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{{short description|Scottish mathematician (1550–1617)}}
{{For|other people with the same name|John Napier (disambiguation)}}
{{Use dmy dates|date=December 2020}}
{{Infobox scientist
| name              = John Napier
| image             = File:John Napier of Merchiston, 1616.jpg
| image_size        = 
| caption           = 1616 portrait of Napier
| birth_date        = {{birth date|1550|2|1|df=yes}}
| birth_place       = [[Edinburgh]], [[Kingdom of Scotland|Scotland]]
| death_date        = {{death date and age|1617|4|4|1550|2|1|df=yes}}
| death_place       = Edinburgh, Scotland
| nationality       = [[Scotland|Scottish]]
| field             = [[Mathematician]]
| alma_mater        = [[University of St Andrews]]
| known_for         = [[Logarithm]]s<br>[[Napier's bones]]<br>[[decimal point|Decimal notation]]<br>[[Spherical trigonometry]]
}}
[[File:Statue of John Napier, Scottish National Portrait Gallery.jpg|thumb|upright|Statue of John Napier, Scottish National Portrait Gallery]]
'''John Napier of Merchiston''' ({{IPAc-en|ˈ|n|eɪ|p|i|ər}} {{respell|NAY|pee|ər}};<ref>[http://dictionary.reference.com/browse/napier "Napier"] {{Webarchive|url=https://web.archive.org/web/20160305013747/http://dictionary.reference.com/browse/napier |date=5 March 2016 }}. ''[[Random House Webster's Unabridged Dictionary]]''.</ref> [[Latinisation of names|Latinized]] as '''Ioannes Neper'''; 1 February 1550&nbsp;– 4 April 1617), nicknamed '''Marvellous Merchiston''', was a Scottish landowner known as a [[mathematician]], [[physicist]], and [[astronomer]]. He was the 8th [[Laird]] of [[Merchiston]].

John Napier is best known as the discoverer of [[logarithm]]s.  He also invented the so-called<!-- nothing to do with slide rule --> "[[Napier's bones]]" and made common the use of the [[decimal point]] in arithmetic and mathematics.

Napier's birthplace, [[Merchiston Castle|Merchiston Tower]] in [[Edinburgh]], is now part of the facilities of [[Edinburgh Napier University]]. There is a memorial to him at [[St Cuthbert's Parish Church|St Cuthbert's]] at the west side of Edinburgh.<ref name="ReferenceA">Monuments and monumental inscriptions in Scotland: The Grampian Society, 1871</ref>

==Life==
Napier's father was [[Archibald Napier (landowner)|Sir Archibald Napier]] of Merchiston Castle, and his mother was Janet Bothwell, daughter of the politician and judge [[Francis Bothwell]],<ref name=EB1911>{{cite EB1911 |wstitle=Napier, John |volume=19 |pages=171—175}}</ref> and a sister of [[Adam Bothwell]] who became the [[Bishop of Orkney]]. Archibald Napier was 16 years old when John Napier was born.<ref name=EB1911/>
[[File:Merchiston castle.jpg|thumb|Merchiston Castle from an 1834 woodcut]]
There are no records of Napier's early learning, but many believe that he was privately tutored during early childhood. At age 13, he was enrolled in [[St Salvator's College, St Andrews]]. Near the time of his matriculation the quality of the education provided by the university was poor, owing in part to the Reformation's causing strife between those of the old faith and the growing numbers of Protestants. There are no records showing that John Napier completed his education at St Andrews.<ref name=":0" /> It is believed he left Scotland to further his education in mainland Europe, following the advice given by his uncle Adam Bothwell in a letter written to John Napier's father on 5 December 1560, saying, "I pray you, sir, to send John to the schools either to [[France]] or [[Flanders]], for he can learn no good at home".<ref>{{Cite book|title=Memoirs of John Napier of Merchiston : his lineage, life, and times, with a history of the invention of logarithms|last=Napier, Mark, 1798-1879.|date=1834|publisher=W. Blackwood|pages=67|oclc=5167671}}</ref> It is not known which university Napier attended in Europe, but when he returned to Scotland in 1571 he was fluent in Greek, a language that was not commonly taught in European universities at the time. There are also no records showing his enrollment in the premier universities in Paris or Geneva during this time.<ref name=":2">{{Cite book|title=John Napier : life, logarithms, and legacy|last=Havil, Julian, 1952-|isbn=9781400852185|location=Princeton|pages=13–17|oclc=889552514|date = 5 October 2014}}</ref><ref name=":0">{{Citation|last1=Rice|first1=Brian|title=John Napier|date=2017|work=The Life and Works of John Napier|pages=12|publisher=Springer International Publishing|isbn=9783319532813|last2=González-Velasco|first2=Enrique|last3=Corrigan|first3=Alexander|doi=10.1007/978-3-319-53282-0_1}}</ref>

In 1571, Napier, aged 21, returned to Scotland, and bought a castle at [[Gartness, Stirling|Gartness]] in 1574. On the death of his father in 1608, Napier and his family moved into [[Merchiston Castle]] in Edinburgh, where he resided the remainder of his life. He had a property within Edinburgh city as well on Borthwick's Close off the [[Royal Mile]].<ref>Cassell's Old and New Edinburgh vol.II p. 242.</ref>

[[Image:St Cuthbert Napier.jpg|thumb|upright|Memorial to John Napier in [[St Cuthbert's Church, Edinburgh]]]]
On 7 June 1596 Napier wrote a paper ''Secret inventions, profitable and necessary in these days for defence of this island''. He describes two kinds of [[Burning glass|burning mirror]] for use against ships at a distance, a special kind of artillery shot, and a musket-proof metal chariot.<ref>[[Thomas Birch]], ''Memoirs of Thomas Birch'', vol. 2 (London, 1754), pp. 28-9.</ref>

Napier died from the effects of [[gout]] at home at [[Merchiston Castle]] at the age of 67. He was buried in the [[kirkyard]] of [[High Kirk of St Giles|St Giles]] in Edinburgh. Following the loss of the kirkyard of St Giles to build [[Parliament House, Edinburgh|Parliament House]], his remains were transferred to an underground vault on the north side of [[St Cuthbert's Parish Church]] at the west side of Edinburgh.<ref name="ReferenceA"/> There is also a wall monument to Napier at St Cuthbert's.<ref name="ReferenceA"/>

Many mathematicians at the time were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. Napier was famous for his devices to assist with these issues of computation. He invented a well-known mathematical artefact, the ingenious numbering rods more quaintly known as "Napier's bones",<ref name="NapiersBones">{{cite journal |last1=Caulfield |first1=Michael J. |title=John Napier: His Life, His Logs, and His Bones - Introduction |journal=Convergence |date=July 2010 |url=https://www.maa.org/press/periodicals/convergence/john-napier-his-life-his-logs-and-his-bones-introduction |access-date=10 March 2019 |publisher=Mathematical Association of America |location=Washington, D.C. |language=en |archive-date=2 May 2023 |archive-url=https://web.archive.org/web/20230502083708/https://www.maa.org/press/periodicals/convergence/john-napier-his-life-his-logs-and-his-bones-introduction |url-status=live }}</ref> that offered mechanical means for facilitating computation.

In addition, Napier recognized the potential of the recent developments in mathematics, particularly those of [[prosthaphaeresis]], decimal fractions, and symbolic index arithmetic, to tackle the issue of reducing computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry. Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant.

==Mathematics==
===Napier's Bones===
[[File:Bones of Napier (board and rods).png|thumb|[[Napier's bones|Napier's Bones]] ]]

His work ''[[Mirifici Logarithmorum Canonis Descriptio]]'' (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables listing the [[natural logarithm]]s of [[trigonometric function]]s.<ref name="discriptio-trans" />{{rp|Ch. III}} The book also has a discussion of theorems in [[spherical trigonometry]], usually known as Napier's Rules of Circular Parts.

Modern English translations of both Napier's books on logarithms and their description can be found on the web, as well as a discussion of [[Napier's bones]] and [[Promptuary]] (another early calculating device).<ref>{{cite web |url=http://www.17centurymaths.com/ |title=17th Century Maths.com |publisher=17centurymaths.com |date=25 March 2011 |accessdate=April 3, 2022 |archive-date=17 May 2011 |archive-url=https://web.archive.org/web/20110517094956/http://www.17centurymaths.com/ |url-status=live }}</ref>

His invention of logarithms was quickly taken up at [[Gresham College]], and prominent English mathematician [[Henry Briggs (mathematician)|Henry Briggs]] visited Napier in 1615. Among the matters they discussed were a re-scaling of Napier's logarithms, in which the presence of the [[e (mathematical constant)|mathematical constant now known as ''e'']] (more accurately, ''e'' times a large power of 10 rounded to an integer) was a practical difficulty. Neither Napier nor Briggs actually discovered the constant ''e''; that discovery was made decades later by [[Jacob Bernoulli]].

Napier delegated to Briggs the computation of a revised table. The computational advance available via logarithms, the inverse of powered numbers or [[exponential notation]], was such that it made calculations by hand much quicker.<ref name = DNB>{{cite DNB|wstitle=Napier, John}}</ref> The way was opened to later scientific advances, in [[astronomy]], [[dynamics (mechanics)|dynamics]], and other areas of [[physics]].

Napier made further contributions. He improved [[Simon Stevin]]'s decimal notation, introducing the [[full stop]] (.) as the delimiter for the fractional part.<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|quote=In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.}}</ref>{{rp|p. 8, archive p. 32)}} [[Lattice multiplication]], used by [[Fibonacci]], was made more convenient by his introduction of [[Napier's bones]], a multiplication tool using a set of numbered rods.

Napier may have worked largely in isolation, but he had contact with [[Tycho Brahe]] who corresponded with his friend [[John Craig (physician)|John Craig]]. Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itself came later); there is a story from [[Anthony à Wood]], perhaps not well substantiated, that Napier had a hint from Craig that [[Longomontanus]], a follower of Brahe, was working in a similar direction. Craig had notes on a method of [[Paul Wittich]] that used [[trigonometric identities]] to reduce a multiplication formula for the [[sine function]] to additions.<ref>{{ODNBweb|id=19758|title=Napier, John|first=George|last=Molland}}</ref>

===Trigonometry===
[[File:Spherical_trigonometry_Napier_right-angled.svg|center|thumb|300x300px]]

When one of the angles, say ''C'', of a spherical triangle is equal to ''{{pi}}''/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set ''a'', ''b'', ''c'', ''A'', ''B''.

Napier<ref>{{cite book |last=Napier |first=J |url=https://books.google.com/books?id=VukHAQAAIAAJ |title=Mirifici Logarithmorum Canonis Constructio |year=1614 |page=50 |author-link=John Napier |access-date=2016-05-14 |archive-url=https://web.archive.org/web/20130430105056/http://books.google.com/books?id=VukHAQAAIAAJ |archive-date=2013-04-30 |url-status=live}} An 1889 translation ''The Construction of the Wonderful Canon of Logarithms'' is available as en e-book from [https://www.abebooks.co.uk/servlet/SearchResults?tn=Construction+Wonderful+Canon+Logarithms Abe Books] {{Webarchive|url=https://web.archive.org/web/20200303190642/https://www.abebooks.co.uk/servlet/SearchResults%3Ftn%3DConstruction%2BWonderful%2BCanon%2BLogarithms|date=2020-03-03}}</ref> provided an elegant [[Mnemonic|mnemonic aid]] for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).

First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with  a  gives ''aCbAcB''. Next replace the parts that are not adjacent to C (that is ''A, c, B'') by their complements and then delete the angle C from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the ''middle'' part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by

* sine of the middle part = the product of the tangents of the adjacent parts
* sine of the middle part = the product of the cosines of the opposite parts

The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine. For an example, starting with the sector containing <math>a</math> we have:

:<math> \sin a = \tan(\pi/2-B)\,\tan b = \cos(\pi/2-c)\, \cos(\pi/2-A)
=\cot B\,\tan b = \sin c\,\sin A. </math>
The full set of rules for the right spherical triangle is (Todhunter,<ref name=todhunter>{{cite book
|last = Todhunter
|first = I.
|author-link = Isaac Todhunter
|title = Spherical Trigonometry
|year = 1886
|publisher = MacMillan
|edition = 5th
|url = http://www.gutenberg.org/ebooks/19770
|access-date = 2013-07-28
|archive-date = 2020-04-14
|archive-url = https://web.archive.org/web/20200414233849/http://www.gutenberg.org/ebooks/19770
|url-status = live
}}</ref> Art.62)

<math>
\begin{alignat}{4}
&\text{(R1)}&\qquad  \cos c&=\cos a\,\cos b,
&\qquad\qquad
&\text{(R6)}&\qquad  \tan b&=\cos A\,\tan c,\\
&\text{(R2)}&  \sin a&=\sin A\,\sin c,
&&\text{(R7)}&  \tan a&=\cos B\,\tan c,\\
&\text{(R3)}&  \sin b&=\sin B\,\sin c,
&&\text{(R8)}&   \cos A&=\sin B\,\cos a,\\
&\text{(R4)}&  \tan a&=\tan A\,\sin b,
&&\text{(R9)}&  \cos B&=\sin A\,\cos b,\\
&\text{(R5)}&  \tan b&=\tan B\,\sin a,
&&\text{(R10)}&   \cos c&=\cot A\,\cot B.
\end{alignat}
</math> 

[[File:Spherical_trigonometry_Napier_quadrantal_01.svg|center|thumb|300x300px|A quadrantal spherical triangle together with Napier's circle for use in his mnemonics]]
A quadrantal spherical triangle is defined  to be a spherical triangle in which one of the sides subtends an angle of ''{{pi}}''/2 radians at the centre of the sphere: on the unit sphere the side has length ''{{pi}}''/2. In the case that the side  ''c'' has length ''{{pi}}''/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle ''A'B'C'<nowiki/>''  with sides ''a',b',c'<nowiki/>''  such that ''A'<nowiki/>'' = ''{{pi}}''&nbsp;−&nbsp;''a'',&nbsp; ''a'''  =&nbsp;''{{pi}}''&nbsp;−&nbsp;''A'' etc. The results are:

<math>
\begin{alignat}{4}
&\text{(Q1)}&\qquad  \cos C&=-\cos A\,\cos B,
&\qquad\qquad
&\text{(Q6)}&\qquad \tan B&=-\cos a\,\tan C,\\
&\text{(Q2)}&  \sin A&=\sin a\,\sin C,
&&\text{(Q7)}&  \tan A&=-\cos b\,\tan C,\\
&\text{(Q3)}&  \sin B&=\sin b\,\sin C,
&&\text{(Q8)}&   \cos a&=\sin b\,\cos A,\\
&\text{(Q4)}&  \tan A&=\tan a\,\sin B,
&&\text{(Q9)}&  \cos b&=\sin a\,\cos B,\\
&\text{(Q5)}&  \tan B&=\tan b\,\sin A,
&&\text{(Q10)}&   \cos C&=-\cot a\,\cot b.
\end{alignat}
</math>

'''Logarithm''' 

Given a positive [[real number]] ''b'' such that ''b'' ≠ 1, the ''[[logarithm]]'' of a positive real number ''x'' with respect to base ''b'' is the exponent by which ''b'' must be raised to yield ''x''. In other words, the logarithm of ''x'' to base ''b'' is the unique real number ''y'' such that ''b''<sup>''y''</sup> = ''x''.

The logarithm is denoted "log<sub>''b''</sub> ''x''" (pronounced as "the logarithm of ''x'' to base ''b''", "the base-''b'' logarithm of ''x''", or most commonly "the log, base ''b'', of ''x''").

An equivalent and more succinct definition is that the function <math>f \colon x\to log_{b} x</math> is the [[inverse function]] to the function <math>f \colon x\to b^{x}.</math>

==Theology==
[[File:Logarithms book Napier.jpg|thumb|upright|Cover of ''Mirifici logarithmorum canonis descriptio'' (1614)]]
[[File:Napier log table 45 degrees.agr.jpg|thumb|upright|Page from ''Mirifici logarithmorum canonis descriptio,'' for 45 degrees]]
Napier had an interest in the ''[[Book of Revelation]]'', from his student days at [[St Salvator's College, St Andrews]]. Under the influence of the sermons of [[Christopher Goodman]], he developed a strongly anti-papal reading, going as far as to say that the Pope was the [[Antichrist]] in some of his writings.<ref name="DNB"/><ref name="Rice 2017 65">{{Citation|last1=Rice|first1=Brian|title=John Napier|date=2017|work=The Life and Works of John Napier|pages=1–60|publisher=Springer International Publishing|isbn=9783319532813|last2=González-Velasco|first2=Enrique|last3=Corrigan|first3=Alexander|doi=10.1007/978-3-319-53282-0_1}}</ref>

Napier regarded ''A Plaine Discovery of the Whole Revelation of St. John'' (1593) as his most important work. It was written in English, unlike his other publications, in order to reach the widest audience and so that, according to Napier, "the simple of this island may be instructed".<ref name="Rice 2017 65"/> ''A Plaine Discovery'' used mathematical analysis of the ''Book of Revelation'' to attempt to predict the date of the [[Apocalypse]]. Napier identified events in chronological order which he believed were parallels to events described in the ''Book of Revelation'' believing that ''Revelation''<nowiki/>'s structure implied that the prophecies would be fulfilled incrementally.<ref name="Rice 2017 65"/> In this work Napier dated [[the seventh trumpet]] to 1541, and predicted the end of the world would occur in either 1688 or 1700. Napier did not believe that people could know the true date of the Apocalypse, but claimed that since the Bible contained so many clues about the end, God wanted the Church to know when the end was coming.<ref name="Rice 2017 65"/>

In his dedication of the ''Plaine Discovery'' to [[James VI]], dated 29 Jan 1594,<ref>{{cite encyclopedia|url=https://www.britannica.com/topic/Plaine-Discovery-of-the-Whole-Revelation-of-Saint-John|title=Plaine Discovery of the Whole Revelation of Saint John, work by Napier|encyclopedia=Encyclopædia Britannica|access-date=2024-06-12}}</ref> Napier urged the king to see "that justice be done against the enemies of God's church," and counselled the King "to reform the universal enormities of his country, and first to begin at his own house, family, and court." The volume includes nine pages of Napier's English verse. It met with success at home and abroad. In 1600 [[Michiel Panneel]] produced a Dutch translation, and this reached a second edition in 1607. In 1602 the work appeared at [[La Rochelle]] in a French version, by [[Georges Thomson]], revised by Napier, and that also went through several editions (1603, 1605, and 1607). A new edition of the English original was called for in 1611, when it was revised and corrected by the author, and enlarged by the addition of ''With a resolution of certain doubts, moved by some well affected brethren.'';<ref>[http://lib.ugent.be/en/catalog/rug01:001591742 ''A plaine discovery ...''] {{Webarchive|url=https://web.archive.org/web/20150207235346/http://lib.ugent.be/en/catalog/rug01:001591742 |date=7 February 2015 }}, Universiteitsbibliotheek Gent</ref> this appeared simultaneously at Edinburgh and London. The author stated that he still intended to publish a Latin edition, but it never appeared. A German translation, by [[Leo de Dromna]], of the first part of Napier's work appeared at Gera in 1611, and of the whole by [[Wolfgang Meyer]] at [[Frankfurt-am-Main]], in 1615.<ref name = DNB/> Among Napier's followers was Matthew Cotterius ([[Matthieu Cottière]]).<ref>Arthur H. Williamson, ''Scottish National Consciousness in the Age of James VI'' (1979), p.&nbsp;34.</ref>

==Interest in the occult==
In addition to his mathematical and religious interests, Napier was often perceived as a magician, and is thought to have dabbled in [[alchemy]] and [[necromancy]]. It was said that he would travel about with a black spider in a small box, and that his black [[cockerel]] was his [[familiar spirit]].<ref>{{cite news |last=Johnston |first=Ian |url=http://news.scotsman.com/scitech.cfm?id=523542005 |title=Scots genius who paved way for Newton's discoveries |newspaper=[[The Scotsman]] |date=14 May 2005 |access-date=29 March 2011 |archive-date=27 December 2005 |archive-url=https://web.archive.org/web/20051227004113/http://news.scotsman.com/scitech.cfm?id=523542005 |url-status=live }}</ref><ref>{{cite news |last=Springer |first=Will |url=http://heritage.scotsman.com/myths.cfm?id=41962005 |title=Napier's wizard roots&nbsp;– article about Napier's interest in the occult |newspaper=The Scotsman |date=14 February 2005 |access-date=29 March 2011 |archive-date=21 October 2007 |archive-url=https://web.archive.org/web/20071021012042/http://heritage.scotsman.com/myths.cfm?id=41962005 |url-status=live }}</ref><ref>{{cite web|url=https://www.nms.ac.uk/explore-our-collections/stories/science-and-technology/napiers-bones/|title=Napier's bones|publisher=National Museums Scotland|access-date=23 December 2017|archive-date=24 September 2023|archive-url=https://web.archive.org/web/20230924002353/https://www.nms.ac.uk/explore-our-collections/stories/science-and-technology/napiers-bones/|url-status=live}}</ref>

Some of Napier's neighbours accused him of being a sorcerer and in league with the devil, believing that all of the time he spent in his study was being used to learn the black art. These rumours were stoked when Napier used his black cockerel to catch a thief. Napier told his servants to go into a darkened room and pet the cockerel, claiming the bird would crow if they were the one who stole his property. Unbeknownst to the servants, Napier had covered the bird with soot and when the servants emerged from the room, Napier inspected their hands to find the one who had been too afraid to touch the rooster.<ref>{{Cite book|title=John Napier : life, logarithms, and legacy|last=Havil, Julian, 1952-|isbn=9781400852185|location=Princeton|pages=21|oclc=889552514|date = 5 October 2014}}</ref>

Another act which Napier is reported to have carried out, which may have seemed mystical to the locals, was when Napier removed the pigeons from his estate since they were eating his grain. Napier caught the pigeons by strewing grain laced with alcohol throughout the field and then capturing the pigeons once they were too drunk to fly away.<ref>{{Cite book|title=John Napier|last=Havil|first=Julian|date=2014-01-31|publisher=Princeton University Press|isbn=9781400852185|location=Princeton|pages=20|doi = 10.1515/9781400852185}}</ref>

A contract still exists for a [[Treasure hunting|treasure hunt]], made between Napier and [[Robert Logan of Restalrig]]. Napier was to search [[Fast Castle]] for treasure allegedly hidden there, wherein it is stated that Napier should "do his utmost diligence to search and seek out, and by all craft and ingine to find out the same, or make it sure that no such thing has been there."<ref name = DNB/><ref name=":1" /> This contract was never fulfilled by Napier, and no gold was found when the Edinburgh Archaeological Field society excavated the castle between 1971 and 1986.<ref name=":1">{{Cite book|title=John Napier : life, logarithms, and legacy|last=Havil, Julian, 1952-|isbn=9781400852185|location=Princeton|pages=22, 23|oclc=889552514|date = 5 October 2014}}</ref>

==Influence==
Among Napier's early followers were the instrument makers [[Edmund Gunter]] and [[John Speidell]].<ref>{{cite book|title=The London encyclopaedia: or Universal dictionary of science, art, literature, and practical mechanics, comprising a popular view of the present state of knowledge|url=https://books.google.com/books?id=u1IPAQAAMAAJ&pg=PA498|access-date=23 May 2012|date=1829|page=498}}</ref><ref>{{cite book|author=Florian Cajori|title=A History of Mathematics|url=https://books.google.com/books?id=mGJRjIC9fZgC&pg=PA152|access-date=23 May 2012|date=1991|publisher=American Mathematical Soc.|isbn=978-0-8218-2102-2|page=152}}</ref><ref>{{cite book|author=Ivor Grattan-Guinness|title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences|url=https://books.google.com/books?id=f5FqsDPVQ2MC&pg=PA1129|access-date=23 May 2012|date=1 August 2003|publisher=JHU Press|isbn=978-0-8018-7397-3|page=1129|author-link=Ivor Grattan-Guinness}}</ref> The development of logarithms is given credit as the largest single factor in the general adoption of [[decimal arithmetic]].<ref>{{cite book|author=David Eugene Smith|title=History of Mathematics|url=https://archive.org/details/historyofmathema0002smit|url-access=registration|access-date=23 May 2012|date=1 June 1958|publisher=Courier Dover Publications|isbn=978-0-486-20430-7|page=[https://archive.org/details/historyofmathema0002smit/page/244 244]}}</ref> The ''Trissotetras'' (1645) of [[Thomas Urquhart]] builds on Napier's work, in [[trigonometry]].<ref>{{cite book|author1=Garrett A. Sullivan|author2=Alan Stewart|title=The Encyclopedia of English Renaissance Literature|url=https://books.google.com/books?id=R7UeL_0Pu3oC&pg=PA995|access-date=23 May 2012|date=1 February 2012|publisher=John Wiley & Sons|isbn=978-1-4051-9449-5|page=995}}</ref>

Henry Briggs was an early adopter of the Napierian logarithm. He later computed a new table of logarithms to base 10, accurate to 14 decimal places.<ref>{{Citation|last1=Rice|first1=Brian|title=John Napier|date=2017|work=The Life and Works of John Napier|pages=37|publisher=Springer International Publishing|isbn=9783319532813|last2=González-Velasco|first2=Enrique|last3=Corrigan|first3=Alexander|doi=10.1007/978-3-319-53282-0_1}}</ref>

==Eponyms==
An alternative unit to the [[decibel]] used in [[electrical engineering]], the [[neper]], is named after Napier, as is [[Edinburgh Napier University]] in Edinburgh, Scotland.

The crater [[Neper (crater)|Neper]] on the Moon is named after him.<ref>[http://planetarynames.wr.usgs.gov/jsp/FeatureNameDetail.jsp?feature=64377 Neper] {{Webarchive|url=https://web.archive.org/web/20100616095731/http://planetarynames.wr.usgs.gov/jsp/FeatureNameDetail.jsp?feature=64377 |date=16 June 2010 }} Gazetteer of Planetary Nomenclature&nbsp;– USGS Astrogeology</ref>

In French, Spanish and Portuguese, the [[natural logarithm]] is named after him (respectively, ''Logarithme Népérien'' and ''Logaritmos Neperianos'' for Spanish and Portuguese).  In Finnish and Italian, the mathematical constant [[E (mathematical constant)|''e'']] is named after him (''Neperin luku'' and ''Numero di Nepero'').

==Family==
In 1572, Napier married 16-year-old Elizabeth, daughter of James Stirling, the 4th [[Laird]] of [[Keir House|Keir]] and of [[Cadder]].<ref>{{Citation|last1=Rice|first1=Brian|title=John Napier|date=2017|work=The Life and Works of John Napier|pages=12–14|publisher=Springer International Publishing|isbn=9783319532813|last2=González-Velasco|first2=Enrique|last3=Corrigan|first3=Alexander|doi=10.1007/978-3-319-53282-0_1}}</ref> They had two children. Elizabeth died in 1579, and Napier then married Agnes Chisholm, with whom he had ten more children.

Napier's father-in-law, Sir James Chisholm of Cromlix, was one of many excommunicated by the General Assembly of the Presbyterian party following the [[Spanish blanks plot]]. Napier sat on the General Assembly that excommunicated the plotters, and petitioned the King [[James VI and I]] to enforce the punishment on the plotters, but was ultimately ignored since the King believed the ministers were acting cruelly, and was in favor of pursuing policies of more appeasement.<ref name="Rice 2017 65"/><ref>{{Cite journal|last=Clouse|first=Robert G.|date=April 1974|title=John Napier and Apocalyptic Thought|journal=Sixteenth Century Journal|volume=5|issue=1|pages=105|doi=10.2307/2539589|issn=0361-0160|jstor=2539589}}</ref>

His half-brother (through his father's remarriage) was [[Alexander Napier, Lord Laurieston]].

==List of works==
*(1593) ''A Plaine Discovery of the Whole Revelation of St. John''
*(1614) ''[https://www.digitale-sammlungen.de/en/view/bsb11110567?page=11 Mirifici logarithmorum canonis descriptio] {{Webarchive|url=https://web.archive.org/web/20230427013211/https://www.digitale-sammlungen.de/en/view/bsb11110567?page=11 |date=27 April 2023 }}'' – [[Edward Wright (mathematician)|Edward Wright]]'s English translation was published in 1616.<ref name=discriptio-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|archive-date=22 April 2023|archive-url=https://web.archive.org/web/20230422184237/http://17centurymaths.com/contents/napier/ademonstratiobookone.pdf|url-status=live}}</ref>
*(1617) ''[https://archive.org/details/rabdologiaseunu00napi Rabdologiæ seu Numerationis per Virgulas libri duo]'' ([https://books.google.com/books?id=Vhc1AAAAcAAJ Google Books link])(published posthumously) [[Rabdology|Rabdology (Wikipedia)]]
*(1619) ''[https://books.google.com/books?id=VukHAQAAIAAJ Mirifici logarithmorum canonis constructio]'' –written before the ''Descriptio'', but published posthumously by his son Robert. Translated into English in 1889.<ref name=constructio-en1889IA>{{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=1619|publisher=Blackwood & Sons|publication-place=Edinburgh|via=Internet Archive}}</ref>
*(1839) ''[https://books.google.com/books?id=Iu0HAAAAQAAJ&pg=3 De arte logistica]''

==See also==
*[[List of colleges and universities named after people]]
*[[Location arithmetic]]
*[[Spherical trigonometry#Napier's analogies|Napier's analogies]]
*[[Napierian logarithm]]
*[[Rabdology]]

==Notes==
{{Reflist|30em}}

==References==
*{{MacTutor Biography|id=Napier}}
*Diploudis, Alexandros. [https://web.archive.org/web/20070203233116/http://www.cee.hw.ac.uk/~greg/calculators/napier/index.html Undusting Napier's Bones.] Heriot-Watt University, 1997
*"John Napier." ''Math & Mathematicians: The History of Math Discoveries around the World.'' 2 vols. U*X*L, 1999
*[http://www.thocp.net/biographies/napier_john.html John Napier] {{Webarchive|url=https://web.archive.org/web/20150908025125/http://www.thocp.net/biographies/napier_john.html |date=8 September 2015 }} ''The History of Computing Project''
*[http://johnnapier.com/ John Napier—Short biography and translation of work on logarithms] {{Webarchive|url=https://web.archive.org/web/20081228100129/http://www.johnnapier.com/ |date=28 December 2008 }}
*[http://www.rwgrayprojects.com/rbfnotes/trig/strig/strig.html Intro to Spherical Trig.] {{Webarchive|url=https://web.archive.org/web/20060329153606/http://www.rwgrayprojects.com/rbfnotes/trig/strig/strig.html |date=29 March 2006 }} Includes discussion of The Napier circle and Napier's rules
*[http://eebo.chadwyck.com/home EEBO (Early English Books Online)] {{Webarchive|url=https://web.archive.org/web/20171228233712/http://eebo.chadwyck.com/home |date=28 December 2017 }} has electronic copies of some of his work, in facsimiles of editions of Napier's time (subscription or Athens login required)
*[http://www.17centurymaths.com/contents/napiercontents.html Mirifici logarithmorum canonis descriptio] {{Webarchive|url=https://web.archive.org/web/20190911095649/http://www.17centurymaths.com/contents/napiercontents.html |date=11 September 2019 }} ''Descriptio'' ''De Arte Logistica'' English Translation by Ian Bruce

'''Attribution'''
* {{DNB|wstitle=Napier, John}}

==Further reading==
* {{cite wikisource|first=John|last=Napier|wslink=The Construction of the Wonderful Canon of Logarithms}}, the 1889 English translation.
*{{cite book|title=Memoirs of John Napier of Merchiston, his lineage, life and times, with a history of the invention of logarithms|author=Napier, Mark|publisher=William Blackwood|place=Edinburgh|date=1834|url=https://archive.org/details/memoirsofjohnnap00napi }}

==External links==
*{{commons category-inline|John Napier (mathematician)}}
*{{wikisource author-inline}}
*{{wikiquote-inline}}

{{Scientists whose names are used as non SI units}}
{{Authority control}}

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[[Category:Burials at the kirkyard of St Giles]]
[[Category:Clan Napier|John]]
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[[Category:Writers from Edinburgh]]