The history of logarithms is the story of a correspondence (in modern terms, a group isomorphism) 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 (base 10) 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 (invented around 1620–1630), 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.
Development of the logarithm function The method of logarithms was publicly propounded for the first time by John Napier in 1614, in his book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms). 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 (Construction of the Wonderful Canon of Logarithms), 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 , 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:Ian Bruce (2000) "Napier’s Logarithms", American Journal of Physics 68(2):148, doi: 10.1119/1.19387
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 (1-20,000 and 90,001 to 100,000). 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 ("The First Thousand Logarithms"), 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,In 1647, Gregoire de Saint-Vincent published his book, Opus geometricum quadraturae circuli et sectionum coni (Geometric work of squaring the circle and conic sections), vol. 2 (Antwerp, (Belgium): Johannes and Jakob Meursius, 1647). On page 586, Proposition CIX, he proves that if the abscissas of points are in geometric proportion, then the areas between a hyperbola and the abscissas are in arithmetic proportion. This finding allowed Saint-Vincent's former student, Alphonse Antonio de Sarasa, to prove that the area between a hyperbola and the abscissa of a point is proportional to the abscissa's logarithm, thus uniting the algebra of logarithms with the geometry of hyperbolas. See: Alphonse Antonio de Sarasa, Solutio problematis a R.P. Marino Mersenne Minimo propositi ... [Solution to a problem proposed by the reverend father Marin Mersenne, member of the Minim order ... ], (Antwerp, (Belgium): Johannes and Jakob Meursius, 1649). Sarasa's critical finding occurs on page 16 (near the bottom of the page), where he states: "Unde hae superficies supplere possunt locum logarithmorum datorum ... " (Whence these areas can fill the place of the given logarithms ... ). [In other words, the areas are proportional to the logarithms.]See also: Enrique A. González-Velasco, Journey through Mathematics: Creative Episodes in Its History (New York, New York: Springer, 2011), page 118. related logarithms to the quadrature of the hyperbola, by pointing out that the area under the hyperbola from to satisfiesAlphonse Antonio de Sarasa, Solutio problematis a R.P. Marino Mersenne Minimo propositi ... [Solution to a problem proposed by the reverend father Marin Mersenne, member of the Minim order ... ], (Antwerp, (Belgium): Johannes and Jakob Meursius, 1649). Sarasa realized that given a hyperbola and a pair of points along the abscissa which were related by a geometric progression, then if the abscissas of the points were multiplied together, the abscissa of their product had an area under the hyperbola which equaled the sum of the points' areas under the hyperbola. That is, the logarithm of an abscissa was proportional to the area, under a hyperbola, corresponding to that abscissa. This finding united the algebra of logarithms with the geometry of hyperbolic curves. Sarasa's critical finding occurs on page 16 (near the bottom of the page), where he states: "Unde hae superficies supplere possunt locum logarithmorum datorum ... " (Whence these areas can fill the place of the given logarithms ... ). [In other words, the areas are proportional to the logarithms.] See also: Enrique A. González-Velasco, Journey through Mathematics: Creative Episodes in Its History (New York, New York: Springer, 2011), pp. 119–120.
At first the reaction to Saint-Vincent's hyperbolic logarithm was a continuation of studies of quadrature as in Christiaan Huygens (1651)Christiaan Huygens (1651) Theoremata de quadratura hyperboles, ellipsis, et circulari and James Gregory (1667).James Gregory (1667) Quadraturii di Circuli et Hyperbole Subsequently, an industry of making logarithms arose as "logaritmotechnia", the title of works by Nicholas Mercator (1668),Nicholas Mercator (1668) Logarithmo-technia from HathiTrust Euclid Speidell (1688),Euclid Speidell (1688) and John Craig (1710).John Craig (1710) Logarithmotechnia Generalis (Method of making logarithms), Philosophical Transactions of the Royal Society via Biodiversity Heritage Library
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 (0,2). Since the series is negative in (0,1), 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:Derek Thomas Whiteside (1961) "Patterns of mathematical thought in the later seventeenth century", Archive for History of Exact Sciences 1(3):179–388, § III.1 The logarithm as a type-function pp 214–231, quote p 231
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 such that an area of one square unit lies beneath the hyperbola, right of (1,1) 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 (or the knowledge of a sufficiently simple algorithm to generate 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 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.See, e.g., .
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.
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