GREGORY (James), an eminent mathematical genius of Scotland, was born at Aberdeen in 1639, and educated at that university. He made a good progress in classical learning: but being more delighted with philosophical researches, the works of Des Cartes and Kepler were his principal study; and he began early to make improvements on their discoveries in optics. The first of these improvements was the invention of the reflecting telescope, which still bears his name; and which was so happy a thought, that that has given occasion to the most considerable improvements made in optics since the invention of the telescope.
He published the construction of this instrument in 1663, at the age of 24; and coming next year or the year after that to London, he became acquainted with Mr John Collins, who recommended him to the best optic glass grinders there, in order to have it executed. But as this could not be done for want of skill in the artists to grind a plate of metal for the object speculum into a true parabolic concave, which the design required, he was much discouraged thereby; and after a few imperfect trials made with an ill polished spherical one, which did not succeed to his wish, he dropped the pursuit, and resolved to make the tour of Italy, then the mart of mathematical learning, in the view of prosecuting his favourite study with greater advantage.
He had not been long abroad, when the same inventive genius which had before shewed itself in practical mathematics, carried him to some new improvements in the speculative part. The sublime geometry on the doctrine of curves was then hardly past its infant state; and the famed problem of squaring the circle still continued a reproach to it, when our author discovered a new analytical method of summing up an infinite converging series, whereby the area of the hyperbola, as well as of the circle, may be computed to any degree of exactness. He was then at
Padua; and getting a few copies of his invention printed there in 1667, he sent one to his friend Mr Collins, who communicated it to the Royal Society, where it met with the commendations of lord Brounker and Dr Wallis. Our author printed it at Venice, and published it the following year 1668; together with another piece, wherein he first of any one entertained the public with an account of the transformation of curves. An account of this piece was also read by Mr Collins before the Royal Society, of which Mr Gregory, being returned from his travels, was chosen a member, and communicated to them an account of the controversy in Italy about the motion of the earth, which was denied by the famous astronomer Riccioli and his followers.
The same year his quadrature of the circle being attacked by the celebrated Mr Huygens, a controversy arose between these two eminent mathematicians, in which our author produced some improvements of his series.
In 1672, Sir Isaac Newton, in his wonderful discoveries on the nature of light, having contrived a new reflecting telescope, made several objections to Mr Gregory's. This gave occasion to a controversy betwixt these two philosophers, which was carried on this and the following year in the most amicable manner on each side. Mr Gregory defended his own construction, but gave his antagonist the whole honour of having made the catoptric telescopes preferable to the dioptric ones; shewing that the imperfections in these instruments were not so much owing to a defect in the object speculum, as to the different refrangibility of the rays of light. In the course of this dispute our author described a burning concave mirror, which was approved by Sir Isaac, and is still in good esteem.
All this while he attended the proper business of his professorship with great diligence; which, taking up the greatest part of his time, especially in the winter season, hindered him in the pursuit of his proper studies. These, however, led him to farther improvements in the invention of infinite series, which he occasionally communicated to his friend and correspondent Mr Collins, who might have had the pleasure of receiving many more, had not our professor's life been cut short by a fever in December 1675, at the age of 36 years.
Besides the inventions already mentioned, he was the first who gave a geometrical demonstration of lord Brounker's series for squaring the hyperbola, as it had been explained by Mercator in his Logarithmotechnia. He was likewise the first who demonstrated the meridian line to be analogous to a scale of logarithmic tangents of the half complement of latitude. He also invented, and demonstrated geometrically, by the help of the hyperbola, a very swift converging series for making the logarithms, and therefore recommended by Dr Halley as very proper for practice. He also sent to Mr Collins the solution of the famous Keplerian problem by an infinite series. He found out a method of drawing tangents to curves geometrically, without any previous calculations. He gave a rule for the direct and inverse method of tangents, which stands upon the same principle (of exhaustion) with the fluxions, and differs not much from it in the method of
Gregory. of application. He likewise gave a series for the length of the area of a circle from the tangent, and vice versa; as also for the secant and logarithmic tangent and secant, and vice versa. These, with others for certifying or measuring the length of the elliptic and hyperbolic curves, were sent to Mr Collins in return for some received from him of Sir Isaac Newton's; and their elegance being admirable, above whatever he had produced before, and after the manner of Sir Isaac Newton, gave room to think that he had improved himself greatly by that master, whose example he followed in giving his series in simple terms, independent of each other. These several inventions are contained, 1. In his Optica promota, &c. 4to. edit. 1663. 2. Vera circuli & hyperbole quadratura, Padua 1667. 3. Geometrie pars universalis, &c. 1667, 4to. 4. Several letters and papers printed in the Philosophical Transactions; the Journal des Savans; the Commerce. epistol. Jo. Collins & alior. 1715, 8vo. and in the Appendix to the English edition of Dr David Gregory's Elements of Optics, 1735, 8vo. by Dr Desaguliers.