James Gregory (astronomer and mathematician)

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For other people with the same name, see James Gregory.

James Gregory
James Gregory (1638-1675)
Born	1638 Drumoak, Aberdeenshire, Scotland
Died	1675 Edinburgh, Scotland
Residence	Scotland
Nationality	Scottish
Fields	Mathematician and Astronomer
Institutions	University of St. Andrews
Alma mater	University of Padua
Doctoral advisor	Stefano degli Angeli
Known for	Gregorian telescope Diffraction grating
Notes Uncle of David Gregory.

James Gregory (November 1638 – October 1675), was a Scottish mathematician and astronomer.

He was born at Drumoak, Aberdeenshire, and died at Edinburgh. He was successively professor at the University of St Andrews and the University of Edinburgh.

In 1663 he published his Optica Promota, in which the compact reflecting telescope known by his name, the Gregorian telescope, is described. His system of Gregorian optics is also used in radio telescopes such as Arecibo, which features a "Gregorian dome".^[1]

The telescope design attracted the attention of several people in the scientific establishment: Robert Hooke, the Oxford physicist who eventually built the telescope, Sir Robert Moray, polymath and founding member of the Royal Society and Isaac Newton, who was at work on a similar project of his own.

The Gregorian telescope was the first practical reflecting telescope and remained the standard observing instrument for a century and a half. However, the Gregorian telescope design is rarely used today, as other types of reflecting telescopes are known to be more efficient for standard applications.

In the Optica Promota 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.

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

In 1667 he 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.

In 1671, or perhaps earlier, he rediscovered the theorem that 14th century Indian mathematician Madhava of Sangamagrama had originally discovered, the arctangent series

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

for θ between −π/4 and π/4. This formula was used by Madhava to calculate digits of π and later used in Europe for the same purpose, 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. 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.

A crater on the moon is named for him, see Gregory (lunar crater). The mathematician David Gregory was his nephew.