Corol. Since CT is 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. THEOREM 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. theref. by div. TA For, by theor. 7, TC: AC :: AC: DC, AD: TC: AC or CB, and by comp. and by sim. tri. Corol. Hence TA, TD, TC, TB and TG, TE, TH, TI are also proportionals.. For these are as AG, DE, CH, BI, by similar triangles. } THEOREM IX. 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. That is, B AFD For, draw the ordinate DE, and fe parallel to FE. By cor. 1, theor. 5, CA: CD :: CF : CA + FE, and by th. 7, CA CD: CT: CA; therefore + For, from the Dor: craw any ime TEE to cut the curve Two points and, from which er fal. the perps. ED, BG; Fa Tangent and Ordinate the drawn from any Point in the Carre, meeting the Trustee Axis; the Semi-transverse will been Niom potional barween the Distances of the said Two Intersections from the Centre. That is amand cr; or cals a mean proportional between Cacare con BCTA D timel sportionals. CD. Γ CT), CT, + CT. TD; Q.E. D. Corol Corol. Since CT is 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. THEOREM 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 are also proportionals.. For these are as AG, DE, CH, BI, by similar triangles. THEOREM IX. 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. AF draw the ordinate DE, and fe parallel to FE. theor. 5, CA: CD :: CF CAFE, CA: CD: CT: CA; E therefore therefore CT CF CA: CA + FE; and by add. and sub. TF: Tf :: FE: 2CA + FE or fe by th. 5. But by sim. tri. therefore But, because therefore the TF: Tf :: FE : fE; fE fe, and conseq. Le=LfEe. FE is parallel to fe, the 4e4FET; Q. E. D. 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. this is the reason why the points F, f, are called foci, or burning points. THEOREM X. ་ And All the Parallelograms inscribed between the four Conjugate Hyperbolas are equal to one another, and each equal to the Rectangle of the two Axes. That is, the parallelogram PQRS == a R Bd Let xo, 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 cê perpendicular to PQ; and let the axis produced meet the sides of the parallelograms, produced, if necessary, in T and t. Then, by theor. 7, CT: CA :: or, by division, and, by composition, CD: DB :: DA: DT; conseq. the rectangle CD *Corol. Because cd2 therefore CA2 In like manner ca2 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. For, draw the ordinates ED, ed. Then, by cor. to theor. 10, CA2 = CD2 cd2, -ca2 de2 DE2; theref, the difference CE2 - ¢e2 = CD2 + DE2 ce2; eg2. Q. E. D. THEOREM XU. All the Parallelograms are equal which are formed between the Asymptotes and Curve, by Lines drawn Parallel to the Asymptotes. For, |