Page images
PDF
EPUB

and the segment

INH seg. INK ;

consequently the sector ciu= sec. cik.

Corol. If the geometrical proportionals H, 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 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, co, &c. ; namely, CHI or DHIE the log. of the ratio of

CD to CE, or of ce to co, &c.; or of Er to DH, or of GK to EI, &c.; and CHк or DнKG the log. of the ratio of CD to co, &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 com. mon intersection of the two planes, or the axis of the para. bola, and FG, HI ordinates per. pendicular to it.

K

G

M

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,

therefore

[ocr errors]

KF = MH;

AF AH: KF. FL: MH. HN.

But, by the circle, KF. FL = FG2, and мH. HN = HI3 ;
Therefore
AF AH: FG2: HI2

[ocr errors]
[ocr errors]

Q. E. D.

Corol. Hence the third proportional

AF

FG3 HI2
AH

or

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.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

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

[ocr errors]

KI. IH,

GH3,

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 is,

AF = FE = {P,

where F is the focus.

A

E

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

FE =
= P ;

But, by definition 17, therefore also

[ocr errors][ocr errors]
[blocks in formation]

THEOREM IV.

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.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Corol. 1. If, through the point &, 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.

[blocks in formation]

E

[blocks in formation]

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.

OF THE PARABOLA.

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, measured from the Vertex.

[blocks in formation]

Let cc, an indefinitely small portion of a parabolic curve, be produced to meet the prolongation of the axis in T; and let cm be drawn parallel to cм, and cs parallel to AG the axis. Let, also, p= parameter of the parabola.

Then, by sim. tri. cs: sc :: CM: MA + AT = MT,

.*.cs=

MT. CS

CM

Also, th. 1. cor. p. am = mc3 = ms2 + 2ms. sc + sc3,
MC2+2мc. sc + sc3,

and p . AM = MC2.

Consequently, omitting sc3 as indefinitely small, and subtracting the latter equa. from the former, we have

P. (Am AM) = p. cs = 2cs. MC :

or, substituting for cs its value above,

[blocks in formation]

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.

VOL. I.

67

[blocks in formation]

For, draw the ordinate pc to the point of contact c.

Then, by theor. 5, ar = AD ;

therefore

But, by theor. 4.

FT

AF + ad.

[blocks in formation]

theref. by equality, FC

[blocks in formation]

Corol. 1. If ca be drawn perpendicular to the curve, or to the tangent, at c; then shall FG = FC = Ft. For, draw rн perpendicular to rc, which will also bisect TC, because FT FC; and therefore, by the nature of the parallels, Fn also bisects To in F. And consequently y =

FT FC.

So that is the centre of a circle passing through T, C, G.

Corol. 2. The subnormal DG is a constant quantity, and equal to half the parameter, or to 2AF, double the focal distance. For, since TCG is a right angle,

therefore TD or 2AD: DC :: DC : DG ;

but by the def AD: DC DC : parameter ;

therefore DG = half the parameter = 2AF.

Corcl. 3. The tangent at the vertex AH, is a mean proportional between AF and AD.

For, because FиT is a right angle,

therefore or between Likewise,

Al is a mean between AF, at,

[ocr errors]

AF, AD, because AD = AT.

[ocr errors]

FH is a mean between FA, FT,
or between Fa, ¿C.

Corol. 4. The tangent rc makes equal angles with FC and the axis FT; as well as with Fc and c1.

For, because FT = FC,

Therefore the FCT FTC.

Also, the angle acr the angle GCK,

drawing Ick parallel to the axis AG.

Corol. 5. 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.

« PreviousContinue »