PROBLEM XII. Any two conjugate diameters A B and C D, being given, and a right line G H passing through the centre F, to find a diameter which will be conjugate to G H, without drawing any part of the ellipsis. 1. Through D, draw E K parallel to A B, and produce the given line H G, to cut the tangent in E. 2. From D, make D I perpendicular to E F, and equal to F A, or F B. 3. Join EI; from I, draw I K perpendicular to I E, cutting the tangent E K, at K; through the centre F, draw F K. 4. Through the points g, and m, where the lines EI, and I K, cut the circle; draw g G, and m M, parallel to and K F, at the points G and M; make G, and F L, equal to F. M; then M L, I F, cutting E F, and G H, will be the two other conjugate diameters. PROBLEM XIII. Any two conjugate diameters A B and C D, being given, to find the two axes, from thence, to describe the ellipsis. 1. Through D, draw E F, parallel to AB; draw DI perpendicular to E F and equal to M A, or M B. 2. Upon I, with the radius I D, describe the arc g DI. 3. Join I M, and bisect it by a perpendicular, meeting the tangent E F at N. 4. On N, as a centre, with the distance N I, describe a semicircle E IF, cutting E F, at the points E and F. 5. Through the centre M, draw F K, and E H. 6. Join IF, and I E, cutting the arc g Dl, at g, and l. 7. Draw IL, and g G, parallel to I M, cutting K F, and H E, at G and L; make M K equal to M L, and M H equal to M G, then E H, and K L, will be the two axes required. |