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understanding the parameter of any diameter, as a third proportional to any absciss and its ordinate. Some of the most material of which are demonstrated in the four following theorems.

THEOREM XII.

The Parameter of any Diameter is equal to four Times the Line drawn from the .ocus to the Vertex of that Diame. ter.

'That is, 4FC = p,

the param. of the diam. cm.

F

מן

M

N

For. draw the ordinate MA parallel to the tangent CT: also CD, MN perpendicular to the axis AN, and ru perpendicular to the tangent CT.

Then the abscisses av, cm or aT, being equal, by theor. 5, the parameters will be as the squares of the ordinates CD, MA or CT, by the definition ;

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Corol. Hence the parameter p of the diameter cm is equal to 4FA + 4AD, or to P + 4AD, that is, the parameter of the axis added to 4AD.

THEOREM XIII.

If an Ordinate to any Diameter pass through the Focus, it will be equal to Half its Parameter; and its Absciss equal to One Fourth of the same Parameter.

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therefore

CM == ip.

Again, by the defin. cм or ¦p: ME :: ME: P,
and consequently ME= P = 2CM. Q. E. D.

Corol. 1. Hence, of any diameter, the double ordinate which passes through the focus, is equal to the parameter, or to quadriple its absciss.

C rol. 2. Hence, and from cor 1, to theor. 4, and theor. 6 and 12, it appears, that if the directrix GH be drawn, and any lines HE, HE, parallel to the axis; then every parallel HE will be equal to EF, or ¦ of the parameter of the diameter to the point E.

THEORIM XIV.

HIGH H

E

E

E

If there be a Tangent, and any Line drawn from the Point of Contact and meeting the Curve in some other Point, as also another Line parallel to the Axis, and limited by the First Line and the Tangent: then shall the Curve divide this Second Line in the same Ratio as the Second Line divides the First Line.

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For, draw LP parallel to IK, or to the axis.

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therefore by equality, IE: IK :: CK. CL: CL2;

or

and, by division,

IE IK :: CK CL;

IE: EK :: CK KL. Q. E. D.

Corol. When сK = KL, then IE = EK = IK.

THEOREM XV.

If from any Point of the Curve there be drawn a Tangent, and also Two Right Lines to cut the Curve; and Diameters be drawn through the Points of Intersection E and L., meeting those Two Right Lines in two other Points G and K then will the Line KG joining these last Two Points be parallel to the Tangent.

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For, by theor. 14, ck : KL :: EI : EK ; and by composition, CK: ct. ;; EI : KI ; and by the parallels CK CL :: GH : LH. But, by sim. tri. theref. by equal. consequently

and therefore

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CK CL KI: LH;

KI: LI :: GH: LH:

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KI = GH,

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KG is parallel and equal to IH.

THEOREM XVI.

Q. E. D.

If an ordinate be drawn to the point of contact of any tangent, and another ordinate produced to cut the tangent; it will be, as the difference of the ordinates

Is to the difference added to the external part,

So is double the first ordinate

To the sum of the ordinates.

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For, by cor. 1, theor. 1, P: DC :: DC: DA,

and

But, by sim. triangles, therefore, by equality,

or,

Again, by theor. 2,

P: 2DC :: DC: DT or 2DA.
KI: KC :: DC: DT;

P: 2DC :: KI: KC,

P: KI KL: KC.

P: KH: KG: KC;

therefore by equality, KH KI :: KL: KG.

Corol. 1. Hence, by composition and division, it is, KH KI :: GK : GI,

and HI: HK:: HK: KL,

also IH IK :: IK IG;

4. E. D.

that is, IK is a mean proportional between IG and IH. Corol. 2. And from this last property a tangent can easily be drawn to the curve from any given point 1. Namely, draw IHG perpendicular to the axis, and take IK a mean pro. portional between IH, IG; then draw KC parallel to the axis, and c will be the point of contact, through which and the given point I the tangent ic is to be drawn.

THEOREM XVII.

If a tangent cut any diameter produced, and if an ordinate to that diameter be drawn from the point of contact; then the distance in the diameter produced, between the vertex and the intersection of the tangent, will be equal to the absciss of that ordinate.

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Corol. 1. The two tangents cI, LI, at the extremities of any double ordinate CL, meet in the same point of the diameter of that double ordinate produced. And the diameter drawn through the intersection of two tangents, bisects the line connecting the points of contact.

Corol. 2. Hence we have another method of drawing a tangent from any given point 1 without the curve. Namely, from I draw the diameter IK, in which take EK EI, and through K draw CL parallel to the tangent at E; then c and L are the points to which the tangents must be drawn from 1.

THEOREM XVIII.

If a line be drawn from the vertex of any diameter, to cut the curve in some other point, and an ordinate of that VOL. I.

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diameter be drawn to that point, as also another ordinate any where cutting the line, both produced if necessary: The three will be continual proportionals, namely, the two ord.nates and the part of the latter limited by the said line drawn from the vertex.

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Corol. 1. Or their equals GK, GH, GI, are proportionals; where EK is parallel to the diameter AD.

Corol. 2. Hence it is DE AG :: P: GI, where p is

the parameter, or
For, by the defin.

AG GI :: DE: P.

AG GH GH: P.

Corol. 3. Hence also the three MN, MI, MO, are proportionals, where мo is parallel to the diameter, and am parallel to the ordinates.

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If a diameter cut any parallel lines terminated by the curve; the segments of the diameter will be as the rectangle of the segments of those lines.

That is, EK EM:: CK. KL: NM. MO.

Or, Ek is as the rectangle CK

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