## Sunday 9 October 2022

### #14 Logarithm ....from old math terms notes

Logarithm is the combination of two Greek roots, Logos(reason or ratio) + artihmus(number).

The ratio refers to the original method of constructing logarithms by geometric sequences. The name was introduced by John Napier (1550-1617), the inventor of logarithms, in his 1614 work on logarithms, Mirifici logarithmorum canonis descriptio, [Description of the wonderful canon of logarithms .... but it is usually called "The Descripto"]. It was originally written in Latin and subsequently translated into English. Here is a site where you can find a digital copy of the English text

It seems that Pietro Mengoli (1625-1686) was the first to use the term "natural logarithm". Boyer writes, "Mercator took over from Mengoli the name 'natural logarithm' for values that are derived by means of this series." The term Mengoli and Mercator actually used was "logarithmus naturalis". In a discussion group, Jeff Miller suggested that it might be this use of noun before adjective that prompted the use of the symbol "ln" for natural log rather than "nl". According to Cajori, the symbol "ln" was first used for the natural logarithm (log base e) in 1893 by Irving Stringham (1847-1909). Stringham introduces the notation without comment in a list of symbols following the table of contents, then uses it for the first time on page 41, shown below.

Thanks to Dave Renfro for help in getting this digital pic.

I also recently heard in a correspondence from George Zeliger that when he was a student in Russia (around 1989) it was common to use "lg" for the common logarithm (log base ten).

When Napier constructed his tables he used a base that was slightly smaller than one (1-10-7) and so as the number, n, got bigger, the logarithm, l, got smaller. It was common at the time in trigonometry tables to divide the radius of a circle into 10,000,000 parts. Because the main intention of his creation was focused on addressing the difficulty in performing trigonometric computations, Napier also divided his basic unit into 107 parts. Then to avoid having to use fractions, he multiplied each value by 107. In notation of today's mathematics, the form of Napier's logs would look like :
107 (1-10-7)L=N. Then L is the Naperian logarithm of N.

According to e: The Story of a Number by Eli Maor,

In the second edition of Edward Wright's translation of Napier's Descripto (London, 1618), in an appendix probably written by William Oughtred, there appears the equivalent of the statement that loge10 = 2.302585.
Since the actual tables contains no decimals it was probably given as 2302585 without the decimal point.

In a famous meeting between Napier and Henry Briggs, Briggs suggested the use of a base of 10 instead of 1- 10-7 and to have the logarithm of one equal to zero. This Napier agreed to but the task of constructing tables of "common" logarithms fell to Briggs, and they were often called Brigg's Logarithms in his honor.

Robin Wilson, in his Gresham College lecture on the number e, that "Early ideas of logarithms are given in works of Chuquet and Stifel around the year 1500. They listed the first few powers of 2 and noticed that to multiply any two of them it is enough to add their exponents." Maor notes that Joost Burgi of Switzerland probably created a table of logarithms before Napier by several years, but did not publish until later, and he is almost forgotten today. Burgi may also have independently discovered the method of Prosthaphaeresis and gave it to Tycho Brahe. Burgi is also remembered as the person who taught Kepler Algebra.

The impact of logartihms on the working scientist of the period is hard to appreciate, but one may get an idea from this quote by Pierre Laplace, "Logarithms, by shortening the labors, doubled the life of the astronomer." While it is Napier's work on logartihms that he is remembered for today, in his own time he was famous for the calculating method called Napiers rods and a method of calculating spherical right triangle trigonometry. He thought his most important work had been published 21 years earlier in 1593. In that year he published a mathematical analysis of the book of Revelations in the Bible, A Plaine Discovery of the Whole Revelation of Saint John. In the book he revealed that the Pope was the antichrist, and that the world would end in the year 1786. Fortunately for us, he was wrong on at least that one point. To his credit, he more accurately predicted the development of the machine gun, the submarine, and the tank.

Gordon Fisher recently posted a time line of the development of the use of the abbreviation "log" for lograrithms. Here is his post with a few notes thrown in
Log. (with a period, capital "L") was used by Johannes Kepler (1571-1630) in 1624 in Chilias logarithmorum (Cajori vol. 2, page 105)
log. (with a period, lower case "l") was used by Bonaventura Cavalieri (1598-1647) in Directorium generale Vranometricum in 1632 (Cajori vol. 2, page 106).
log (without a period, lower case "l") appears in the 1647 edition of Clavis mathematicae by William Oughtred (1574-1660) (Cajori vol. 1, page 193).
Kline (page 378) says Leibniz introduced the notation log x (showing no period), but he does not give a source.
loga was introduced by Edmund Gunter (1581-1626) according to an Internet source. [I do not see a reference for this in Cajori.]
Many students (and teachers) have heard colorful legends about the reasoning behind the use of "ln" for the natural logarithm (from the French for something, or something about the name Napier). Most of them seem to me to be more myth than fact. The facts, as best I know them, is that the first use of the terms "natural" and "logarithm" together was by Nicholas Mercator (not the cartographer) in 1668 in his logarithmo technica in which he used the Latinized "log naturalis". [[[In early 2005 a post from Jeff Miller pointed out that, according to Carl Boyer, Pietro Mengoli used the term before Mercator. Both were working with values derived from a series, Mercator with the expansion of log(1+x)]]] The first use of "ln" as a symbol was, as Gordon points out(below), by Stringham (I have not seen this book and do not know if he gives an explanation). As to the correct pronunciation of "ln(x)", whatever your teacher says is correct, but high school students should be aware that many college mathematicians find the symbol disturbing. In his 1984 biography, Paul Halmos described the symbol as "childish". It is, however, very commonly used in computer science.
ln (for natural logarithm) was used in 1893 by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107).
The same note from Jeff Miller mentioned above pointed out that Anton Steinhauser used the abbreviation "log.nat." in 1875
William Oughtred (1574-1660) used a minus sign over the characteristic of a logarithm in the Clavis Mathematicae (Key to Mathematics), "except in the 1631 edition which does not consider logarithms" (Cajori vol. 2, page 110). The Clavis Mathematicae was composed around 1628 and published in 1631 (Smith 1958, page 393). Cajori shows a use from the 1652 edition.

I also recently saw a post that suggested that in computer classes it is sometimes common to use "lg" for the base two log.

In 1647 the French mathematician Saint-Vincent showed that the area under the hyperbola y = 1/x were like the logarithm function, that is, the area from 1 to 2 plus the area from 1 to 3 was equal to the area from 1 to 6, 2x3.