Editing Isaac Barrow

{{short description|English Christian theologian, and mathematician}}
{{about|the mathematician|his uncle, the bishop|Isaac Barrow (bishop)}}
{{Use dmy dates|date=October 2018}}
{{Infobox scientist
| honorific_prefix  = [[The Reverend]]
| name              = Isaac Barrow
| image             = Isaac Barrow by Mary Beale.jpg
| caption           = Portrait of Barrow by [[Mary Beale]]
| birth_date        = October 1630
| birth_place       = London, England
| nationality       = English
| death_date        = {{death date and age|df=y|1677|5|4|1630|10|01}}
| death_place       = London, England
| field             = [[Mathematics]]
| work_institution  = [[Trinity College, Cambridge]], [[Gresham College]]
| education         = [[Felsted School]], [[Trinity College, Cambridge]]
| doctoral_advisor  = <!--No doctorate at Cambridge until 1919-->
| academic_advisors = [[James Duport]]
| doctoral_students = <!--No doctorate at Cambridge until 1919-->
| notable_students  = [[Isaac Newton]]
| known_for         = [[Fundamental theorem of calculus]]<br>[[Optics]]
| awards            =
| signature         = <!--(filename only)-->
| footnotes         = <small>His mentor was [[James Duport]] who was a classicist, but Barrow really learned his mathematics by working under [[Gilles Personne de Roberval]] in Paris and [[Vincenzo Viviani]] in Florence.</small> 
}}
'''Isaac Barrow''' (October 1630 – 4 May 1677) was an English [[Christian theologian]] and mathematician who is generally given credit for his early role in the development of [[infinitesimal calculus]]; in particular, for proof of the [[fundamental theorem of calculus]].<!--'discovery of the...' is not true. Barrow himself attributed it to James Gregory: "This extremely useful theorem is due to that most learned man, Gregory of Aberdeen"//--><ref name=BarrowGeometricLectures>{{cite book| last1=Child | first1= James Mark | last2=Barrow | first2=Isaac| title= The Geometrical Lectures of Isaac Barrow | year=1916 | url=https://archive.org/details/geometricallectu00barr| publisher= Chicago: [[Open Court Publishing Company]]}}</ref> His work centered on the properties of the [[tangent]]; Barrow was the first to calculate the tangents of the [[kappa curve]]. He is also notable for being the inaugural holder of the prestigious [[Lucasian Professor of Mathematics|Lucasian Professorship of Mathematics]], a post later held by his student, [[Isaac Newton]].

== Life ==
=== Early life and education ===
[[Image:Barrow - Lectiones habitae in scholiis publicis academiae Cantabrigiensis AD 1664, 1683 - 48461.jpg|thumb|''Lectiones habitae in scholiis publicis academiae Cantabrigiensis AD 1664'']]
Barrow was born in London. He was the son of Thomas Barrow, a linen [[draper]] by trade. In 1624, Thomas married Ann, daughter of William Buggin of North Cray, Kent and their son Isaac was born in 1630. It appears that Barrow was the only child of this union—certainly the only child to survive infancy.  Ann died around 1634, and the widowed father sent the lad to his grandfather, Isaac, the Cambridgeshire J.P., who resided at [[Spinney Abbey]].<ref>'The Abbey Scientists' Hall, A.R. p12: London; Roger & Robert Nicholson; 1966</ref>  Within two years, however, Thomas remarried; the new wife was Katherine Oxinden, sister of Henry Oxinden of Maydekin, Kent. From this marriage, he had at least one daughter, Elizabeth (born 1641), and a son, Thomas, who apprenticed to Edward Miller, skinner, and won his release in 1647, emigrating to Barbados in 1680.<ref>{{cite book|last1=Cheesman|first1=Francis|title=Isaac Newton's Teacher|date=2005|publisher=Trafford Publishing|location=Victoria, BC, Canada|isbn=1-4120-6700-6|page=[https://archive.org/details/isaacnewtonsteac0000chee/page/115 115]|edition=first|url=https://archive.org/details/isaacnewtonsteac0000chee/page/115}}</ref>

=== Early career ===
Isaac went to school first at [[Charterhouse School|Charterhouse]] (where he was so turbulent and pugnacious that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to [[Felsted School]], where he settled and learned under the brilliant [[puritan]] Headmaster Martin Holbeach who ten years previously had educated [[John Wallis]].<ref>{{cite book |first=M. R. |last=Craze |title=A History of Felsted School, 1564–1947 |publisher=Cowell |year=1955 }}</ref> Having learnt Greek, Hebrew, Latin and logic at Felsted, in preparation for university studies,<ref>{{cite web |first1=J. J. |last1=O'Connor |first2=E. F. |last2=Robertson |work=School of Mathematics and Statistics [[University of St Andrews]] |url=http://www.gap-system.org/~history/Biographies/Barrow.html |title=gap-system |access-date=1 February 2012 |url-status=dead |archive-url=https://web.archive.org/web/20101226042236/http://www.gap-system.org/~history/Biographies/Barrow.html |archive-date=26 December 2010 }}</ref> he continued his education at [[Trinity College, Cambridge]]; he enrolled there because of an offer of support from an unspecified member of the [[Walpole family]], "an offer that was perhaps prompted by the Walpoles' sympathy for Barrow's adherence to the [[Royalist]] cause."<ref>{{cite book |first=Mordechai |last=Feingold |title=Before Newton: The Life and Times of Isaac Barrow |publisher=Cambridge University Press |year=1990 |page=256 |isbn=9780521306942 |url=https://books.google.com/books?id=jR1rxaob2PUC&pg=PA256 }}</ref> His uncle and namesake [[Isaac Barrow (bishop)|Isaac Barrow]], afterwards [[Bishop of St Asaph]], was a Fellow of [[Peterhouse, Cambridge|Peterhouse]].  He took to hard study, distinguishing himself in classics and mathematics; after taking his degree in 1648, he was elected to a fellowship in 1649.<ref>{{acad|id=BRW643I|name=Barrow, Isaac}}</ref> Barrow received an MA from Cambridge in 1652 as a student of [[James Duport]]; he then resided for a few years in college, and became candidate for the Greek Professorship at Cambridge, but in 1655 having refused to sign the [[Engagement controversy|Engagement to uphold the Commonwealth]], he obtained travel grants to go abroad.<ref>{{cite book |title=A Portrait of Isaac Newton |url=https://archive.org/details/portraitofisaacn00manu |url-access=registration |first=Frank E. |last=Manuel |year=1968 |publisher=Belknap Press, MA |page=[https://archive.org/details/portraitofisaacn00manu/page/92 92] }}</ref>

==== Travel ====
He spent the next four years traveling across France, Italy, and Turkey.  In Turkey he lived in Izmir and studied in Istanbul (then called Smyrna and Constantinople), and after many adventures returned to England in 1659. He was known for his courageousness. Particularly noted is the occasion of his having saved the ship he was upon, by the merits of his own prowess, from capture by [[pirate]]s. He is described as "low in stature, lean, and of a pale complexion," slovenly in his dress, and having a committed and long-standing habit of tobacco use (an ''[[Wikt:inveterate|inveterate]] smoker''). In respect to his courtly activities his aptitude to wit earned him favour with [[Charles II of England|Charles II]], and the respect of his fellow courtiers. In his writings one might find accordingly, a sustained and somewhat stately eloquence. He was an altogether impressive personage of the time, having lived a blameless life in which he exercised his conduct with due care and conscientiousness.<ref>D.R. Wilkins –  [[Trinity College, Dublin]] [http://www.maths.tcd.ie/pub/HistMath/People/Barrow/RouseBall/RB_Barrow.html School of Mathematics]. Retrieved 1 February 2012</ref>

=== Later career ===
==== Work ====
On the [[Restoration (England)|Restoration]] in 1660, he was ordained and appointed to the [[Regius Professor of Greek (Cambridge)|Regius Professorship]] of [[Greek language|Greek]] at the [[University of Cambridge]]. In 1662, he was made professor of [[geometry]] at [[Gresham College]], and in 1663 was selected as the first occupier of the [[Lucasian Professor of Mathematics|Lucasian chair]] at Cambridge. During his tenure of this chair he published two mathematical works of great learning and elegance, the first on geometry and the second on optics. In 1669 he resigned his professorship in favour of [[Isaac Newton]].<ref>For a summary of the Barrow–Newton relationship, see {{cite book |author=Gjersten, Derek |title=The Newton Handbook |year=1986 |location=London |publisher=Routledge & Kegan Paul | pages = 54–55}}</ref> About this time, Barrow composed his ''Expositions of the Creed, The Lord's Prayer, Decalogue, and Sacraments''.  For the remainder of his life he devoted himself to the study of [[divinity]]. He was made a [[Doctor of Divinity]] by Royal mandate in 1670, and two years later Master of [[Trinity College, Cambridge|Trinity College]] (1672), where he founded the library, and held the post until his death.

[[Image:StatueOfIsaacBarrow.jpg|thumb|right|Statue of Isaac Barrow in the chapel of [[Trinity College, Cambridge]]]]
His earliest work was a complete edition of the ''Elements'' of [[Euclid]], which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of the ''Data''. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title ''Lectiones Mathematicae''; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which [[Archimedes]] was led to his chief results. In 1669 he issued his ''Lectiones Opticae et Geometricae''. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of the ''On Conic Sections'' of [[Apollonius of Perga]], and of the extant works of Archimedes and [[Theodosius of Bithynia]].

In the optical lectures many problems connected with the reflection and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflection or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it.  Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the [[René Descartes|Cartesian]] explanation of the [[rainbow]].

Barrow was the first to find the [[integral of the secant function]] in [[Closed-form expression|closed form]], thereby proving a conjecture that was well-known at the time.

==== Death and legacy ====
Barrow died unmarried in London at the early age of 46, and was buried at [[Westminster Abbey]]. [[John Aubrey]], in the [[Brief Lives]], attributes his death to an opium addiction acquired during his residence in Turkey.

Besides the works above mentioned, he wrote other important treatises on mathematics, but in literature his place is chiefly supported by his sermons,<ref>Isaac Barrow, John Tillotson, Abraham Hill – The works of the learned Isaac Barrow ... Printed by J. Heptinstall, for Brabazon Aylmer, 1700 [https://books.google.com/books?id=ZmVZAAAAYAAJ&dq=isaac+barrow+priest&pg=PA217  Published by DR JOHN TILLOTSON THE LORD ARCHBISHOP OF CANTERBURY ]  {&}   Isaac Barrow – The theological works of Isaac Barrow, Volume 1 [https://books.google.com/books?id=_8ctAAAAYAAJ&dq=isaac+barrow+priest&pg=PA318 The University Press, 1830] {&} Isaac Barrow, Thomas Smart Hughes 1831 – ''The Works of Dr. Isaac Barrow: With Some Account of His Life, Summary of Each Discourse, Notes, &c (1831)''- ''Fourth Volume''  [https://archive.org/stream/worksdrisaacbar01hughgoog#page/n5/mode/2up A.J. Valpy]. Retrieved 1 February 2012</ref> which are masterpieces of argumentative eloquence, while his ''Treatise on the Pope's Supremacy'' is regarded as one of the most perfect specimens of controversy in existence. Barrow's character as a man was in all respects worthy of his great talents, though he had a strong vein of eccentricity.

==Calculating tangents==
The geometrical lectures contain some new ways of determining the areas and [[tangent]]s of curves. The most celebrated of these is the method given for the determination of tangents to [[curve]]s, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, [[Johann Hudde|Hudde]] and Sluze were working on the lines suggested by [[Pierre de Fermat|Fermat]] towards the methods of the [[differential calculus]].

<!-- FIGURE: BARROW DIAGRAM goes here
-->
Fermat had observed that the tangent at a point ''P'' on a curve was determined if one other point besides ''P'' on it were known; hence, if the length of the subtangent ''MT'' could be found (thus determining the point ''T''), then the line ''TP'' would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point ''Q'' adjacent to ''P'' were drawn, he got a small [[triangle]] ''PQR'' (which he called the
differential triangle, because its sides ''QR'' and ''RP'' were the differences of the abscissae and ordinates of ''P'' and ''Q''), so that
K
:''TM'' : ''MP'' = ''QR'' : ''RP''.

To find ''QR'' : ''RP'' he supposed that ''x'', ''y''
were the co-ordinates of ''P'', and ''x'' − ''e'', ''y''
− ''a'' those of ''Q'' (Barrow actually used ''p'' for ''x'' and ''m'' for ''y'', but this article uses the standard modern notation).  Substituting the co-ordinates of ''Q'' in the equation of the curve, and neglecting the squares and higher powers of ''e'' and ''a'' as compared with their first powers, he obtained ''e'' : ''a''. The [[ratio]] ''a''/''e'' was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point.

Barrow applied this method to the curves
#''x''<sup>2</sup> (''x''<sup>2</sup> + ''y''<sup>2</sup>) = ''r''<sup>2</sup>''y''<sup>2</sup>, the [[kappa curve]];
#''x''<sup>3</sup> + ''y''<sup>3</sup> = ''r''<sup>3</sup>;
#''x''<sup>3</sup> + ''y''<sup>3</sup> = ''rxy'', called ''[[Folium of Descartes|la galande]]'';
#''y'' = (''r'' − ''x'') tan π''x''/2''r'', the [[quadratrix]]; and
#''y'' = ''r'' tan π''x''/2''r''.

It will be sufficient here to take as an illustration the simpler case of the parabola ''y''<sup>2</sup> = ''px''.
Using the notation given above, we have for the point ''P'', ''y''<sup>2</sup> = ''px''; and for the point ''Q'':
:(''y'' − ''a'')<sup>2</sup> = ''p''(''x'' − ''e'').

Subtracting we get
:2''ay'' − ''a''<sup>2</sup> = ''pe''.
But, if ''a'' be an infinitesimal quantity,
''a''<sup>2</sup> must be infinitely smaller and therefore may be neglected
when compared with the quantities 2''ay'' and ''pe''.  Hence
:2''ay'' = ''pe'', that is, ''e'' : ''a'' = 2''y'' : ''p''.
Therefore,
:''TM'' : ''y'' = ''e'' : ''a'' = 2''y'' : ''p''.
Hence
:TM = 2''y''<sup>2</sup>/''p'' = 2''x''.

This is exactly the procedure of the differential calculus, except that there we
have a rule by which we can get the ratio ''a''/''e'' or ''dy''/''dx'' directly without the labour of going through a calculation similar to the above for every separate case.

== Publications ==
* ''Epitome Fidei et Religionis Turcicae'' (1658)
* "De Religione Turcica anno 1658" (poem)
* ''[https://archive.org/details/b30337094/ Euclidis Elementorum]'' (1659) [in Latin] ''[https://books.google.com/books?id=oTcDAAAAQAAJ&pg=PP5 Euclide's Elements]'' (1660) [in English] translations of [[Euclid's Elements|Euclid's ''Elements'']]
* [https://archive.org/details/lectionesoptic00barr/page/n17/ ''Lectiones Opticae''] (1669)
* [https://archive.org/details/lectionesoptic00barr/page/n175/ ''Lectiones Geometricae''] (1670), translated as [https://books.google.com/books?id=dfA2AAAAMAAJ&pg=PP9 ''Geometrical Lectures''] (1735) by [[Edmund Stone]], later translated as [https://archive.org/details/geometricallectu00barr/page/n9/ ''The Geometrical Lectures of Isaac Barrow''] (1916) by James M. Child <ref>{{cite journal|author=Dresden, Arnold|author-link=Arnold Dresden|title=Review: ''The Geometrical Lectures of Isaac Barrow'', translated, with notes and proofs, by James Mark Child|journal=Bull. Amer. Math. Soc.|year=1918|volume=24|issue=9|pages=454–456|url=https://www.ams.org/journals/bull/1918-24-09/S0002-9904-1918-03122-4/S0002-9904-1918-03122-4.pdf |archive-url=https://web.archive.org/web/20140427025451/http://www.ams.org/journals/bull/1918-24-09/S0002-9904-1918-03122-4/S0002-9904-1918-03122-4.pdf |archive-date=2014-04-27 |url-status=live|doi=10.1090/s0002-9904-1918-03122-4|doi-access=free}}</ref>
* [https://archive.org/details/apolloniiconicam00apol/ ''Apollonii Conica''] (1675) translation of ''[[Apollonius of Perga|Conics]]''
* [https://archive.org/details/archimedisoperaa00arch/ ''Archimedis Opera''] (1675) translation of [[Archimedes]]’s works
* [https://archive.org/details/theodosiisphaeri00theo/ ''Theodosii Sphaerica''] (1675) translation of [[Theodosius' Spherics|Theodosius' ''Spherics'']]
* [https://books.google.com/books?id=MGI9AAAAcAAJ ''A Treatise on the Pope's Supremacy, to which is Added a Discourse Concerning the Unity of the Church''] (1680) ([https://archive.org/details/treatiseofpopess0000barr 1859 edition])
* [https://archive.org/details/mathematicalwor00whewgoog/page/n25/ ''Lectiones Mathematicae''] (1683) translated as [https://archive.org/details/usefulnessofmath0000barr/ ''The Usefulness of Mathematical Learning''] (1734) by John Kirkby 
* [https://quod.lib.umich.edu/e/eebo/a31073.0001.001/1:5.1?view=toc ''Of Contentment, Patience, and Resignation to the Will of God''] (1685)
* ''The works of the learned Isaac Barrow, D.D.'' (1700) [https://archive.org/details/worksoflearnedis01barr/ Vol. 1], [https://archive.org/details/worksoflearnedis23barr/ Vol. 2–3]
* ''The Works of Dr. Isaac Barrow'' (1830), [https://archive.org/details/worksofdrisaacba01barr/ Vol. 1], [https://archive.org/details/worksofdrisaacba02barr_0/ Vol. 2], [https://archive.org/details/worksofdrisaacba03barr_0/ Vol. 3], [https://archive.org/details/worksofdrisaacba04barr_0/ Vol. 4], [https://archive.org/details/worksofdrisaacba05barr/ Vol. 5], [https://archive.org/details/worksofdrisaacba06barr/ Vol. 6], [https://archive.org/details/worksofdrisaacba07barr/ Vol. 7] [sermons and theological essays]

==See also==
* The [[lunar crater]] [[Barrow (crater)|Barrow]] is named after him
* [[Gresham Professor of Geometry|Gresham Professors of Geometry]]

==References==
{{Reflist|30em}}

==Further reading==
* {{Cite SBDEL|wstitle=Barrow, Isaac |short=x}}
* [[W. W. Rouse Ball]]. ''A Short Account of the History of Mathematics'' (4th edition, 1908)
* [[Clinton Bennett]], ''Promise, Predicament and Perplexity: Isaac Barrow (1630–1677) on Islam'' ([[Gorgias Press]], 2022)
* {{cite book |title=Isaac Newton's Teacher |last=Cheesman |first=Francis W. |year=2005 |publisher=Trafford |isbn=9781412067003 |url=https://archive.org/details/isaacnewtonsteac0000chee/ |url-access=registration}}
* {{cite book |title=Before Newton: The life and times of Isaac Barrow |editor-last=Feingold |editor-first=Mordechai |year=1990 |publisher=Cambridge University Press |isbn=9780521306942 |url=https://archive.org/details/beforenewtonlife0000unse/ |url-access=registration }}
* {{cite book |contributor-last=Hill |contributor-first=Abraham |contributor-link=Abraham Hill |editor-last=Hughes |editor-first=Thomas Smart |editor-link=Thomas Smart Hughes |contribution=Biographical Memoir of Dr. Isaac Barrow |title=The Works of Dr. Isaac Barrow |last=Barrow |first=Isaac |volume=1 |contribution-url=https://archive.org/details/worksofdrisaacba01barr/page/n14 |pages=ix–xcii |year=1830 |orig-year=1683 |publisher=A.J. Valpy}}

==External links==
*{{Commons category-inline}}
{{wikiquote}}
{{EB1911 poster|Barrow, Isaac}}
* {{MacTutor Biography|id=Barrow}}
* {{MathGenealogy|id=67643}}
* {{Gutenberg author |id=3430| name=Isaac Barrow}}
* {{Internet Archive author |sname=Isaac Barrow}}
* [http://www.trin.cam.ac.uk/index.php?pageid=172 The Master of Trinity] at [[Trinity College, Cambridge]]
* {{Google books|oBnP1vRoAOcC|Correspondence of Scientific Men of the Seventeenth Century}}
* {{Google books|MwVbAAAAQAAJ|The Usefulness of Mathematical Learning Explained and Demonstrated}}

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{{Succession box|title=[[Regius Professor of Greek (Cambridge)|Regius Professor of Greek]] [[University of Cambridge]]|years=1660–1663|before=[[Ralph Widdrington (academic)|Ralph Widdrington]]|after=[[James Valentine (classicist)|James Valentine]]}}
{{succession box |
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{{Lucasian Professors of Mathematics}}
{{Isaac Newton}}
{{Masters of Trinity College, Cambridge}}
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