Corol. 3. From the proposition, and the last corollary, it follows that the constant rectangle HEK or EHE is = IL2. And the equal constant rect. Heк or eнe = MLN or IM2 — IL2.

Corol. 4. And hence IL = the parallel semi-diameter cs.
For, the rect. EHE = IL2,

and the equal rect. eнe = IM*
theref. ILIM2 — 112, or Im2

- IL3,

= 2112 ;
2c82,

but, by cor. 4 theor. 23, IM2 =

and therefore

IL = C8.

And so the asymptotes pass through the opposite angles of all the inscribed parallelograms.

THEOREM XXVII.

The rectangle of any two lines drawn from any point in the curve, parallel to two given lines, and limited by the asymptotes, is a constant quantity.

That is, if AP, EG, DI be parallels,

as also

AQ, EK, DM parallels,

then shall the rect. PAQ

E

rect. GEK = rect. IDM.

D

E

P

For, produce KE, MD to the other asymptote at н, L.
Then, by the parallels, HE: GE :: LD : ID;

but
EK EK DM: DM;
theref, the rectangle HEK: GEK:: LDM:IDM.
But, by the last theor. the reсt. HK = LDM ;
and therefore the rect. GEK IDM = PAQ.

THEOREM XXVIII.

Q. E. D.

Every inscribed triangle, formed by any tangent and the two intercepted parts of the asymptotes is equal to a constant quantity; namely, double the inscribed parallelogram.

That is, the triangle crs = 2 paral. oк.

OF THE HYPERBOLA.

For, since the tangent rs is bisected by the point of contact E, (th. 26, cor. 2), and EK is parallel to TC, and GE to CK; therefore CK, KS, GE, are all equal, as are also co, GT, KE. Consequently the triangle GTE

the triangle KES, and each equal to half the constant in. scribed parallelogram GK. And therefore the whole triangle ors, which is composed of the two smaller triangles and the parallelogram, is equal to double the constant inscribed parallelogram GK.

THEOREM XXIX.

Q. E. D.

If from the point of contact of any tangent, and the two intersections of the curve with a line parallel to the tangent, three parallel lines be drawn in any direction, and terminated by either asymptote; those three lines shall be in continued proportion.

That is, if нKм and the tangenti be parallel, then are the parallels DH, EI, GK in continued proportion.

H

C D E

K

LG M

For, by the parallels, EI: IL :: DH: HM;

and, by the same,
theref. by compos.
but, by theor. 26, the
and theref. the rect.

EI IL GK KM;

EI: IL3 :: DH. GK : HMK ;
rect., HMK = IL2;

DH GK = EI2,

or

DH

EI:: EI: GK.

Q. E. D.

THEOREM XXX.

Draw the semi-diameters CH, CIN, CK;

Then shall the sector CHI the sector CIK.

For, because нK and all its parallels are bisected by cix,

therefore the triangle CNH tri. CNK,

and the segment

INH seg. INK;

consequently the sector ciu= sec. cik.

Corol. If the geometrical proportionals DH, EI, GK be parallel to the other asymptote, the spaces DHIE, Eikg will be equal; for they are equal to the equal sectors CHI, CIK.

So that by taking any geometrical proportionals CD, CE, co, &c. and drawing DH, EI, GK, &c. parallel to the other asymptote, as also the radii CH; CI, CK ;

then the sectors CHI, CIK, &c.

or the spaces DHIE, EIKG, &c.
will be all equal among themselves.
Or the sectors CHI, CHK, &c.

or the spaces DḤIE, DHKG, &c.

will be in arithmetical progression.

And therefore these sectors, or spaces, will be analogous to the logarithms of the lines or bases CD), CE, CG, &c. ; namely, CHI or DHIE the log. of the ratio of

CD to CE, or of ce to CG, &c.; or of Et to DH, or of GK to EI, &c. ; and CHк or DнKG the log. of the ratio of CD to CG, &c. or of GK to DH, &c.

OF THE PARABOLA.

THEOREM I.

The Abscisses are proportional to the Squares of their
Ordinates.

Let AVM be a section through the axis of the cone, and AGIH a parabolic section by a plane per. pendicular to the former, and parallel to the side vм of the cone; also let AFH be the com. mon intersection of the planes, or the axis of the para. bola, and FG, HI ordinates per. pendicular to it.

two

K

M

N

Then it will be, as AF: AH :: FG2: HI'.

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 as to the axis of the parabola.

Then, by similar triangles, AF: AH:: FL: HN; but, because of the parallels,

KF = MH;

FL = FG2, and мH. HN HI2;

AF AH: FG2: H13 Q E. D.

Corol. Hence the third proportional

is a con.

stant quantity, and is equal to the parameter of the axis, by defin. 16.

Or AF: FG :: FG: P the parameter.

Or the rectangle P. AF = FG3.

THEOREM II.

As the Parameter of the Axis :

Is to the Sum of any Two Ordinates ::
So is the Difference of those Ordinates:
To the Difference of their Abscisses.

So that any diameter EI is as the rectangle of the segments KI, IH of the double ordinate Kн.

THEOREM III.

The Distance from the Vertex to the Focus is equal to of the Parameter, or to Half the Ordinate at the Focus.

For, the general property is aF; FE :: FE: P.

But, by definition 17,

therefore also

A Line drawn from the Focus to any Point in the Curve. is equal to the Sum of the Focal Distance and the Absciss of the Ordinate to that Point.

Corol. 1. If, through the point G, the HHH line GH be drawn perpendicular to the axis, it is called the directrix of the parabola.* The property of which, from this theorem, it appears, is this: That drawing any lines HE parallel to the axis, HE is always equal to FE the distance of the focus from the point E.

F
E

E

FE

D E

Corol. 2. Hence also the curve is easily described by points. Namely, in the axis produced take AG = AF the focal distance, and draw a number of lines EE perpendicular to the axis AD; then with the distances GD, GD, GD, &c. as radii, and the centre F, draw arcs crossing the parallel ordinates in E, E, F, &c. Then draw the curve through all the points E, E, E.

* Each of the other conic sections has a directrix; but the consideration of it does not occur in the mode here employed of investigating the general properties of the curves,