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

and the segment

INH seg. INK;

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, CHк, &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 ce to co, &c. ; or of EI to DH, or of GK to EI, &c.; and CHK 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 Fe, HI ordinates perpendicular to it.

M

Then it will be, as AF: AH :: FG3 : HI2.

N

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

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: KF. FL: MH. HN.

Q. E. D.

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

FG2 HI2

Corol. Hence the third proportional- -or- is a constant

AF

AH

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

So that any diameter EI 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 Ar: FE: FE : P.

But by definition 17

therefore also

REP;

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.

That is,

FEFA+AD = GD,

taking AG AF.

A

D

For, since FD=AD AF,

theref. by squaring, FD2 AF2-2AF. AD + AD2, But, by cor. theor. 1, DE2 P. AD4AF. AD;

=

theref. by addition, FD2+ DEAFS + 2AF. AD + AD2, But, by right-ang. tri. FD2+ DE2 = FE2;

therefore

FE2 AF22AF. AD + AD3,

and the root or side is FEAF + AD,

or

FEGD, by taking AG AF.

Q. E. D.

Corol. 1. If, through the point &, the HHH G HER 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.

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, ares 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;

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 approach towards each other, the point E descending and the point н 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 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

AM.

contact.

Q. E. D.

THEOREM VI.

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

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

therefore

But, by theor. 4,

theref. by equality,

FTAFAD.

FC = AF+AD ;

FC FT.

Corol. 1. If co 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 FG FT

=FC.

So that F is the centre of a circle passing through т, c, G.

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

For, because FHT is a right angle, therefore

or between

Likewise,

AH is a mean between AF, AT,

AF, AD, because ADAT.

FH is a mean between FA, FT,

or between FA, FC.

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

For, because FT=FC,

therefore the FCT FTC.

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 That is, all rays parallel to the axis, are reflected to the focus, or burning point.

F.

THEOREM