GEOMETRY. 371 



tions of circles put together can ever form an el- 

 lipsis. But by this means a figure may be drawn, 

 which approaches nearly to an ellipsis, and, there- 

 fore, may be often substituted for it when a tram- 

 mel cannot be had, or when the ellipsis is too 

 small to be drawn by it. At the joining of the 

 portions of circles in this operation, the defect is 

 perceivable ; and the best way is not to join them 

 quite, and to help the curve by hand. 



Prob. 29. An ellipsis, A C D B, being given, to 

 find the transverse and conjugate axes. 



Draw any two parallel lines, A B and C D, cut- 

 ting the ellipsis at the points A, B, C, D ; bisect 

 them in e and^ Through e and fy draw G H, 

 cutting the ellipsis at G and H ; bisect G H at 1 5 

 and it will give the centre. 



Upon I, with any radius, describe a circle, cut^ 

 ting the ellipsis in the four points /^, /, ???, n /join 

 ky /, and w, n ; bisect k /, or m 7Z, at or p. 

 Through the points 0, 1, or I, ^, draw Q R, cut- 

 ting the ellipsis at Q and R ; then Q R will be the 

 transverse axis. Through I draw T S, parallel to 

 k /, cutting the ellipsis at T and S ; and T S will 

 be the conjugate axis. 



Prob, 30. To describe an ellipsis similar to a 

 given one A D B C, to any given length I K, or to 

 a given width M L. 



Let A B and C D be the two axes of the given 

 ellipsis. Through the points of contact A,D, B, C, 

 complete the rectangle G E H F ; draw the dia- 

 gonals E F and G H : they will pass through the 

 centre at R. Through I and K draw P N and 

 O Q parallel to C D, cutting the diagonals E F 

 and GH, at P, N, Q, O. Join PO and N Q, 

 cutting CD at L and M ; then I K is the trans-. 

 vei'sCj and M L the conjugate axis of an ellipsis, 

 p B 2 



