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Isaac Barrow

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This article is about the mathematician. For his uncle, the bishop, seeIsaac Barrow (bishop).

The Reverend
Isaac Barrow
Portrait of Barrow byMary Beale
BornOctober 1630
London, England
Died4 May 1677(1677-05-04)(aged 46)
London, England
NationalityEnglish
EducationFelsted School,Trinity College, Cambridge
Known forFundamental theorem of calculus
Optics
Scientific career
FieldsMathematics
InstitutionsTrinity College, Cambridge,Gresham College
Academic advisorsJames Duport
Notable studentsIsaac Newton
Notes
His mentor wasJames Duportwho was a classicist, but Barrow really learned his mathematics by working underGilles Personne de Robervalin Paris andVincenzo Vivianiin Florence.

Isaac Barrow(October 1630 – 4 May 1677) was an EnglishChristian theologianand mathematician who is generally given credit for his early role in the development ofinfinitesimal calculus;in particular, for proof of thefundamental theorem of calculus.[1]His work centered on the properties of thetangent;Barrow was the first to calculate the tangents of thekappa curve.He is also notable for being the inaugural holder of the prestigiousLucasian Professorship of Mathematics,a post later held by his student,Isaac Newton.

Life

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Early life and education

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Lectiones habitae in scholiis publicis academiae Cantabrigiensis AD 1664

Barrow was born in London. He was the son of Thomas Barrow, a linendraperby 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 atSpinney Abbey.[2]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.[3]

Early career

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Isaac went to school first atCharterhouse(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 toFelsted School,where he settled and learned under the brilliantpuritanHeadmaster Martin Holbeach who ten years previously had educatedJohn Wallis.[4]Having learnt Greek, Hebrew, Latin and logic at Felsted, in preparation for university studies,[5]he continued his education atTrinity College, Cambridge;he enrolled there because of an offer of support from an unspecified member of theWalpole family,"an offer that was perhaps prompted by the Walpoles' sympathy for Barrow's adherence to theRoyalistcause. "[6]His uncle and namesakeIsaac Barrow,afterwardsBishop of St Asaph,was a Fellow ofPeterhouse.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.[7]Barrow received an MA from Cambridge in 1652 as a student ofJames 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 theEngagement to uphold the Commonwealth,he obtained travel grants to go abroad.[8]

Travel

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He spent the next four years travelling across France, Italy, 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 bypirates.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 (aninveteratesmoker). In respect to his courtly activities his aptitude to wit earned him favour withCharles 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.[9]

Later career

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Work

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On theRestorationin 1660, he was ordained and appointed to theRegius ProfessorshipofGreekat theUniversity of Cambridge.In 1662, he was made professor ofgeometryatGresham College,and in 1663 was selected as the first occupier of theLucasian chairat 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 ofIsaac Newton.[10]About this time, Barrow composed hisExpositions of the Creed, The Lord's Prayer, Decalogue, and Sacraments.For the remainder of his life he devoted himself to the study ofdivinity.He was made aDoctor of Divinityby Royal mandate in 1670, and two years later Master ofTrinity College(1672), where he founded the library, and held the post until his death.

Statue of Isaac Barrow in the chapel ofTrinity College, Cambridge

His earliest work was a complete edition of theElementsofEuclid,which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of theData.His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the titleLectiones 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 whichArchimedeswas led to his chief results. In 1669 he issued hisLectiones 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 theOn Conic SectionsofApollonius of Perga,and of the extant works of Archimedes andTheodosius 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 theCartesianexplanation of therainbow.

Barrow was the first to find theintegral of the secant functioninclosed form,thereby proving a conjecture that was well-known at the time.

Death and legacy

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Barrow died unmarried in London at the early age of 46, and was buried atWestminster Abbey.John Aubrey,in theBrief 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,[11]which are masterpieces of argumentative eloquence, while hisTreatise on the Pope's Supremacyis 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

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The geometrical lectures contain some new ways of determining the areas andtangentsof curves. The most celebrated of these is the method given for the determination of tangents tocurves,and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow,Huddeand Sluze were working on the lines suggested byFermattowards the methods of thedifferential calculus.

Fermat had observed that the tangent at a pointPon a curve was determined if one other point besidesPon it were known; hence, if the length of the subtangentMTcould be found (thus determining the pointT), then the lineTPwould be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a pointQadjacent toPwere drawn, he got a smalltrianglePQR(which he called the differential triangle, because its sidesQRandRPwere the differences of the abscissae and ordinates ofPandQ), so that K

TM:MP=QR:RP.

To findQR:RPhe supposed thatx,y were the co-ordinates ofP,andx−e,y −athose ofQ(Barrow actually usedpforxandmfory,but this article uses the standard modern notation). Substituting the co-ordinates ofQin the equation of the curve, and neglecting the squares and higher powers ofeandaas compared with their first powers, he obtainede:a.Theratioa/ewas 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

  1. x2(x2+y2) =r2y2,thekappa curve;
  2. x3+y3=r3;
  3. x3+y3=rxy,calledla galande;
  4. y= (r−x) tan πx/2r,thequadratrix;and
  5. y=rtan πx/2r.

It will be sufficient here to take as an illustration the simpler case of the parabolay2=px. Using the notation given above, we have for the pointP,y2=px;and for the pointQ:

(y−a)2=p(x−e).

Subtracting we get

2ay−a2=pe.

But, ifabe an infinitesimal quantity, a2must be infinitely smaller and therefore may be neglected when compared with the quantities 2ayandpe.Hence

2ay=pe,that is,e:a= 2y:p.

Therefore,

TM:y=e:a= 2y:p.

Hence

TM = 2y2/p= 2x.

This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratioa/eordy/dxdirectly without the labour of going through a calculation similar to the above for every separate case.

Publications

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See also

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References

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  1. ^Child, James Mark; Barrow, Isaac (1916).The Geometrical Lectures of Isaac Barrow.Chicago:Open Court Publishing Company.
  2. ^'The Abbey Scientists' Hall, A.R. p12: London; Roger & Robert Nicholson; 1966
  3. ^Cheesman, Francis (2005).Isaac Newton's Teacher(first ed.). Victoria, BC, Canada: Trafford Publishing. p.115.ISBN1-4120-6700-6.
  4. ^Craze, M. R. (1955).A History of Felsted School, 1564–1947.Cowell.
  5. ^O'Connor, J. J.; Robertson, E. F."gap-system".School of Mathematics and StatisticsUniversity of St Andrews.Archived fromthe originalon 26 December 2010.Retrieved1 February2012.
  6. ^Feingold, Mordechai (1990).Before Newton: The Life and Times of Isaac Barrow.Cambridge University Press. p. 256.ISBN9780521306942.
  7. ^"Barrow, Isaac (BRW643I)".A Cambridge Alumni Database.University of Cambridge.
  8. ^Manuel, Frank E. (1968).A Portrait of Isaac Newton.Belknap Press, MA. p.92.
  9. ^D.R. Wilkins –Trinity College, DublinSchool of Mathematics.Retrieved 1 February 2012
  10. ^For a summary of the Barrow–Newton relationship, seeGjersten, Derek (1986).The Newton Handbook.London: Routledge & Kegan Paul. pp. 54–55.
  11. ^Isaac Barrow, John Tillotson, Abraham Hill – The works of the learned Isaac Barrow... Printed by J. Heptinstall, for Brabazon Aylmer, 1700Published by DR JOHN TILLOTSON THE LORD ARCHBISHOP OF CANTERBURY{&} Isaac Barrow – The theological works of Isaac Barrow, Volume 1The 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 VolumeA.J. Valpy.Retrieved 1 February 2012
  12. ^Dresden, Arnold(1918)."Review:The Geometrical Lectures of Isaac Barrow,translated, with notes and proofs, by James Mark Child "(PDF).Bull. Amer. Math. Soc.24(9): 454–456.doi:10.1090/s0002-9904-1918-03122-4.Archived(PDF)from the original on 27 April 2014.

Further reading

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Wikiquote has quotations related toIsaac Barrow.
Wikisourcehas the text of the1911Encyclopædia Britannicaarticle "Barrow, Isaac".
Academic offices
Preceded by Regius Professor of GreekUniversity of Cambridge
1660–1663 		Succeeded by
Preceded by Master of Trinity College, Cambridge
1672–1677 		Succeeded by
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