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If a Tangent and Ordinate be drawn from any Point in the Curve, meeting the Transverse Axis; the Semi-transverse will be a Mean Proportional between the Distances of the said Two Intersections from the Centre.
Corol. Since cris always a third proportional to CD, CA; if the points D, A, remain constant, then will the point T be constant also ; and therefore all the tangents will meet in this point T, which are drawn from the point E, of every hyperbola described on the same axis AB, where they are cut by the common ordinate DEE drawn from the point D.
The OREM VIII.
If there be any Tangent meeting Four Perpendiculars to the Axis drawn from these four Points, namely, the Centre, the two Extremities of the Axis, and the Point of Contact; those Four Perpendiculars will be Proportionals.
Corol. Hence TA, TD, Tc, TB and TG, TE, TH, TI For these are as AG, DE, cH, Bi, by similar triangles.
are also proportionals.
If there be any Tangent, and two Lines drawn from the Foci to the Point of Contact; these two Lines will make equal Angles with the Tangent.
Corol. As opticians find that the angle of incidence is equal to the angle of reflexion, it appears, from this proposition, that rays of light issuing from the one focus, and meeting the curve in every point, will be reflected into lines drawn from the other focus. So the ray fe is reflected into Fe. And this is the reason why the points F, f, are called foci, or burning points.
* THEOREM X.
All the Parallelograms inscribed between the four Conjugate Hyperbolas are equal to one another, and each equal to the Rectangle of the two Axes.
Let EG, eg be two conjugate diameters parallel to the sides of the parallelogram, and dividing it into four less and equal parallelograms. Also, draw the ordinates De, de, and ck perpendicular to po ; and let the axis produced meet the sides of the parallelograms, produced, if necessary, in T and t. Then, by theor. 7, CT : ca :: cA : cd, and - - ‘ct : ca : ; ca : cq; theref. by equality, cr : ct :: cd : cD; but, by sim. triangles, cor : ct :: TD : cd, theref, by equality, TD : cd ; c.d : CD,
* Corol. Because cd’ = AD . D B = cD” – c.A". therefore CA.” E CI)” – cd”. In like manner ca” – de” – DE”. But,
But the rect. CK • ce = the parallelogram cepe, theref. the rect. CA ce = the parallelogram cepe, conseq. the rect. AB . ab = the paral. PQRs. Q: E. D.
- THEOREM XI.
The Difference of the Squares of every Pair of Conjugate Diameters, is equal to the same constant Quantity, namely
the Difference of the Squares of the two Axes.
All the Parallelograms are equal which are formed between the Asymptotes and Curve, by Lines drawn Parallel to the Asymptotes.
vol. FI. I
But the parallelograms coek, cpAQ, being equiangular, are as the rectangles GE - Ek and PA . Aq.
Therefore the parallelogram GK = the paral. PQ.
That is, all the inscribed parallelograms are equal to one another. Q. E. D.
Corol. 1. Because the rectangle GLK or cGE is constant, therefore GE is reciprocally as cc, or co : cf. :: PA : G.E. And hence the asymptote continually approaches towards the curve, but never meets it : for GE decreases continually as cG increases; and it is always of some magnitude, except when cq is supposed to be infinitely great, for then GE is infinitely small, or nothing. So that the asymptote co, may be considered as a tangent to the curve at a point infinitely distant from c.
Corol. 2. If the abscisses cD, CE, cG, &c, taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates DH, EI, GK, &c, parallel to the ot, crasymptote, be a decreas$ng geometrical progression, having the same ratio. For, all the rectangles CDH, CEI, c.9K, &c, being equal, the ordinates pH, Ei, GK, &c, are reciprocally as the abscisses cd, ee, cc, &c, which are geometricals. And the reciprocals of geonetricals are also geometricals, and in the same ratio, but decreasing, or in converse order.