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==History== {{Main|History of logarithms}} The history of logarithms in seventeenth-century Europe saw the discovery of a new [[function (mathematics)|function]] that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by [[John Napier]] in 1614, in a book titled ''[[Mirifici Logarithmorum Canonis Descriptio]]'' (''Description of the Wonderful Canon of Logarithms'').<ref>{{citation<!-- CB don't touch--> |first=John |last=Napier |author-link=John Napier |title=Mirifici Logarithmorum Canonis Descriptio |trans-title=The Description of the Wonderful Canon of Logarithms |language=la |location=Edinburgh, Scotland |publisher=Andrew Hart |year=1614 |url=https://archive.org/details/mirificilogarit00napi/ }} {{pb}} The sequel ''... Constructio'' was published posthumously: {{pb}} {{citation |first1=John |last1=Napier |author-link=John Napier |first2=Henry |last2=Briggs |author2-link= Henry Briggs (mathematician) |title=Mirifici Logarithmorum Canonis Constructio |trans-title=The Construction of the Wonderful Canon of Logarithms |language=la |location=Edinburgh |publisher=Andrew Hart |year=1619 |url=https://archive.org/details/bub_gb_vgq2sG0g8BAC }} {{pb}} Ian Bruce has made an [http://17centurymaths.com/contents/napiercontents.html annotated translation of both books] (2012), available from 17centurymaths.com.</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> Prior to Napier's invention, there had been other techniques of similar scopes, such as the [[prosthaphaeresis]] or the use of tables of progressions, extensively developed by [[Jost Bürgi]] around 1600.<ref name="folkerts">{{citation | last1 = Folkerts | first1 = Menso | last2 = Launert | first2 = Dieter | last3 = Thom | first3 = Andreas | arxiv = 1510.03180 | doi = 10.1016/j.hm.2016.03.001 | issue = 2 | journal = [[Historia Mathematica]] | mr = 3489006 | pages = 133–147 | title = Jost Bürgi's method for calculating sines | volume = 43 | year = 2016| s2cid = 119326088 }}</ref><ref>{{mactutor|id=Burgi|title=Jost Bürgi (1552 – 1632)}}</ref> Napier coined the term for logarithm in Middle Latin, {{lang|la|logarithmus}}, literally meaning {{gloss|ratio-number}}, derived from the Greek {{Transliteration|grc|logos}} {{gloss|proportion, ratio, word}} + {{Transliteration|grc|arithmos}} {{gloss|number}}. The [[common logarithm]] of a number is the index of that power of ten which equals the number.<ref>William Gardner (1742) ''Tables of Logarithms''</ref> Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by [[Archimedes]] as the "order of a number".<ref>{{citation | last = Pierce | first = R. C. Jr. | date = January 1977 | doi = 10.2307/3026878 | issue = 1 | journal = [[The Two-Year College Mathematics Journal]] | jstor = 3026878 | pages = 22–26 | title = A brief history of logarithms | volume = 8}}</ref> The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities.<ref>Enrique Gonzales-Velasco (2011) ''Journey through Mathematics – Creative Episodes in its History'', §2.4 Hyperbolic logarithms, p. 117, Springer {{isbn|978-0-387-92153-2}}</ref> Such methods are called [[prosthaphaeresis]]. Invention of the [[function (mathematics)|function]] now known as the [[natural logarithm]] began as an attempt to perform a [[quadrature (mathematics)|quadrature]] of a rectangular [[hyperbola]] by [[Grégoire de Saint-Vincent]], a Belgian Jesuit residing in Prague. Archimedes had written ''[[The Quadrature of the Parabola]]'' in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a [[geometric progression]] in its [[argument of a function|argument]] and an [[arithmetic progression]] of values, prompted [[A. A. de Sarasa]] to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in [[prosthaphaeresis]], leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by [[Christiaan Huygens]], and [[James Gregory (mathematician)|James Gregory]]. The notation {{math|Log ''y''}} was adopted by [[Gottfried Wilhelm Leibniz]] in 1675,<ref>[[Florian Cajori]] (1913) "History of the exponential and logarithm concepts", [[American Mathematical Monthly]] 20: 5, 35, 75, 107, 148, 173, 205</ref> and the next year he connected it to the [[integral calculus|integral]] <math display="inline">\int \frac{dy}{y} .</math> Before Euler developed his modern conception of complex natural logarithms, [[Roger Cotes#Mathematics|Roger Cotes]] had a nearly equivalent result when he showed in 1714 that<ref>{{citation |last1=Stillwell |first1=J. |title=Mathematics and Its History |date=2010 |publisher=Springer |edition=3rd}}</ref> <math display="block">\log(\cos \theta + i\sin \theta) = i\theta.</math>
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