Cara. 1. The two seri-axes, and the focal distance from *** *****, are the odes of a right-angled triangle cra; and *** ***** *z from the focus to the extremity of the con3-gate 2xit, is - acthe semi-transverse. Carol. 2. The corogate semi-axis ca is a mean propororz: *ween AP, FE, or between af, f=, the distances of exo-er focus from the two vertices. The Sum of two Lines drawn from the two Foci to meet at any Point in the Curve, is equal to the Transverse Axis. And the root or side of this square is FE = cA — c1 = AI. Corol. 1. Hence c1 or ea – Fe is a 4th proportional to ca, CF, CD. Corol. 2. And fe – Fe = 2c1 ; that is, the difference between two lines drawn from the foci, to any point in the curve, is double the 4th proportional to CA, CF, CD. Corol. 3. Hence is derived the common method of describing this curve mechanically by points, or with a thread, thus: In the transverse take the foci F, f, and any point I. Then with the radii AI, B1, 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. Or, take a thread of the length AB of the transverse axis, and fix its two ends in the foci F, f, by two pins. Then carry a pen or pencil round by the thread, keeping it always stretched, and its point will trace out the curve line. THEOREM VI. If from any Point 1 in the Axis produced, 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 Middle of AB : Then shall cM be to cI as the Square of AM to the Square of AI. For from the point 1 draw any other line IEH to cut the curve in two points E and H3 from which let fall the perpendiculars ED and HG; and bisect DC in K. But, when the line IH, by revolving about the point 1, comes into the position of the tangent il, then the points E and H meet in the point L, and the points D, K, G, coincide with the point M ; and then the last proportion becomes cM : ci :: AM*: A1*. Q: E. D. THEOREM vii. 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 listances of the said Two Intersections from the Centre. or cD, CA, CT, are conti- T ATI, TC nued proportionals. 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 ellipse described on the same axis AB, where they are cut by the common ordinate DEE drawn from the point D. THEoREM Theor EM VIIIs 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, B1, by similar triangles. } are also proportionals. 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. Corol. As opticians find that the angle of incidence is equal to the angle of reflexion, it appears from this theorem, that rays of light issuing from the one focus, and meeting the curve in every point, will be reflected into lines drawn from those points to the other focus. So the ray fe is reflected into Fe. And this is the reason why the points F, f, are called the foci, or burning points. theorem x. All the Parallelograms circumscribed about an Ellipse 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 5 and let the axis ca, produced meet the sides of the parallelogram, produced if necessary, in T and t. Then, by theor 7, CT : CA : : CA : CD, and ct : CA :: CA : că ; theref, by equality, CT : ct :: cd : CD : but, by sim. triangles, ct: ct :: TD : cl, theref. by equality, TD : cd :: Cd : CD, * Corol. Because cq* = AD . DB E CA2 — cI)”, therefore cA* = cl,” + cd”. In like manner, ca’ = DE” + de”. I n |