E

That is, AB': ab2 or

AC2: ac2:: AD. DB : DE⭑.

B

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.

Therefore

AC2: AD. DB:: ac2 : DE2;

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

or, by doubling,

AB2: ab2:: AD. DB : DE2.

ab2

Coral. Or, by div. AB: :: AD. DB or CA2

AB

that is, AB :p :: AD. DB or CA2 CD: DE2;

where p is the parameter

ab2

AB

by the definition of it

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: de2.

G

For, draw the ordinate ED to the transverse AB.

Then, by theor. 1, ca2: CA2:: DE: AD. DB or Ca2 - CD2,

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, ca2: ca2 :: AD. DB or DG2 : DE2,

In like manner

Also, by equality,
Therefore cgG is a

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.

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.

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 therefore

CA2: Ca2:: CA2 CF2: FE2;

-

Ca2

: FE2;

ca2

CA2 CF2;

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 — CF2 = CA + CE. 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

H

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, CA: AG :: CA- CD: DE2;

and, by sim. tri.

consequently

Also FD= CF

CA: AG: CA
DE2 = AG2

CD2: AG DH2;

DH2 ca2 - DH2.

CD, and FD2 = CF2 - 2CF. CD + CD2;

And, by right-angled triangles, Fe2 = fd2 + de2; therefore FE = CF2 + Ca2 2CF. CD + CD2 — DH2. But by theor. 4, CF+ Ca CA2,

and by supposition,

theref. FE2 = CA2

2CF. CD = 2CA. CI;

2CA. CI + CD2 DH2.

Again, by supp. CA2: CD2:: CF2 or CA2 -- AG2 : CI2;

and, by sim. tri. CA2: CD2 :: CA2

And the root or side of this square is FE = CA

DH2;

CI= AL.

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

CF, CD.

Corol. 2. And fɛ

FE is a 4th proportional to ca,

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.

Coral. 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 I, 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.

E
ADMK C

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. DB: AG • GB: D E*: G H2,

and by sim. triangles, ID:

IG2:: DE2: GH2;

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

CG2CK+ AD;

But DBCB+CD=AC+CD = AG + DC CG2CK+ AG, and GB CBCG ACCG AD + DC theref. AD. 2CK + AD.AG: AG. 2CK + AD. AG :: ID2:IG2, and, by div. DG. 2CK: IG2 ID2 or DG. 2IK :: AD. 2CK + AD. AG : ID2,

or

or .

or

2CK: 21K :: AD. 2CK + AD. AG : ID",

AD. 2CK: AD. 21K :: AD. 2CK + AD. AG: ID2; theref. by div. CK: IK :: AD. AG : ID2

and, by comp. CK: IC :: AD. AG: ID2 CK: CI: AD. AG: AI2.

or

-

AD. 2IK,

AD. ID IA,

But, when the line IH, by revolving about the point 1, comes into the position of the tangent IL, then the points E and meet in the point L, and the points D, K, G, coincide with the point м; and then the last proportion becomes CM: CI :: AM2 : AI2. Q. E. D.

THEOREM VII.

If a Tangent and Ordinate be drawn from any Point in the Curve, meeting the Transverse Axis; the Semi-transverse will be a Mean Proportional between the Distances of the said Two Intersections from the Centre.

That is,

CA is a mean proportional

between CD and CT;

or CD, CA, CT, are conti

nued proportionals.

For, by theor. 6, CD: CT :: AD2: AT2. that is, CD: CT :: (CA CD)2: (CT

or

and

or

or

hence

and

CD : CT : : CD2 + CA2 : CA2 + CT2,
CD; DT:: CD2 + CA2: CT2 — CD2,

CD: DT:: CD2 + CA2: (CT + CD) DT,

CD2: CD. CT :: CD2 + CA2: CD. DT + CF. DT,
CD2: CA2:: CD. DT: CT. DT,

CD2: CA2:: CD: CT.

therefore (th. 78, Geom.) CD: CA :: CA: CT.

Q. E. D.

Corol. Since CT is always a third proportional to CD, CA; if the points D, 4, remain constant, then will the point T be constant also; and therefore all the tangents will meet in this point T, which are drawn from the point E, of every ellipse described on the same axis AB, where they are cut by the common ordinate DEE drawn from the point D.

THEOREM