ordinates being equal to the rectangle under the difference of the abscisses and the parameter of that diameter, or a third proportional to any absciss and its ordinate. THEOREM X. If a Line be drawn parallel to any Tangent, and cut the Curve in two Points; then if two Ordinates be drawn to the Intersections, and a third to the Point of Contact, these three Ordinates will be in Arithmetical Progression, or the Sum of the Extremes will be equal to Double the Mean. That is, EGHI 2cv. For, draw EK parallel to the axis, and produce Hi tô L. Then, by sim. triangles, EK but, by theor. 2, theref. by equality, But, by the defin. EK HK :: TD or 2AD: CD; : P. 2AD Q. E. D. theref. the 2d terms are equal, KL = 2CD, that is, EGHI 2CD. Cord. When the point E is on the other side of AI; then Any Diameter bisects all its Double Ordinates, or Lines parallel to the Tangent at its Vertex. That is, M FOR, FOR, to the axis AI draw the ordinates EG, CD, HI, and MN parallel to them, which is equal to CD. Then, by theor. 10, 2MN or 2CD = EG + HI, therefore м is the middle of EH. And, for the same reason, all its parallels are bisected. Q. E. D. SCHOL. Hence, as the abscisses of any diameter and their ordinates have the same relations as those of the axis, namely, that the ordinates are bisected by the diameter, and their squares proportional to the abscisses; so all the other properties of the axis and its ordinates and abscisses, before demonstrated, will likewise hold good for any diameter and its ordinates and abscisses. And also those of the parameters, understanding the parameter of any diameter, as a third proportional to any absciss and its ordinate. Some of the most material of which are demonstrated in the following theorems: THEOREM XII. The Parameter of any Diameter is equal to four Times the Line drawn from the Focus to the Vertex of that Diameter. FOR, draw the ordinate MA parallel to the tangent CT: as also CD, MN perpendicular to the axis AN, and Fн perpendicular to the tangent CT. Then the abscisses AD, CM or AT, being equal, by theor. 5, the parameters will be as the squares of the ordinates CD, MA or CT, by the definition But, by theor. 3, P = 4FA, and therefore P=4FT or 4FC. E. B. Corol. Hence the parameter p of the diameter CM is equal to 4FA+4AD, or to P + 4AD, that is, the parameter of the axis added to 4AD. THEOREM XIII, If an Ordinate to any Diameter, pass through the Focus, it will be equal to Half its Parameter; and its Absciss equal to One Fourth of the same Parameter. Again, by the defin. cм or p: ME :: ME: P2 and consequently Corol. 1. Hence, of any diameter, the double ordinate which passes through the focus, is equal to the parameter, or to quadruple its absciss. Corol. 2. Hence, and from cor. 1 to theor. 4, and theor. 6 and 12, it appears, that if the directrix GH be drawn, and any lines HE, HE, parallel to the axis; then every parallel HE will be equal to EF, or of the parameter of the diameter to the point E. HH G }} E JE E THEOREM THEOREM XIV. If there be a Tangent, and any Line drawn from the Point of Contact and meeting the Curve in some other Point, as also another Line parallel to the Axis, and limited by the First Line and the Tangent: then shall the Curve divide this Second Line in the same Ratio, as the Second Line divides the First Line. FOR, draw LP parallel to IK, or to the axis. therefore by equality, IE IK :: CK. CL: CL2; Coral. When CK = KL, then IE EK IK. THEOREM XV. If from any Point of the Curve there be drawn a Tangent, and also Two Right Lines to cut the Curve; and Diameters be drawn through the Points of Intersection E and L, meeting those Two Right Lines in two other Points G and K: Then will the Line Kd joining these last Two Points be parallel to the Tangent. FOR, If a Rectangle be described about a Parabola, having the same Base and Altitude; and a diagonal Line be drawn from the Vertex to the Extremity of the Base of the Parabola, forming a right-angled Triangle, of the same Base and Altitude also; then any Line or Ordinate drawn across the three Figures, perpendicular to the Axis, will be cut in Continual Proportion by the Sides of those Figures. theref. by Geom. th. 78, EF, EG, EH are proportionals, H or EF EG EG EH. Q. E. D. THEOREM XVII. The Area or Space of a Parabola, is equal to Two-Thirdsof its Circumscribing Parallelogram. That is, the space ABCGA = ABCD; or, the space ADCGA ABCD. FOR, conceive the space ADCGA to be composed of, or divided into, indefinitively small parts, by lines parallel to DC or AB, such as IG, which divide AD into like small and equal parts, the number or sum of which is expressed by the line AD. Then, by the parabola, BC2: EG :: AB AE, Hence |