80 ' rHILOSOPHICAL TRANSACTIONS. [aNNO 1684. 



James Gregory, printed at Padua, in the year 1667, entitled Vera Circuli et 

 Hyperbolge Quadratura, &c. Wherein he shows, that any sector of the circle, 

 hyperbola, or ellipse, is the limit of a certain converging series; whose first two 

 terms are a and b, of which a is a triangle, which, as to the circle or ellipse is 

 inscribed, but as to the hyperbola, circumscribed to the said sector ; and b, a 

 trapezium, which contrarywise, as to the circle or ellipse is circumscribed, but 

 as to the hyperbola, inscribed to the said sector; the two second terms, }/ ab 



and 7- : the two thirds in like manner derived from the seconds as these 



a + -v/afi 



are from the first. And so infinitely, with other things appertaining to the 

 same, and to other such like approximations. 



He then mentions another method, different from that former, published at 

 London the year following, by Mr. Nicholas Mercator, in his Logarithmo- 

 technia, for squaring the hyperbola by the help of infinite series. Approved 

 also and demonstrated afterwards by Mr. James Gregory, apagogically. But, 

 that a general method for such cases was yet wanting. 



That about the beginning of the year 1 670, he understood from Mr. John 

 Collins, that Mr. Isaac Newton, professor of mathematics at Cambridge, had 

 before that time a general method for such quadratures, and other like cases. 

 Which, as an instance, Mr. Collins sent him an example of such an infinite 

 series accommodated to a circular zone; namely, if the radius be r, and the breadth 



of the zone b, the zone is equal io ibr — — -r -jr^ — T?=-=r, &c. 



That Mr. James Gregory was in pursuit of like methods of infinite series, 

 but was prevented by death; and, except some particular examples, left nothing 

 in his papers, yet found, that might declare his method and way of finding such 

 examples. 



That himself, Mr. David Gregory, in this treatise explains a method, which 

 may suit such examples of his uncle. This he performs chiefly by the principles 

 of indivisibles and the arithmetic of infinites, already known and received by 

 geometricians as sufficiently demonstrated ; applying them to particular cases, in 

 parabolas, hyperbolas, ellipses, spirals, cycloids, conchoids, cissoids, &c. 



Together with divers expedients or preparative observations, by division and 

 extraction of roots in species, for reducing compound quantities into infinite 

 series, thereby rendering them capable of having the method of infinites ap- 



gcript, a short Treatise on the Nature and Arithmetic of Logarithms, which is printed at the end of 

 Keill's translation of Commandine's Euclid; and a Treatise on Practical Geometry, which was after- 

 wards translated and published in 1745, by the celebrated Mr. Maclaurin. He left also in manuscript 

 a Commentary on Newton's Principia, which Newton valued and kept by him for many years after 

 the death of the author. 



