That is, the triangle CTS = 2 paral. GK.

G

T

E

For, since the tangent Ts is bisected by the point of contact E, and Ek is parallel to rc, and GE to CK; therefore cK, as GE are all equal, as are also eG,GT, KE Consequently the triangle GTE = the triangle KES, and each equal to half the constant inscribed parallelogram CE. And therefore the whole triangle CTs, which is composed of the two smaller triangles and the parallelogram, is equal to double the constant inscribed paralleiogram GK.

C

THEOREM XXIX.

K

S

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.

Draw the semi-diameters CH, CIN, CK;
Then shall the sector CHI = the sector CIK.

H

Q. E. D

For, because нк and all its parallels are bisected by CIN, therefore the triangle CNH = tri CNK,

consequently the sector CIH

seg. INK;

Corol. If the geometricals DH, EI, GR. be parallel to the other asymptote, the spaces DHIE, EIRG will be equal; for they are equal to the equal sectors CHI, CIK.

So that by taking any geometricals CD, CE, CG, &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, CHA, &c.

or the spaces DHIE, 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 CF to CG, &c; or of EI to DH, or of GK to EI, &c; and CHк or DHKG 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 perpendicular to the former, and parallel to the side vм of the cone; also let AFH be the common intersection of the two planes, or the axis of the parabola, and FG, HI ordinates perpendicular to it.

K

Then it will be, as ar: AH :: FG2 : H12.

For, through the ordinates FG, HI draw the circular sections, KGL, MIN, parallel to the base of the cone, having «L,

MN

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,

therefore

KF = MH;

AF: AH: FG2: HI2.

AF: AH: KF. FLMH

HN.

E D.

FG2

Corol. Hence the third proportional

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

therefore

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

GH2: KI, IH.

THEOREM III.

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

That

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.

But, by cor. theor. 1, DE2 = P. AD = 4AF, AD;

theref. by addition,

But, by right-ang. tri. FD2 + DE2 = FE2;

therefore

and the root or side is FE

2AF. AD + AD2,

FD2 + DE AF2 + 2AF AD + AD,

FE 2 =

AF22AF. AD + AD2,

AF + AD,

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 line HE parallel to the axis.E HE is always equal to FE the distance of the focus from the point E.

E

E

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, E, &c. Then draw the curve through all the points,

E, E, E.

THEOREM

THEOREM V.

If a Tangent be drawn to any Point of the Parabola, meeting the Axis produced; and if an Ordinate to the Axis be drawn from the Point of Contact; then the Absciss of that Ordinate will be equal to the External Part of the Axis.

For, from the point T, draw any line cutting the curve in the two points E, H: to which draw the ordinates DE, GH; also draw the ordinate мc to the point of contact c.

Then, by th. 1, AD: AG: DE2: GH2;

and by sim. tri.

TD2: TG2:: DE2 : GH2;

theref.by equality,AD AG :: TD2 : TG2;

and, by division, AD

or

and, by division,

and again by div.

or

:

DGTD: TG-TD2 or DG. (TV+TG), AD TD : TD: TD+TG;

AD: AT:: TD: TG,

AD AT ATAG;

AT is a mean propor. between AD, AG.

Now if the line тH be supposed to revolve about the point T; then, as it recedes farther from the axis, the points E and н approach towards each other, the point E descending and the point a ascending, till at last they meet in the point c, when the line becomes a tangent to the curve at c. And then the points D and G meet in the point м, and the ordinates DE, GH in the ordinates CM. Consequently AD, AG, becoming each equal to AM, their mean proportional AT will be equal to the absciss AM. That is the external part of the axis, cut off by a tangent, is equal to the absciss of the ordinate to the point of contact.

THEOREM VL

Q. E. D.

If a tangent to the Curve meet the Axis produced; then the Line drawn from the Focus to the Point of Contact, will be equal to the Distance of the Focus from the Intersection of the Tangent and Axis.

That