Mr Gregory went to London about the year 1664 or 1665, and was introduced to Mr Collins, the secretary to the Royal Society, who introduced him to some of the best practical opticians, for the purpose of having his reflecting telescope executed. A Mr Rives was employed for this purpose, but he could not polish the speculum upon the tool, and was therefore obliged to do it with a cloth and putty. The success of this trial was so little, that Gregory was discouraged from making any farther attempts, and a tube was never even made to hold the mirrors. He after wards, however, made some trials with a little concave and convex speculum ; but, to use his own words, " they wer but rude, seeing I had but transient views of the object ; being so possessed with the fancie of the defective figure, that I wold not be at the pains to fix every thing in its due distance." After these unsuccessful attempts to construct a reflect ing telescope, Gregory England, and fixed his resi dence at Padua, which was then in high repute as a seat of mathematical learning. Ilere he published, in 1667, 1668, his work, entitled, Vera Circuli ct Hyperbola Quadratura in propria sea proportion's specie invents, et demonstrata, which contains his discovery of an infinitely converging se ries for the areas of the circle, ellypsis, and hyperbola. A copy of this work was laid before the Royal Society by his friend Mr Collins, and was honoured with the approbation of Lord Brouncker and Mr Wallis. In the following year he reprinted it at Venice, and added a new work, entitled, Geometrice pars universalis quantitatunz curvarum trans mutationi et mensure, inserviens, which contains a new me thod for the transmutation of curves. This work had pre viously appeared at Padua in 1668 ; and, upon its arrival in England, was read by Mr Collins to the Royal Society.
In the year 1670, Mr Gregory received, in a letter from Mr Collins, a series for the area of the zone of a circle, and being informed that Newton had invented an universal me thod by which he could square all cut ves geometrical and mechanical by infinite series of that kind, Gregory applied himself to the investigation of the subject, and discovered an universal method of series, which he communicated to Newton, and the other English mathematicians, by a letter to Collins, dated February 1671. His brother David urged him to publish this method without delay ; hut he declined this from the most honourable motives ; for Newton having been the first inventor, he thought himself bound to wait till his method should be published.
Upon Mr Gregory's return to London, We believe in 1668, ho was elected a fellow of the Royal Society, and laid befotie them an account of a dispute in Italy relative to the earth's motion, which Riccioli and his followers had de nied. About this time also he engaged in a dispute with the celebrated Huygens through the medium of the Phi losophical Transactions. in the Journal
des Scavans, July 2d 1668, some animadversions on Gre gory's quadrature of the circle, and particularly objected to the proposition which stated the impossibility of express ing perfectly the area of a circle in any known algebraical form besides that of an infinite converging series. Grego ry defended himself in the 37th Number of the Philosophi cal Transactions, and the dispute was carried on with con siderable warmth by both parties. The whole of the con troversy will be found in Huygens' Opera Varia, vol. ii. p. 463.
In 1668, Gregory published in London his Exercitationes Geometricx, a small work of twenty-six pages, which con tains the following subjects : Appendicula ad verum Circuli et Hyperbolae Quadra turam.
N. Mercatoris Quadratura Hyperbolm Gebmetricc de monstrata.
Analogia inter Lineam Meridianam Planispherii Nautici et Tangentes Artificiales 0 eometrice demonstrata ; seu quod secantium Naturalium additio efficiat Tangentes Artifi ciales.
Item, Quod Tangentium Naturalium additio efficit Secan tes Artificiales.
Quadratura Conchoidis.
Quadratura Cissoidis.
Mcthodus Facilis et Accurata componendi Secantes et Tangentcs Artificialcs.
The preface to this work, and the introduction to the rippendicula, &c. are remarkably interesting, in so far as they throw considerable light on the dispositions of our au thor. He speaks with great severity of the jealousy and injustice of his contemporaries, and alludes to the treat mentwhich he had received from Huygens. In the introduc tion to the ilppendicula, he resumes this subject with more keenness. He declares, that Huygens had accused him of ignorance and plagiarism ; and after arguing against Huy gens' claim to the discovery, he concludes with this re markable passage : " At parum refert quis sit cjus primus inventor, satin enim constat me primum esse publicato rem ; neque mihi csset difficile affirmarc (si modo mentiri vellem) me ante 20 annos illam cognovisse : utcunque sit, conabor hic circuli et hyperbolae quadraturam ad talem per fectionem prornovcre, tit llugenius prolem suam vix cog noseat." Mr Gregory was about this time elected professor of mathematics in the university of St Andrew's. In 1669, he married the daughter of George Jamieson, the celebrated Scottish painter, by whom he had a son, James, the lather of Dr John Gregory, (the subject of a succeeding article,) and two daughters. In August 1672, he began a corre spondence with his friend Mr Collins, relative to the com parative merits of his own telescope, and that of Sir Isaac Newton. The sentiments of the two philosophers were communicated to each other by their respective friends, and the dispute was thus carried on in the most amicable man ner. The correspondence has been published by Dr Desagu liers, at the end of his edition of Dr David Gregory's Ca toptrics and Dioptrics.