THEOREM VIII

If there be any Tangent meeting Four Perpendiculars to the Axis drawn from these four Points, namely, the Centre, the two Extremities of the Axis, and the Point of Contact; those Four Perpendiculars will be Proportionals.

That is,

AG: DE:: CH: BI.

E

B.

For, by theor. 7, TC: AC :: AC : DC,

TA: AD :: TC: AC or CB,

theref. by div.

Corol. Hence TA, TD, TC, TB

and TG, TE, TH, TI

For these are as AG, DE, CH, BI, by similar triangles.

THEOREM IX.

If there be any Tangent, and two Lines drawn from the Foci to the Point of Contact; these two Lines will make equal Angles with the Tangent.

That is,

the FET fɛe.

B

For, draw the ordinate DE, and fe parallel to FE.

By cor. 1, theor. 5, CA: CD :: CF: CA

- FE,

TF: Tf :: FE: 2CA -
TF: Tf :: Fe: fe;

FE;

FE or fe by th. 5.

fEfe, and conseq. Le fee. FE is parallel to fe, the Le LFET; FET=4fɛe.

Q. E. D.

Cordl

Corol. As opticians find that the angle of incidence is equal to the angle of reflexion, it appears from this theorem, that rays of light issuing from the one focus, and meeting the curve in every point, will be reflected into lines drawn from those points to the other focus. So the ray fe is reflected into FE. And this is the reason why the points F, f, are called the foci, or burning points.

THEOREM X.

All the Parallelograms circumscribed about an Ellipse are equal to one another, and each equal to the Rectangle of the two Axes.

That is,'

the parallelogram PQRS = T

the rectangle AB. ab

D

R

Let EG, eg, be two conjugate diameters parallel to the sides of the parallelogram, and dividing it into four less and equal parallelograms. Also, draw the ordinates DE, de, and CK perpendicular to PQ; and let the axis CA produced meet the sides of the parallelogram, produced if necessary, in T

and t.

*Corol. Because cd'AD. DB CA2 CD',

therefore CA2CD2 + cd'.
ca2 DE2 + de2.

In like manner,

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.

Note. All these theorems in the Ellipse, and their demonstrations, are the very same, word for word, as the corresponding number of those in the Hyperbola, next following, having only sometimes the word sum changed for the word difference.

OF

OF THE HYPERBOLA.

THEOREM I.

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 vertex and axis of the opposite cones; AGIH another section of them perpendicular to the plane of the former; AB the axis of the hyperbolic sections; and FG, HI, ordinates perpendicular to it. Then it will be, as FG2: HI:: AF. FB: AH.HB.

For, through the ordinates FG, HI, draw the circular sections KGL, MIN, parallel to the base of

M

R

'B

D

P

E

K

Q

the cone, having KL, MN, for their diameters, to which FG, HI, are ordinates, as well as to the axis of the hyperbola. Now, by the similar triangles AFL, AHN, and EFK, 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 мH . HN = HI2; Therefore the rect. AF. FB : AH. HB :: FG2: HI2.

For, by theor. 1, AC. CB: AD. DB :: Ca2: DE2;
But, if c be the centre, then AC. CB= Ac2, and ca is the

As the Square of the Conjugate Axis
To the Square of the Transverse Axis ::

The Sum of the Squares of the Semi-conjugate, and
Distance of the Centre from any Ordinate of the Axis:
The Square of their Ordinate.

That is,

Ca2: CA2 :: ca2 + cd2 : dâ2,

For, draw the ordinate ED to the transverse Ab.

and by this theor. CA2: ca2 :: CD2 + CA2 : De2,
therefore

In like manner,

DE2: De2:: CD2-CA2: CD2+ CA2. de2: dɛ2 :: cd2-ca2: cd2+ ca2.

THEOREM