The Difference of the Squares of every Pair of Conjugate Diameters, is equal to the same constant Quantity, namely the Difference of the Squares of the two Axes. All the Parallelograms are equal which are formed between the Asymptotes and Curve, by Lines drawn Parallel to the Asymptotes. 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. CA: AL:: CD2 CA2: DE2, and, by the same, QA Al:: EK: Eh; theref. by comp. PA AQ AL: GE. EK HE. Eh; and, because AL2 = HE . Eh, theref. PA. AQ = GE. EK. 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 GLK 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. 1 K G E Corol. 2. If the abscisses CD, ce, CG, &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 rectangles CDH, CEI, CGK, &c, being equal, the ordinates DH, EI, 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 XIIf, 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 — PAEG = QAEK, all standing on the same arc AE. P FOR, by theor. 12, CPAQ CGEK; 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. GLET AVM be a section through the axis of the cone, and AGIH a parabolic section by a plane per pendicular to the former, and parallel to the side VM of the K cone; also let AFH be the com-MA I 2 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, therefore KF = MH; AF AH: KF. FL: MH. HN, But, by the circle, KF. FL = FG2, and MH. HN = HI2; Corol. Hence the third proportional FG2 HI2 AF AH Q.E. D. or is a con stant quantity, and is equal to the parameter of the axis by As the Parameter of the Axis : Is to the Sum of any Two Ordinates :: So that any diameter E1 is as the rectangle of the segments KI, IH of the double ordinate Kн. 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. That is, AF = FE = P, where F is the focus. For, the general property is AF: FE :: FE : P. E 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 FE D Corol. 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. |