OF THE ELLIPSE.

THEOREM I.

The Squares of the Ordinates of the Axis are to each other as the Rectangles of their Abscisses.

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 = HI3; Therefore the rect, AF FB: AH. HB :: FG2: HI2.

Q. E. D.

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.

Therefore

AC2: AD DB:: ac2 DE2;

or, by permutation, AC2 : ac2 :: AD. DB : DE2;

or, by doubling,

Corol. Or, by div.

AB2: ab2:: AD

that is, AB ::: AD.

AB

DB or CA2 - CD2 : DE2;
ab2

where p is the parameter

-, by the definition of it.

AB

That is, As the transverse,
Is to its parameter,

So is the rectangle of the abscisses,

To the

square of their ordinate.

THEOREM III.

As the Square of the Conjugate Axis :

Is to the Square of the Transverse Axis ::

So is the Rectangle of the Abscisses of the Conjugate, or 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.

That is,

ca2: CB2:: ad. db or ca2- cd2 : dɛ2.

A

For, draw the ordinate ED to the transverse AB.

Then, by theor. 1, ca2 : CA2 :: DE2 : AD. DB or ca2 — CD2,

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 : dɛ.

For, by the nature of the circle, ad

by the nature of the ellipse, ca2 : ca2 :: AD. DB or DG2 : DE2,

In like manner
Also, by equality,

or CA Ca2:: DG: DE.

ca: CA dg dɛ.

DG: DE or cd :: dE or DC : dg.
Therefore cgG is a continued straight line.

Carol. 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.

THEOREM IV.

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, CF CA2 - ca2,
= ·

or Ff2 = AB2 - ab2.

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 CA2 Ca2:: CA2 CF2: FE2; : FE2;

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

Corol. 1. The two semi-axes, and the focal distance from the centre, are the sides of a right-angled triangle CFa; 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-CF2CA + CF. CA- CF AF. FB.

THEOREM V.

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

That is,
FE+fe AB.

G

E

H

FDI 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,

and, by sim. tri.

consequently

Also FD CF

CD, and FD2 = cf2-2cf. CD + CD2;

And, by right-angled triangles, fe2 = fd2 + de2; therefore FE2 = CF+ ca2 2CF. CD + CD2 — DH2.

But by theor. 4, CF+ ca2 CA2,

and, by supposition, 2CF. CD = 2CA. CI;

Again, by supp.

CA2: CD2:: CF2 or CA2 - AG2: cr2; and, by sim. tri. CA2: CD :: CA2 AG2: CD2 therefore Cr2 CD2 DH2;

"consequently FE2 CA22CA. CI + cr2. And the root or side of this square is FE = CA

= CÀ + CI = BI.

In the same manner it is found that fe
Conseq. by addit. FE + fE AI +BI AB.

Corol. 1. Hence CI or CAFE is a 4th proportional to CA, CF, CD.

Corol. 2. And fɛ · -- FE= 2c1; that is, the difference between two lines drawn from the foci, to any point in the curve, is double the 4th proportional to CA, CF, CD.

Corol. 3. Hence is derived the common method of describing this curve mechanically by points, or with a thread, thus:

In the transverse take the foci F, f, and any point 1. Then with the radii AI, BI, and centres F, f, describe arcs intersecting in E, which will be a point in the curve. In like manner, assuming other points 1, as many other points will be found in the

a

FI

curve. Then with a steady hand, the curve line may be drawn through all the points of intersection E.

Or, take a thread of the length AB of the transverse axis, and fix its two ends in the foci F, f, by two pins. Then carry a pen or pencil round by the thread, keeping it always stretched, and its point will trace out the curve line.

THEOREM VI.

If from any Point 1 in the Axis produced, a Line IL be drawn touching the Curve in one Point L; and the Ordinate LM be drawn; and if c be the Centre or Middle of AB: Then shall cм be to ci as the Square of AM to the Square of AI.

That is,

CM CI :: AM2 : AI2.

H

E

ADMK CG

For, from the point I draw any other line IEH to cut the curve in two points E and H; from which let fall the perpendiculars ED and HG; and bisect DG in K.

Then, by theo. 1, AD •

and by sim. triangles,

DB: AG. GB :: DE2: GH2,

ID2: IG2:: DE: GH";

theref. by equality, AD. DB: AG. GB :: ID2 : IG2.

But DB = CB + CD = AC + CD = AG + DC-CG=2CK+AG, and GB CB CG AC- CG AD + DC-CG2CK + AD; theref. AD. 2CKAD. AG: AG. 2CK + AD AG :: ID2: IG2, and, by div. DG. 2CK: IG2 ID or DG. 2IK;; AD. 2CK + AD. AG: ID2,

or