332 A Theory of Fluxions. 



generate the superficies xy. Let this superficies be conceiv- 

 ed to move from the point, at which it commenced its exis- 

 tence, downward, until the solid intended to be generated is 

 completed: at that instant xy becomes an invariable quan- 

 tity, and bounds the solid at the bottom. Conceive the point 

 generating the line z to move yet onward, carrying along 

 with it the superficies xy, the result will be xyz-, the first 

 term of the fluxion. Proceeding in like manner, xz repre- 

 sents the superficies that bounds the side opposite to the com- 

 mencing point, and xzy will be the second term of the flux- 

 ion. Also yz represents the superficies, that bounds the end, 

 opposite to the commencing point ; this multiplied by x', the 

 fluxion of the length will be yzx', the remaining term. Col- 

 lecting the three terms together, xyz'-\-xzy-\-yzx' is the 

 fluxion of a solid, whose dimensions are expressed by the 

 product xyz. Corresponding to these three terms, the fluent 

 consists of three pyramids. The first is a pyramid, whose 

 apex lies at the corner at which the solid originated, and 

 whose base is at the bottom of the solid : the second has its 

 apex at the same place, and base at the opposite side : the 

 third, its apex at the same place, and base at the opposite 

 end. When the three variable quantities become equal to 

 each other, the solid becomes a cube, and the fluxional ex- 

 pression is x 2 x' -\-x 2 x- -\-x 2 x'=3x 2 x\ But, inasmuch as the 

 philosophical idea of motion is not essential to the method, 

 and is introduced merely for the purpose of illustration, we 

 may conceive quantities of the first, second, third, fourth 

 powers, and of any power whatever, to be generated in a 

 manner somewhat analogous to those which are geometrical, 

 that is, by passing successively through every assignable 

 magnitude. Here we proceed in a manner purely mathe- 

 matical, for we may suppose a quantity to assume any mag- 

 nitude, at pleasure, out of the endless variety of magnitudes, 

 contained within the limits of the greatest and the least. 



By attending to the manner in which fluxions are obtain- 

 ed, the following things will be evident. 1. That each of 

 the variable quantities, in its turn, after the generation of the 

 quantity, termed the fluent, is completed, flows still onward 

 for a given interval of time, producing, by an uniform motion, 

 a line of a finite length ; which may be termed the fluxional 

 base. 2. That all the other variable quantities, at the instant 

 in which the fluent is completed, become invariable, and ac- 

 company the point producing the fluxional base, as it moves 



