History of logarithms


The history of logarithms is the story of a correspondence between multiplication on the positive real numbers and addition on real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer. The Napierian logarithms were published first in 1614. E. W. Hobson called it "one of the very greatest scientific discoveries that the world has seen." Henry Briggs introduced common logarithms, which were easier to use. Tables of logarithms were published in many forms over four centuries. The idea of logarithms was also used to construct the slide rule, which was ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, and required the assimilation of a new function into standard mathematics.

Napier's wonderful invention

The method of logarithms was publicly propounded for the first time by John Napier in 1614, in his book entitled Mirifici Logarithmorum Canonis Descriptio. The book contains fifty-seven pages of explanatory matter and ninety pages of tables of trigonometric functions and their natural logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central to astronomy and celestial navigation and which typically include products of sines, cosines and other functions. Napier described other uses, such as solving ratio problems, as well.
John Napier wrote a separate volume describing how he constructed his tables, but held off publication to see how his first book would be received. John died in 1617. His son, Robert, published his father's book, Mirifici Logarithmorum Canonis Constructio, with additions by Henry Briggs, in 1619 in Latin and then in 1620 in English.
Napier conceived the logarithm as the relationship between two particles moving along a line, one at constant speed and the other at a speed proportional to its distance from a fixed endpoint. While in modern terms, the logarithm function can be explained simply as the inverse of the exponential function or as the integral of 1/x, Napier worked decades before calculus was invented, the exponential function was understood, or coordinate geometry was developed by Descartes. Napier pioneered the use of a decimal point in numerical calculation, something that did not become commonplace until the next century.
Napier's new method for computation gained rapid acceptance. Johannes Kepler praised it; Edward Wright, an authority on navigation, translated Napier's Descriptio into English the next year. Briggs extended the concept to the more convenient base 10.

Common logarithm

As the common log of ten is one, of a hundred is two, and a thousand is three, the concept of common logarithms is very close to the decimal-positional number system. The common log is said to have base 10, but base 10,000 is ancient and still common in East Asia. In his book The Sand Reckoner, Archimedes used the myriad as the base of a number system designed to count the grains of sand in the universe. As was noted in 2000:
In 1616 Henry Briggs visited John Napier at Edinburgh in order to discuss the suggested change to Napier's logarithms. The following year he again visited for a similar purpose. During these conferences the alteration proposed by Briggs was agreed upon, and on his return from his second visit to Edinburgh, in 1617, he published the first chiliad of his logarithms.
In 1624, Briggs published his Arithmetica Logarithmica, in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places.
This table was later extended by Adriaan Vlacq, but to 10 places, and by Alexander John Thompson to 20 places in 1952.
Briggs was one of the first to use finite-difference methods to compute tables of functions.
He also completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places and the tangents and secants for the same to ten places, all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica; this work was probably a successor to his 1617 Logarithmorum Chilias Prima, which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th decimal place.

Natural logarithm

In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent, related logarithms to the quadrature of the hyperbola, by pointing out that the area A under the hyperbola from to satisfies
At first the reaction to Saint-Vincent's hyperbolic logarithm was a continuation of studies of quadrature as in Christiaan Huygens and James Gregory. Subsequently, an industry of making logarithms arose as "logaritmotechnia", the title of works by Nicholas Mercator, Euclid Speidell, and John Craig.
By use of the geometric series with its conditional radius of convergence, an alternating series called the Mercator series expresses the logarithm function over the interval. Since the series is negative in, the "area under the hyperbola" must be considered negative there, so a signed area determines the hyperbolic logarithm.
Historian Tom Whiteside described the transition to the analytic function as follows:
Leonhard Euler treated a logarithm as an exponent of a certain number called the base of the logarithm. He noted that the number 2.71828, and its reciprocal, provided a point on the hyperbola xy = 1 such that an area of one square unit lies beneath the hyperbola, right of and above the asymptote of the hyperbola. He then called the logarithm, with this number as base, the natural logarithm.
As noted by Howard Eves, "One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use." Carl B. Boyer wrote, "Euler was among the first to treat logarithms as exponents, in the manner now so familiar."

Pioneers of logarithms

Predecessors

The Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of quarter squares. Thus, such a table served a similar purpose to tables of logarithms, which also allow multiplication to be calculated using addition and table lookups. However, the quarter-square method could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers.
The Indian mathematician Virasena worked with the concept of ardhaccheda: the number of times a number of the form 2n could be halved. For exact powers of 2, this equals the binary logarithm, but it differs from the logarithm for other numbers and it gives 2-adic order rather than the logarithm.
Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a table of binary logarithms.
In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity
or similar to convert the multiplications to additions and table lookups. However, logarithms are more straightforward and require less work. It can be shown using Euler's formula that the two techniques are related.

Bürgi

The Swiss mathematician Jost Bürgi constructed a table of progressions which can be considered a table of antilogarithms independently of John Napier, whose publication was known by the time Bürgi published at the behest of Johannes Kepler. We know that Bürgi had some way of simplifying calculations around 1588, but most likely this way was the use of prosthaphaeresis, and not the use of his table of progressions which probably goes back to about 1600. Indeed, Wittich, who was in Kassel from 1584 to 1586, brought with him knowledge of prosthaphaeresis, a method by which multiplications and divisions can be replaced by additions and subtractions of trigonometrical values. This procedure achieves the same result that logarithms will a few years later.

Napier

The method of logarithms was first publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio.
Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier, remarked:
Napier imagined a point P travelling across a line segment P0 to Q. Starting at P0, with a certain initial speed, P travels at a speed proportional to its distance to Q, causing P to never reach Q. Napier juxtaposed this figure with that of a point L travelling along
an unbounded line segment, starting at L0, and with a constant speed equal to that of the initial speed of point P. Napier defined the
distance from L0 to L as the logarithm of the distance from P to Q.
By repeated subtractions Napier calculated for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10−5. Napier then calculated the products of these numbers with for L from 1 to 50, and did similarly with and. These computations, which occupied 20 years, allowed him to give, for any number N from 5 to 10 million, the number L that solves the equation
Napier first called L an "artificial number", but later introduced the word "logarithm" to mean a number that indicates a ratio: λόγος meaning proportion, and ἀριθμός meaning number. In modern notation, the relation to natural logarithms is:
where the very close approximation corresponds to the observation that
The invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri, Edmund Wingate, Xue Fengzuo, and Johannes Kepler's Chilias logarithmorum helped spread the concept further.