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 therefore Ab and De are only parts of two diameters. 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 HI, paral lel to the tangent at A, is the conjugate to AB.

Hence the conjugate HI of the axis AB is perpendicular to it. And hence there is no conjugate to a diameter of the parabola, unless it be considered as at an infinite distance from the vertex.

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, crdinates to the diameter DE.

Hence the ordinates to 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 abscisses; but in the parabola, only one; the other vertex of the diameter being infinitely distant.

16. The

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 ordihate is equal to half the parameter; as K and L, where DK or EL is equal to the semiparameter.

Hence, the ellipse and hyperbola have each two foci ; but the parabola only one.

18. If DAE, FBG be two opposite hyperbolas, having AB 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 a contrary order, namely, ab their first axis, and AB their second; then these two latter curves dae, fbg, are called the conju gate hyperbolas to the two former DAE, FBG; and each pair of opposite curves mutually conjugate to the other.

19. 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.

SCHOLIUM. The rectangle, inscribed between the four conjugate hyperbolas, is similar to a rectangle, circumscrib ed 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 the pair of equal conjugate diameters. Moreover, 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.

In the ellipse, the semiconjugate axis CD, or CE, is a mean proportional between CO and CP, the parts of the diameter OP of a circle, drawn through the centre C of the ellipse, and parallel to the base of the cone. For DE is a double ordinate, or diameter, in this circle, being perpendicular to OP as well as to AB.

In like manner, in the hyperbola, the length of the semiconjugate axis CD, or CE, is a mean proportional between CO and CP, drawn parallel to the base, and meeting the sides of the cone in O and P. Or, if AỔ be drawn parallel to the side VB, and meet PC produced in O′, making CO' CO; and on this diameter O'P a circle be drawn parallel to the base: then the semiconjugate CD, or CE, will be an ordinate of this circle, being perpendicular to O'P as well as to AB.

Or, in both figures, the whole conjugate axis DE is a mean proportional between QA and BR, parallel to the base of the cone. For, because AB is double AC, or CB, therefore, by similar triangles, QA is double OC, and BR double CP; consequently

DE or 2CD 2CE, or 2CO 2CP is QA BR, or QA: DE :: DE : BR,

In the parabola, both the transverse and conjugate are infinite; for AB and BR are both infinite.

COR. 2. In all the sections AG will be equal to the parameter of the axis, if QG be drawn making the angle AQG equal to the angle BAR.

For, by the definition, AB DE :: DE: p the parameter. But, by Cor. 1, BR : DE :: DE : AQ__;

Therefore, AB : BR :: AQ : p.

But, by similar triangles, AB : BR :: AQ: AG ;
And therefore AG the parameter.

AG=;

In

In like manner Bg will be equal to the parameter, if Rg be drawn to make the angle BRg the angle ABQ ; since here also AB : AQ :: BR ; Bg=p.

COR. 3. Hence the upper hyperbolic section, or section of the opposite cone, is equal and similar to the lower section. For the two sections have the same transverse or first axis AB, and the same conjugate or second axis DE, which is the mean proportional between AQ and RB; they have also equal parameters AG, Bg. So that the two opposite sections make, as it were, but the two opposite ends of one entire section or hyperbola, the two being every where mutually equal and similar; like the two halves of an ellipse, with their ends turned the contrary way.

COR. 4. And hence, although both the transverse and conjugate axis in the parabola be infinite, yet the former is infinitely greater than the latter, or has an infinite ratio to it. For the transverse has the same ratio to the conjugate, as the conjugate has to the parameter, that is, as an infi, nite to a finite quantity, which is an infinite ratio.

ELLIPSE.

PROPOSITION I,

THE squares of the ordinates of the axis are to each other as the rectangles of their abscisses.

Let RVB be a plane passing through the axis of the cone; AEBD another section of the cone perpendicular to the plane RVB, but oblique to another plane passing through the axis perpendicularly to this; AB the axis of