# 1 Review of the Logarithm

The elementary calculations performed using slide rules utilize the concept of the logarithm, invented in the early 1600’s by John Napier of Scotland.^{1} Napier introduced the name *logarithm* as a combination of two ancient Greek terms *logos*, meaning proportion, and *arithmos*, meaning number. He was looking for a relationship between an arithmetic and a geometric progression, so that tables could be generated that would aid in performing multiplication calculations. At that time, calculations in astronomy and, perhaps most importantly, in navigation required the multiplication of numbers of up to 6-7 digits each.

To illustrate the issue suppose we want to multiply the two numbers 4873 and 382. This can be performed on paper in the following way:

\[ \begin{matrix} ~ & ~ & ~ & {\tiny 2} & {\tiny 2} & ~ & ~ \\ ~ & ~ & ~ & {\tiny 6} & {\tiny 5} & {\tiny 2} & ~ \\ ~ & ~ & ~ & {\tiny 1} & {\tiny 1} & ~ & ~ \\ ~ & ~ &~ & {\bf 4} & {\bf 8} & {\bf 7} & {\bf 3} \\ ~ & ~ & \times & ~ & {\bf 3} & {\bf 8} & {\bf 2} \\ \hline ~ & {\tiny 1} & {\tiny 2} & {\tiny 2} & ~ & ~ & ~ \\ ~ & ~ & ~ & 9 & 7 & 4 & 6 \\ ~ & 3 & 8 & 9 & 8 & 4 & ~\\ 1 & 4 & 6 & 1 & 9 & ~ & ~\\ \hline {\bf 1} & {\bf 8} & {\bf 6} & {\bf 1} & {\bf 4} & {\bf 8} & {\bf 6}\\ \end{matrix} \]

Of course one needs to know the “times tables” very well. And after a few “carries” (indicated above in small type, such as in “8 times 7 is 56, carry the 5”), keeping things lined up and keeping track of what is being added requires discipline and concentration, especially for situations involving many more digits than the above. Napier’s concept was to find for each number a corresponding *logarithm* such that multiplying two numbers was reduced to *adding* their logarithms. Adding two numbers, even if the numbers have many digits, is a simpler task and much less error prone.

Napier developed ratios of geometric sequences which allowed him to successfully compute tables of numbers that could be added together in order to perform a multiplication or a division. In essence, he was describing the addition and subtraction of *exponents* of a particular *base* number, which in turn, is equivalent to multiplication and division. It is interesting to note that his development of logarithms came at a time before exponential notation (\(b^x\), say) had been developed. Nonetheless, by creating tables of logarithms and producing scales that were proportional to their values, the invention of the slide rule followed within a few short years. Its user was able to perform quick multiplication and division calculations with sufficient accuracy for a wide variety of computational applications.

A nice discussion of Napier’s development of his early tables of logarithms can be found in Denis Roegel’s,

*Napier’s ideal construction of the logarithms*[Research Report], ffinria-00543934f (2010). See https://hal.inria.fr/inria-00543934/en .↩︎