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Isaac Barrow

born October 1630, London, England

died May 4, 1677, London

English classical scholar, theologian, and
mathematician who was the teacher of Isaac
Newton. He developed a method of determining
tangents that closely approached the methods of
calculus, and he first recognized that what
became known as the processes of integration and
differentiation in calculus are inverse
operations.

Barrow entered Trinity College, Cambridge, in
1643. There he distinguished himself as a
classical scholar as well as a mathematician,
earning his bachelor’s degree in 1648. He was
elected a fellow of the college in 1649 and
received his master’s degree in 1652. Such
precociousness helped to shield him from Puritan
rule, for Barrow was an outspoken Royalist and
Anglican. By the mid-1650s he contemplated the
publication of a full and accurate Latin edition
of the Greek mathematicians, yet in a concise
manner that utilized symbols for brevity.
However, only Euclid’s Elements and Data
appeared in 1656 and 1657, respectively, while
other texts that Barrow prepared at the time—by
Archimedes, Apollonius of Perga, and Theodosius
of Bythnia—were not published until 1675. Barrow
embarked on a European tour before the Elements
was published, as the political climate in
England deteriorated and the Regius
professorship of Greek at the University of
Oxford, to which he had been elected, was given
to another. He spent four years in France,
Italy, and Constantinople, returning to England
with the restoration of the Stuart monarchy in
1660. On his return to England, Barrow was
ordained in the Anglican Church and appointed to
a Greek professorship at Cambridge. In 1662 he
was also elected professor of geometry, but he
resigned both positions after his election as
Lucasian Professor of Mathematics at Cambridge
in 1663.

Barrow was instrumental in institutionalizing
the study of mathematics at Cambridge. From 1664
to 1666, he delivered a set of mathematical
lectures—predominantly on the foundations of
mathematics—that were published posthumously as
Lectiones mathematicae (1683). These lectures
treated such basic concepts as number,
magnitude, and proportion; delved into the
relationship between the various branches of
mathematics; and considered the relation between
mathematics and natural philosophy—most notably
the concept of space. Barrow followed these with
a series of lectures on geometry, Lectiones
geometricae (1669), that were far more technical
and novel. In investigating the generation of
curves by motion, Barrow recognized the inverse
relationship between integration and
differentiation and came close to enunciating
the fundamental theorem of calculus. His last
series of lectures, on optics, Lectiones opticae
(1670), built on the work of Johannes Kepler
(1571–1630), René Descartes (1596–1650), and
Thomas Hobbes (1588–1679), among others. In
these lectures Barrow made major contributions
to determining image location after reflection
or refraction; opened new vistas for the study
of astigmatism and caustics (a collection of
rays that, emanating from a single point, are
reflected or refracted by a curved surface); and
made suggestions toward a theory of light and
colours.

Barrow’s tenure as mathematics professor
coincided with the maturation of Newton’s
mathematical studies, and scholars often debate
the exact nature of their relationship. Barrow
was not Newton’s official tutor, though they
were both members of Trinity College. Newton
attended Barrow’s lectures, and it is clear that
Barrow encouraged and furthered Newton’s
studies. Fully cognizant of the young man’s
talents, Barrow resigned his professorship in
1669 in Newton’s favour and accepted a position
as royal chaplain in London. In 1673 Barrow was
appointed master of Trinity College by King
Charles II.

Although Barrow was regarded by his
mathematical contemporaries in England as second
only to Newton, he was more widely esteemed for
his sermons and other writings on behalf of the
Church of England, and these were often
reprinted well into the 19th century.

*Mordechai Feingold*