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Ellipse. Hyperbolas. Parabola.
Hence the centre of a parabola is infinitely distant from the vertex. And of an ellipse, the axis and centre lie within the curve; but of an hyperbola, without.
12. A Diameter is any right line, as AB or DE, drawn through the centre, and terminated on each side by the curve; and the extremities of the diameter, or its intersections with the curve, are its vertices. Hence all the diameters of a parabola are parallel to the axis, and infinite in length. And hence also every diameter of the ellipse and hyperbola have two vertices; but of the parabola, only one; unless we consider the other as at an infinite distance.
13. The Conjugate to any diameter, is the line drawn. through the centre, and parallel to the tangent of the curve at the vertex of the diameter. So, FG, parallel to the tangent at D, is the conjugate to DE; and H1, parallel to the tangent at A, is the conjugate to AB. Hence the conjugate HI, of the axis AB, is perpendicular to 1t. 14. An Ordinate to any diameter, is a line parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter and curve. So DK, EL, are ordinates to the axis AB; and MN, No, ordinates to the diameter DE. Hence the ordinates of the axis are perpendicular to it.
15. An Absciss is a part of any diameter contained between its vertex and an ordinate to it; as AK or BK, or DN or EN. *
Hence, in the ellipse and hyperbola, every ordinate has two determinate abscisses; but in the parabola, only one; the other vertex of the diameter being infinitely distant.
16. The Parameter of any diameter, is a third proportional to that diameter and its conjugate.
17. The Focus is the point in the axis where the ordinate is equal to half the parameter. As K and L, where pro or EL is equal to the semi-parameter. The name focus being given to this point from the peculiar property of it mentioned in the corol. to theor. 9 in the Ellipse and Hyperbola following, and to theor. 6 in the Parabola. *. Hence, the ellipse and hyperbola have each two foci, but the parabola only one. - H a. I
E 2 b .* N. 18. If DAE, FBG, be two opposite hyperbolas, having As for their first or transverse axis, and ab for their second or conjugate axis. And if dae, fbg, be two other opposite hyperbolas having the same axes, but in the contrary order, namely, ab their first axis, and AB their second; then these two latter curves dae, fbg, are called the conjugate hyperbolas to the two former DAE, FBG, and each pair of opposite curves mutually conjugate to the other; being all cut by one plane, from four conjugate cones, as in page 94, def. 8. 13. And if tangents be drawn to the four vertices of the curves, or extremities of the axes, forming the inscribed rectangle HIKL ; the diagonals HCK, IcL, of this rectangle, are called the asymptotes of the curves. And if these asymptotes intersect at right angles, or the inscribed rectangle be a square, or the two axes AB and ab be equal, then the hyperbolas are said to be right-angled, or equilateral. SCHOLIUM. The rectangle inscribed between the four conjugate hyperbolas, is similar to a rectangle circumscribed about an ellipse, by drawing tangents, in like manner, to the four extremities of the two axes; and the asymptotes or diagonals in the hyperbola, are analogous to those in the ellipse, cutting this curve in similar points, and making that pair of conjugate diameters which are equal to each other. Also, the whole figure formed by the four hyperbolas, is, as it were, an ellipse turned inside out, cut open at the extremities D, E, F, G, of the said equal conjugate diameters, and those four points drawn out to an infinite distance; the curvature being turned the contrary way, but the axes, and the rectangle passing through their extremities, continuing fixed.
OF THE ELLIPSE.
The Squares of the Ordinates of the Axis are to each other as the Rectangles of their Abscisses.
LET Ave be a plane passing through the axis of the cone; AGIH another section of the cone perpendicular to the plane of the former; AB the axis of this elliptic section; and FG, HI, ordinates perpendicular to it. Then it will be, as FG% : HI*::AF. FB : AH . HB.
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 ellipse.
hence, taking the rectangles of the corresponding terms, it is, the rect. AF. FB : AH. HB :: KF. FL: MH. H.N.
As the Square of the Transverse Axis: * ... Is to the Square of the Conjugate
So is the Rectangle of the Abscisses :
To the Square of their Ordinate.
Vol. II, H That
That is, As the transverse,
As the Square of the Conjugate Axis:
Carol. 1. If two circles be described on the two axes as diameters, the one inscribed within the ellipse, and the other circumscribed about it; then an ordinate in the circle will be to the corresponding ordinate in the ellipse, as the axis of this ordinate, is to the other axis.
Corol. 2. Hence also, as the ellipse and circle are made up of the same number of corresponding ordinates, which are all in the same proportion of the two axes, it follows that the areas of the whole circle and ellipse, as also of any like parts of them, are in the same proportion of the two axes, or as the square of the diameter to the rectangle of the two axes; that is, the areas of the two circles, and of the ellipse, are as the square of each axis and the rectangle of the two ; and therefore the ellipse is a mean proportional between the two circles. -
The Square of the Distance of the Focus from the Centre, is equal to the Difference of the Squares of the Semiaxes;
Or, the Square of the Distance between the Foci, is equal to the Difference of the Squares of the two Axes.
For, to the focus F draw the ordinate FE; which, by the definition, will be the semi-parameter. Then, by the nature of the curve - CA*: Cao :: cA* – cF* : FE”;