View source for History of logarithms

{{Short description|Development of the mathematical function}}
[[File:Mirifici Logarithmorum canonis Descriptio.jpg|thumb|Title page of John Napier's ''[[Mirifici Logarithmorum Canonis Descriptio]]'' from 1614, the first published table of logarithms]]
[[File:A page from Mirifici logarithmorum canonis descriptio.jpg|thumb|A page from Napier's ''Mirifici logarithmorum'' tables, with trigonometric and log trig data for 34 degrees]]
The '''history of [[logarithm]]s''' 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 logarithm]]s were published first in 1614. [[E. W. Hobson]] called it "one of the very greatest scientific discoveries that the world has seen."<ref name=Hobson/>{{rp|p.5}} [[Henry Briggs (mathematician)|Henry Briggs]] introduced [[common logarithm|common (base 10) logarithms]], which were easier to use. [[mathematical table|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 (mathematics)|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]]'' (''Description of the Wonderful Canon of Logarithms'').<ref name=Hobson>{{Citation|author=Ernest William Hobson|title=John Napier and the invention of logarithms, 1614|year=1914|publisher=The University Press|location=Cambridge|url=https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34187/31151005337641.pdf}}</ref> The book contains fifty-seven pages of explanatory matter and ninety pages of tables of trigonometric functions and their [[natural logarithm]]s. 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.<ref name= descriptio-trans>{{cite book|url=http://www.17centurymaths.com/contents/napier/ademonstratiobookone.pdf|title=The Description of the Wonderful Canon of Logarithms|first=John|last=Napier|translator-last1=Wright|translator-first1=Edward|translator-last2=Bruce|translator2-first=Ian|year=1614|publisher=17centurymaths.com|access-date=March 14, 2022}}</ref>

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 (mathematician)|Henry Briggs]], in 1619 in Latin<ref>{{Cite web |title=Mirifi ci logarithmorum canonis constructi, 1619, Edinburgh |url=https://sfdf6bd4395e3be71.jimcontent.com/download/version/1509993513/module/13914029622/name/Napier%2C%20John.%20Constructio.1619.Edinburgh.pdf}}</ref> and then in 1620 in English.<ref name=constructionIA>{{cite book|title=[[Commons:File:The_Construction_of_the_Wonderful_Canon_of_Logarithms.djvu|The Construction of the Wonderful Canon of Logarithms]]|first=John|last=Napier|translator-last1=Macdonald|translator-first1= William Rae|date=1889|orig-date=1620|publisher=Blackwood & Sons|publication-place=Edinburgh|via=Internet Archive}}</ref>

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]].<ref name=Hobson/>{{rp|pp.6–8}} Napier pioneered the use of a decimal point in numerical calculation, something that did not become commonplace until the next century.<ref name=Hobson/>{{rp|pp.21–23}}

Napier's new method for computation gained rapid acceptance. [[Johannes Kepler]] praised it; [[Edward Wright (mathematician)|Edward Wright]], an authority on navigation, translated Napier's ''Descriptio'' into English the next year.<ref name= descriptio-trans/> Briggs extended the concept to the more convenient base 10.<ref name=Hobson/>{{rp|pp.16–18}}

==Common logarithm==
{{main|Common logarithm}}
[[Image:Briggs - Canon logarithmorum pro numeris serie naturali crescentibus ab 1. ad 20000., s.d. - 72507.jpg|thumb|''Canon logarithmorum'']]
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 [[radix|base]] 10, but base 10,000 is ancient and still common in [[myriad#East Asia|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:<ref>Ian Bruce (2000) "Napier’s Logarithms", [[American Journal of Physics]] 68(2):148, doi: 10.1119/1.19387</ref>
:In antiquity Archimedes gave a recipe for reducing multiplication to addition by making use of [[geometric progression]] of numbers and relating them to an [[arithmetic progression]].
In 1616 [[Henry Briggs (mathematician)|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 number]]s 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 method]]s to compute tables of functions.<ref name="ReferenceB">{{cite journal | doi = 10.2307/3621843 | last1 = Bruce | first1 = I. | year = 2002 | title = The Agony and the Ecstasy: The Development of Logarithms by Henry Briggs | journal = The Mathematical Gazette | volume = 86 | issue = 506 | pages = 216–227 | jstor = 3621843 | s2cid = 125835646 }}</ref><ref name="jacques-laporte.org">{{cite web|url=http://www.jacques-laporte.org/The%20method%20of%20Henry%20briggs.htm|title=The Difference Method of Henry Briggs|access-date=2012-04-24|archive-url=https://web.archive.org/web/20120329033744/http://www.jacques-laporte.org/The%20method%20of%20Henry%20briggs.htm|archive-date=2012-03-29|url-status=dead}}</ref>
He also completed a table of [[logarithmic sine]]s and [[logarithmic tangent|tangents]] for the hundredth part of every [[degree (angle)|degree]] to fourteen decimal places, with a table of [[sine|natural sines]] to fifteen places and the [[tangent (trigonometric function)|tangents]] and [[secant (trigonometric function)|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==
{{main|Natural logarithm}}
[[File:hyperbola E.svg|thumb|237px|Graph of the equation {{Math|1=''y'' = 1/''x''}}. Here, [[Euler's number]] '''e''' makes the shaded area equal to 1.|left]]
[[Image:Saint-Vincent - Opus geometricum posthumum ad mesolabium per rationum proportionalium novas proprietates, 1668 - 877986.jpg|thumb|''Opus geometricum posthumum'', 1668]]
In 1649, [[Alphonse Antonio de Sarasa]], a former student of [[Grégoire de Saint-Vincent]],<ref>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).  [https://books.google.com/books?id=xMBhhRtupQsC&pg=PT35 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, [https://books.google.com/books?id=p9E-AAAAcAAJ&pg=PA1 ''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 [https://books.google.com/books?id=p9E-AAAAcAAJ&pg=PT17 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.]<br>See also:  Enrique A. González-Velasco, ''Journey through Mathematics: Creative Episodes in Its History'' (New York, New York:  Springer, 2011), [https://books.google.com/books?id=0sTd4qJgOmsC&pg=PA118 page 118.]</ref> related logarithms to the [[Quadrature (geometry)|quadrature]] of the hyperbola, by pointing out that the [[area]] ''A''(''t'') under the hyperbola from {{nowrap|''x'' {{=}} 1}} to {{nowrap|''x'' {{=}} ''t''}} satisfies<ref>Alphonse Antonio de Sarasa, [https://books.google.com/books?id=p9E-AAAAcAAJ&pg=PA1 ''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).<br><br>
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 [https://books.google.com/books?id=p9E-AAAAcAAJ&pg=PT17 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), [https://books.google.com/books?id=0sTd4qJgOmsC&pg=PA119 pp. 119–120.]</ref>

:<math>A(tu) = A(t) + A(u).</math>

At first the reaction to Saint-Vincent's [[hyperbolic logarithm]] was a continuation of studies of quadrature as in [[Christiaan Huygens]] (1651)<ref>[[Christiaan Huygens]] (1651) ''Theoremata de quadratura hyperboles, ellipsis, et circulari''</ref> and [[James Gregory (mathematician)|James Gregory]] (1667).<ref>[[James Gregory (mathematician)|James Gregory]] (1667) ''Quadraturii di Circuli et Hyperbole''</ref> Subsequently, an industry of making logarithms arose as "logaritmotechnia", the title of works by [[Nicholas Mercator]] (1668),<ref>[[Nicholas Mercator]] (1668) [https://catalog.hathitrust.org/Record/009310290?type%5B%5D=title&lookfor%5B%5D=logarithmotechnia&ft=ft Logarithmo-technia] from [[HathiTrust]]</ref> [[Euclid Speidell]] (1688),<ref>[[Euclid Speidell]] (1688) {{Google books|9l6zSrUQL0UC|Logarithmotechnia: the making of numbers called logarithms}}</ref> and [[John Craig (mathematician)|John Craig]] (1710).<ref>[[John Craig (mathematician)|John Craig]] (1710) ''[https://www.biodiversitylibrary.org/item/196063#page/215/mode/1up Logarithmotechnia Generalis (Method of making logarithms)]'', [[Philosophical Transactions of the Royal Society]] via [[Biodiversity Heritage Library]]</ref>

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:<ref>[[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</ref>
:By the end of the 17th century we can say that much more than being a calculating device suitably well-tabulated, the logarithm function, very much on the model of the hyperbola-area, had been accepted into mathematics. When, in the 18th century, this geometric basis was discarded in favour of a fully analytical one, no extension or reformulation was necessary – the concept of "hyperbola-area" was transformed painlessly into "natural logarithm".

[[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 (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."<ref>H. Eves (1976) ''Introduction to the History of Mathematics'', 4th edition, page 250, [[Holt, Rinehart & Winston]]</ref> [[Carl B. Boyer]] wrote, "Euler was among the first to treat logarithms as exponents, in the manner now so familiar."<ref>C.B. Boyer & Uta C. Merzbach (1989) ''A History of Mathematics'', 2nd edition, page 496 [[John Wiley & Sons]]</ref>

==Pioneers of logarithms==

===Predecessors===
The [[Babylonian mathematics|Babylonians]] sometime in 2000–1600 BC may have invented the [[Multiplication algorithm#Quarter square multiplication|quarter square multiplication]] algorithm to multiply two numbers using only addition, subtraction and a table of quarter squares.<ref>{{citation |title= Quarter Tables Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers |last=McFarland |first=David |url=http://escholarship.org/uc/item/5n31064n |page=1 |year=2007}}</ref><ref>{{cite book| title=Mathematics in Ancient Iraq: A Social History | url=https://archive.org/details/mathematicsancie00robs_842 | url-access=limited |last=Robson |first=Eleanor |page=[https://archive.org/details/mathematicsancie00robs_842/page/n257 227] |year=2008 |isbn= 978-0691091822 }}</ref> 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 [[Multiplicative inverse#Algorithms|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.{{citation needed|date=December 2015}}

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 [[power of two|powers of 2]], this equals the binary logarithm, but it differs from the logarithm for other numbers and it gives [[p-adic order|2-adic order]] rather than the logarithm.<ref>See, e.g., {{citation|title=Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness|volume=22|series=Progress in Computer Science and Applied Logic|first=Igor|last=Shparlinski|publisher=Birkhäuser|year=2013|isbn=978-3-0348-8037-4|page=35|url=https://books.google.com/books?id=z635BwAAQBAJ&pg=PA35}}.</ref><ref>{{citation| contribution=History of Mathematics in India|title=Students' Britannica India: Select essays|editor1-first=Dale|editor1-last=Hoiberg|editor-link1=Dale Hoiberg|editor2-first=Indu|editor2-last=Ramchandani|first=R. C.|last=Gupta|author-link=R.C Gupta|page=329|publisher=Popular Prakashan|year=2000| contribution-url=https://books.google.com/books?id=-xzljvnQ1vAC&q=Virasena+logarithm&pg=PA329}}</ref>

[[Michael Stifel]] published ''Arithmetica integra'' in [[Nuremberg]] in 1544, which contains a table<ref>{{Citation|first=Michaele|last=Stifelio|publisher=Iohan Petreium|location=Nuremberg|year=1544|title=Arithmetica Integra|url = https://books.google.com/books?id=fndPsRv08R0C&pg=RA1-PT419}}</ref> of integers and powers of 2 that has been considered an early version of a table of [[binary logarithm]]s.<ref name="ReferenceA">{{springer  | title=Arithmetic  | id= A/a013260 | last=Bukhshtab  | first=A.A.   | last2=Pechaev | first2=V.I.}}</ref><ref name="Vivian Shaw Groza and Susanne M. Shelley 1972 182">{{Citation|title = Precalculus mathematics|author = Vivian Shaw Groza and Susanne M. Shelley|publisher = Holt, Rinehart and Winston|location=New York|year=1972|isbn=978-0-03-077670-0|page = 182|url = https://books.google.com/books?id=yM_lSq1eJv8C&q=stifel&pg=PA182}}</ref>

In the 16th and early 17th centuries an algorithm called [[prosthaphaeresis]] was used to approximate multiplication and division. This used the trigonometric identity
:<math>\cos\alpha\cos\beta = \frac12[\cos(\alpha+\beta) + \cos(\alpha-\beta)]</math>
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 [[antilogarithm]]s<ref>Jost Bürgi, ''Arithmetische und Geometrische Progress Tabulen'' … [Arithmetic and Geometric Progression Tables … ],  (Prague, (Czech Republic):  University [of Prague] Press, 1620).  Available on-line at:  [http://daten.digitale-sammlungen.de/~db/0008/bsb00082065/images/index.html?id=00082065&fip=eayaxsewqxsfsdreayasdassdassdasen&no=5&seite=7 Bavarian State Library, Germany]<br>  Unfortunately, Bürgi did not include, with his table, instructions for using the table. Neither the table nor the instructions were published, apparently only proof sheets of the table were printed. The contents of the instructions were reproduced in:  Hermann Robert Gieswald, ''Justus Byrg als Mathematiker, und dessen Einleitung zu seinen Logarithmen'' [Justus Byrg as a mathematician, and an introduction to his logarithms] (Danzig, Prussia:  St. Johannisschule, 1856), [https://books.google.com/books?id=xPhSAAAAcAAJ&pg=PA26 pages 26 ff.]</ref> independently of [[John Napier]], whose publication (1614) 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 [[multiplication]]s and [[division (mathematics)|division]]s can be replaced by [[addition]]s and [[subtraction]]s of trigonometrical values.  This procedure achieves the same result that logarithms will a few years later.

===Napier===
[[File:John Napier of Merchiston, 1616.jpg|alt=A baroque picture of a sitting man with a beard.|thumb|[[John Napier]] (1550–1617), the inventor of logarithms]]
{{Main article|Napierian logarithm}}
The method of logarithms was first publicly propounded by [[John Napier]] in 1614, in a book titled ''Mirifici Logarithmorum Canonis Descriptio''.<ref>{{citation |first=John |last=Napier |author-link=John Napier |title=Mirifici Logarithmorum Canonis Descriptio |trans-title=The Description of the Wonderful Rule of Logarithms |language=la |location=Edinburgh, Scotland |publisher=Andrew Hart |year=1614 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN527914568&DMDID=DMDLOG_0001&LOGID=LOG_0001&PHYSID=PHYS_0001 }}</ref>
<ref name= Mirifici-trans>{{cite book|url=http://www.17centurymaths.com/contents/napier/ademonstratiobookone.pdf|title=The Description of the Wonderful Canon of Logarithms|first=John|last=Napier|translator-last1=Wright|translator-first1=Edward|translator-last2=Bruce|translator2-first=Ian|year=1614|publisher=17centurymaths.com|access-date=March 14, 2022}}</ref>
<ref>{{Citation|first=Ernest William |last=Hobson|title=John Napier and the invention of logarithms, 1614|year=1914|publisher=The University Press|location=Cambridge|url=https://archive.org/details/johnnapierinvent00hobsiala}}</ref>

[[Johannes Kepler]], who used logarithm tables extensively to compile his ''Ephemeris'' and therefore dedicated it to Napier,<ref>{{Citation |title=John Napier: Logarithm John |first=Lynne |last=Gladstone-Millar |publisher=National Museums Of Scotland |year=2003 |isbn=978-1-901663-70-9}}, p. 44</ref> remarked:
{{blockquote| ... the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ... instead of rearing up his child for the public benefit he deserted it in the birth.|Johannes Kepler<ref>{{Citation |last=Napier |first=Mark |author-link=Mark Napier (historian) |title=Memoirs of John Napier of Merchiston |publisher=William Blackwood |location=Edinburgh |year=1834 |url=https://books.google.com/books?id=husGAAAAYAAJ&pg=PA1}}, p. 392.</ref>|Rudolphine Tables (1627)}}

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.<ref>{{cite web| url=https://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/john-napier/content-section-1.3| title=Napier's approach to logarithms| access-date=2019-11-22| archive-date=2021-11-22| archive-url=https://web.archive.org/web/20211122134722/https://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/john-napier/content-section-1.3| url-status=dead}}</ref>

By repeated subtractions Napier calculated {{nowrap|(1 − 10<sup>−7</sup>)<sup>''L''</sup>}} for ''L'' ranging from 1 to 100. The result for ''L''=100 is approximately {{nowrap begin}}0.99999 = 1 − 10<sup>−5</sup>{{nowrap end}}. Napier then calculated the products of these numbers with {{nowrap|10<sup>7</sup>(1 − 10<sup>−5</sup>)<sup>''L''</sup>}} for ''L'' from 1 to 50, and did similarly with {{nowrap|0.9998 ≈ (1 − 10<sup>−5</sup>)<sup>20</sup>}} and {{nowrap|0.9 ≈ 0.995<sup>20</sup>}}.<ref>{{cite magazine |url=http://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms |title=Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms |last1=Clark |first1=Kathleen M. |last2=Montelle |first2=Clemency |author2-link= Clemency Montelle |date=2015|magazine=Convergence |publisher=Mathematical Association of America |access-date=2015-12-12}}</ref> 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

:<math>N=10^7 (1-10^{-7})^L. </math>

Napier first called ''L'' an "artificial number", but later introduced the word ''"logarithm"'' to mean a number that indicates a ratio: {{lang|grc|λόγος}} (''[[logos]]'') meaning proportion, and {{lang|grc|ἀριθμός}} (''arithmos'') meaning number. In modern notation, the relation to [[natural logarithm]]s is:
<ref>{{Citation
| title = The Encyclopædia Britannica: a dictionary of arts, sciences, and general literature; the R.S. Peale reprint
| volume = 17
| edition = 9th
| author = William Harrison De Puy
| publisher = Werner Co.
| year = 1893
| page = 179
| url = http://babel.hathitrust.org/cgi/pt?seq=7&view=image&size=100&id=nyp.33433082033444&u=1&num=179
}}</ref>

:<math>L = \log_{(1-10^{-7})} \left( \frac{N}{10^7} \right) \approx 10^7 \log_{1/e} \left( \frac{N}{10^7} \right) = -10^7 \log_e \left( \frac{N}{10^7} \right),</math>

where the very close approximation corresponds to the observation that

:<math>(1-10^{-7})^{10^7} \approx \frac{1}{e}.  </math>

The invention was quickly and widely met with acclaim. The works of [[Bonaventura Cavalieri]] (in Italy), [[Edmund Wingate]] (in France), Xue Fengzuo (in China), and [[Johannes Kepler]]'s ''Chilias logarithmorum'' (in Germany) helped spread the concept further.<ref>{{Citation |last1=Maor |first1=Eli |title=e: The Story of a Number |publisher=[[Princeton University Press]] |isbn=978-0-691-14134-3 |year=2009}}, section 2</ref>

===Euler===
Around 1730, [[Leonhard Euler]] defined the [[exponential function]] and the natural logarithm by<ref name="Whitley2009">
{{Harvard citations
 |last1=Maor |year=2009 |nb=yes |loc=sections 1, 13}}</ref><ref>{{Citation |last1=Eves |first1=Howard Whitley |author1-link=Howard Eves |title=An introduction to the history of mathematics |publisher=Saunders |location=Philadelphia |edition=6th |series=The Saunders series |isbn=978-0-03-029558-4 |year=1992}}, section 9-3</ref><ref>{{Citation | last1=Boyer | first1=Carl B. | author1-link=Carl Benjamin Boyer | title=A History of Mathematics | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-54397-8 | year=1991 | url-access=registration | url=https://archive.org/details/historyofmathema00boye }}, p. 484, 489</ref>
: <math>
\begin{align}
e^x & = \lim_{n \rightarrow \infty} \left( 1 + \frac x n \right)^n, \\[6pt]
\ln(x) & = \lim_{n \rightarrow \infty} n(x^{1/n} - 1).
\end{align}
</math>
In his 1748 textbook [[Introduction to the Analysis of the Infinite]], Euler published the now-standard approach to logarithms via an [[inverse function]]: In chapter 6, "On exponentials and logarithms", he begins with a constant base ''a'' and discusses the [[transcendental function]] <math>y = a^z .</math> Then its inverse is the logarithm:
: ''z'' = log<sub>''a''</sub> ''y''.

==Tables of logarithms==
[[File:Logarithmorum Chilias Prima page 0-67.jpg|thumb|A page from [[Henry Briggs (mathematician)|Henry Briggs]]' 1617 ''Logarithmorum Chilias Prima'' showing the base-10 (common) logarithm of the integers 1 to 67 to fourteen decimal places.]]
[[File:Abramowitz&Stegun.page97.agr.jpg|thumb|Part of a 20th-century table of [[common logarithm]]s in the reference book [[Abramowitz and Stegun]].]]
[[File:APN2002-table3-30deg.tiff|thumb|A page from a table of logarithms of [[trigonometric function]]s from the 2002 [[American Practical Navigator]]. Columns of differences are included to aid [[interpolation]].]]
[[Mathematical table]]s containing [[common logarithm]]s (base-10) were extensively used in computations prior to the advent of [[computer]]s and [[calculator]]s, not only because logarithms convert problems of multiplication and division into much easier addition and subtraction problems, but also for an additional property that is unique to base-10 and proves useful: Any positive number can be expressed as the product of a number from the interval {{closed-open|1,10}} and an integer power of {{math|10.}} This can be envisioned as shifting the decimal separator of the given number to the left yielding a positive, and to the right yielding a negative exponent of {{math|10.}} Only the logarithms of these ''normalized'' numbers (approximated by a certain number of digits), which are called [[significand|mantissas]], need to be tabulated in lists to a similar precision (a similar number of digits). These mantissas are all positive and enclosed in the interval {{closed-open|0,1}}. The common logarithm of any given positive number is then obtained by adding its mantissa to the common logarithm of the second factor. This logarithm is called the ''characteristic'' of the given number. Since the common logarithm of a power of {{math|10}} is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For positive numbers less than {{math|1,}} the characteristic makes the resulting logarithm negative, as required.<ref>E. R. Hedrick, [https://archive.org/details/logarithmictrigo00hedriala Logarithmic and Trigonometric Tables] (Macmillan, New York, 1913).</ref> See [[common logarithm]] for details on the use of characteristics and mantissas.

===Early tables===
[[Michael Stifel]] published ''Arithmetica integra'' in [[Nuremberg]] in 1544 which contains a table<ref>{{Citation|first=Michaele|last=Stifelio|publisher=Iohan Petreium|location=London|year=1544|title=Arithmetica Integra|url = https://books.google.com/books?id=fndPsRv08R0C&pg=RA1-PT419}}</ref> of integers and powers of 2 that has been considered an early version of a logarithmic table.<ref name="ReferenceA"/><ref name="Vivian Shaw Groza and Susanne M. Shelley 1972 182"/>

The first published table of logarithms was in [[John Napier]]'s 1614,  ''Mirifici Logarithmorum Canonis Descriptio''.<ref name= Hobson /> The book contained fifty-seven pages of explanatory matter and ninety pages of tables of trigonometric functions and their [[natural logarithm]]s.<ref name= Mirifici-trans/>

The English mathematician [[Henry Briggs (mathematician)|Henry Briggs]] visited Napier in 1615, and proposed a re-scaling of [[Napier's logarithm]]s to form what is now known as the [[common logarithm|common]] or base-10 logarithms. Napier delegated to Briggs the computation of a revised table, and they later published, in 1617, ''Logarithmorum Chilias Prima'' ("The First Thousand Logarithms"), which gave a brief account of logarithms and a table for the first 1000 integers calculated to the 14th decimal place.

In 1624, Briggs' ''Arithmetica Logarithmica'' appeared in folio as a work containing the logarithms of 30,000 [[natural number]]s 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 method]]s to compute tables of functions.<ref name="ReferenceB"/><ref name="jacques-laporte.org"/>

Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error."<ref>''Athenaeum,'' 15 June 1872.  See also the ''[[Monthly Notices of the Royal Astronomical Society]]'' for May 1872.</ref>  An edition of Vlacq's work, containing many corrections, was issued at [[Leipzig]] in 1794 under the title ''Thesaurus Logarithmorum Completus'' by [[Jurij Vega]].

[[François Callet]]'s seven-place table ([[Paris]], 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of [[interpolation]], which were greatest in the early part of the table, and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by [[Edward Sang]] in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.

Briggs and Vlacq also published original tables of the logarithms of the [[trigonometric function]]s. Briggs completed a table of [[logarithmic sine]]s and [[logarithmic tangent]]s for the hundredth part of every [[degree (angle)|degree]] to fourteen decimal places, with a table of [[sine|natural sines]] to fifteen places and the [[tangent (trigonometric function)|tangents]] and [[secant (trigonometric function)|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''. Tables logarithms of trigonometric functions simplify hand calculations where a function of an angle must be multiplied by another number, as is often the case.

Besides the tables mentioned above, a great collection, called ''Tables du Cadastre,'' was constructed under the direction of [[Gaspard de Prony]], by an original computation, under the auspices of the [[France|French]] republican government of the 1790s.  This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris.  It was begun in 1792, and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years."<ref name="Pronybio">''English Cyclopaedia, Biography,'' Vol. IV., article "Prony."</ref> [[Cubic function|Cubic]] [[interpolation]] could be used to find the logarithm of any number to a similar accuracy.

For different needs, logarithm tables ranging from small handbooks to multi-volume editions have been compiled:<ref>{{citation|title=Orbital Motion|first=A. E.|last=Roy|edition=4th|publisher=CRC Press|year=2004|isbn=9781420056884|page=236|url=https://books.google.com/books?id=Hzv7k2vH6PgC&pg=PA236|quote=In G. Darwin's day, logarithm tables came in different sizes}}</ref>
{| class="wikitable"
|-
! Year !! Author !! Range !! Decimal places !! Note
|-
| 1614|| [[John Napier]], ''Mirifici Logarithmorum Canonis Descriptio'' || 0°–90°, by minutes || 7 || sin(Θ) & ln(sin(Θ)), see image
|-
| 1617|| [[Henry Briggs (mathematician)|Henry Briggs]], ''Logarithmorum Chilias Prima'' || 1–1000 || 14 || see image
|-
| 1624|| Henry Briggs ''Arithmetica Logarithmica'' || 1–20,000, 90,000–100,000 || 14 ||
|-
| 1628|| [[Adriaan Vlacq]] || 20,000–90,000 || 10 || contained only 603 errors<ref>"this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error.", ''Athenaeum,'' 15 June 1872.  See also Glaisher,  in ''Monthly Notices of the Royal Astronomical Society'' for May 1872, pp255-262.</ref>
|-
| 1792–94|| [[Gaspard de Prony]] ''Tables du Cadastre'' || 1–100,000 and 100,000–200,000 || 19 and 24, respectively  || "seventeen enormous folios",<ref name="Pronybio"/> never published
|-
| 1794 || [[Jurij Vega]] ''Thesaurus Logarithmorum Completus'' ([[Leipzig]]) ||  || || corrected edition of Vlacq's work
|-
| 1795 || François Callet ([[Paris]]) || 100,000–108,000 || 7  ||
|-
| 1827 || [[Georg Frederik Ursin]] ||1-100,000 || 6 || Based on earlier works by Briggs, Vlacq, and Vega, but with corrections made and rounded (not truncated) to 6 figures. Widely used by astronomers in 19th century. Full digitised book available <ref name=Ursin>{{Citation|author=Georg Frederik Ursin|title=Ursini Logarithmi |year=1827|publisher=The University of Copenhagen|location=Copenhagen|url=https://books.google.com/books?id=IH04AAAAMAAJ}}</ref>
|-
| 1827 || [[Charles Babbage]] ||1-108,000 || 7 || Based on earlier works by Callet, Gardiner,<ref name=Gardiner1742>{{Citation|author=William Gardiner|title=Tables of logarithms, for all numbers from 1 to 102100, and for the sines and tangents to every ten seconds of each degree in the quadrant; as also, for the sines of the first 72 minutes to every single second: with other useful and necessary tables.|year=1742|publisher=G.Smith|location=London}}</ref> Hutton,<ref name=Hutton1785>{{Citation|author=Charles Hutton|title=Mathematical tables: containing common, hyperbolic, and logistic logarithms, also sines, tangents, secants, and versed-sines, etc. |year=1785|publisher=G. G. J., J. Robinson, and R. Baldwin|location=London}}</ref> Vega, and Vlacq, but carefully corrected. Widely considered at the time to be the most error-free set of tables.<ref name=Roegel2012>{{Citation|author=Denis Roegel|title=A reconstruction of Charles Babbage's table of logarithms (1827) |year=2012|publisher=Loria|location=Lorraine|url=https://locomat.loria.fr/babbage1827/babbage1827doc.pdf}}</ref><ref name=Campbell-Kelly1987>{{Citation|author=M. Campbell-Kelly|title=Charles Babbage's table of logarithms (1827) |year=1987|publisher=University of Warwick. Department of Computer Science.|location=Warwick|url=http://wrap.warwick.ac.uk/60802}}</ref> Later editions had further corrections applied.   Full digitised book available <ref name=Babbage1827>{{Citation|author=Charles Babbage|title=Table of Logarithms of the Natural Numbers, from 1 to 108000 |year=1827|publisher=J.Mawman|location=London|url=https://books.google.com/books?id=piNNAAAAMAAJ}}</ref>
|-
| 1871 || Edward Sang || 1–200,000 || 7  ||
|}

==Slide rule==
{{main article|Slide rule}}
[[File:oughtred.jpg|thumb|right|[[William Oughtred]] (1575–1660), inventor of the circular slide rule.]]
[[File:Slide rules - Museum of the History of Science, Oxford - England.jpg|thumb|A collection of slide rules at the [[Museum of the History of Science]], Oxford]]
The [[slide rule]] was invented around 1620–1630, shortly after [[John Napier]]'s publication of the concept of the [[logarithm]]. [[Edmund Gunter]] of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by [[Edmund Wingate]] (c.1593–1656), an English mathematician, in a book entitled ''L'usage de la reigle de proportion en l'arithmetique & geometrie''. The book contains a double scale, logarithmic on one side, tabular on the other. In 1630, [[William Oughtred]] of Cambridge invented a circular slide rule, and in 1632 combined two handheld [[Gunter's scale|Gunter rule]]s to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, [[Isaac Newton]], Oughtred taught his ideas privately to his students.  Also like Newton, he became involved in a vitriolic controversy over priority, with his one-time student [[Richard Delamain]] and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.

In 1677, [[Henry Coggeshall]] created a two-foot folding rule for timber measure, called the [[Coggeshall slide rule]], expanding the slide rule's use beyond mathematical inquiry.

In 1722, Warner introduced the two- and three-decade scales, and in 1755 Everard included an inverted scale; a slide rule containing all of these scales is usually known as a "polyphase" rule.

In 1815, [[Peter Mark Roget]] invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers.

In 1821, [[Nathaniel Bowditch]], described in the ''[[American Practical Navigator]]'' a "sliding rule" that contained scales trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.

In 1845, Paul Cameron of Glasgow introduced a Nautical Slide-Rule capable of answering navigation questions, including [[right ascension]] and [[declination]] of the sun and principal stars.<ref>"Cameron's Nautical Slide Rule", The Practical Mechanic and Engineer's Magazine, April 1845, p187 and Plate XX-B</ref>

===Modern form===
[[File:Slideruleinuse.wpafb.chap2-2.jpg|thumb|Engineer using a slide rule, with [[mechanical calculator]] in background, mid 20th century]]
A more modern form of slide rule was created in 1859 by French artillery lieutenant [[Amédée Mannheim]], "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around this time that [[engineering]] became a recognized profession, resulting in widespread slide rule use in Europe–but not in the United States.  There Edwin Thacher's cylindrical rule took hold after 1881. The duplex rule was invented by William Cox in 1891, and was produced by [[Keuffel and Esser Co.]] of New York.<ref name="fn_3">{{cite book|url=http://www.mccoys-kecatalogs.com/K&EManuals/4081-3_1943/4081-3_1943.htm |title=The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual |publisher=Keuffel & Esser |first1=Lyman M. |last1=Kells |first2=Willis F. |last2=Kern |first3=James R. |last3=Bland |year=1943 |page=92 |archive-url=https://web.archive.org/web/20090214020502/http://www.mccoys-kecatalogs.com/K%26EManuals/4081-3_1943/4081-3_1943.htm |archive-date=14 February 2009 |url-status=dead }}</ref><ref name="fn_4">''The Polyphase Duplex Slide Rule, A Self-Teaching Manual'', Breckenridge, 1922, p. 20.</ref>

==Impact==
Writing in 1914 on the 300th anniversary of Napier's tables, [[E. W. Hobson]] described logarithms as "providing a great labour-saving instrument for the use of all those who have occasion to carry out extensive numerical calculations" and comparing it in importance to the "Indian invention" of our decimal number system.<ref name=Hobson/>{{rp|p. 5}} 
Napier's improved method of calculation was soon adopted in Britain and Europe. Kepler dedicated his 1620 ''Ephereris'' to Napier, congratulating him on his invention and its benefits to astronomy.<ref name=Hobson/>{{rp|p. 16}} [[Edward Wright (mathematician)|Edward Wright]], an authority on celestial navigation, translated Napier's Latin ''Descriptio'' into English in 1615, shortly after its publication.<ref name= descriptio-trans/> Briggs extended the concept to the more convenient base 10, or [[common logarithm]].<ref name=Hobson/>{{rp|pp.16–18}}

“Probably no work has ever influenced science as a whole, and mathematics in particular, so profoundly as this modest little book [the Descriptio]. It opened the way for the abolition, once and for all, of the infinitely laborious, nay, nightmarish, processes of long division and multiplication, of finding the power and the root of numbers.”<ref>{{cite book|last=Waters|title=The Art of Navigation in England in Elizabethan and Early Stuart Times|date=1958|page=402|url=https://www.sophiararebooks.com/pages/books/4638/john-napier/mirifici-logarithmorum-canonis-constructio-et-eorum-ad-naturales-ipsorum-numeros-habitudines}}</ref>

The logarithm function remains a staple of mathematical analysis, but printed tables of logarithms gradually diminished in importance in the twentieth century as [[mechanical calculator]]s and, later, electronic pocket calculators and computers took over computations that required high accuracy.<ref>[https://www.britannica.com/science/logarithm Logarithms, Encyclopedia Britanica] online edition, accessed June 17, 2022</ref> The introduction of hand-held [[scientific calculator]]s in the 1970s ended the era of slide rules.<ref>[https://www.britannica.com/science/slide-rule Slide rule, Encyclopedia Britanica] online edition, accessed June 17, 2022</ref> [[Logarithmic scale]] graphs are widely used to display data with a wide range. The [[decibel]], a logarithmic unit, is also widely used.

==References==
{{Reflist}}

===Original sources===
* [[Henry Briggs (mathematician)|Henry Briggs]] (1624) ''Arithmetica Logarithmica''
* [[Grégoire de Saint-Vincent]] (1647) ''Opus Geometricum Quadraturae Circuli et Sectionum Coni''
* [[Christiaan Huygens]] (1651) [https://archive.org/stream/uvrescompltespu00haargoog#page/n319/mode/2up Theoremata de quadratura hyperboles, ellipsis et circuli], in ''Oeuvres Complètes'', Tome XI, link from [[Internet Archive]].
* [[James Gregory (mathematician)|James Gregory]] (1667) [https://archive.org/details/ita-bnc-mag-00001357-001/page/n10 Vera Circuli et Hyperbolae Quadratura], Padua: Patavii, via Internet Archive
* [[William Brouncker (mathematician)|William Brouncker]] (1667) [http://www.biodiversityheritagelibrary.org/item/75194#page/267/mode/1up The Squaring of the Hyperbola] {{Webarchive|url=https://web.archive.org/web/20160403195809/http://www.biodiversityheritagelibrary.org/item/75194#page/267/mode/1up |date=2016-04-03 }}, [[Philosophical Transactions of the Royal Society of London]], abridged edition 1809, v. i, pp 233–6, link form [[Biodiversity Heritage Library]].
* [[Nicholas Mercator]] (1668) ''Logarithmitechnia'', London

===Secondary sources===
* Frances Maseres (1791) [https://books.google.com/books?id=FZciAQAAMAAJ Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms], link from [[Google Books]].
* Karl Bopp (1907) "Die Kegelschnitte der Gregorius a St. Vincentio", ''Abhandlungen zum Geschichte der mathematische Wissenschaft'', XX Heft. 
* [[Florian Cajori]] (1913) "History of the exponential and logarithm concepts", [[American Mathematical Monthly]] 20: [https://www.jstor.org/stable/2973509 pages 5 to 14], [https://www.jstor.org/stable/2974078 pages 35 to 47], [https://www.jstor.org/stable/2973441 pages 75 to 84], [https://www.jstor.org/stable/2972960 pages 107 to 117], [https://www.jstor.org/stable/2972412 pages 148 to 151], [https://www.jstor.org/stable/2973069 pages 173 to 182], [https://www.jstor.org/stable/2974104 pages 205 to 210], links from [[Jstor]]
* George A. Gibson (1922) "James Gregory’s mathematical work", [[Proceedings of the Edinburgh Mathematical Society]] 41: 2 to 25 & (second series) 1: 1 to 18.
* [[Christoph J. Scriba]] (1983) "Gregory’s converging double sequence: a new look at the controversy between Huygens and Gregory over the 'analytical' quadrature of the circle", [[Historia Mathematica]] 10: 274 to 85.
* R.C. Pierce (1977) "A brief history of logarithm", [[Two-Year College Mathematics Journal]] 8(1):22–6.
* K.M. Clark (2012) "Priority, parallel discovery, and pre-eminence: Napier, Burgi and the early history of the logarithm relation", ''Revue d’histoire de Mathematique'' 18(2): 223–70.

==External links==
{{wikiquote}}
* Rafael Villareal-Calderon (2008) [https://web.archive.org/web/20160527234454/http://www.math.umt.edu/tmme/vol5no2and3/tmme_vol5nos2and3_a14_pp.337_344.pdf Chopping Logs: A Look at the History and Uses of Logs], ''The Montana Mathematical Enthusiast'' 5(2,3): 237 to 44, link from [[University of Montana]]
* Martin Flashman [http://users.humboldt.edu/flashman/Presentations/HSU%20Colloquia/colloq2_4_99.html The History of Logarithms] from [[Humboldt State University]]
{{History of mathematics}}

[[Category:History of mathematics|Logarithms]]
[[Category:Logarithms]]