8o O/i the PRINCIPLES of the 



ed, and that it is equally geometrical with the method of ex- 

 hauftions of the ancients, who never fiippofed lines, furfaces, 

 or folids, to be refolved into indefinitely fmall or infinitely little 

 elements. The expreflion infnitely little magnitude indeed im- 

 plies a contradidlion. For what has magnitude cannot be infi- 

 nitely little. 



This geometrical calculus, though it has no connection with 

 the various modifications of motion, is equally convenient in its 

 application with tlie method of fluxions, (which is unquefliona- 

 bly a branch of general arithmetical proportion, in which i or 

 unit is the common ftandard of comparifon, as well as the con- 

 fequent of every ratio compounded, or decompounded). 



EXAMPLE I. 



In the circle ATB, (Fig. I. PI. I.) let the diameter AB be re- 

 prefented by D, TE perpendicular to it by Y, and AE by X. 

 Then (13. E. 6.) ^ is equal to the redangle DX — X\ But the 



antecedental of Y' is 2YY, and that of DX-X' is DX— 2XX, 

 (p. 6. Antecedental Calculus). Wherefore D — 2X is to 2Y as 



a a 



Y to X, tliat is, as TE to CE, (p. 9. Ant. Cal.). Confequently 

 CE is a third proportional to EO and TE. 



EXAMPLE IT. 



To find the furface of the fphere of which ATBA is a great 

 circle, (Fig. I. PI. L). 



The furface of the fpherical fegment, cut off by the circle, 

 of which TE is the radivis, has to the fquare on any given line 

 B, a ratio compounded of the circumference of faid circle to B, 

 and of the antecedental of the curve AT to B, [Ant. On I. p. 9.) 

 But the antecedental of the curve is a fourth proportional to 2YD 



and 



