For, let A be the vertex of the curve, or extremity of the semi-transverse axis Ac, perp, to which draw AL or Al, which will be equal to the semi-conjugate, by definition 19. Also, draw HEDeh parallel to Ll, Then, by theor. 2. CA2: AE2 :: CD2 CA2: DE2, and, by parallels, theref. by subtract. conseq. the square AL2 CA2 CA2 AL2:: CD2: DH2; AL2 :: CA2: DH2 rect. HE. Eh; the rect. HE Eh. : But, by sim. tri, and, by the same, theref. by comp. PA AL GE; EH, AL and, because PA AQ AL GE. EK: HE. Eh; But the parallelograms CGEK, CPAQ, being equiangular, are as the rectangles GE. EK and PA. AQ. Therefore the parallelogram GK = the paral. PQ That is, all the inscribed parallelograms are equal to one another. Q. E. D. Corol. 1. Because the rectangle GEK or CGE is constant, therefore GE is reciprocally as CG, or CG: CP :: PA : GE. And hence the asymptote continually approaches towards the curve, but never meets it for GE decreases continually as CG increases; and it is always of some magnitude, except when CG is supposed to be infinitely great, for then GE is infinitely small, or nothing. So that the asymptote CG may be considered as a tangent to the curve at a point infinitely distant from c.. Corol. 2. If the abscisses CD, ce, GG, &c, taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates DH, EI, GK, &c, parallel to the other asymptote, be a decreasing geometrical progression, having the same ratio. For, all the 44 K rectangles CDH, CEI, CGK, &c, being equal, the ordinates DH, El, GK, &c, are reciprocally as the abscisses CD, CE, CG, &c, which are geometricals. And the reciprocals of geometricals are also geometricals, and in the same ratio, but decreasing, or in converse order. THEOREM THEOREM XIII. The three following Spaces, between the Asymptotes and the Curve, are equal; namely, the Sector or Trilinear Space contained by an Arc of the Curve and two Radii, or Lines drawn from its Extremities to the Centre; and each of the two Quadrilaterals, contained by the said Arc, and two Lines drawn from its Extremities parallel to one Asymptote, and the intercepted Part of the other Asymp tote. That is, The sector CAE = PAEĠ = QÁEK, all standing on the same arc AE. FOR, by theor. 12, CPAQ = CGEK; subtract the common space CGIQ, P there remains the paral. PI = the par. IK; to each add the trilineal IAE, then take the equal triangles CAQ, CEK, and there remains the sector CAE QAEK. Q. E. D. OF THE PARABOLA. THEOREM I. The Abscisses are Proportional to the Squares of their Ordinates. LET AVM be a section through the axis of the cone, and AGIH a parabolic section by a plane perpendicular to the former, and parallel to the side VM of the cone; also let AFH be the common intersection of the two planes, or the axis of the parabola, and FG, HI ordinates perpendicular to it. I 2 K D M Then Then it will be, as AF: AH :; FG2 : HI2. For, through the ordinates FG, HI draw the circular sections, KGL, MIN, parallel to the base of the cone, having KL, MN for their diameters, to which FG, HI are ordinates, as well as to the axis of the parabola. Then, by similar triangles, AF : AH :: FL: HN;' but, because of the parallels, KF = MH; AF AH: FG2: HI2. Q. E. D. FG2 HI2 But, by the circle, KF. FLFG2, and MH. HN = HI2; Corol. Hence the third proportional or is a conАН AF 1 stant quantity, and is equal to the parameter of the axis by As the Parameter of the Axis : So that any diameter EI is as the rectangle of the seg ments KI, IH of the double ordinate KH. THEOREM THEOREM III. The Distance from the Vertex to the Focus is equal to of the Parameter, or to Half the Ordinate at the Focus. For, the general property is AF: FE :: FE : P. But, by definition 17, therefore also THEOREM IV. A Line drawn from the Focus to any Point in the Curve, is equal to the Sum of the Focal Distance and the Absciss of the Ordinate to that Point. A E E D E E D Coro!. 1. If, through the point G, the HHH G line GH be drawn perpendicular to the axis, it is called the directrix of the parabola. The property of which, from this theorem, it appears, is this: That drawing any lines HE parallel to the axis, HE is always equal to FE the distance of the focus from the point E. Corol. Corol. 2. Hence also the curve is easily described by points. Namely, in the axis produced take AG AF the focal distance, and draw a number of lines EE perpendicular to the axis AD; then with the distances GD, GD, GD, &c, as radii, and the centre F, draw arcs crossing the parallel ordinates in E, E, E, &c. Then draw the curve through all the points E, E, E. THEOREM V. If a Tangent be drawn to any Point of the Parabola, meeting the Axis produced; and if an Ordinate to the Axis be drawn from the Point of Contact; then the Absciss of that Ordinate will be equal to the External Part of the Axis. L For, from the point T, draw any line cutting the curve in the two points E, H: to which draw the ordinates DE, GH; also draw the ordinate MC to the point of contact c. Then, by th. 1, AD: AG :: DE2 : GH2; and, by sim. tri. TD2: TG2:: DE2: GH2; theref. by equality, AD : AG :: TD2: TG2; and, by division, AD: DG:: TD: TG-TD2 or DG. (TD+TG), and again by div. AD AT: AT: AG; or AT is a mean propor. between AD, AG. Now, if the line TH be supposed to revolve about the point T; then, as it recedes farther from the axis, the points E and H approach towards each other, the point E descending, and the point H ascending, till at last they meet in the point c, when the line becomes a tangent to the curve at c. And then the points D and G meet in the point M, and the ordinates DE, GH in the ordinate CM. Consequently AD, AG, becoming each equal to AM, their mean proportional AT will be equal to the absciss AM. That is, the external part of the axis, cut off by a tangent, is equal to the absciss of the ordinate to the point of contact. Q.E. D. THEOREM |