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For, produce KE, MD to the other asymptote at H, L.
theref. the rectangle
EK : EK DM: DM;"
HEK GEK: LDM: IDM.
But, by the last theor. the rect. HEK = LDM; and therefore the rect. GEK = IDM = PAQ
THEOREM XXVIII (27).
Q. E. D.
Every Inscribed Triangle, formed by any Tangent and the two Intercepted Parts of the Asymptotes, is equal to a Constant Quantity; namely Double the Inscribed Parallelogram.
That is, the triangle CTS 2 paral. GK. For, since the tangent Ts is bisected by the point of contact E, and EK is parallel to TC, and GE to CK; therefore CK, KS, GE are all equal, as are also CG, GT, KE. Consequently the triangle GTE = the triangle KES, and each equal to half the constant inscribed parallelogram GK. And therefore the whole triangle CTs, which is composed of the two smaller triangles and the parallelogram, is equal to double the constant inscribed parallelogram GK. Q. E. D.
THEOREM XXIX (29).
If from the Point of Contact of any Tangent, and the two Intersections of the Curve with a Line parallel to the Tangent, three parallel Lines be drawn in any Direction, and terminated by either Asymptote; those three Lines shall be in continued Proportion.
For, by the parallels, EI: IL :: DH: HM;
and, by the same,
theref. by compos.
EI IL GK: KM;
EI: IL:: DH, GK: HMK;
but, by theor. 26, the rect. HMK = IL2;
and theref. the rect.
DH. GK = EI2,
THEOREM XXX (30).
Draw the semi-diameters CH, CIN, CK;
For, because HK and all its parallels are bisected by CIN, therefore the triangle CNH = tri. CNK,
and the segment
INH seg. INK;
consequently the sector CIH = sec. CIK.
Corol. If the geometricals DH, EI, GK be parallel to the other asymptote, the spaces DHIE, EIKG will be equal; for they are equal to the equal sectors CHI, CIK.
So that by taking any geometricals CD, CE, CG, &c, and drawing DH, EI, GK, &c, parallel to the other asymptote, as also the radii CH, CI, CK;
then the sectors CHI, CIK, &c,
or the spaces DHIE, EIKG, &c,
or the spaces DHIE, DHKG, &c,
will be in arithmetical progression.
And therefore these sectors, or spaces, will be analogous to the logarithms of the lines or bases CD, CE, CG, &c; namely CHI or DHIE the log. of the ratio of
CD to CE, or of CE to CG, &c; or of EI to DH, or of GK to EI, &c; and CHK or DHKG the log. of the ratio of CD to CG, &c, or of GK to DH, &c.
OF THE PARABOLA.
THEOREM XX (7).
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, 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.