the triangle

LQH becomes the triangle ceM,

and the space TELK becomes the triangle TEC;

and theref. the A cem = A TEC = ▲ ANC = ▲ IRC.

THEOREM XX.

Any Diameter bisects all its Double Ordinates, or the Lines drawn Parallel to the Tangent at its Vertex, or to its Conjugate Diameter.

That is, if ag be parallel

to the tangent TE, or to ce,

then shall LQ = Lg.

H

C

For, draw qн, qh perpendicular to the transverse. Then by cor. 3 theor. 19, the ▲ LQH ⇒ ALqh ; but these triangles are also equiangular;

consequently their like sides are equal, or LQ = L9.

Corol. Any diameter divides the ellipse into two equal parts.

For, the ordinates on each side being equal to each other, and equal in number; all the ordinates, or the area, on one side of the diameter, is equal to all the ordinates, or the area, on the other side of it.

THEOREM XXI.

As the Square of any Diameter :
Is to the Square of its Conjugate ::

So is the Rectangle of any two Abscisses:
To the Square of their Ordinate.

Then, similar triangles

M

being as the squares of their like sides, it is,

by

by sim. triangles, ▲ CET: A CEK :: CE2: CL2; or, by division, A CET trap. TELK:: CE2; CR2 — CL2. Again, by sim. tri. ▲ ceм: ▲ LQH :: Ce2 : LQ2. But, by cor. 5 theor. 19, the ▲ ceм = ▲ CET, and, by cor. 4 theor. 19, the A LQH = trap. TELK; theref. by equality, CE2; ce2:: CE2 CL2: LQ2, CE2; ce2:: EL. LG: LQ3.

or

-

Q. E. D.

Corol. 1. The squares of the ordinates to any diameter, are to one another as the rectangles of their respective abscisses, or as the difference of the squares of the semidiameter and of the distance between the ordinate and centre, For they are all in the same ratio of CE to ce2.

Corol. 2. The above being the same property as that be longing to the two axes, all the other properties before laid down, for the axes, may be understood of any two conjugate diameters whatever, using only the oblique ordinates of these diameters, instead of the perpendicular ordinates of the axes; namely, all the properties in theorems 6, 7, 8, 14, 15, 16,

18 and 19.

THEOREM XXII.

If any Two Lines, that any where intersect each other, meet the Curve each in Two Points; then

The Rectangle of the Segments of the one :

Is to the Rectangle of the Segments of the other ::
As the Square of the Diam. Parallel to the former:
To the Square of the Diam. Parallel to the latter.

For, draw the diameter CHE, and the tangent TE, and its parallels PK, RI, MH, meeting the conjugate of the diameter CR in the points T, K. I, M. Then, because similar triangles are as the squares of their like sides, it is,

VOL. I.

Rrr

by

by sim. triangles,
and
theref. by division,
Again, by sim. tri
and by division,

But, by cor. 5 theor.

CR2: Gr2:: A▲ CRI: ▲ GPK,
CR2: GH2: A CRI: A GHM;
CR2: GP2 — GH2:: CRI: KPHм.
CE2: CH2:: ▲ CTE: ▲ CMH;

CE2: CE2 CH2: A CTE: TEHM. 19, the ▲ CTE = ▲ CIR,

and by cor. 1 theor. 19, TEHC = KPHG, or TEHM = KPHM; theref. by equ. OE2: CE2 — CH2 :: CR2: GP2 —GH2 oг PH. MQ. In like manner CE2: CE2 — câ2:: cr2; pн . 69.

Theref. by equ. CR2: cr2:: PH. HQ: pH. нq.

Q. E. Di

Corol. 1. In like manner, if any other lines p'H'q', parallel to cr or to hq, meet PHQ; since the rectangles Pн'Q, 'n'q' are also in the same ratio of CR2 to cr2; therefore rect PHQHq PH'q: 'H'q'.

Also, if another line r'hq' be drawn parallel to PQ or CR; because the rectangles p'hq', p'hq' are still in the same ratio, therefore, in general, the rect. PHQ : pHq :: P'hq' ; p'hq'.

That is, the rectangles of the parts of two parallel lines, are to one another, as the rectangles of the parts of two other parallel lines, any where intersecting the former.

Corol. 2. And when any of the lines only touch the curve, instead of cutting it, the rectangles of such become squares, and the general property still attends them.

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 perpendi cular 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: H12:: AF. FB: AH. HB. For, through the ordinates FG, HI, draw the circular sections KGL, MIN, parallel to the base of the cone, having KL, MN, for their

R

L

I

E

diameters, to which FG, HI, are ordinates, as well to the axis of the hyperbola.

Now, by the similar triangles AFĻ, AHN, and BFK, BHM, it is AF AH :: FL: HN,

and FB: HB :: KF : MH;

HN.

hence, taking the rectangles of the corresponding terms, it is, the rect. AF. FB: AH. HB:: KF. FL: MH. But, by the circle, KF . FL = FG, and мH. HN = UI2; Therefore the rect. AF. FB: AH HB;: FG2: HI2.

Q. 2. D.

Ac2: ac2:: AD. DB : DE2.

B

C

b

For

For, by theor. 1, AC. CB: AD. DB:: Ca2: DE2;

But, if c be the centre, then ac. CB AC2, and ca is the

Therefore

semi-conj.

AC2: AD. DB:: ac2: DE2;

or, by permutation,

AC2: ac2:: AD. DB: DE2;

or, by doubling,

AB2: ab2: AD. DB: DE2.
ab2

Q. E D.

Corol. Or, by div. AB :-:: AD. DB or CD2-CA3: DE*,

AB

that is, AB::: AD. DB or CD2 - CA2: DE2;

where is the parameter

ab2

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

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:: DE2: AD. DE OF CD2 —CA2,

or

But

Ca2: CA2:: cd2 : dr2 -CA2.
ca2: CA2:: c 2 : CA2.

theref by compos. ca2: ca2:: ca2 + cd2: dɛ2.

In like manner,

CA2: ca2:: CA2 + CD2 : De2.

Q. E. D.

Corol. By the last theor. cA2: ca3:: CD2-CA2: DE3, 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