46 LIMITS AND FLUXIONS 



It is this ; That a Quantity Infinitely Great, a Finite or 

 any given Quantity, an Infìnitesimal, a7id Nihilum 

 Geornetricum, are in Geometrical Troportion. I 

 confess I cannot discover the truth of this. . . . 

 Let in denote an infinite Quantity, d any finite one ; 

 then is d / ;;/ the Infinitesimal of d, according to 

 Mr. Neiwentiit. Now his Assertion is, that m : d : : 

 d I m : o ; therefore since from the nature of Geo- 

 metrical Proportion, 'tis aiso m : d : : d / m : dd / mm ; 

 it foUows that dd I mm is = o . . . then d / m = o. 

 Now Mr. Neiwentiit will hardly allow his Infinitesimal 

 to be nothing ; and yet ... I think it must follow, 

 that d=o.'' Proceeding geometrically, Ditton ex- 

 plains the fluxions of lines, areas, solids, and surfaces. 

 Next he takes up algebraical expressions. To find 

 the fluxion of x'% he lets x flow uniformly and re- 

 presents the augment of ;r in a given particle of time 

 by the symbol o. While x becomes x-\-o, x" becomes 

 (x-\-oy. Expanding the binomial, he finds that the 

 two augments are as i to nx''~'^-{-(n'^ — n)ox'''^ / 2 + 

 etc. " And the Ratio of them (making o to vanish) 

 will be that of i to nx""'^." According to his nota- 

 tion i" is a fluxion of x, and .ir is a fluxion of a:. 

 Taking ^ as a very small quantity, he lets the ex- 

 pressions oà, oy represent the moinents, or increments 

 of the flowing quantity z, y generated in a very small 

 part of time. " If therefore now, at the present 

 Moment, the flowing Quantities are z, y, x ;. the next 

 Moment (when augmented by these Increments) 

 they will become x+oò; y-\-oy, x-\-ox.'' He ex- 

 presses the general mode of procedure for finding 



