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That is, KH KI :: KL; KG.
For, by cor. 1 theor. 1, P: DC DC DA,
P: 2DC: DC: DT or 2DA,
But, by sim. triangles, KI: KC DC: DT;
P: 2DC:: KI : KC,
KH:: KG: KC;
KI KL KG.
Again, by theor. 2,
Corol. 1, Hence, by composition and division,
it is, KH
KI :: GK: GI,
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 proportional 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 XXI (16).
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.
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 XXII (18).
If a Line be drawn from the Vertex of any Diameter, to cut the Curve in some other Point, and an Ordinate of that 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 Ordinates and the Part of the Latter limited by the said Line drawn from the Vertex.
Corol. 1. Or their equals, GK, GH, GI, are proportionals; where EK is parallel to the diameter AD.
AG :: P
Corol. 2. Hence it is DE
: GI, where p is
the parameter, or AG
Corol. 3. Hence also the three MN, MI, MO, are proportionals, where MO 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,
GI, GH, GK,
or of their equals
which are proportionals by cor. 1.
THEOREM XXIII (19).
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.
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
NM 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, ar: as :: CK.KL: NM. MO,
that is Corol. 1. The rect. CK. KL = rect. EK and the param. of Ps. For the rect. CK. KL = rect. QR and the param. of Ps.
EK EM:: CK. KL NM. MO.
Corol. 2. If any line CL be cut by two diameters, EK, GH; the rectangles of the parts of the line, are as the segments of the diameters.
For EK is as the rectangle CK. KL.
therefore EK: GH:: CK. KL: CH. HL.
Corol. 3. If two parallels, CL, NO, be cut by two diameters, EM, GI; the rectangles of the parts of the parallels, will be as the segments of the respective diameters.
theref. by equal. EM GH 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.
EV PV :: PV : P the param.
EV: GW:: PV2: PW2,
EV GH:: Py2; CH. HL.
THEOREM XXIV (20).
If two Parallels intersect any other two Parallels; the Rectangles of the Segments will be respectively Proportional. That is, CK. KL: DK. KE :: GỊ. IH: NI.IO.
3 theor. 23,
QI:: CK. KL: GI. IH;
QI DK.KE: NI.IO;
For, by cor. and by the same, theref. by equal. CK. KL: DK. KE:: GI. IH: NI. 10. Corol. When one of the pairs of intersecting lines comes into the position of their parallel tangents, meeting and limiting 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:: tang2. parallel to c; tang2, parallel to DE.
THEOREM XXv (21).
If there be Three Tangents intersecting each other; their Segments will be in the same Proportion.
That is, GI IH :: CG GD :: DH HE.
For, through the points G, I, D, H, draw the diameters GK, IL, DM, HN; 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;
But the 3d terms KL, CK, MN are all equal;
as also the 4th terms LN, KM, ne.
Therefore the first and second terms, in all the lines, are proportional, namely GI IH :: CG : GD :; DH; HE. Q. E.D.
ON THE CONIC SECTIONS AS EXPRESSED BY ALGEBRAIC EQUATIONS, CALLED THE EQUATIONS OF THE CURVE.
Let d denote AB, the transverse, or any diameter;
x = AK, any absciss, from the extremity of the diam.
Then, theor. 2, AB2: HI2:: AK. KÈ : DK2,
that is, d2: c2:: x(d-x): y2, hence d'y2 = c2(dx- x2), or dy = c(dx-2), the equation of the curve.
And from these equations, any one of the four letters or quantities, d, c, x, y, may easily be found, by the reduction of equations, when the other three are given.
Or, if p denote the parameter, = cd by its definition; then, by cor. th. 2, d: p :: x(d-x) : y2, or dy2=p(dx—x2), which is another form of the equation of the curve.
Or, if d AC the semiaxis; c = CH the semiconjugate; pc2d the semiparameter; x = CK the absciss counted from the centre; and y = DK the ordinate as before.
Then is AK=d-−x, and кB=d+x, and AK.KB = (d−x)× (d+x) = d2 — x2
Then, by th. 2, d2: c:: d2-x:y, and d2y2=c2(d2 — x2), or dy = c(d2x2), the equation of the curve.
Or, d:p :: d2x2 : y2, and dy2 = p(d2 — x2), another form of the equation to the curve; from which any one of the quantities may be found, when the rest are given.
2. For the Hyperbola.
Because the general property of the opposite hyperbolas, with respect to their abscisses and ordinates, is the same as