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genius, which had before shewn itself in practical mathe. matics, carried him to some new improrements in the speculative part. The sublime geometry on the doctrine of curves was then hardly passed 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, by which the area of the hyperbola, as well as 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 under the title “ Vera Circuli et Hyperbolæ Quaclratura,” he sent one to his friend Mr. Collins, who communicated it to the royal society, where it met with the cornmendation of lord Brounker and Dr. Wallis. He reprinted it at Venice, and published it the following year, 1668, together with another piece entitled “ Geometria pars universalis, inserviens quantitatum curvarum trausmu. tationi et mensuræ,” in which he is allowed to have shewn, for the first time, a method for the transmutation of curves. These works engaged the notice, and procured the author the correspondence of the greatest mathematicians of the age, Newton, Huygens, Wallis, and others. An account of this piece was also read by Mr. Collins before the royal society, of which Gregory, being returned from his travels, was chosen a member, admitted the 14th of January this year, and communicated to them an account of the controversy in Italy about the motion of the earth, which was denied by Riccioli and bis followers. ; The same year, bis quadrature of the circle being attacked by Mr. Hoygens, a controversy arose between those two eminent mathematicians, in which our author produced some improvements of his series. But in this dispute it happened, as it generally does in most others, that the antagonists, though setting out with decent tenper, yet grew too much heated in the combat. This was the case here, especially on the side of Gregory, whose de fence was, at his own request, inserted in the “ Philosophical Transactions ;” but Leibnitz, who allows Gregory the highest merit for his genius and discoveries, is of opi. nion, that Huygens has pointed out, though not errors, some considerable deficiencies in the treatise above-men: tioned, and has shewn a much simpler method of attaining the same end. Gregory also received from Mr. Collins, about this time, an account of the series invented by sir

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Isaac Newton ; who in that had actually effected what our author was 'stiffy contending against Huygens to be utterly impossible: that is, the ratio of the diameter of a circumference, expressed in a series of simple termis, independent of each other, and entirely freed from the magic vinculum of surds, in which they had till then been indissolubly held.

In 1668 our author published at London another work, entitled “ Exercitationes Geometricæ,” which contributed still much farther to extend his reputation. About this time he was elected professor of mathematics in the univer: sity of St. Andrew's, an office which he held for six years, During his residence there he married, in 1669, Mary, the daughter of George Jameson, the celebrated painter, whom Mr. Walpole has termed the Vandyke of Scotland, and who was fellow disciple with that great artist in the school of Rubens at Antwerp. His fame placed him in so great esteem with the royal academy at Paris, that, in the beginning of 1671, it was resolved by that academy to recommend him to their graud monarch for a pension ; and the design was approved even by Mr. Huygens, though he said be had reason to think himself im. properly treated by Mr. Gregory, on account of the controversy between them. Accordingly, several members of that academy wrote to Mr. Oldenburg, desiring him to acquaint the council of the royal society with their proposal ; informing him likewise, that the king of France was willing to allow pensions to one or two learned Englishmen, wbom they should recommend. But no answer was ever made to that proposal; and our author, with respect to this particular, looked upon it as nothing more than a compliment,

In 1672 he published “ The great and new art of weighing Vanity: or a discovery of the ignorance and arrogance of the great and new artist, in his pseudo-pbilosophical writings. By M. Patrick Mathers, arch-bedal to the uni. versity of St. Andrew's. To which are annexed some tentamina de motu penduli et projectorum." Under this assumed name, our author wrote ibis little piece to expose the ignorance of Mr. Sinclare, professor at Glasgow, in his hydrostatical writings, and in return for some ill-usage of that author to a colleague of Mr. Gregory's. In the same year, sir Isaac Newton, on bis wonderful discoveries in the nature of light, having contrived a new reflecting telescope, and made several objections to Mr. Gregory's, this gave birth to a dispute between those two philosophers,

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which was continued during that and the following year, in the most amicable manner on each side ; Mr. Gregory de. fending his own construction so far, as to give his antagonist the whole honour of having made the catoptric telescopes preferable to che dioptric; and 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. Several letters that passed in this dispute are printed by Dr. Desaguliers, in an appendix to the English edition of Dr. David Gregory's “ Elements of Catoptrics and Dioptrics." 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, interrupted 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 intimate, friend and correspondent Mr. Collins. In 1674 Mr. Gregory was called to Edinburgh, to fill the chair of mathematics in that university. This place he had held but little more than a year, when, in October 1675, being employed in shewing the satellites of Jupiter through a telescope to some of his pupils, he was suddenly struck with total blindness, and died a few days after, to the great loss of the mathematical world, at only thirty-seven years of age. . The most shining part of Gregory's character is that of his mathematical genius as an inventor. In this view, particularly, he merits a place in these memoirs; and therefore we shall conclude this article with a list of the most remarkable of his inventions. His reflecting telescope; burn, ing concave mirror; his quadrature of the circle, by an infinite copverging series; and his method for transformation of curves, have been already mentioned. Besides these, 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 compliment of latitude *. He also invented and

" * This invention is of great use in navigation ; and his just merit as the

inyentor of the demonstration of it was afterwards asserted by Dr. Halley, who

demonstrated geometrically, by the help of the hyperbola, a very simple 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 exhaustions) with that of fluxions, and differs not much from it in the manner of application. He likewise gave a series for the length of the arc 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, and above whatever he had produced before, and after the manner of sir Isaac, gave room to think he had improved himself greatly by that master, whose example he followed, in delivering his series in simple terms, independent on each other.

We are assured, that at his death he was in pursuit of a general method of quadrature, by infinite series, like that of sir Isaac. This appeared by his papers, which came into the hands of his nephew, Dr. David Gregory, who published several of them; and he himself assured Mr. Collins, he had found out the method of making sir Isaac's series; who thereupon concluded he must have written a treatise upon it. This encouraged Mr. Stewart, professor of mathematics in Aberdeen, to take the trouble of examining his papers, then in the hands of Dr. David Gregory, the late dean of Christ church, Oxford; but no such treatise could be found, nor any traces of it, and the same had been declared before by Dr. David Gregory; wbence it happens, that it is still unknown what his method was of making those serieses. However, Mr. Stewart affirms, that, in turning over his papers, he saw several curious ones upon particular subjects, not yet printed. On the contrary, some letters which he saw confirmed Dr. David Gregory's remark, and made it evident, that our author had never compiled any treatise, containing the foundations of this general method, a very short time before his death ; so that all that can be known about his method can only be collected from his letters, published in the short history of his " Mathematical Discoveries," compiled by Mr. Collins, and his letters to that gentleman in the Commercium Epistolicum.” From these it appears, that, in the beginning of 1670, when Mr. Collins sent him sir Isaac Newton's series for squaring the circular zone, it was then so much above every thing he comprehended in this way, that after having endeavoured in vain, by comparing it with several of his own, and combining them together, to discover the method of it, he concluded it to be no legitimate series; till, being assured of his mistake by his friend, he went again to work, and after almost a whole year's indefatiga. ble pains, as he acknowledges, he discovered, at last, that it might be deduced from one of his own, upon the subject of the logarithms, in which he had given a method for finding the power to any given logarithm, or of turning the root of any pure power into an infinite series; and in the same manner, viz. by comparing and combining his own series together, or else by deduction therefrom, he fell upon several more of sir Isaac's, as well as others like them, in which he became daily more ready by continual prac. tice; and this seems to have been the utmost he ever actually attained to, in the progress towards the discovering any universal method for those series.'

however, at the same time observes, that it was performed, not without a long train of consequences, and complications of proportions, whereby the evidence of the demonstration was in a great measure lost, and the reader wearied before he attains it. Miscel.

Curios. vol. II, 1727. The truth is, complication, tediousness, and intricacy, were faults complained of in all his series, before be had learned to im. prove them by a sight of those of sir Isaac Newton. Commerc. Epistol. Na. 53.

GREGORY (David), elder brother of the preceding, was born in 1627 or 1628, and although he possessed all the genius of the other branches of his family, was edu. cated by his father for trade, and served an apprenticeship to a mercantile house in Holland. Having a stronger pas. sion, however, for knowledge than for money, he abandoned trade in 1655, and returning to his own country, he succeeded, upon the death of an elder brother, to the estate of Kivardie, situated about forty miles north of Aberdeen, where he lived many years, and where thirty-two children were born to him by two wives. Of these, three sons made

I Biog, Brit.--Hutton's Dict. Martin's Biog. Philos.--Preface to Dr. Joha Gregory's Works, edit. 1788, 4 vols. 12mo,

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