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 FG: 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

K

L

G

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 = FG, and MH. HN Therefore the rect. AF. FB: AH. HB :: FG2: HI2.

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 semi-conj.

AC2: AD DB: : ac2 : DE2;

Ac2: ac2:: AD . DB : DE2;

AB2: ab2:: AD. DB : DE2.

Q.E. D.

CA2: DE2;

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.

That is,

la

:

Ca2 CA: ca2 + cd2: de'.

For, draw the ordinate ED to the transverse AB.

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

CA2: Ca2 :: CA2 + CD2 : De2.

Corol. By the last theor. cA2: Ca2:: CD2 - CA2: DE2,

CD2

and by this theor. CA: ca:: CD2+ CA2: De2,

Q. E. n.

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 CA: Ca2:: CF2 CA2: FE2;

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

G

Ca2

: FE2;

therefore

and by addition,

or, by doubling,

Corol. 1. The two semi-axes, and the focal distance from the centre, are the sides of a right-angled triangle c^a; 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, fв, the distances of either focus from the two vertices.

For Ca2 CF2 - CA2 CF + CA. CF

CA AF. FB.

THEOREM V.

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

That is, fEFE AB.

a

G...

FDI

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

Then,

Then, by th. 2, CA: AG2:: CD2 - CA2: DE2;

ag2 AG2

and, by sim. As, CA: AG2:: CD-CA: DH2 - AG2; DE DH2 - AG2 = DH2

consequently

-

ca2.

Also, FD CF CD, and FD2 = CF2 2CF. CD + CD2; and, by right-angled triangles, fe2 = fd2 + de2.

CF2

therefore FE2
But, by theor. 4,
and, by supposition,
theref. FE2 = CA2
Again, by suppos.
and, by sim. tri.
therefore

consequently

2CF.CD+CD2 + DH2.

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

2CA. CI+ cr2.

CA = AL.

Q. E. D.

In the same manner, it is found that fE = CI + CA =`BI.
Conseq. by subtract. fe FE BI

CD.

AI AB.

Corol. 1. Hence CHCI is a 4th proportional to CA, CF,

Corol. 2. And fE + FE = 2CH or 2c1; or FE, CH, fE, are in continued arithmetical progression, the common difference being CA the semi-transverse,

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

In the transverse AB, produced, 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 curve.

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

In the same manner are constructed the other two or conjugate hyperbolas, using the axis ab instead of AB.

THEOREM VI.

If from any Point I in the Axis, 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 the Middle of AB: Then shall Cм be to ci as the Square of AM to the Square of AI.

That is,

CM CI: AM2: A12.

B

H

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

Then, by theor. 1, and by sim. triangles, theref. by equality, But DB

AD. DB: AG. GB :: DE2 : GH2,

ID2 : IG2 :: DE*: GH2;
AD. DB: AG. GB :: ID2 : IG2,

-

CB + CD = CA + CD = CG + CD — AG =2CK — AG, and GB CB+CG CA+CGCG + CD-AD=2CK — AD; theref. AD. 2cK AD. AG AG. 2CK and, by div. DG. 2CK: IG2

AD AG: ID2.

AD. AG :: ID2 : IG2,

ID2 or DG

2IK: AD. 2CK

But, when the line IH, by revolving about the point 1, comes into the position of the tangent IL, then the points E and H meet in the point L, and the points D, K, G, coincide with the point M; 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.

theref. (th. 78, Geom.) CD: CA :: CA : CD.

DT + CT

E

TD;

Q. E. D.

Corol