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This Proposition being well understood, 'twill be very easily apply'd to Practice, supposing the Focal Distance given, or any other Data by which it may be found. Thus draw any Right Line to represent the Parabola's Axis, and from its vertical Point, as at S, set off the Focal Distance both upwards and downwards, viz. make Sf=SF, the Distance of the given Focus from the Vertex; as in the Scheme : Then by the Propofition’tis, evident, that, if never so many Lines be drawn Ordinately at Right Angles to the Axis, the true Distance between the Point f out of the Parabola, and any of those Lines (or Ordinates) being measur’d or set off from the Focus F to the same Line or Ordinate, 'twill assign the true Point in that Line through which the Curve must pass; that is, it will shew the true Limits or Length of that Ordinate ; as at B in the laft Scheme.
Proceeding on in the very fame Manner from Ordinate to Ordinate, you may with great Expedition and Exactness find as many Ordinates (or rather their Points only, like B) as may be thought convenient, which, being all join'd together with an even Hand, will form the Parabola requir’d.
N. B. The more Ordinates (or their Points) there are found, and the nearer they are to one another, the easier and exaéter may the Curve of the Parabola be drawn. The fame is to be observ'd when any other Curve is requir'd to be drawn by Points.
Sect. 5. To draw a Tangent to any given Point in the
Curve of a Parabola. Tangents are very easily drawn to the Curve of any Parabola ; For, fuppofing S to be its Vertex, B the Point of Contact (viz. the Point where the Tangent must touch the Curve) and P the Point where the Tangent will intersect (or meet with)
À the Parabola's Axis produced: Then if from the Point of Contaxt B there be drawn the Semi-ordinate B A at Right Angles to the Axis S A, wheresoever the Point A falls in the Axis, 'twill be SP=SA.
CH A P. IV. Concerning the chief Properties of the Hyperbola. TOTE, any Part of the Axis of an Hyperbola, which is inter
cepted between its Vertex and any Ordinate (viz. any intercepted Diameter) is call'd an Abscisá; as in the Parabola.
Sect. I. The Plain of every Hyperbola is proportion'd by this general
Let the following Figure HVG represent a Right Cone cut into two Parts by the Right Line SA; then will the Plain of that Section be an Hyperbola (by Sect. 5. Chap. 1.) in which let S A be its Axis, or intercepted Diameter, b ab and B AB Ordinates rightly apply'd (as before in the Parabola) and TS its Transverse Diameter. Again, if the Cone is suppos’d to be cut by hg, parallel to its Base HG, it will also be the Diameter of a Circle, 90. as in the Ellipsis and Parabola. Then will the A Sga and A SGA be alike, also the A Tab' and ATAH will be alike; therefore it
These Proportions are the common Property of every Hyperbola, and do only differ from those of the Ellipsis in the Signs +
and —-; as plainly appears in the fol-
(in a manner) be found as in the Ellipfis, due Regard being had to changing of the Sines.
Sca. 2. To find the Latus Kecum, or Kight Parameter,
of any Hyperbola.