^NTECEDENTJL CALCULUS. 8i 



and X, {ibidem). Wherefore, if lYXp reprefent faid circum- 



a 



ference, the antecedental of the fpherlcal fegment is /'DX, of 

 which the antecedeiat is /'DX. 



EXAMPLE III. 



If it be required to draw a tangent to the parabola (Fig. t. 

 PL I.) ATG at the point T ; let the latus redlum be reprefented 



by L. Then L.X is equal to Y\ and L.X to aYY. Where- 



a a 



fore L is to 2Y (2TE) as Y to X, tliat is, [Ant. Cal. p. 9.) as 

 TE to CE, which is confequently equal to twice AE. 



EXAMPLE IV. 



In finding the area of the parabola, fince X is equal to 



aYY " 



"■■ L ^e get the antecedental of the area, or YX, equal to 



sY^Y . aY3 2 



— Y — , the antecedent of which is — — , or its equal — XXY. 



otherwise; 



The ratios of the antecedentals of the area AET, and the 

 re<Slangle under AE, and any given fine B to the fquare on B, 



a 

 a a a aY^Y 



are YX and BX to B\ But YX is equal to , the antece- 



dent of which is ,* or its equal — X XY ; and the antece- 



3L ^3 



dent of BX is BX. Wherefore the area of the parabola is 



two thirds of the redlangle AE, ET. 



Vol. IV. . L EX- 



