ANl'ECEDENTAL CALCULUS. 8i 



and X, (ibidem). Wherefore, if 2YXp reprefent faid circum- 



a 



ference, the antecedental of the fpherical fegment is pDX, of 

 which the antecedent is pDX.. 



EXAMPLE III. 



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

 PI. I.) ATG at the point T ; let the latus re<5luni be reprefented 



a a 



by L. Then L.X is equal to Y\ and L.X to 2YY. Where- 



ei a ' 



fore L is to 2Y (2TE) as Y to X, that is, (Jnt. Cal. p. 9.) a« 

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



EXAMPLE IV. 



a 



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



a 



aYY " 



— L~" ^^ S^t the antecedental of the area, or YX, equal to 



a 



aY^Y 2Y3 2 



— ;. — , the antecedent of which is — f-, or its equal — XXY. 

 L ' 3L ' ^ 3 



otherwise: 



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

 redlangle under AE, and any given line B to the fquare on B, 



a 

 a , a a <>Y*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- 



a 3L 3 



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



two thirds of the redlangle AE, ET. 



Vol. IV. . L E X« 



