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La Lg. Or if the tangent TE be produced, then EV=EW. Also the diameter GCEH bisects all lines drawn parallel to TE or aq, and limited either by one hyperbola, or by its two conjugate hyperbolas.


As the Square of any Diameter:
Is to the Square of its Conjugate ::
So is the Rectangle of any two Abscisses:
To the Square of their Ordinate.

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For, draw the tangent TE, and produce the ordinate QL to the transverse at K. Also draw aн, eм perpendicular to the transverse, and meeting EG in H and M. Then, similar triangles being as the squares of their like sides, it is,

by sim. triangles,
or, by division,
Again, by sim. tri.
But, by cor. 5 theor.

CE: LQ2.

A CET trap. TELK :: CE2: CL2
A Ceм: A LQH :: Ce2: LQ2.
21, the A CeMA CET,

and, by cor. 4 theor. 21, the ▲ LQH = trap. TELK;
theref. by equality, CE: ce :: CL CE: LQ3,
CE ce2:: EL. LG: LQ2.



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Corol. 1. The squares of the ordinates to any diameter, are to one another as the rectangles of their respective abscisses, or as the difference of the squares of the semi-diameter and of the distance between the ordinate and centre. For they are all in the same ratio of CE2 to ce2.

Corol. 2. The above being the same property as that belonging to the two axes, all the other properties before laid down, for the axes, may be understood of any two conjugate diameters whatever, using only the oblique ordinates of these `diameters instead of the perpendicular ordinates of the axes; namely, all the properties in theorems 6, 7, 8, 16, 17, 20, 21. Corol. 3. Likewise, when the ordinates are continued to the conjugate hyperbolas at a', q', the same properties still obtain, substituting only the sum for the difference of the squares of CE and CL,

That is, CE2: Ce2 :: CL2 + CE2 : LQ”.

And so La2: La"2:: CL2. CE2: CL2+ CE2.


Corol. 4. When, by the motion of La' parallel to itself, that line coincides with EV, the last corollary becomes CE: Ce2 2CE2: Ev2,

or Ce2: Ev2 : : 1 : 2,
or ce : EV :: 1 :√2,

or as the side of a square to its diagonal.

That is, in all conjugate hyperbolas, and all their diameters, any diameter is to its parallel tangent, in the constant ratio of the side of a square to its diagonal.


If any Two Lines, that any where intersect each other, meet the Curve each in Two Points; then

The Rectangle of the Segments of the one :
Is to the Rectangle of the Segments of the other ::
As the Square of the Diam. Parallel to the former :
To the Square of the Diam. Parallel to the latter.

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For, draw the dia.neter CHE, and the tangent TE, and its parallels PK, RI, MH, meeting the conjugate of the diameter CR in the points T, K, I, M. Then, because similar triangles are as the squares of their like sides, it is,

by sim. triangles, CR2: GP2 :: A CRI : ▲ GPK,



theref. by division, CR2 : GP2 GH CRI: KPHM. Again, by sim. tri. CE2; CH2:: ACTE: A CMH; and by division, CE2: CH2

CE2 : : A CTE : ТЕНМ.

But, by cor. 5 theor. 21, the A CTE = ▲ CIR,


and by cor. 1 theor. 21, TEHGKPHG, or TEHM = KPHM; theref. by equ. CE2 : CH2 CE: CR2: GP2- GH2 or PH.HQ. In like manner CE2: CH2 ÇE::cr2: pH. нq. PH. HQ: рH. нq. C 2

Theref. by equ. CR2: cr2::

Q.É. D.


Corol. 1. In like manner, if any other line p''q', paralles to cr or to pq, meet PHQ; since the rectangles PH'a, р'H'q′ are also in the same ratio of CR2 to cr2; therefore the rect. PHQ: рHq :: PH'Q : p'H'q.

Also, if another line P'ha' be drawn parallel to PQ or CR; because the rectangles p'ho', p'hq' are still in the same ratio, therefore, in general, the rectangle PHQ: pHq :: P'ho' : p'hq'. That is, the rectangles of the parts of two parallel lines, are to one another, as the rectangles of the parts of two other parallel lines, any where intersecting the former.

Corol. 2. And when any of the lines only touch the curve, instead of cutting it, the rectangles of such become squares, and the general property still attends them.

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If a Line be drawn through any Point of the Curves, Parallel to either of the Axes, and terminated at the Asymptotes; the Rectangle of its Segments, measured from that Point, will be equal to the Square of the Semi-axis to which it is.. parallel.

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For, draw AL parallel to ca, and aL to CA.

by the parallels,

and, by theor. 2, theref. by subtr.


CA2: ca2 or AL2:: CD2: DH2;
CA2: ca2:: CD2 CA2: DE2;

CA2: ca2:: CA2: DH2 DE2 or HEK.

But the antecedents CA2, CA2 are equal,
theref. the consequents ca2, HEK must also be equal.

In like manner it is again,

by the parallels, and, by theor 3,

theref. by subtr.

CA2: ca2 or AL2 :: CD2: DH2;
CA2 : ca2 :: CD2 + CA2: De2;

CA2: ca2:: CA2 : De2 - DH2 or HeK.

But the antecedents CA2, CA2 are the same,

theref. the conseq. ca2, HeK must be equal.

In like manner, by changing the axes, is hek or hek = CA2. Corol. 1. Because the rect. HEK = the rect. HеK. therefore EH: CH: eK : EK.

And consequently HE is always greater than He.
Corol. 2. The rectangle hEK = the rect. HEK.
For, by sim. tri. Eh: EH :: Ek : EK.


It is evident that this proposition is general for any line oblique to the axis also, namely, that the rectangle of the segments of any line, cut by the curve, and terminated by the asymptotes, is equal to the square of the semi-diameter to which the line is parallel. Since the demonstration is drawn from properties that are common to all diameters.


All the Rectangles are equal which are made of the Segments of any Parallel Lines cut by the Curve, and limited by the Asymptotes.

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For, each of the rectangles HEK or Hek is equal to the square of the parallel semi-diameter cs; and each of the rectangles hɛk or hek is equal to the square of the parallel semidiameter cr. And therefore the rectangles of the segments of all parallel lines are equal to one another.


Q. E. D.


Corol. 1. The rectangle HEK being constantly the same, whether the point E is taken on the one side or the other of the point of contact 1 of the tangent parallel to HK, it follows that the parts HE, KE, of any line HK, are equal.

And because the rectangle Heк is constant, whether the point e is taken in the one or the other of the opposite hyperbolas, it follows, that the parts He, Ke, are also equal.

Corol. 2. And when HK comes into the position of the tangent DIL, the last corollary becomes IL ID, and IMIN, and LM DN.

Hence also the diameter CIR bisects all the parallels to DL which are terminated by the asymptote, namely RH = RK.

Corol. 3. From the proposition, and the last corollary, it follows that the constant rectangle HEK or EHE is = IL2. And the equal constant rect. Hek or eнe MLN or IM2 — IL2. Corol. 4. And hence IL = the parallel semi-diameter cs, For, the rect. EHE = IL2,

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And so the asymptotes pass through the opposite angles of all the inscribed parallelograms.


The Rectangle of any two Lines drawn from any Point in the Curve, Parallel to two given Lines, and Limited by the Asymptotes, is a Constant Quantity.

That is, if AP, EG, DI be parallels,

as also

AQ, EK, DM parallels,

then shall the rect. PAQ rect. GEK = rect. IDM.





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