1
The History and Origin of Logarithms
The logarithm is notorious among high school students as a pain and a nuisance. Each
year they learn them and each year at least one student questions them, “When are we going to
have to use this in life?” “Why is it called a logarithm?” “What does a logarithm even do?”
“Who came up with something this complicated?” The origin of the logarithm dates back to the
late sixteenth century where the demands and developments of society inspired two men, John
Napier and Joost Bürgi, to create the logarithm (Clark and Montelle).
The modern definition of a logarithm is “the exponent of power to which a stated
number, called the base, is raised to yield a specific number.” It is “nothing more than an
exponent. The basic concept of logarithms can be expressed as a shortcut.” Multiplication is a
shortcut for addition, exponents for multiplication, and logarithms for exponents (Outt). In
school, students do not learn why logarithms are used. They only learn how to solve them. There
are many reasons why logarithms are used. Logarithms can be used to find the number of
payments on a loan needed or the time to reach an investment goal; to model natural processes
including living systems, loudness, or the brightness of light; to measure the pH of chemical
solutions; or to measure the intensity of an earthquake on the Richter scale (Outt).
During the late sixteenth century, develops in many sciences, including observation
astronomy, long-distance navigation, and geodesy science (“efforts to measure and represent the
earth”), required a lot from mathematics. The foundation of these sciences lay in trigonometry,
and “trigonometric tables, identities, and related calculations were the subject of intensive
enterprise.” The goal for most mathematicians was to reduce the “calculation burden that
resulted from dealing with such large numbers for practitioners in these applied disciplines.”
Most effort was directed towards what was known as the “art of computation.” Mathematicians
2
looked for techniques to replace lengthy processes like long multiplication and division with
addition and subtraction. Prosthaphaereris, the changing of long multiplications or divisions in
to additions and subtractions by trigonometric identities, was formed. The word itself was
derived from the Greek prosthesis, for addition, and aphaeresis, for subtraction. From this,
logarithms were born. The word itself comes from the Greek logos, meaning proportion, and
arithmos, meaning number (Clark and Montelle).
One man, Scottish baron John Napier (1550-1617), recognized the potential in recent
developments, mainly prosthaphaeresis, decimal fractions, and symbolic index arithmetic, to
reduce computations (Clark and Montelle). Before the logarithm, Napier was known for his
contributions to spherical geometry and the designing of a mechanical calculator. He was also
the first person to use and popularize the decimal point as “a means of separating the whole from
the fractional part in a number” (Villareal-Calderon). Napier explains the need for logarithms by
saying:
Seeing there is nothing…that is so troublesome to Mathematicall practise, nor that doth
more modest and hinder Calculators, than the Multiplications, Divisions, square and
cubical extractions of great numbers, which besides the tedious expense of time, are for
the most part subject to many errors, I began therefore to consider in my minde by what
certaine and ready Art I might remove those hindrances (Villareal-Calderon).
Napier used an algebraic approach when creating logarithms (Outt). His “conception” of the
logarithm was based on a “kinematic framework.” He had two particles traveling on two parallel
lines, one infinite and the other finite. The first particle, traveled in uniform motion on the
infinite line. It went an equal distance in equal times. The second particle traveled at a velocity
3
“proportional to the remaining distance.” At any time, sine was the distance not yet covered on
the finite line while the distance already covered on the infinite line was the log of sine. As the
sine decreased, Napier’s logs increased. On his visual representation, sine decreased in a
geometric proportion while logs increased in an arithmetic proportion (Clark and Montelle).
x=Sin(θ)
y=lognap(x)
The relation between the two lines
and the logs and sines.
It took twenty years for Napier to complete his table. In 1614, Napier published the first work on
logarithms, Mirifici logarithmorum canonis descriptio (A Description of the Wonderful Table of
Logarithms). English mathematician, Henry Briggs, visited Napier shortly after his book was
published and was impressed with his work. He said:
4
My lord, I have undertaken this long journey purposely to see your person, and to know
by what engine of wit or ingenuity you came first to think of this most excellent help in
astronomy, viz. the logarithms; but, my lord, being by you found out, I wonder nobody
found it out before, when now known it is so easy.
First page of Napier’s
tables
Locating a logarithm value
using Bürgi’s table
Joost Bürgi, a Swiss craftsman (1552-1632), had a more geometric approach towards
logarithms (Clark and Montelle) (Outt). His main motivation was to “facilitate computation” and
create one table, instead of many, that was applicable to all arithmetical operations. He claimed
that having separate tables for multiplication, division, square roots, and cube roots was “not
only irksome, but also laborious and cumbersome” (Clark and Montelle). Bürgi’s conception of
logarithms was based between to progressions. One table for a “multiplicity of calculations by
considering two ‘self=producing and corresponding progressions’: one arithmetic, one
geometric” (Clark and Montelle). He discovered that successive powers of two increased too fast
5
to be of any use when inserting between values. Bürgi used a common ratio of 1.0001 and his
values were recorded as: bn+1 = bn(1.0001) where b0=108. Each successive value was found by
multiplying the previous by 1.0001. The factor of 108 was used by Bürgi for greater precision
(Clark and Montelle).
When thinking about how there was once a time without calculators, students may begin
to understand how logarithms are more convenient than they are inconvenient. Without logs,
high school students would still be using complicated trig functions to evaluate certain types of
problems. Understanding the history of logarithms helps one appreciate them and not feel that
they have no point.
6
Works Cited
Clark, Kathleen and Clemency Montelle. Mathematical Association of America. Mathematical
Association of America. January 2011. Web. 11 October 2014.
Villarreal-Calderon, Rafael. Department of Mathematical Sciences. The University of Montana.
Montana Council of Teachers of Mathematics & Information Age Publishing. 2008.
Web. 11 October 2014.
Outt. History of Logarithms. West Clermont Local School District. Web. 11 October 2014