PARABOLA. 



times the rectangle M Q B -j- 4 M Q* ; 

 that is, to 4 times the rectangle Q M B. 

 But M Q = Q K = D R, and M B =: 

 I) A ; wherefore Q R J = 4 times the rect- 

 angle 11 D A : and because Q R, E L are 

 ordinates to the dian>eter A D, Q R 1 (by 

 cor. 2, of Prop. 4), : E L* (.- R D : L D) 

 : r 4 times the rectangle R D A : 4 times 

 the rectangle L D A. Therefore E L 1 = 

 4 times the rectangle L D A, or the rect- 

 angle contained under the absciss L D, 

 and the parameter of the diameter A 1) : 

 and from this property Apollonius called 

 the curve a parabola. Q. E. I). 



Prop. 6. If from any point of a para- 

 bola, A. (iig. 6) there be drawn an ordi- 

 nate, A C, to the diameter B C ; and a 

 tangent to the parabola in A, meeting- the 

 diameter in D : then shall the segment of 

 the diameter, C D, intercepted between 

 the ordinate and the tangent, be bisected 

 in the vertex of the diameter B. For let 

 B E be drawn parallel to A D, it will be 

 an ordinate to the diameter A E ; and the 

 absciss B C will be equal to the absciss 

 A E, or B D. Q. E. D. 



Hence, if A C be an ordinate to B C, 

 and A D be drawn so as to make B D = 

 D C, then is AD a tangent to the parabo- 

 la. Also the segment of the tangent, A D, 

 intercepted between the diameter and 

 point of contact, is bisected by a tangent 

 B G, passing through the vertex of D C. 



" To draw Tangents to the Parabola." 

 If the point of contact C be given : (fig. 7) 

 draw the ordinate C B, and produce the 

 axis till A T be = A B ; then join T C, 

 which will be the tangent. Or if the 

 point be given in the axis produced : take 

 A B = A T, and draw the ordinate B C, 

 which will give C the point of contact ; to 

 which draw the line T C as before. If D 

 be any other point, neither in the curve 

 nor in the axis produced, through which 

 the tangent is to pass : draw D E G per- 

 pendicular to the axis, and take D H 

 a mean proportional between D E and 

 I) G, and draw H C parallel to the axis, so 

 shall C be the point of contact through 

 which, and the given point D, the tangent 

 D C T is to be drawn. 



When the tangent is to make a given 

 angle with the ordinate at the point of 

 contact : take the absciss A I equal to half 

 the parameter, or to double the focal dis- 

 tance, and draw the ordinate I E : also 

 draw A H to make with A I the angle 

 H A I equal to the given angle ; then 

 draw H C parallel to the axis, and it will 

 cut the curve in C, the point of contact, 

 where a line drawn to make the given 



angle with C B will IQ the tangent re- 

 quired. 



" To find the Area of a Parabola." Mul- 

 tiply the base E G by the perpendicular 

 height A I, and of the product will be 

 the area of the space A EGA; because 

 the parabolic space is of its circumscrib- 

 ing parallelogram. 



" To find the Length of the Curve 

 A C," commencing at the vertex. Let >j 

 = the ordinate B C, p = the parameter, 



q ^l } an d $ = ^/ 1 -}- y> ; then shall 



I P X (? * H- hyp. log. of q -f s) be the 

 length of the curve A C. 



PAHAUOLA, Cartesian, is a curve of the 

 second order, expressed by the equation 

 x y = a x3 -j- b x 1 - -f- c x -f- d, containing 

 four infinite legs, viz. two hyperbolic 

 ones, M M, B m, (Plate Parabola, fig. 8), 

 (A E being the asymptote) tending con- 

 trary ways, and two parabolic legs B N, 

 M N joining them, being the sixty-sixth 

 species of lines of the third order, accord- 

 ing to Sir Isaac Newton, called by him a 

 trident : it is made use of by Des Cartes, 

 in the third book of his Geometry, for find- 

 ingthe roots of equations of six dimensions 

 by its intersections with a circle. Its most 

 simple equation is x y = xi -}- ai, and the 

 points through which it is to pass, may be 

 easily found by means of a common para- 

 bola, whose absciss is a x- -f- ^ x + c, 



and an hyperbola, whose absciss is ; 



for y will be equal to the sum or differ- 

 ence of the correspondent ordinates of 

 this parabola and hyperbola. 



PARABOLA, diverging, a name given by 

 Sir Isaac Newton to five different lines 

 of the third order, expressed by the equa- 

 tion y y = a x3 -j- b x* -|- c x -}- d. 



PARABOLIC asymptote, in geometry, 

 is used for a parabolic line approaching 

 to a curve, so that they never meet ; yet, 

 by producing both indefinitely, their dis- 

 tance from each other becomes less than 

 any given line. Maclaurin observes, that 

 there may be as many different kinds of 

 these asymptotes as there are parabolas 

 of different orders. 



When a curve has a common parabola 

 for its asymptote, the ratio of the sub- 

 tangent to the absciss approaches con- 

 tinually to the ratio of two to one, when 

 the axis of the parabola coincides with 

 the base ; but this ratio of the subtan- 

 gent to the absciss approaches to that of 

 one to two, when the axis is perpendicu- 

 lar to the base. And by observing the 

 limit to which the ratio of the subtangent 



