334 



A Theory of Fluxions. 



are different, and therefore not a straight line. Hence a cir- 

 cle cannot be a polygon of an infinite number of sides. 



Fig. 4. 



Prin. IV. The fluxion GnKE is not an 

 ^elementary part of the fluent AKL ; that 

 is, the fluent AKL is not compounded of 

 any number of any such small spaces as 

 GnKE whatever. For, however small the 

 curvilineal space GLKE may have been 

 made by repeated divisions, the fluxion 

 GnKE still differs from it by the triangle 

 GLn ; and, notwithstanding we may con- 

 tinue to subdivide it, there will always re- 

 Bmain the triangular space ILa, by which 

 the fluxion differs from an elementary part. 



Prin. V. If the difference between two quantities, contin- 

 ually approaching towards each other, becomes less than 

 any assignable quantity, those two quantities are then equal. 



For if it be said that they are unequal, let their difference 

 be represented by D, then if the approximation be continued, 

 their difference will, at last, become less than D, and so D 

 is not their difference, which is absurd. Therefore the pro- 

 position advanced is true. 



Prin. VI. When the difference between any two quantities 

 is less than any assignable quantity, the one may be taken 

 for the other. 



Sec. 3. The grounds of Fluxions. 



Every variable quantity, which occurs in the process of a 

 mathematical calculation, may be considered as an isolated 

 term in a series of fluents, that may be conceived to arise, 

 when that variable quantity is expanded. 



