In like Manner an indefinite Number of Points may be found; through which, if a Curve Line be fuppofed drawn, it will comprehend a Space called an Ellipfe. DEFINITIONS. 1. The Points H and K are called the Foci. 2. A Diameter is a right Line which paffes through C, the Middle of AB, and bifects all Lines within the Curve, that are parallel to the Tangent touching its Vertex, and the Lines fo bifected are called Ordinates to that Diameter; fo FY is a Diameter; XOOZ are Ordinates, being parallel to the Tangent touching the Curve in F, the Vertex of the Diameter. 3. The Point of Interfection C, of all the Diameters, is called the Center. 4. That Diameter on which the Ordinates ftand at right Angles, is called the tranfverfe Axe, as AB; and that which paffes through the Center, cutting it at right Angles, is called the conjugate. Axe, as ED. 5. The Point, where the Ordinates interfect the Diameter, is called the Point of Application, as G and O. 6. The Segment of the Diameter, intercepted. between the Vertex and the Point of Application, is called the Abfciffa; as FO, OY; or BG, AG. PROPOSITION I. As the Square of any Ordinate to the transverse Axe is to the Rectangle of the Abfciffas which it divides, fo is the Square of the Conjugate to the Square of the tranfverfe Axe. DEMONSTRATION. Let AC, CE=c, KC=b, CG=x, FG =y, and z equal to the Difference between the Line KF and the Semi-tranfverfe Axe AC; then KH 2b, KGb+x, and GHx-b or b-x, according as the Point G falls on this or that Side the Focus H; alfo, by the Genefis, KF: =+ and FH-t-z; whence (by Eu. 47. 1.) HF2=HG2+GF2; or t2 — 2 tz+z2±b2 → 2 b x + x2 + y2, and KF2= KG2+GF2; or t2 + 2 tx+z2 = b2 + 2 b x + x2 + y2; hence, the former of these Equations taken from the latter gives, 4tz=4bx; therefore z= 46x bx ; which being fubftituted in Place of 4t t z, in either of the foregoing Equations, there will 4 2 come out ++b2 x2 = t2 b2 + t2x2 + t 2 y2; 2 but b2=t2 — c2, by the Genefis; therefore 2x2 xc2=t2y2; which reduced to an Analogy, gives y2: ¿ + x x = x :: c2:t; that is, FG2: AGXGB 2 ::DE': AB'. Q.E. D. COROLLARY. AG× Let any Abfciffa be x, and its Ordinate y, the tranfverfe Axis t, and the Conjugate c; (which Symbols represent the fame Things in all the following Demonftrations) then by this Theorem, t2: c2::t-xxx: y2; or t2 y2 = c2 tx — c2x2; which generally is called the Equation of the Curve. 2 2 2 DEFINITION. A third Proportional to the tranfverfe and conjugate Axis, is called the Parameter of the Axe; that is, if for the Parameter be put p, then t:c cp; therefore tpc2. PROPOSITION II. As the tranfverfe Axe, is to its Parameter, fo is the Rectangle of any two Abfciffas, to the Square of the Ordinate which divides them. DE DEMONSTRATION. By the Definition of the Parameter pc', and by putting tp in the Equation of the Curve for c2, a new Equation of the Curve will be produced in Terms of the Parameter, &c. viz. ty' 2 2 =tpx-px; or y1 t:pt-xxx:y'. QE. D. COROLLARY. As the Rectangle of any two Abscissas, is to the Square of the Ordinate which divides them, fo is the Rectangle of any other two Abfciffas, to the Square of the Ordinate which divides them. For (by this Prop.) t-xxx:yt:p::t-XxX: Y'. 2 PROPOSITION III. The tranfverfe Axe into one fourth of its Parameter, is equal to the Rectangle of the greatest and least Distance of either Focus from the Vertex; that is, pxAB=AH× HB = BK × KA. DEMONSTRATION. Let HB-q, then HA=t-q, and CH=4-q: 2 2 But HE EC2+CH; that is, t2=t1 = t q + q2 + 4 c 2; or t-qxq=c2 = 4pt; or 4px ABAH x HB. QE. D. COROLLAR Y. The femi-conjugate Axe, is a mean Proportional between the greatest and least Distance of either Focus from the Vertexes: For fince 7-qxq= c2; therefore t―q:cc:q; that is, AH: CD :: CD: HB. PROPOSITION IV. The Parameter of the Axe, is double the Ordinate applied to the Focus. DEMONSTRATION. Let the focal Distance be q, and the Ordinate paffing through the Focus y; then (by Prop. 2.) t:p :: t = qxq: y'; but (by Prop. 3.) — qxq= 2 2 pt; therefore t:p :: 4 pt : 4 p2 = y2, and 1⁄2 p y; or p2y. Q.E. D. PRO |