by Niccolò Guicciardini

© Oxford University Press 2004 All rights reserved

**Gregory, James** (1638-1675), mathematician, was born at the manse of Drumoak, near Aberdeen, in November 1638, the youngest son of John and Janet Gregory. His father, minister of Drumoak, was fined, deposed, and imprisoned by the covenanters. Somewhat sickly as a child, Gregory received his early education (including an introduction to geometry) from his mother, who belonged to the scholarly Anderson family. His older brother David, an enthusiastic amateur mathematician, sent him to Aberdeen, first to a grammar school and later to Marischal College. After graduating there, Gregory moved to London in 1662 where he published his first work, *Optica promota* (1663). In the epilogue Gregory describes the revolutionary design of a telescope based not on lenses but on mirrors, the idea that Newton developed later; Newton's design, however, is different and easier to construct. In 1663 the London optician Richard Reeve was commissioned by Gregory to construct a telescope according to his design, but failed to polish its mirrors correctly, so Newton's telescope was the first reflector actually built. The first Gregorian telescope was presented to the Royal Society by Robert Hooke in 1674.

During his stay in London, Gregory established a friendship with the influential Robert Moray. In order to improve his mathematical education he moved to Italy in 1664, where in Padua he studied under Stefano degli Angeli, a pupil of Evangelista Torricelli. During his four years in Italy Gregory came into contact with the discoveries and methods of the Galileian school. These methods are generally labelled as pre-calculus techniques. The Italians were interested in problems of quadratures, rectification of curves, finding tangents, and so on, problems which were later dealt with in much more general terms in the calculi invented by Newton and Leibniz. Recent research shows that Gregory in his manuscripts and letters had anticipated many calculus concepts and methods.

The fruits of the Italian period were the *Vera circuli et hyperbolae quadratura* (1667) and the *Geometriae pars universalis* (1668). In the first work Gregory tried to prove that it is impossible to achieve an algebraic quadrature of the general conic sector--and of the whole circle in particular. His demonstration was attacked by Huygens and a controversy ensued. Some themes of the controversy were dealt with in *Exerciationes geometricae* (1668), where Gregory studied the quadrature of the cissoid and the conchoid and gave a geometrical demonstration of Nicolas Mercator's quadrature of the hyperbola. The *Geometriae pars universalis* is an anthology of contemporary methods in tangent, quadrature, cubature, and rectification problems. Gregory maintains the interesting idea that, in order to solve these problems, general methods of 'transmuting' the properties defining a curve are needed. These methods allow the reduction of the problem to the quadrature of a curve which is already known. Similar techniques of 'transmutation' later played a prominent role in Leibniz's invention of the calculus.

About Easter 1668 Gregory returned to London where, backed by John Collins's reviews of the two treatises written in Italy, he was elected to the Royal Society on 11 June. In late 1668, probably through Moray's recommendation, he was elected to the chair of mathematics in St Andrews. One year later he married Mary, daughter of George Jameson the painter and widow of Peter Burnet. They had two daughters and a son, James, afterwards professor of physics in King's College, Aberdeen. Much of Gregory's time was spent teaching elementary mathematics. His scientific production continued, however, as is evident from his letters to John Collins and from his manuscripts. About 1670, upon reading Mercator's *Logarithmotechnia* (1668), he became interested in series expansions. Through Collins he became aware of recent results achieved by mathematicians such as Barrow, Huygens, and Newton. Early in 1671 Moray tried unsuccessfully to secure for Gregory a post as *pensionnaire* at the French Académie des Sciences.

In his letters to Collins, Gregory demonstrated his advanced knowledge on series expansions; they also show that he discovered the binomial theorem independently of Newton. In February 1671 he communicated to Collins without proof several trigonometric series. The printing of Gregory's manuscripts in 1939 made it clear that he obtained these series by the method now known as the Taylor expansion of a function.

A manuscript entitled *Geometriae propositiones quaedam generales,* published in 1996 by A. Malet, shows furthermore that Gregory was interested in the foundational aspects of his quadrature techniques. He came very close to the theory of prime and ultimate ratios, presented in the first section of Newton's *Philosophiae naturalis principia mathematica* (1687). Thus the analogies with Newton's work relate not only to the binomial theorem (which Newton found about 1665) and the Taylor expansion (which Newton stated in the 1690s), but also to the conceptual foundations of the new techniques of approximation. Newton was indeed disturbed by some similarities between his and Gregory's mathematical work. In 1684, David Gregory, James's nephew, published (without due acknowledgement to James) many of his uncle's results on series in a work entitled *Exercitatio geometrica.* Upon reading this work, Newton was induced, in order to establish his priority over the Gregories, to initiate a mathematical treatise entitled *Matheseos universalis specimina.* James Gregory, trained in pre-calculus techniques, is one of the mathematicians who, before Newton and Leibniz, came close to the discovery of calculus.

Gregory's scientific achievements were not restricted to the telescope and to pure mathematics. In an appendix to a book on hydrostatics, *Great and New Art of Weighing Vanity* (1672), written in collaboration with William Sanders, under the pseudonym Patrick Mathers, Arch-Bedal to the University of St Andrews, in order to disprove the theories maintained by the Glasgow professor George Sinclair, he contributed an important result in dynamics, deducing the infinite series which expresses the time of vibration in a circular pendulum for a small arc of swing. Gregory also contributed to astronomy: in a letter to Oldenburg of 8 June 1675 he suggested the differential method of stellar parallaxes; elsewhere he pointed out the use of transits of Venus and Mercury for determining the distance of the sun, and originated the photometric method of estimating the distances of the stars. The photometric method was utilized by Newton in *The System of the World* (1728), another sign of Newton's debt to Gregory.

In 1673 Gregory travelled to London to purchase telescopes and other instruments and to consult Flamsteed on this topic. During his absence the students at St Andrews had rebelled against the antiquated curriculum. Gregory's attempts to introduce science in the university were now seen as a threat to the old system, and he met with many difficulties at St Andrews--even his salary was not paid. In 1674 he was invited by Edinburgh University to take the newly established chair of mathematics. In October 1675, a few months after his arrival in Edinburgh, a paralysing stroke blinded him while he was showing Jupiter's satellites to his students. He died of apoplexy a few days later.

NICCOLÒ GUICCIARDINI

**Sources **

D. T. Whiteside, 'Gregory, James', *DSB*

*Biographia Britannica, or, The lives of the most eminent persons who have flourished in Great Britain and Ireland,* 4 (1757), 2355-65

H. W. Turnbull, ed., *James Gregory tercentenary memorial volume* (1939)

A. Malet, *From indivisibles to infinitesimals: studies on seventeenth-century mathematizations of infinitely small quantities* (1996)

A. Malet, 'Studies on James Gregorie, 1638-1675', PhD diss., Princeton University, 1989

G. A. Gibson, 'James Gregory's mathematical work', *Proceedings of the Edinburgh Mathematical Society,* 1st ser., 41 (1923), 2-25

P. D. Lawrence, 'The Gregory family: a biographical and bibliographical study', PhD diss., U. Aberdeen, 1971

**Archives **

BL, papers, Sloane MS 3208

U. Edin. L., papers | NRA, priv. coll., corresp. with John Collins

RS, corresp. with John Collins, etc.

U. St Andr. L., corresp. with John Collins

**Likenesses **

eleventh earl of Buchan, chalk drawing (after unknown artist), Scot. NPG

eleventh earl of Buchan, pencil and chalk drawing (after John Scougall), Scot. NPG

W. Holl, stipple, BM, NPG; repro. in Chambers, *Scots.*

portrait; at Marischal College, U. Aberdeen, in 1890

portrait, Fyvie Castle, Aberdeenshire *[see illus.]*

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