But, by Prop. II. CA: Ca :: (ADDB or) Cd : DE*; therefore, CA : Ca :: Cd DE;

in like manner, CA : Ca :: CD: de,

or Ca : de :: CA: CD.

But, by Prop. VIII. CT:

CA :: CA : CD;

therefore, by equality, CT : CA :: Ca
: CK +: Ce : de;

de.

CA :: Ca: Ce,

But, by sim. tri. CT
therefore, by equality, CK:
and the rectangle CK CeCA Ca.
But the rectangle CK Ce
therefore the rect. CA Ca
and consequently the rect.

the parallelogram CEPe,

the parallelogram CEP, AB'ab the paral. PQRS.

Q. E. D.

COR. I. The rectangles of every pair of conjugate diameters are to one another reciprocally as the sines of their included angles. For the areas of their parallelograms, which are all equal among themselves, are equal to the rectangles of the sides, or conjugate diameters, multiplied by the sines of their contained angles, the radius being 1. That is, the rectangle of every two conjugate diameters, drawn into the sine of their contained angle, is equal to the same constant quantity. And therefore the rectangle of the diameters is inversely as the sine of their contained angle.

COR. 2. As it is proved in this proposition, that every circumscribing parallelogram of an ellipse is a constant quantity; so it may hence be shewn, that each of the spaces EAP, EagQ2 gBGR, GbeS, between the curve and the tangents, is equal to a constant quantity. For, since every diameter bisects the ellipse, the conjugate diameters EG, eg, divide the ellipse into four equal sectors CEAe, CEag, CgBG, CGbe; but the same conjugate di

ameters divide also the whole tangential parallelogram PQRS into four equal parts, or small parallelograms CEPe, CEQ g, CgRG, CGSe; and therefore the differences be tween these small parallelograms and the sectors, which are the said external spaces, must be all equal among themselves. And as the ellipse and circumscribing parallelograms both remain constant, the difference of their fourth parts will also be a constant quantity. That is, the said external parts are each equal to the same constant quantity.

The sum of the squares of every pair of conjugate diameters is equal to the same constant quantity, namely, the sum of the squares of the two axes.

That is, AB+ab'EG'+eg', where EG, eg, are any conjugate diameters.

For, draw the ordinates ED, ed.

Then, by step 10, last dem. CAD DB,

in like manner, Ca'DE'+de2;

therefore, the sum CA'+Ca'CD+DE+Cd'+de":

But,

But, by right-angled triangles, CE2=CD2+DE', and CeCd'+de2;

therefore, the sum CE+CeCD+DE+Cd'+de2 ; consequently, CA2+Ca2=CE2+Ce2 ; or, by doubling, AB2+ab2=EG2+eg2.

HYPERBOLA.

Q. E. D.

PROPOSITION I.

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

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 hyperbola.

Now,

Now, by the similar triangles AFL, AHN, and BFK, BHM, we have AF : AH :: FL : HN, and FB : HB :: KF : MH;

hence, taking the rectangles of the corresponding terms, we have the rect. AF·FB: AHHB :: KFFL : MH·HN,

But, by the nature of the circle, KF FLFG', and MH HN HI';

therefore, the rect. AFFB AHHB :: FG* ; HI1.

Q. E. Di

COR. I. All the parallel sections are similar figures, or have their two axes in the same proportion; that is,

AB : ab :: DE : de.

For, by sim. tri. AB: ab; AQ : aq,
: ab :: RB : rb ;

and AB

therefore, by comp. AB

:

ab :: AQRB; aqʻrb,

But AQ RB DE3, and aqrbde2; therefore AB : ab 2 :: DE2 : de

2

or AB : ab ;; DE : de,

2

COR. 2. Hence also, as the property is the same for the ordinates on both sides of the diameter, it follows, that

1. At equal distances from the centre, or from the vertices, the ordinates on both sides are equal, or that the double ordinates are bisected by the axis; and that the whole figure, made up of all the double ordinates, is also bisected by the axis.

2. The two foci are equally distant from the centre, or from either vertex.

COR. 3. When the angle, which the plane of the section makes with the base of the cone, decreases till it become equal to the angle made by the side of the cone and.

the

the base, or till the section be parallel to the opposite side of the base; then the axis becomes infinitely long, and the hyperbola degenerates into a parabola; and because then the infinites FB and HB are in a ratio of equality, the general property,

namely, AF-FB : AH HB :: FG2 : HI*,
becomes AF : AH :: FG*
FG: HI"',

or, in the parabola, the abscisses are to each other as the squares of their ordinates,

As the

PROPOSITION II.

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.

That is, AB; ab2 or AÇ2; aC2 ;; AD·DB : DE2.

or, by doubling, AB1

But, if C be the centre, then ACCB AC,

the semiconjugate ;

or, by permutation, AC: aC :: AD DB: DE';

and Ca is

:: aC2 : DE';