theref. by equality, cK : ca: ; ca : ce, and the rectangle cK - ce = CA. ca. But the rect. ck. ce = the parallelogram cepe, theref. the rect. CA. ca = the parallelogram cepe, conseq. the rect. AB. ab = the parallelogram PQRS. Q: E. D. THEOREM x1. The Sum of the Squares of every Pair of Conjugate Diameters, is equal to the same constant Quantity, namely, the Sum of the Squares of the two Axes. Note. All these theorems in the Ellipse, and their demonstrations, are the very same, word for word, as the corresponding number of those in the Hyperbola, next following, having only sometimes the word sum changed for the word difference. OF THE HYPER BOLA. THEOREM I. The Squares of the Ordinates of the Axis are to each other as the Rectangles of their Abscisses. LEr Ave be a plane passing through the vertex and axis of the opposite cones; AGIH another section of them perpendicular to the plane of the former; AB the axis of the hyperbolic sections; and FG, HI, ordinates perpendicular to it. Then it will be, as FG”: HI*::AF. FB : AH. H.B. For, through the ordinates MoFG, H1, draw the circular sections “... KGL, MIN, parallel to the base of the cone, having Kl, MN, for their diameters, to which FG, H1, are ordinates, as well as to the axis of the hyperbola. • As the Square of the Transverse Axis : That is, As the transverse, THEOREM III. As the Square of the Conjugate Axis : The OREM IV. The Square of the Distance of the Focus from the Centre, is equal to the Sum of the Squares of the Semi-axes. Or, the Square of the Distance between the Foci, is equal to the Sum of the Squares of the two Axes. For, to the focus f draw the ordinate FE; which, by the definition, will be the semi-parameter. Then, by the nature of the curve - cA*: ca”:: CF’ — ca.” : Fe’; and by the def. of the para. cA*: ca’:: ca” : Fe’; Corol. 1. The two semi-axes, and the focal distance from the centre, are the sides of a right-angled triangle cAa; an the distance Aa is = cr the focal distance. Corol. 2. The conjugate semi-axis ca is a mean proportional between AF, FB, or between Af, fB, the distances of either focus from the two vertices. The Difference of two Lines drawn from the two Foci, to meet at any Point in the Curve, is equal to the Transverse Axis. For, draw AG parallel and equal to ca the semi-conjugate; and join cq, meeting the ordinate DE produced in H ; also take CI a 4th proportional to cA, CF, CD. • . Then, w *) Corol. 1. Hence ch = c1 is a 4th proportional to cA, cF, CD, Corol. 2. And fe + FE = 2CH or 2C1; or FE, cH, fe, are in continued arithmetical progression, the common difference being CA the semi-transverse, Corol. 3. Hence is derived the common method of describing this curve mechanically by points, thus: In the transverse AB, produced, take the foci F, f, and . any point I. Then with the radii A1, BI, and centres F, f, describe arcs intersecting in E, which will be a point in the curve. In like manner, assuming other points I, as many other points will be found in the curve. - Then, with a steady hand, the curve line may be drawn through all the points of intersection E. In the same manner are constructed the other two or conjugate hyperbolas, using the axis ab instead of AB. THE OREM VI. If from any Point 1 in the Axis, a Line IL be drawn touching the Curve in one Point L'; and the Ordinate LM be drawn; and if c be the Centre or the Middle of AB : Then shall cM be to ci as the Square of AM to the Square of AI. |