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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 following, 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 hy perbolas 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.
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 hy perbolas, is similar to a rectangle circumscribed about an ellipse, by drawing tangents, in like manner, to the four exa 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 extre mities D, E, F, G, of the said equal conjugate diameters, and those four points drawn out to an infinite distance; the cur vature 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 will be, as FG2: HI2 : : 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.
Now, by the similar triangles AFL, AHN, and BFK, BHM,
it is AF AH :: FL: HN,
and FB: HB :: KF: MH;
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 = FG2, and MH. HN = HI2; Therefore the rect. AF. FB: AH. HB :: FG2: HI2. Q. E. D.
That is, AB: ab2 or
AC2: ac2:: AD. DB : DE2.
For, by theor. 1, AC. CB: AD
. DB:: ca2: DE2; But, if c be the centre, then AC. CB = AC2, and ca is the semi-conjugate.
AC2: AD. DB:: ac2 : DE2; or, by permutation, ac2: ac2 :: AD DB : DE2;
or, by doubling,
AB2: ab2:: AD .
That is, As the transverse,
Is to its parameter,
So is the rectangle of the abscisses,
As the Square of the Conjugate Axis:
So is the Rectangle of the Abscisses of the Conjugate, or
ca2: CB2:: ad. db or ca2
For, draw the ordinate ED to the transverse AB.
Then, by theor. 1, ca2: CA2:: DE2: AD DB or CA2
Ca2: CA2:: Ca2 - cd2 or ad. db : dɛ2.
Q. E. D.
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.
That is, CA: ca :: DG : DE,
and ca: CA :: dg : dɛ.
For, by the nature of the circle, AD. DB = DG2; theref. by the nature of the ellipse, CA: Ca2:: AD. DB or DG2 : DE2,
In like manner
or CA ca :: DG: De.
ca: CA :: dg: dɛ.
DG: DE or cd :: dE or DC : dg. continued straight line.
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 Semi
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 CA2: Ca2:: CA2 CF2: FE2;
of the curve
and by the def. of the para. CA2: ca2 ::
and by addit. and subtr. or, by doubling,
Ca : FE2;
Corol. 1. The two semi-axes, and the focal distance from the centre, are the sides of a right-angled triangle cra; and the distance Fa from the focus to the extremity of the conjugate axis, is AC the semi-transverse.
Corol. 2. The conjugate semi-axis ca is a mean proportional between AF, FB, or between af, fB, the distances of either focus from the two vertices.
For ca2 CA — CE2 = CA + CF. CA
CF AF. FB.
The Sum of two Lines drawn from the two Foci to meet at any Point in the Curve, is equal to the Transverse Axis.
F DI C
For, draw AG parallel and equal to ca the semi-conjugate; and join CG meeting the ordinate DE in H; also take ci a 4th proportional to CA, CF, CD.
Then, by theor. 2, CA2: AG2 :: CA2 - CD2 : DE2;
and, by sim. tri
Also FD CF
CA: AG: CA2 CD2: AG - DH";
And, by right-angled trianglés, FE2 = FD2 + DE2;
Again, by supp. CA2: CD2:: CF2 or CA2 - AG2: CI2; and, by sim. tri. CA2 : CD2 :: CA2 AG2: CD2
And the root or side of this square is FE = CA
In the same manner it is found that fE = CA + CI = BI.
Q. E. D.