1$13.1 Sir Isaac Keivlon. 52 7 



which represent the natural, triangular, and pyramidal numbers. 

 Now this will hold good whether m be a whole number or a 

 fraction. In the case which occasioned the investigation, oaniely, 

 (1 - x *)i' } m = h and > consequently, the numerators deduced 

 from the preceding formulas are J, 1, ^, T V> ttt' ^ c * 

 These, multiplied into the terms of the series, namely, x - 



ff + -II __ ?' + £ , &c, give us the following series : x - 



i! _ » * _ J-_.v - lJ- 9 x', &c., a series which ob- 

 viouslv represents the area of the circular segment, correspond- 

 in- to the abscissa x. This investigation led him likewise to the 

 discover? of the binomial theorem, so celebrated in algebra, 

 and of so much importance in an infinite number of investiga- 



tion? 



Newton had already made these discoveries, and many others, 

 when the Losarithmotechnia of Mercator was published ; which 

 contains only a particular case of the theory just explained, but, 

 from an excess of modesty and of diffidence, he made no attempt 

 to publish his discoveries, expressing his conviction that mathe- 

 maticians would discover them all before he was of an age suffi- 

 ciently mature to appear, with propriety, before the mathema- 

 tical world. But Dr. Barrow having contracted an acquaintance 

 with him soon after, speedily understood his value, and exhorted 

 him not to conceal so many treasures from men of science : he 

 even prevailed upon him to allow him to transmit to some of his 

 friends in London a paper containing a summary view ot some 

 of his discoveries. This paper was afterwards published under 

 the title of Analysis per Equations NumerQ Terminorum Infi- 

 nitas. Besides the method of extracting the roots ot all equations, 

 and of reducing fractional and irrational expressions into mnmte 

 series, it contains the application of all these discoveries to \ne 

 Quadrature, and the rectification of curves ; together with diffe- 

 rent series for the circle and hyperbola. He does not confine 

 him.elf to geometrical curves, but gives some examples of the 

 quadrature of mechanical curves. He speaks oi a method 

 tangents, of which he was in possession, in which he was not 

 Mopped by surd quantities, and which applied equally- well to 

 mechanical and geometrical curves. Finally, we .find in this 

 extraordinary paper the method of fluctiom and of fiumts, 

 explained and demonstrated with sufficient clearness; from 

 which it follows, irresistibly, that before that period he was m 

 ,„ of that admirable calculus: for the editors of this 

 r , which was published in the Commerctum Eptstottcum, 

 Attest that it was faithfully taken from the copy winch Coffins 

 had transcribed, iron, the manuscript sent by Barrow. At the 

 of Dr. Barrow, be drew up a full account ol this method, 



