For, because нx and all its parallels are bisected by cis, therefore the triangle CNH = tri. CNK,

and the segment

INH seg. INK;

sec. CIK.

consequently the sector CIH

Corol. If the geometricals 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 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, CHK, &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

on to ce, or of CE to CG, &c.; or of EI to DH, or of GK to EI, and CHк or DHKG the log. of the ratio of

CD to CG, &c. or of GK to DH, &c.

&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.

M

K

N

H

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

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

NN

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 :: KF. FL MH. HN.

But, by the circle, KF . FL = FG2, and мH HN = HI2;
Therefore

FG3

AF AH: FG : H12.

Q. E. D.

Corol. Hence the third proportional

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

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 E is as the rectangle of the segments KI, IH of the double ordinate кн.

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

For, the general property is AF : FE :: FE : P.

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 right-ang. tri. FD2 + DE2 = fe2;

therefore
and the root or side is FE AF+AD,

FEAF 2AF. AD + AD2,

or

FE

GD, by taking AG = AF.

Q. E. ).

G HHN

E

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, HE is always equal to FE the distance of the focus from the point E.

FE

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,

2, 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.

G

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

and, by division,

or

and, by division,

and again by div.

ог

,AD AG TD2: TG2;

AD

AÐ

ᎪᎠ : ᎪᎢ : : ᎢᎠ : ᎢᏀ,

ᎪᎠ : ᎪᎢ : : ᎪᎢ : AG ;

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 H approach towards each other, the point E descending and the point H 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 x, and the ordinates DE, GH in the ordinates cм. 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 VI.

Q. E. D.

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

That

That is,

FCFT.

K

FOR, draw the ordinate Dc to the point of contact c.
Then, by theor. 5, AT = AD;

Corol. 1. If CG be drawn perpendicular to the curve, or to the tangent, at c; then shall FG FC FT.

For, draw Fн perpendicular to TC, which will also bisect. TC, because FT FC; and therefore, by the nature of the parallels, FH also bisects TG in F. And consequently G =

FTFC.

So that F is the centre of a circle passing through 7, c, &.

Corol. 2. The tangent at the vertex AH is a mean proportional between AF and AD.

Corol. 3. The tangent Tc makes equal angles with rc and the axis FT.

Also, the angle GCF the angle GCK,
drawing ICK parallel to the axis AG.

Corol. 4. And because the angle of incidence GCK is = the angle of reflection GCF; therefore a ray of light falling on the curve in the direction KC, will be reflected to the focus F. That is, all rays parallel to the axis, are reflected to the focus, or burning point.

THEOREM