VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 13^ 



quadratures, where he represents fluxions by pointed letters in the first propo- 

 sition, by ordinates of curves in the last proposition, and by other symbols, in 

 explaining the method and illustrating it with examples, in the introduction. 

 Mr. Leibnitz has no symbols of fluxions in his method, and therefore Mr» 

 Newton's symbols of fluxions are the oldest in the kind. Mr. Leibnitz began 

 to use the symbols of moments or differences etc, dy, dz, in the year l677« Mr. 

 Newton represented moments by the rectangles under the fluxions and the mo- 

 ment 0, when he wrote his Analysis, which was at least 46 years since. Mr. 

 Leibnitz has used the symbols sx, sy, sz, for the sums of ordinates ever since 

 the year l68d; Mr. Newton represented the same thing in his Analysis, by 

 inscribing the ordinate in a square or rectangle. All Mr. Newton's symbols are 

 the oldest in their several kinds by many years. 



And whereas it has been represented that the use of the letter o is vulgar, and 

 destroys the advantages of the difi^erential method; on the contrary, the method 

 of fluxions, as used by Mr. Newton, has all the advantages of the diff^erential 

 and some others. It is more elegant, because in his calculus there is but one 

 infinitely small quantity represented by a symbol, the symbol o. We have no 

 ideas of infinitely small quantities, and therefore Mr. Newton introduced 

 fluxions into his method, that it might proceed with finite quantities as much 

 as possible. It is more natural and geometrical, because founded on the prime 

 ratios of nascent quantities, which have a being in geometry, while indivisibles 

 on which the diff^erential method is founded, have no being, either in geometry 

 or in nature. There are prime ratios of nascent quantities, but no prime nas- 

 cent quantities. Nature generates quantities by continual flux or increase, and 

 the ancient geometricians admitted such a generation of areas and solids, when 

 they drew one line into another by local motion, to generate an area, and the 

 area into a line by local motion, to generate a solid. But the summing up of 

 indivisibles, to compose an area or solid, was never yet admitted into geometry. 

 Mr. Newton's method is also of greater use and certainty, being adapted either 

 to the ready finding out of a proposition, by such approximations as will create 

 no error in the conclusion, or to the demonstrating it exactly ; Mr. Leibnitz's 

 is only for finding it out. When the work succeeds not in finite equations, 

 Mr. Newton has recourse to converging series, and thereby his method becomes 

 incomparably more universal than that of Mr. Leibnitz, which is confined to 

 finite equations; for he has no share in the method of infinite series. Some 

 years after the method of series was invented, Mr. Leibnitz invented a propo- 

 sition for transmuting curvilinear figures into other curvilinear figures, of equal 

 areas, in order to square them by converging series; but the methods of squar- 

 ing those other figures by such series, were not his. By the help of the nev¥ 



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