That is, the triangle cts = 2 paral. GK. T K each equal to half the constant inscribe:l paralit.logram ce. And therefore the whole triangle cts. w ich is composed of the two smaller triangles and the parallelogram, is equal to double the constant inscribed paralleiogram Gk. Q. E. D. THEOREM XXIX. If from the Point of Contaci of any Tangent, and the two Intersections of the Curve with a Line parallel to the Tangent, three parallel Lines be drawn in any Direction, and terminated by either Asymptote ; those three Lines sball be in continued Proportion. H, That is, if h&m and the tangent il be parallel, then are the parallels pH, EI, GK in continued proportion. K C D E LG ML theref by compos. EI? : IL .::DH, GK : HNK ; Q. E. D. DH THEOREM XXX. Draw the semi-diameters cH, CIN, CK; N K C D E L 6 M VOL. I. For; For, because hk and all its parallels are bisected by Cin, therefore the triangle CNH = tri CNK, consequently the sector CiH = sec. CIK, So that by taking any geometricals CD, CE, CG, &c. and drawing DH EI, GK, &c. parallel to the other asymptote, as also the radii cH, CI, CK ; then the sectors chi, CIK, &c. will be in arithmetical progression. and chk Or DHKG the log. of the ratio of co to CG, &c. or of GK to DH, &c. 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 K cone ; also let AFH be the common G intersection of the two planes, or M N the axis of the parabola, and FG, I HII ordinates perpendicular to it. Then it will be, as Ar : AH : : TG?: 11%. For, through the ordinates FG, HI draw the circular sece tions, KGL, MIN, parallel to the base of the cone, having RL, NN un 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 ; therefore AF: AH :: KF. FL : MH . HN. Bui, by the circle, KF . FL + FG%, and mH . AN = Therefore AF: AH :: FG2 : H12, Q. ED. FG2 HI Corol. Hence the third proportional or is a constant hi; AF AH quantity, and is equal to the parameter of the axis by defin. 16. Or av: FG : : FG : P the parameter, DE? For, by cor. theor. 1, P. AG - GR, and P. AD = DE? ; theref. by subtraction, P DC = GH: Or, P. DG = KI. IH, therefore P: KI: :1H: DG or EI. Q. 2. D. Corol. Hence because P . EI = KI . IH, and, by cor. theor. 1, P. AG = GH, therefore AG: EI :: GH? : KI, IH. So that any diameter Ei is as the rectangle of the seg. ments KI, IH of the double ordinate kh. 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 А That is, ar = {PE = 12 where F is the focus. For, the general property is AF : FE : : FE: 1'. But by definition 17, EFP; therefore also AF = FE = R. E. B. 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. = ApS For, since FD AD AF, theref. by squaring. FD? 2AF. AD + AD”, But, by cor. theor. 1, DE? = P. AD = 4AF.AD; theref. by addition, FD? + de = AF? + 2AF, AD + AD, But, by right-ang. tri. FD? + DER = FE?; therefore FE2 = AF2 + 2AF. AD + AD, and the root or side is TE = AF + AD, or FE = GD, by taking AG = AF. Q: E. B. Corol. 1. If, through the point G, the HHH THE 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 line he parallel to the axis.lt E He is always equal to FE the distance of the focus from the point E. EL Corol. 2. Hence also the curve is easily described by points Namely, in the axis produced take AG = Ar the focal distance, and draw a number of lines Es 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. THEOREM THEOREM V. If a Tangent be drawn to any Point of the Parabola, meet ing 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. T or ΑΙ That is, if tc touch the curve D E at the point c; M м then is AT =AM. L G H 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 : Gho; and by sim. tri. TD? : TG2 : : DE? : Gho; theref.by equality,AD : AG : : TDP : TG'; and, by division, Ad:DG::: 1G2-TD? or DG. (TDATG), AD : TD :: TD : TD+TG; and, by division, AD : AT :: TD : TG, and again by div. AD: AT :: AT: AG; AT is a mean propor. between AD, AG: Now if the line tu 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 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 ordinates cm. Consequently AD, AG, becoming each equal to ÅM, their mean proportional Ar 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. P. or THEOREM VI. If a tangent to the Curve meet the Axis produced ; then the Line drawn from the Focus to the point of Contact, will be equal to the Distance of the Focus from the Intersection of the Tangent and Axis. That |