Let P be the given point.

Join S, P; draw PN perpendicular to the directrix.

Bisect the angle SPN by the straight line Tt.

PROP. III.

To draw a tangent to the parabola at any point.

Tt is a tangent at the point P.

For if Tt be not a tangent, let Tt cut the curve in some other point p.

Join S, p; draw pn perpendicular to the directrix; join S, N.

Since SP PN, PO common to the tri

angles SPO, NPO, and angle SPO = angle NPO by construction, .. SO NO, and angle SOP = angle NOP.

Sp = .: pN =

Again, since SO = NO, Op common to the triangles SOp, NOP, and angle SOp angle NOP,

pn

pn.

.: SpNp.

But since p is a point in curve, and pn is drawn perpendicular to the directrix,

n

.. Tt is a tangent to the curve at P.

Cor. 1. A tangent at the vertex A, is perpendicular to the axis.

Cor. 2. SP

= ST

For, since NW is parallel to TX
.: STP = < NPT

Then, Pr = Pv

For, since Qq is parallel to Tt

<Pxv =

XPT

= NPT by construction,
= Pur interior opposite angle,

=

Pv

ΤΑ

That is, the hypothenuse of a right-angled triangle equal to one of the sides, which is impossible, .. p is not a point in the curve; and in the same manmay be proved that no point in the straight line Tt can be in the curve,

ner it except P.

= SPT by construction, SP =

ST

Cor. 3. Let Q q be an ordinate to the diameter PW, cutting SP in x.

T

N

W

-X

W

For, MT =

=

=

=

<SGP

SG

PROP. IV.

The subtangent to the axis is equal to twice the abscissa. That is,

MT = 2 AM

MS + ST

MSSP. Prop. 3. cor. 2.
MS+SA+AM. Prop. 1.

2 AM.

Cor. MT is bisected in A.

For, MG SG - SM

= SP SM.

= AS + AM SM.
= AS+AS+ SM-
= 2 AS

PROP. V.

The subnormal is equal to one half of the latus rectum.

L

MG =

if we denote the latus rectum by L.

Prop. 3. cor.4.

Prop. 1.

SM

Prop. 2.

A

T

PROP. VI.

S

If a straight line be drawn from the focus perpendicular to the tangent at any point, it will be a mean proportional between the distance from the focus to that point, and the distance from the focus to the vertex.

That is, if SY be a perpendicular let fall from S upon Tt the tangent at any point p SP: SY:: SY: SA.

or,

Cor. 1. Multiplying extremes and means,

SP: SY:: SY: SA SP = ST by Prop. 3. cor. 2.

That is, if P be any point in the curve

For,

PM2 SP2 SM2

AS. SP

4 AS. SP

L. SP. Prop. 2

PM2 = L. AM.

Geom. Theor. 34.

= (AM+AS)'—(AM—AS)*

SP AM+AS (Prop. 1), &SM=AM-AS

=

= 4 AS. AM. Geom. Theor. 31 & 32.

= L. AM. Prop. 1.

PROP. VI.

The square of any semi-ordinate to the axis is equal to the rectangle under the latus rectum and the abscissa.

Cor. 1. Since L is constant for the same parabola PM2 AM,

A S

That is, The abscissæ are proportional to the squares of the ordinates.

AS

M

M

If Qq be an ordinate to the diameter PW and Pv, the corresponding abscissa, then,

Qu2 = 4SP X Pv.

Draw PM an ordinate to the axis.

Join S, Q; and through Q draw DQN perpendicular to the axis.

From S let fall SY perpendicular on the Langent at P.

But,
QNo
And, PM2. QN2 =

PM2

= .. (PM+QN). QD = But, 2PM. QD = .: (PM—QN). QD = Or, QD

The triangles SPY, QDv, are similar. Qu2 QD2 :: SP2 :: SP The triangles PTM, QDv, are also similar; QD : Dv :: PM :: PM2

..

PROP. VIII.

2PM. QD =

Px

.: PS

AN SM

: SY2

: SA, Prop. vi. Cor. 2.

:: 4AS, AM: 2PM. AM
:: 4AS
: 2PM

4AS. Dv

: MT

: PM.MT

4AS. AM -4AS. AN = 4AS(AM—AN) = 4AS. MN
(PM+ QN) (PM — QN)
(PM+QN). QD
4AS. MN

4AS. DP

4AS. Dv
4AS. Pv
4AS. Pv

Qu2 4AS. Pv :: SP: SA,
Qu2 = 4SP. Pv.

Cor. 1. In like manner it may be proved, that

2

qv2 = 4SP X Pv.

Cor. 2. Let Rr be the parameter to the diameter PW.

Then, by Prop. 111. Cor. 3.

Pv

PV

Hence, Qu qu; and since the same may be proved for any ordinate, it follows, that

A diameter bisects all its own ordinates.

Now, by the Proposition,

RV2=
= 4SP2

.. 4RV2 or Rr2 16SP2
Rr = 4SP.

Hence the Proposition may be thus enunciated:

The square of the semi-ordinate to any diameter is equal to the rectangle under the parameter and abscissa.

It will be seen, that Prop. vII. is a particular case of the present proposition.

4SP. PV

ELLIPSE.

DEFINITIONS.

1. AN ELLIPSE is a plane curve, such that, if from any point in the curve two straight lines be drawn to two given fixed points, the sum of these straight lines will always be the same.

2. The two given fixed points are called the foci. Thus, let ABa be an ellipse, S and H

the foci.

Take any number of points in the curve
P1, P2, P3,

Join S,P1, H,P1; S,P2, H,P2; S,P3,
H,P3;
then,
SP1+ HP1 = SP2+ HP2 = SP2+
HP =

3

B Pi

3. If a straight line be drawn joining the foci and bisected, the point of bisection is called the centre.

4. The distance from the centre to either focus is called the eccentricity.

Join S,H; bisect the straight line SH in C, and produce it to meet at the curve in A and a.

P2

P3

Through C draw any straight line Pp, terminated by the curve in the points P, p. Through C draw Bb at right angles to

Aa.

p

H

5. Any straight line drawn through the centre, and terminated both ways by the curve, is called a diameter.

6. The points in which any diameter meets the curve are called the vertices of that diameter.

7. The diameter which passes through the foci is called the axis major, and the points in which it meets the curve are called the principal vertices.

a

8. The diameter at right angles to the axis major is called the axis minor. Thus, let ABa be an ellipse, S and H the foci.

B