# Napier's e - Napier

Author(s):
Amy Shell-Gellasch (Beloit College)

The other common definition of e is that of the base of the natural logarithm. Logarithms were invented over a period of several years around the turn of the seventeenth century by John Napier (1550-1617), the Scottish Baron of Merchiston. Napier published two influential works that laid out the new mathematical tool of logarithms. The second of the two, Mirifici logarithmorum canonis descriptio (Description of the Marvelous Rule of Logarithms), was published first, in 1614, while his Mirifici logarithmorum canonis constructio (Construction of the Marvelous Rule of Logarithms) was published posthumously in 1619 by his son Robert [2].

John Napier

The impetus for his invention was to reduce the multiplication of large numbers required in astronomy and more importantly, navigation. In short, Napier noticed that the angle addition formulas for sine and cosine reduced multiplication to addition. His logarithms have the property that the log of a product can be calculated by adding the logs of the multipliers and then subtracting a constant [8]. His insight can be explained in the following manner: consider two points at rest and initiate motion at the same time. Napier compared the distance traveled by one point moving at uniform velocity on an infinite line to the distance yet to be traveled by a point on a finite line segment moving at a velocity inversely proportional to the distance it still must travel. From this he coined the term “logarithm” meaning “ratio number” in his “second” book, the Descriptio [2, 5, 6]. (However, in his “first” book, the Constructio, he referred to them as artificial numbers [2].) To calculate longitude and latitude, astronomical readings of up to seven digits (found using spherical trigonometry) need to be multiplied. By providing practitioners with extensive log and antilog tables, the large multiplications required could be computed by looking up their logs in a log table, performing addition and subtraction using those two numbers, then converting the answer back into a regular number using an antilog table. To allow for great enough accuracy, Napier took the length of his line to be 107, since sine tables commonly used at the time were taken to seven places.

In Napier’s terms, X=Nap log y, where Nap log (107)=0. Using calculus, Napier’s log can be converted to modern logs using the following conversion [1]:

Nap log y = 107 log1/e(y/107)

In the famous 1615 meeting between Henry Briggs (1561-1631) and Napier and the collaboration that followed, Briggs convinced Napier to revise his system of logs to use the base 10 instead [1, 5]. This new base gave the simplification log(10)=1. Briggs revised Napier’s work, and after the latter’s death, published Arithmetica logarithmica in 1624. This work contained log tables for the base 10 (common logs) to 14 decimal places for the numbers 1-20,000 and 90,000-100,000, which complemented the tables in Napier’s earlier works [11].

Those of us who live on “this side of the pond” refer to Euler’s number, natural logarithms, and common logs for log base 10, while many who live on the “other side of the pond” in Europe use the terms Napier’s constant, Napier’s logarithms, and Briggsian logarithms, respectively. Even though Euler did give us the notation e, and the terms natural and common logs are more descriptive, the British nomenclature may be more in line with the historical development of the concepts. These differences in nomenclature could even be late remnants of the British Isles versus the Continent divide that originated with the Newton/Leibniz controversy and isolated British mathematicians from continental ones for many years. Intriguingly, the “Napierians” do use e for the constant in question. So in tracing the origins of this most ubiquitous number, perhaps n for Napier should be taken under consideration.

Speaking of priority disputes, Napier does not hold the title to logs free and clear. An invention (or should we call it a discovery?) of logarithms and compilation of log tables occurred in Prague in 1620 by Swiss watchmaker Joost Bürgi (1552-1632) [9]. Although Napier had published in 1614, notes and correspondence show that he was working on it as early as 1594, while Bürgi had started his researches in 1588 [1]. But publication won out on this one, and Napier has retained his title to priority. Part of the reason for the long development of logarithms by Napier and Bürgi is that they were working at a time prior to the development of exponential notation such as nx. Without this notation, it is much harder to see the connections between exponentials and logarithms. Eves states that, “One of the oddities in the history of mathematics is the fact that logarithms were discovered before exponents were in use.” [6, p. 186]

Just as Leibniz’s notation for the calculus helped spur the acceleration of developments in calculus by continental mathematicians, the relation between logs and exponents is a good example of how good notation can actually generate mathematics. As indicated above in the Euler discussion, this connection was not explicitly made until Euler. Though Napier retains the title of “first” in the discovery of logarithms, in all fairness to Bürgi whose work was done independently, perhaps we should call it the Napier-Bürgi constant and denote it by nb. But before we close the book on Euler, e and Napier, I would like to make one final suggestion for the name of 2.718…--- "o".

Amy Shell-Gellasch (Beloit College), "Napier's e - Napier," Convergence (April 2010), DOI:10.4169/loci003209