122 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



i 



-^ = c, and 7» + n = 6: if ex" = z, or c^x* = z ; then putting x 4- o for z, 

 m+n » • /- . , 



and z + (w, or (which is the same thing) substituting z -{- 01/ for z, you have 

 c» X a:* + pox^'^ &c. = z" + noyz""' &c. viz. by omitting the other terms of the 

 series, which at length would vanish. Now taking away the equal terms c"x^ and 

 z", and dividing the rest by 0, there remains c^px^'^ = nyz~^ (= — = I^^Lp"^ 



car 



or dividing by <f*xff then j&a?" will be = ~ or pcx " = «y ; or by restoring 



a? 

 again -^^ instead of c, and w + w instead of jb, that is, m instead of p — n, and 



m m 



na instead of pc, ax" will be = y. Therefore, h. contra, if ax" be = y, then 



-1- ax " will be = z." q. e. d. 



By the same way of working, the second rule may be also demonstrated. 

 And if any equation whatever be assumed, expressing the relation between the 

 abscissa and area of a curve, the ordinate may be found in the same manner, 

 as is mentioned in the next words of the Analysis. And if this ordinate drawn 

 into an unit be put for the area of a new curve, the ordinate of this new curve 

 may be found by the same method, and so on perpetually. And these ordinates 

 represent the first, second, third, fourth, and following fluxions of the first 

 area. Such then was Mr. Newton's way of working in those days, when he 

 wrote this compendium of his Analysis. And the same way of working he used 

 in his book of quadratures, and still uses to this day. 



Among the examples with which he illustrates the method of series and mo- 

 ments, set down in this compendium, are these. Let the radius of a circle be 

 1, the arc z, and the sine x; then the equations for finding the arc whose sine 

 is given, and the sine whose arc is given, will be 



z = a7 + ^+ -^'-\- -H-r^^ + -H-f^' + &c. 

 a? = z — -fz^ + -rhrz' — ttsVs-z' + aa Afto Z" — &C. 



Mr. Collins informed Mr. Gregory of this method in Autumn 1669; ^nd Mr. 

 Gregory, by the help of one of Mr. Newton's series, after a year's study, found 

 out the method in Dec. 1 67O; and two months after, in a letter dated Feb. 1 5, 

 1671, he sent several theorems, discovered thereby, to Mr. Collins, with leave 

 to communicate them freely. And Mr. Collins was very free in communicating 

 what he had received, both from Mr. Newton and from Mr. Gregory, as appears 

 by his letters printed in the Commercium. Among the series which Mr. Gre- 



