Napier and the logarithm: the marvellous abbreviation

Today, for many students, the logarithm is a bogeyman: a strange word, a $\log$ that appears out of nowhere and rules to memorise. In the seventeenth century it was the opposite. It was Europe’s calculator, the tool that saved astronomers, navigators and accountants months of work. The old texts on this site said the logarithm was the purest example of mirabilis abbreviatio, the marvellous abbreviation. And its inventor agreed: the title of his book begins with the word marvellous.

Two rows of numbers

The idea was already hidden in two rows of numbers that anyone can write:

Pick two numbers from the top row, say $8$ and $64$. Look at the numbers below them: $3$ and $6$. Add: $3+6=9$. Now go back up, in the $9$ column: $512$. And indeed $8\times64=512$. A multiplication on top became an addition below. The German Michael Stifel had already noticed this in 1544, in his Arithmetica integra. Divisions become subtractions; roots become divisions by $2$.

The trouble is that Stifel’s table is full of gaps: between $8$ and $16$, or between $64$ and $128$, there is nothing. For the trick to work with any number, such as $3.1416\times2.7183$, the table had to be filled in with much finer rows.

Twenty years of calculation

That is what John Napier, Laird of Merchiston, did. He was a Scottish nobleman who also wrote on theology and designed war machines. For about twenty years he calculated by hand a table with steps so small that almost any number appeared in it. In 1614 he published the result in Latin: Mirifici logarithmorum canonis descriptio, “a description of the marvellous canon of logarithms”. Canon, from the Greek kanṓn, a ruler, was the name given to a reference table.

The word logarithm was his own invention, made of two Greek words: lógos, which here means ratio, proportion, and arithmós, number. The logarithm is the “number of the ratio”: it counts how many times a fixed ratio has been applied to reach a value. In $8=2\cdot2\cdot2$ the ratio $2$ has been applied three times, and the base-$2$ logarithm of $8$ is $3$.

By shortening the labours, logarithms doubled the life of the astronomer.

— Saying attributed to Pierre-Simon Laplace

Success was immediate. In 1615 the English professor Henry Briggs travelled to Scotland to meet Napier, and the two agreed to use base $10$, more practical for the decimal system. Briggs published his tables in 1617 and 1624. Johannes Kepler, who calculated the orbits of the planets, took up logarithms with enthusiasm and published tables of his own in 1624. Around 1622, William Oughtred, the man of the $\times$ sign, put two logarithmic scales side by side and invented the slide rule, which engineers used until the 1970s.

From logarithmus to log

Like radix quadrati, the logarithm also began as a sentence. The sequence below is schematic, reconstructed in the old texts on this site to show the path of the abbreviation:

The last step is the most important from a semiotic point of view. With the base written in, the sentence “$2$ raised to $3$ gives $8$” and the sentence “the logarithm of $8$ in base $2$ is $3$” are visibly linked:

The same three numbers, the same relation, read in two directions. $\log_b$ undoes “raise $b$ to the power”: $\log_b\left(b^n\right)=n$. The old texts on this site took the idea to the extreme and wrote the base already as a power, as in $\log_{2^6}64$, so that the answer, $6$, was in plain sight. It is a teaching trick, not an official notation, but it shows well what we want from an active notation: that the answer be drawn into the question.

The rules the writing hands over

With the $\log$ notation, the properties Napier and Briggs used in their tables are written in one line each:

They need not be memorised, because they come straight from Stifel’s two rows. In base $2$: $\log_2(4\cdot16)=\log_2 4+\log_2 16=2+4=6$, and indeed $4\cdot16=64=2^6$. Multiplying powers is adding exponents, and the logarithm is the exponent. The notation just puts it in writing.

A script that outlived the calculator

Since the 1970s nobody multiplies with log tables any more. But $\log$ has not disappeared, and that says a lot about it. The notation stays active because the relations it draws are still useful: the Richter scale for earthquakes, pH in chemistry and decibels for sound are logarithmic scales, in which adding $1$ means multiplying by $10$. In 1748 Euler showed that the most natural logarithm of all has as its base a number he called $e$, and $\ln$ entered Leibniz’s calculus as if it had always been there. The calculator died; the marvellous abbreviation remained.

Sources: Florian Cajori, A History of Mathematical Notations (1928–1929) · John Napier, MacTutor · Henry Briggs, MacTutor · for the number $e$ and compound interest, see the chapter on Euler.

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