by sim. triangles,

ACET: ACLK :: CER: @L2; or, by division, ACET: trap. TELK:: CE: CE3 - CL3. Again, by sim. tri. Acem : ALQA :: ce2 : LQ2. But, by cor. 5, theor. 19, the ▲ceм=ACET, and, by cor. 4, theor. 19, the ALQH=trap. TELK ; theref. by equality, CE2: ce2:: CE2 -CL2: LQ2, CE: Cea:: EL. LG: LQ2.

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 CE3 to ce2.

Corol. 2. The above being the same property as that belonging 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 GR in the points T, K, I, M. Then, because similar triangles are as the squares of their like sides, it is,

VOL. I.

63

by

by sim. triangles,
and

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

CH2 ACTE: TENM.

But, by cor. 5, theor. 19, the A▲ CTE= ▲CIR,

and by cor. 1, theor., 19, TEHC=KPHG, or TEHM=KPHM; theref by equ. cɛ2 : ce2 —ch3 :: CR2 : GP2 — GH2 oг PH, HQ. In like manner CE: CE-CH2 :: cr2 : pH. нq.

Theref. by equ. cR2 : cr2 :: PH. HQ: PH. нq.

Q. E. D.

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

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

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 to the axis of the hyperbola.

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

and EB HB :: KF : MH;

hence, taking the rectangles of the corresponding terms,

it is, the rect. AF But, by the circle, KF

FB: AH. HB KF

MH. HN.
HN = B12;

FL

Therefore the rect. AF. FB: AH. HB :: FG2: HIa. Q. E. D.

For, by theor. 1, AC . CB : AD. DB:: ca2: DE2;

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

ab2

Corol. Or by div. ab:

AB

Q. E. D.

:: AD. DB or CD3 —ca2 : De2,

that is, AB p:: AD. DB or CD2 — CA3 : DE2 ;

ab2

where p is the parameter

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

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,

ca: CA :: ca2+cd2: de2.

B IC

b

For, draw the ordinate ED to the transverse AB.

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 CF3-CA2: FE2,

and by the def. of the para. CA3: ca2 ::

therefore

and by addition,

or, by doubling,

ca3CF2

ca2 : FE2;

CA2;

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

Corol. 2. The conjugate semi-axes, ca, is a mean proportional between AF, FB, or between af, fB, the distances of either focus from the two vertices.

For ca2 CF2 CA CF CA. CF

THEOREM V.

CA = AF

FB.

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

II

AFDI

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,