VOL. XLII.] VHILOSOPHICAL TRANSACTIONS. 06Q 



rate than — ; or if the fluxion of a be represented by a, the fluxion of b can- 

 not be less than — . And if the successive differences of b be always less than 



n 



those of — , then surely b cannot be said to increase at a greater rate than — ; 



or the fluxion of b cannot be said to be greater in this case than — . 



From those principles the primary propositions in the method of fluxions, 

 and the rules of the direct method, with the fundamental rules of the inverse 

 method, are demonstrated. We must be brief in our account of the remainder 

 of this book. The rule for finding the fluxion of a power is not deduced, as 

 usually, from the binomial theorem, but from one that admits of a much easier 

 demonstration from the first algebraic elements, viz. that when n is any integer 

 positive number, if the terms e"-', e'-^f, e'-^p*, e'-4f3, . . . . p"-', (wherein 

 the index of e constantly decreases, and that of f increases by the same differ- 

 ence unit) be multiplied by e — p, the sum of the products is e" — p"; from 

 which it is obvious, that when e is greater than f, then e" — f" is less than 

 WE"-' X E — F but greater than ne"-' X e — p. 



The rules are sometimes proposed in a form somewhat different from the 

 usual manner of describing them, with a view to facilitate the computations 

 both in the direct and inverse method. Thus, when a fraction is proposed, 

 and the numerator and denominator are resolved into any factors, it is demon- 

 strated, that the fluxion of the fraction divided by the fraction, is equal to the 

 sum of the quotients, when the fluxion of each factor of the numerator is 

 divided by the factor itself, diminished by the quotients that arise by dividing 

 in like manner the fluxion of each factor of the denominator by the factor. 



The notation of fluxions is described in chap, 2, with the rules of the direct 

 method, and the fundamental rules of the inverse method. The latter are 

 comprehended in 7 propositions, 6 of which relate to fluents that are assignable 

 in finite algebraic terms, and the 7th to such as are assigned by infinite series. 

 It is in this place the author treats of the binomial and multinomial theorems, 

 because of their use on this occasion, and they are investigated by the direct 

 method of fluxions. The same method is applied for demonstrating other 

 theorems, by which an ordinate of a figure being given, and its fluxions deter- 

 mined, any other ordinate and area of the figure may be computed. The most 

 useful examples are described in this chapter, by computing the series that 

 serve for determining the arc from its sine or tangent, and the logarithm from 

 its number, and conversely the sine, tangent, or secant, from the arc, and the 

 number from its logarithm. 

 The inverse method is prosecuted farther in the 3d chapter, by reducing 



