A M G, AN:AM or 9E::NE:MO Or C D. By eollStrUeti011 we have y5:on or nA::co:c n; and theretbre by mul tiplication we have it N N A : C n', which property is known to be that of the ellipsis and hyperbola.
Corollary.—Since, in tire parabola, It N and n C are of infi nite length, and may therefore he said to be equal, n N and B c may tberet'ore he expunged from the first two terms of the analogy in the above general property; then we shidl have Or the truth of the operation may be shown by a particular demonstration for the parabola thus: See Figure 9.
Beeause of the similar triangles A N K and A M g, A N : AE::NE:my or CD; by construction we have AM:Ac: : C 0 or NK: CD; and consequently, by 11011tipli•llti011, A N: A C : : N : CD', w lri 'h is the property of the parabola.
Figures S, 9.—fa a runic section, are given the abscissa, A C, an ordinate, D. and a point, K, irn the curve : to deter mine the species', and thence to describe the curve.
Draw 1.: y 1) parallel 1.0 A C, and A E parallel tm C 0; through the points A ands draw A y, cutting E D at y ; make H c : O C : 1)1( : q E. and 1111'0001 the points K turn 0 dlaW K 0 B Or o K u, Which, ir not parallel to A C, produce it until it meet A C or C A iu B ; [hell .111 W ill be a diameter. In this case the curve is ellipsis or hyperbola. It is an ellipsis when the extremities or the diameter are on different sides of the ordinate, as in Figure 5 ; but when the extremities of the diameter are on the same side of the ordinate, the curve is an hyperbola. If E o be parallel to A C. the curve is a parabola. A diameter. A 13, and all ordinate, c I), being thus ascertained, the curve will be described as in Flores Other pa• ticula•s relating to these curves will he 10111131 Under the arti cles ELL1P1.11S, HYPERBOLA, and PARABOLA.