3Kct. i.]. DISSERTATION SECOND. 5 



The notion of Infinite Quantity had, as we have already 

 seen, been for some time introduced into Geometry, and 

 having become a subject of reasoning and calculation, had, 

 in many instances, after facilitating the process of both, led 

 to conclusions from which, as if by magic, the idea of infinity 

 had entirely disappeared, and left the geometer or the alge- 

 braist in possession of valuable propositions, in which were 

 involved no magnitudes but such as could be readily exhibit- 

 ed. The discovery of such results had increased both the 

 interest and extent of mathematical investigation. 



It was in this state of the sciences, that Newton began his 

 mathematical studies, and, after a very short interval, his 

 mathematical discoveries. 1 The book, next to the elements, 

 which was put into his hands, was Wallis's Arithmetic of 

 Infinites, a work well fitted for suggesting new views in 

 geometry, and calling into activity the powers of mathe- 

 matical invention. Wallis had effected the quadrature of all 

 those curves in which the value of one of the co-ordinates 

 can be expressed in terms of the other, without involving 

 either fractional or negative exponents. Beyond this point 

 neither his researches, nor those of any other geometer, 

 had yet reached, and from this point the discoveries of 

 Newton began. The Savilian Professor had himself been 

 extremely desirous to advance into the new region, where, 

 among other great objects, the quadrature of the circle 

 must necessarily be contained, and he made a very noble 

 effort to pass the barrier by which the undiscovered coun- 

 try appeared to be defended. He saw plainly, that if the 

 equations of the curves which he had squared were ranged 

 in a regular series, from the simpler to the more com- 

 plex, their areas would constitute another corresponding 



1 He entered at Trinity College, Cambridge, in June 1660. 

 The date of his first discoveries is about 1663. 



