time step; $s(1)=A\beta$, $s(2)=A\beta^2$, $s(3)=A\beta^3$ and so on. Distance is behaving like a geometric sequence $ar^n$, hence the term “geometric motion”.

Meanwhile of course, the arithmetic particle has moved a distance $d=ut$, the distances $s$ and $d$ are linked by a common time. Now here comes the big step. For simplicity, let $A = 1$, and consider the linked distance pairs
s_1&= \beta^{-t_1} & d_1=&u t_1\\
s_2&= \beta^{-t_2} & d_2=&u t_2
Next consider the product $s_1 s_2=\beta^{-(t_1+t_2)}=s_3$. This distance $s_3$ is also linked to a corresponding $d_3$ with
s_3&= \beta^{-(t_1+t_2)} & d_3=&u (t_1+t_2)

Motion of geometrical particle
Motion of arithmetical particle

If we had some way of translating between these corresponding $s$ and $d$ values then we could do the following procedure: take $s_1$ and $s_2$; lookup the corresponding $d_1$ and $d_2$; add these to obtain $d_3=d_1+d_2$; then reverse lookup to find $s_3$, which is then the desired product $s_1 s_2$. The whole process hinges on being able to translate between $s$ and $d$ values. In short, what we need is a set of tables for the functions
d= -u \log_\beta \left( \frac{s}{A} \right) \qquad \text{and} \qquad s=A \beta^{-(d/u)}
And this is just what Napier produced, out of nowhere in 1614 in the book Mirifici Logarithmorum Canonis Descriptio, containing 57 pages of exposition and 90 pages of detailed tables. This set of tables allowed you to hop back and forth between $s$ and $d$, and hence allowed you to turn multiplication in addition, division into subtraction, and all the rest which follows.

It took him the guts of 20 years to create.

A page of tables from the Mirifici Logarithmorum Canonis Descriptio

A page of tables from the Mirifici Logarithmorum Canonis Descriptio, published by Napier in 1614. The logarithms are actually given for the sine of a related angle $\theta$.

Another cause of confusion in the histories is the seemingly strange base Napier chose. He apparently picked the equivalents of $A=u=10^7$, and $\beta= ( 1-10^{-7} )^{-10^7}$ — because, why not? Actually, $( 1-10^{-7} )^{-10^7} \cong e$, so Napier was quite close to what we now call natural logarithms, with $d=10^7 \ln ( s/10^7 )$, even though the number $e$ had not been discovered. But to cap it all off, Napier decided to make $\sin(\theta)=A-s$ and to give the logarithms corresponding to angles $\theta$ from $0^\circ$ to $90^\circ$ — because, why not? Most likely, he had uses in astronomy and geometry in mind for his tables.

In any event, logarithms turned out to be such an instant hit that all conceptual difficulties were immediately forgiven. Within a year of the ‘Descriptio‘ being published, Napier had collaborated with the English mathematician Henry Briggs to begin producing a set of tables in base $10$. This set of common (or Briggsian) logarithms was eventually published by Briggs in the Arithmetica Logarithmica in 1624, seven years after Napier death in 1617. Over the next 100 years or so, logarithms would be gradually refined into the form we know today, with new tables and slide rules being invented along the way.

But logarithms would not have to wait that long to make their first big impact on the sciences.

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