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THEOREM XIII. The three following Spaces, between the Asymptotes and

the Curve, are equal; namely, the Sector or Trilinear Space contained by an Arc of the Curve and two Radii, or Lines drawn from its Extremities to the Centre ; and each of the two Quadrilaterals, contained by the said Arc, and two Lines drawn from its Extremities parallel to oné Asymptote, and the intercepted Past of the other Asymptoté.

P

That is,
The sector CAE = PAEĜ = QAEK,
all standing on the same arc AE.

Q
For, by theor. 12, CPAQ=CGEK;
subtract the common space CGIQ,
there remains the paral. PI = the par. IK;
to each add the trilineal IAE, then
the sum is the quadr. PAEG = QAEK.

Again, from the quadrilateral CAEK
take the equal triangles CAQ, CEK,
and there remains the sector CAE = QAEK.
Therefore CAE = QAEK = PAEG.

Q. I. D.

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

K parallel to the side vm of the cone; also let A FH be the com

DC

M mon intersection of the two planes, or the axis of the parabola, and FG, Hí ordinates perpendicular to it.

I 2

Then

N

Then it will be, as AF : AH :: FG? : 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; therefore

AF : AH :: KF FL : MH . HN, But, by the circle, KF . FL = FGʻ, and MH . HN = HI?; Therefore

AF : AH :: FG? : HI?.

Q. E. D.

FG? HI ? Corol. Hence the third proportional

is a con

AF
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 = FG”.

or

AH

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FOR, by cor. theor. 1, P AG = GH, and

P. AD = DEʻ; theref. by subtraction, P. DG = GH? · DE. Or,

P.DG = KI therefore

P: KI :: IH : DG or EI.

IH,

Q. E. D.

P.

Corol. Hence, because p. El = KI. IH, and, by cor. theor. 1,

AG = GH”, therefore

AG : EI :: GH’: KI. JH. So that any diameter Ei is as the rectangle of the seg. ments KI, IH of the double ordinate KH.

THEOREM

THEOREŃ 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.

E

For, the general property is AF : FE :: FE : P.
But, by definition 17, FE = {P;
therefore also

AF = FE = P.

R. E. D.

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.

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For, since FD = AD AF, theref, by squaring,

FD? = AF 2AF . AD + AD, But, by cor. theor. 1, DE? =P AD = 4AF. AD; theref. by addition, FD? + DE’ = AF’ + 2AF. AD + AD, But, by right-ang. tri. FD? + DE? = Fe”; therefore

FE? = AF+ 2AF . AD + AD’, and the root or side is FE = AF + AD, FE = GD, by taking AG = AF.

Q. E, D.

or

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Coro!. 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, E from this theorem, it appears, is this: E That drawing any lines HE parallel to

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

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

'THEOREM V. If a Tangent be drawn to any Point of the Parabola, meet

ing 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.

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or

or

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

Then, by th. 1, AD: AG :: DE : Gho; and, by sim. tri. TD’: TG? :: DEĽ : GH”; theref. by equality, AD : AG :: TD’: TG2; and, by division, AD:DG::TD?: TG-TD' or DG.(TD+TG),

AD : TD:: TD : TD + TG; and, by division, AD : AT:: TD: TG, and again by div. AD : AT :: AT : AG;

AT is a mean propor. between An, AG. Now, if the line Th 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 x descend, ing, and the point H ascending, tillsat 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 M, and the ordinates DE, GH in the ordinate 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.

Q. E. Q.

THEOREM

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.

T

A

That is,
FC = FT.

For, draw the ordinate bc to the point of contact 6.

Then, by theor. 5, AT = AD; therefore

FT = AF + AD. But, by theor. 4, FC = AF + AD; theref. by equality, FC = FT.

R. E. 'D. Corol. 1. If co be drawn perpendicular to the curve, or to

the tangent, at c; then shall FG = FC = FT. For, draw fh 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

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

- FC.

Corol. 2. The tangent at the vertex A'm, is a mean propor

tional between Af and AD.

For, because FHT is a right angle, therefore AH is a mean between AF, AT, or between AF, AD, because AD = AT. Likewise, FH is a mean between FA, FT,

or between FA, FC.

Corol. 3. The tangent to makes equal angles with Fc and

the axis FT.

FT = FC,

LFTC.

For, because
therefore the L FCT =
Also, the angle GCF = the angle GCK,
drawing Ick parallel to the axis AG.

Corel

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