therefore by equality, IE: IK :: CK.

or

and, by division,

CL: CL';

IE IK CK CL;

Corol. When CK = 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 в and 1., 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.

H

For, by theor. 14, CK: KL :: EI : EK; and by composition, CK : Ct. ;; EI : KI ;

and by the parallels

But, by sim. tri. theref. by equal. consequently

CK: CL :: GH : LH.

CK CL KI: LH;

KI: LII :: GU: LH:
KI=GH,

and therefore

KG is parallel and equal to in.

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.

That is, KI KI :: KL: KG.

IH K

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,

Q. E. D.

that is, IK is a mean proportiona! 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 proportional between 1H, 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.

That is, IE EK.

For, by the last th. IE: EK :: CK: KL.

But, by theor. 11, CK = KL, and therefore

IEEK.

K

Corol. 1. The two tangents cr, 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.

68

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.

that is,

:: DE2: GH2,

1st: 2d

:: 1st2: 2d3,
:: 2d : 3d,

DE: GII :: GH GI.

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 is

P

the parameter, or
For, by the defin.

AG GH: GH: P.

AG: GI :: DE : P.

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

For, by theor. 9,

or their equals

are as the squares of PN, GH, de,
or of their equals GI, GH, GK,
which are proportionals by cor. 1.

THEOREM XIX.

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. KL.

For, draw the diameter Ps to which the parallels CL, NO are ordinates, and the ordinate EQ parallel to them.

Then CK is the difference, and KL the sum of the ordinates EQ, CR; also

M the difference, and мo the sum of the ordinates EQ, NS. And the differences, of the abscisses, are QR, Qs, or EK, EM.

Then by cor. theor. 9, an: as :: CK.KL: NM. MO,

EK EM CK. KL NM. MO.

Corol 1. The rect. CL . KL = rect, EK and the param. of rs. For the rect. CK. KL = rect. Quand the param of ps. Corol. 2. If any line CL be cut by two dimeters, EK, GI ; the rectangles of the parts of the line, are as the segments of the diameters.

For EK is as the rectangle cx. KL,

and GH is as the rectangle cut . HL;

therefore EK: GH:: CK. KL: CH. HL.

Corol. 3. If two parallels, ct, No, be cut by two diame. ters, EM, GI; the rectangles of the parts of the parallels will be as the segments of the respective diameters.

For and

EK GH CK. KL CH. HL,

theref. by equal. EM: GHI :: NM. MO: CH. HL.

Corol. 4. When the parallels come into the position of the tangent at P, their two extremities, or points in the curve, unite in the point of contact P; and the rectangle of the parts becomes the square of the tangent, and the same properties still follow them.

So that, EV: PV :: PV: p the param.

GW: PW:: PW: P,

EV: GW:: PV2: pw2,
EV: GH: PV2: CH. HL.

THEOREM XX.

If two parallels intersect any other two parallels; the rectangles of the segments will be respectively proportional. That is, CK. KL : DK . KE ;; G1, IH: NI. 10.

Corol. When one of the pairs of intersecting lines comes into the position of their parallel tangents, meeting and limit ing each other, the rectangles of their segments become the squares of their respective tangents. So that the constant ratio of the rectangles, is that of the square of their parallel tangents, namely,

CK KL: DK KE: tang3, parallel to CL: tang'. parallel to DE.

THEOREM XXI.

If there be three tangents intersecting each other; their segments will be in the same proportion.

That is, GI IH: CG: GD :: DH : NE.

For, through the points G, I, D, H, draw the diameters GK, IL, DM, IN; as also the lines CI, EI, which are double ordinates to the diameters GK, HN, by cor. 1 theor. 16; therefore the diameters GK, DM, HN, bisect the lines CL, CE, LE; hence KM = CMCKCE - CL and MN ME — NE = 'CE — ¦ LE

= =

D

E

MN

LE LN or NE,

CLCK or KL,

as also the 4th terms LN, KM NE.

Therefore the first and second terms, in all the lines, are proportional, namely, GI II :: CG: GD. DII: HE. Q. E. D.

THEOREM XXII.

The Area or Space of a Parabola, is equal to Two-Thirds of its Circumscribing Parallelogram.

Let ACB be a semi-parabola, CB the axis, F the focus, ED the directrix; then if the line AF H

be supposed to revolve about F as a centre, while the line AE moves along the directrix perpendicularly to it, the area gene. rated by the motion of AE, will always be equal to double the area generated by FA; and consequently the whole external area

ACF.

E

I

A

G

D

B

AEGD double the area CA D