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11. The Centre c is the middle of the axis.
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 -tlie conjugate to De ; and hi, parallel to the tangent at A, is the conjugate to AB.
Hence the conjugate Hi, of the axis AB, is perpendicular to it.
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
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 dk 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 follows ing, and to theor. 6 in the Parabola.
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 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 opposite curves mutually conjugate to the other; being all cut by one plane, from four conjugate cones, as in page 94, def. 8.
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 bé a square, or the two axes AB and ab be equal, then the hyperbolas are said to be right-angled, or equilateral.
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 ex: tremities 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 extres mities 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 AVB 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 M/H. will be, as FG?:H::AF. FB: AH . HB. BY
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.
Now, by the similar triangles AFL, AHN, and BFK, BHM,
it is AF : AH :: FL : HN,
hence, taking the rectangles of the corresponding terms, it is, the rect. AF, FB : AH . HB :: KF , FL : MH. HN.
But, by the circle, KF. FL = FG>, and MH . HN = ni'; Therefore the rect. AF. FB: AH . HB :: FG? : HI?. Q. E. D.
For, by theor. 1, AC. CB : AD. DB :: ca’ : DE> ;
CB = AC
AC? : AD, DB :: ac? : DE2; or, by permutation, ac- : ac? :: AD DB : DE; or, by doubling, AB? : ab? : : AD
DB : DE?.
Q. E. D. ab? Coral. Or, by div. AB :
DB or CA?
CD : DE, that is, AB :p :: AD . DB or CA? CD': de;
ab? where p is the parameter
by the definition of it.
As the Square of the Conjugate Axis :
the Difference of the Squares of the Semi-conjugate and
Distance of the Centre from any Ordinate of that Axis : To the Square of their Ordinate.
For, draw the ordinate ED to the transverse AB. Then, by theor. 1, ca?::: DE’: AD. DB or CA– CD’,
ca? : CA:: cd?: CA? de?. But
ca? : CA? :: ca? : CA”, theref. by subtr. ca? : CA? :: ca? cd? or ad . db : de’.
Q. E, D.
Corol. 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.
That is, CA : ca :: DG : DE,
and ca : CA :: dg : de. For, by the nature of the circle, AD DB = DG?; theref. by the nature of the ellipse, CAề: ca’ :: AD. DB or DG? : DE”,
or ÇA : ca :: DG : DE. In like manner
ca : CA :: dg : de. Also, by equality, DG : DE or cd :: de or DC : dg. Therefore cgG is a continued straight line.
. Corol. 2. Hence also, as the ellipse and circle are made 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
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 Semi
axes ; Or, the Square of the Distance between the Foci, is equal to
the Difference of the Squares of the two Axes.
That is, CF2 = CA? - ca,
or Ff? = AB? ab?.
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? : ca’ :: CACF : FE';
? and by the def. of the para. CA? : cao ::
: FE; therefore
CF> ; and by addit, and subtr. or, by doubling,
ff? AB? - ab?. H2
CF2 = CAP
Q. E. D.