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.

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.

Then, by theor 7, and

theref. by equality, but, by sim. triangles, theref. by equality, and the rectangle Again, by theor. 7, or, by division,

and by composition,

conseq. the rectangle But, by theor. 1, therefore

CT:CA:: CA: CD,
ct:CA::CA: cd;
CT:ct::cd : CD;
CT: ct :: TD: cd,
TD:cd::cd : CD,
TD. DC is = the square cd2.
CD:CA:: CA: CT,

CD:CA:: DA: AT,
CD: DB:: AD: DT;
CD. DT = cd2 = AD. DB*.
CA2: ca2:: (AD. DB or) cd2: DE,

2

CA: Ca::cd: DE;

* Corol. Because cd2 = AD.DB = CA - CD,

therefore CA = CD2 + cd2.

In like manner,

or

But, by theor. 7, theref. by equality, But, by sim. tri. theref. by equality, and the rectangle CK But the rect.

theref. the rect.

conseq. the rect.

ск. се = the parallelogram CEPE,

CA

• ca = the parallelogram CEPE,

AB. ab = the parallelogram PQRS. Q. E. D.

THEOREM XI.

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.

For, draw the ordinates ED, ed.

2

Then, by cor. to theor. 10, CA2 = CD2 + cd2,

and

ca2 = DE2 + de2;

therefore the sum CA2 + Ca2 = CD + DE2 + cd2 + de2.

Note. All these theorems in the Ellipse, and their demon-strations, 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 perpendieular 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: HI2:: AF. FB:AH.HB.

R

B

D

r

V

E

Q

A

K

L

G

M

11

N

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, 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.

2

HN = HỈ2;

But, by the circle, KF. FL = FG2, and MH
Therefore the rect. AF. FB: AH.HB:: FG2 : HỈ2.

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-conj.

Q. E. D.

Therefore AC: AD.DB:: ac2: DE2; or, by permutation, Ac2: ac2 :: AD. DB: DE2; or, by doubling, AB2: ab2:: AD. DB: DE2. ab2 Corol. Or, by div. AB : :: AD. DB or CD - CA2: DE2, that is, AB:P :: AD. DB or CD2 CA2: DE2; ab2 where p is the parameter

AB

AB

That is, As the transverse,

Is to its parameter,

2

by the definition of it.

So is the rectangle of the abscisses,
To the square of their ordinate.

THEOREM III.

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.

For, draw the ordinate ED to the transverse AB.

Then, by theor. 1. ca2: CA2:: DE: AD. DB or CD2 - CA2,

Ca3: CA2:: cd2: de2

Ca2: CA2:: ca2: CA2.

- CA2.

theref. by compos. ca2: CA2:: ca2 + cd2 : de2. In like manner,

CA2: Ca2:: CA2 + CD2: De2.

Corol. By the last theor. CA2: ca2 :: CD2 - CA2 : DE2, and by this theor. CA2: Ca2 :: CD2 + CA2: De2, therefore

In like manner,

/

Q. E. D.

a

THEOREM IV.

The Square of the Distance of the Focus from the Centre, is equal to the Sum of the Squares of the Semi-axes. Or, the Square of the Distance between the Foci, is equal to the Sum 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

of the curve

and by the def. of the para. CA2: Ca2:: ca2

therefore

and by addition,

or, by doubling,

CA: Ca2:: CF2 - CA: FE;

Corol. 1. The two semi-axes, and the focal distance from the centre, are the sides of a right-angled triangle cha; and the distance Aa is = CF the focal distance.

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.

The Difference of two Lines drawn from the two Foci, to meet at any Point in the Curve, is equal to the Transverse Axis.

For, draw AG parallel and equal to ca the semi-conjugate; and join co, meeting the ordinate de produced in H; also take ci a 4th proportional to CA, CF, CD.

Then,