THEOREM XIX (15).

If from any Point in the Curve there be drawn an Ordinate, and a Perpendicular to the Curve, or to the Tangent at that Point: Then the

Dist, on the Trans. between the Centre and Ordinate, CD :

For, by theor. 2, CA2: ca2:: AD. DB: De2,
But, by rt. angled A s, the rect. TD. DP = DE2;
and, by cor. 1 theor. 16,

therefore

or

CD. ᎠᎢ = ᎪᎠ . ᎠᏴ ;
CA2: ca2:: TD. DC: TD. DP,
CA2: ca2:: DC

: DP.

Q. E. D.

THEOREM XX (18).

If there be Two Tangents drawn, the One to the Extremity of the Transverse, and the other to the Extremity of any other Diameter, each meeting the other's Diameter produced; the two Tangential Triangles so formed, will be equal.

The two triangles CET, CAN have then the angle c common, and the sides about that angle reciprocally proportional; those triangles are therefore equal, viz. the A CETACAN. q.e.d.

Core

Corol. 1. Take each of the equal tri. CET, CAN, from the common space CAPE,

and there remains the external ▲ PAT = A PNE.

Corol. 2. Also take the equal triangles CET, CAN, from the common triangle

CED,

and there remains the ▲ TED = trapez. ANED.

THEOREM XXI (19).

The same being supposed as in the last Proposition; then any Lines кQ, GQ, drawn parallel to the two Tangents, shall also cut off equal Spaces.

That is,

the A KQG = trapez. ANHG. and ▲ Kqg trapez. ANhg.

19

T KAS D

For, draw the ordinate DE. Then

The three sim. triangles CAN, CDE, CGH,

are to each other as

CA, CD, CG2;

th. by div. the trap. ANED: trap. ANHG :: CD2- Ca2: CG2— ca2.

But, by theor. 1, DE2:

GQ2: CD2-CA2: CG2-CA2;

theref. by equ. trap. ANED: trap. ANHG ::

But, by sim. As, tri. TED: tri. KQG :: DE2 : GQ2;

Corol. 1. The three spaces ANHG, TEHG, KQG are all

equal.

Corol. 2.

From the equals ANHG, KQG,

take the equals ANhg, Kqg,

and there remains ghHG = gqQG.

Corol. 3. And from the equals ghнG, gqaG, take the common space gqLHG,

and there remains the ▲ LOH = A Lgh.

Corol. 4. Again, from the equals KQG, TEHG,

take the common space KLHG,

and there remains

TELKA LQH.

Corol

Corol. 5. the lines Ka, GH, moving with a parallel motion, Ka comes into the position IR, where CR is the conjugate to CA; then

And when, by

the triangle KQG becomes the triangle IRC,
and the space ANHG becomes the trianglė ANC;
and therefore the AIRCA ANCA TEC.

Corol. 6. Also when the lines KQ and нQ, by moving with a parallel motion, come into the position ce, Me,

the triangle LQH becomes the triangle cem,

and the space TELK becomes the triangle TEC;
and theref. the A ceм = A TECA ANC = A IRC.

THEOREM XXII (20):

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 aq be parallel to the tangent TE, or to ce, then shall LQ=Lq.

For, draw QH, qh perpendicular to the transverse.

Then by cor. 3 theor. 21, the ▲ LOH =

A Lgh;

but these triangles are also equiangular;

conseq. their like sides are equal, or Lo = £q.

Corol. 1. 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.

Corol. 2. In like manner, if the ordinate be produced to the conjugate hyperbolas at q, q', it may be proved that VOL. III.

C

LQ'

La Lg. Or if the tangent TE be produced, then EV=EW. Also the diameter GCEH bisects all lines drawn parallel to TE or aq, and limited either by one hyperbola, or by its two conjugate hyperbolas.

THEOREM XXIII (21).

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.

For, draw the tangent TE, and produce the ordinate QL to the transverse at K. Also draw aн, eм perpendicular to the transverse, and meeting EG in H and M. Then, similar triangles being as the squares of their like sides, it is,

by sim. triangles,
or, by division,
Again, by sim. tri.
But, by cor. 5 theor.

CE: LQ2.

A CET A CLK :: CE: CL2;
A CET trap. TELK :: CE2: CL2
A Ceм: A LQH :: Ce2: LQ2.
21, the A CeMA CET,

and, by cor. 4 theor. 21, the ▲ LQH = trap. TELK;
theref. by equality, CE: ce :: CL CE: LQ3,
CE ce2:: EL. LG: LQ2.

or

K

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 semi-diameter and of the distance between the ordinate and centre. For they are all in the same ratio of CE2 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, 16, 17, 20, 21. Corol. 3. Likewise, when the ordinates are continued to the conjugate hyperbolas at a', q', the same properties still obtain, substituting only the sum for the difference of the squares of CE and CL,

That is, CE2: Ce2 :: CL2 + CE2 : LQ”.

And so La2: La"2:: CL2. CE2: CL2+ CE2.

Corol.

Corol. 4. When, by the motion of La' parallel to itself, that line coincides with EV, the last corollary becomes CE: Ce2 2CE2: Ev2,

or Ce2: Ev2 : : 1 : 2,
or ce : EV :: 1 :√2,

or as the side of a square to its diagonal.

That is, in all conjugate hyperbolas, and all their diameters, any diameter is to its parallel tangent, in the constant ratio of the side of a square to its diagonal.

THEOREM XXIV (22).

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 dia.neter 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,

by sim. triangles, CR2: GP2 :: A CRI : ▲ GPK,

Vand

CR2: GH2: A CRI: A GHM;

theref. by division, CR2 : GP2 GH CRI: KPHM. Again, by sim. tri. CE2; CH2:: ACTE: A CMH; and by division, CE2: CH2

CE2 : : A CTE : ТЕНМ.

But, by cor. 5 theor. 21, the A CTE = ▲ CIR,

-

and by cor. 1 theor. 21, TEHGKPHG, or TEHM = KPHM; theref. by equ. CE2 : CH2 CE: CR2: GP2- GH2 or PH.HQ. In like manner CE2: CH2 ÇE::cr2: pH. нq. PH. HQ: рH. нq. C 2

Theref. by equ. CR2: cr2::

Q.É. D.

Corol.