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.2 Napier introduced the name logarithm as a combination of two ancient Greek terms logos, meaning “divine reason”, 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. And the repeated multiplication or division of several multi-digit numbers can become very tedious very quickly. 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.
The following table illustrates the type of table being sought:
| \(p\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| \(x\) | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 16384 | 32768 |
The top row – the numbers designated “\(p\)” – run sequentially from 0 to 16. The bottom row – designated “\(x\)” – starts with the number 1. Then, a base number is chosen. For our example we chose “2” as our base. Next, each consecutive number in the \(x\) row is the previous result multiplied by the base number. The list created in the second row is called a geometric sequence. Now notice the following. If we take two numbers in the \(x\) list and multiply them, the result falls under the sum of the corresponding values in the \(p\) list. For instance, take 2 times 8; the result we know is 16. The corresponding values of \(p\) in the table for the numbers 2 and 8 are 1 and 3. We see that the sum of these \(p\) values is 1 + 3 = 4, and the number in the \(x\) row that falls below this result is \(x\) = 16, the result of multiplying our original two numbers. The values of \(p\) are called logarithms of the values of \(x\).
Now suppose we wanted to quickly find 32 times 256. We could take out pencil and paper and write
\[ \begin{matrix} ~ & ~ & ~ & ~ & {\tiny 1} &{\tiny 1} & ~ \\ ~ & ~ & ~ & ~ & {\tiny 1} &{\tiny 1} & ~ \\ ~ & ~ &~ & {\bf } & {\bf 2} & {\bf 5} & {\bf 6} \\ ~ & ~ & \times & ~ & {\bf } & {\bf 3} & {\bf 2} \\ \hline ~ & ~ & ~ & {\tiny 1} & & & ~ \\ ~ & ~ & ~ & & 5 & 1 & 2 \\ ~ & ~ & & 7 & 6 & 8 & \\ \hline ~ & ~ & & 8 & 1 & 9 & 2 \\ \end{matrix} \]
or, from our table above, we could look up the logarithm of 32 which is 5, and the logarithm of 256 which is 8. Adding the logarithms we find 5 + 8 = 13. We then look at the table to find which number \(x\) has a logarithm of \(p\) = 13. The answer found from the table is \(x\) = 8192. Thus, 64 times 512 is equal to 8192, with only simple addition being required.
In our simple example above, the value of \(x\) that can be used from the table is a rather sparse list. The goal of Napier was to find a technique for computing a set of logarithms of all integers in order to fill out a complete table for any value of \(x\). From our table above it can be surmised, for example, that using 2 as our base, the logarithm \(p\) for the number \(x\) = 7 is somewhere between 2 and 3. If we knew its logarithm more precisely, we might be able to perform similar computations using the number 7.
Napier’s goal was to determine a set of logarithms of the natural numbers. His concepts were introduced in 1594 or before, and he spent the next 20 years developing his computational techniques. All of this took place many decades prior to the development of calculus and involved approximations using geometrical figures and rules of proportions. Through years of effort he examined different base numbers to use, and 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 modern mathematical terms, what Napier was calling “logarithms” were related to exponents of a particular base number, which in turn, could be added or subtracted to perform multiplication and division. His development of logarithms came at a time before the concept of exponents and exponential notation (\(b^x\), say) came into being. Nonetheless, by 1614 he was able to create fairly accurate tables of his logarithms and use them to perform accurate multiplication calculations. After his death in 1617, Henry Briggs, a colleague who had been working with Napier, took over the work and completed that same year the first publication of a set of logarithms in what we would today call “base 10.” The details of this mathematical construction would be published in 1619.
Within a year Edmund Gunter began producing scales that were proportional to logarithms, with the invention of the slide rule following just a few years later by William Oughtred by about 1624. 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 .↩︎