# James Gregory (mathematician)

James Gregory

James Gregory (1638–1675)
Born 1638
Drumoak, Aberdeenshire, Scotland
Died 1675
Edinburgh
Residence Scotland, England, Venice
Citizenship Scotland
Nationality Scottish
Fields Mathematics
Astronomy
Institutions University of St. Andrews, University of Edinburgh
Alma mater Marischal College (University of Aberdeen), University of Padua
Known for Gregorian telescope
Diffraction grating, Calculus
Influences Stefano degli Angeli
Influenced David Gregory
Notes
Nephew of Alexander Anderson. Uncle of David Gregory.

James Gregory FRS (November 1638 – October 1675) was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.

## Biography

The youngest of the 3 children of John Gregory, an Episcopalian Church of Scotland minister, James was born in the manse at Drumoak, Aberdeenshire, and was initially educated at home by his mother, Janet Anderson (~1600-1668). It was his mother who endowed Gregory with his appetite for geometry, her uncle – Alexander Anderson (1582-1619) – having been a pupil and editor of French mathematician Viète. After his father's death in 1651 his elder brother David took over responsibility for his education. He was sent to Aberdeen Grammar School, and then to Marischal College, graduating in 1657.

In 1663 he went to London, meeting John Collins and fellow Scot Robert Moray, one of the founders of the Royal Society. In 1664 he departed for the University of Padua, in the Venetian Republic, passing through Flanders, Paris and Rome on his way. At Padua he lived in the house of his countryman James Caddenhead, the professor of philosophy, and he was taught by Stefano Angeli.

Upon his return to London in 1668 he was elected a Fellow of the Royal Society, before travelling to St Andrews in late 1668 to take up his post as the first Regius Chair of Mathematics, a position created for him by Charles II, probably upon the request of Robert Moray.

He was successively professor at the University of St Andrews and the University of Edinburgh.

He had married Mary, daughter of George Jameson, painter, and widow of Peter Burnet of Elrick, Aberdeen; their son James was Professor of Physics at King's College Aberdeen. He was the grandfather of John Gregory (FRS 1756); uncle of David Gregorie (FRS 1692) and brother of David Gregory (1627–1720), an inventor.[1]

About a year after assuming the Chair of Mathematics at Edinburgh, James Gregory suffered a stroke while viewing the moons of Jupiter with his students. He died a few days later at the age of 37.

## Published works

### Optica Promota

In the Optica Promota, published in 1663, Gregory described his design for a reflecting telescope, the "Gregorian telescope". He also described the method for using the transit of Venus to measure the distance of the Earth from the Sun, which was later advocated by Edmund Halley and adopted as the basis of the first effective measurement of the Astronomical Unit.[2]

### Vera Circuli et Hyperbolae Quadratura

In 1667, Gregory issued his Vera Circuli et Hyperbolae Quadratura, in which he showed how the areas of the circle and hyperbola could be obtained in the form of infinite convergent series. This work contains a remarkable geometrical proposition to the effect that the ratio of the area of any arbitrary sector of a circle to that of the inscribed or circumscribed regular polygons is not expressible by a finite number of terms. Hence he inferred that the quadrature of the circle was impossible; this was accepted by Montucla, but it is not conclusive, for it is conceivable that some particular sector might be squared, and this particular sector might be the whole circle. Nevertheless Gregory was effectively among the first to speculate about the existence of what are now termed transcendental numbers. In addition the first proof of the fundamental theorem of calculus and the discovery of the Taylor series can both be attributed to him.

The book also contains series expansions of sin(x), cos(x), arcsin(x) and arccos(x). (The earliest enunciations of these expansions were made by Madhava in India in the 14th century). It was reprinted in 1668 with an appendix, Geometriae Pars, in which Gregory explained how the volumes of solids of revolution could be determined.

## Gregorian telescope

Diagram of a Gregorian reflecting telescope.

In his 1663 Optica Promota, James Gregory described his reflecting telescope which has come to be known by his name, the Gregorian telescope. Gregory pointed out that a reflecting telescope with a parabolic mirror would correct spherical aberration as well as the chromatic aberration seen in refracting telescopes. In his design he also placed a concave secondary mirror with an elliptical surface past the focal point of the parabolic primary mirror, reflecting the image back through a hole in the primary mirror where it could be conveniently viewed. According to his own confession, Gregory had no practical skill and he could find no optician capable of actually constructing one.[3]

The telescope design attracted the attention of several people in the scientific establishment such as Robert Hooke, the Oxford physicist who eventually built the telescope 10 years later, and Sir Robert Moray, polymath and founding member of the Royal Society.

The Gregorian telescope design is rarely used today, as other types of reflecting telescopes are known to be more efficient for standard applications. Gregorian optics are also used in radio telescopes such as Arecibo, which features a "Gregorian dome".[4]

## Other work

In 1671, or perhaps earlier, he established the theorem that

$\theta = \tan \theta - (1/3) \tan^3 \theta + (1/5) \tan^5 \theta - \ldots$,

the result being true only if θ lies between −(1/4)π and (1/4)π. This formula was later used to calculate digits of π, although more efficient formulas were later discovered.

James Gregory discovered the diffraction grating by passing sunlight through a bird feather and observing the diffraction pattern produced.[5] In particular he observed the splitting of sunlight into its component colours – this occurred a year after Newton had done the same with a prism and the phenomenon was still highly controversial.

Gregory, an enthusiastic supporter of Newton, later had much friendly correspondence with him and incorporated his ideas into his own teaching, ideas which at that time were controversial and considered quite revolutionary.

The crater Gregory on the Moon is named after him. He was the uncle of mathematician David Gregory.