PROBLEM III.

A diameter A B, and a double ordinate CD to that diameter, being given, to find the parameter.

1. Join AC, A D, and B C, BD; bisect AB in H, through H draw H I parallel to D C, cutting B C in I.

2. From A, make A F equal to HI, through F draw G H parallel to C D, cutting AC in G, and A D in H, then G H is the parameter sought.

To describe an ellipsis by finding points in the curve, having the two conjugate diameters A B, and C D given.

1. Find FG half the parameter; through G, draw HH parallel to A B.

2. Draw E H parallel to C D, cutting H H, at H.

3. Set off any number of equal divisions, from H, towards G, set the same parts from E, towards C.

4. From the point B, through the points 1, 2, 3, in EC, draw the lines Bi, Bk, Bl.

5. From A, through the points in H G, draw the lines A i, Ak, Al, intersecting the former lines in i, k, l, they will be in the periphery of the ellipsis.

PROBLEM V.

Having a diameter, and a double ordinate to that diameter, to describe the ellipsis, by finding points in the curve.

This problem will be completed in the same manner as in the last problem, as is plainly shewn by the figure.

PROBLEM

PROBLEM VI.

To describe an ellipsis, or any segment of an ellipsis, having a diameter and a double ordinate, by means of points being found in the curve, without finding the parameter.

Let A B be a diameter or double ordinate, let C D be its conjugate, and let ED be the height of the segment.

1. Through D, draw F G parallel to A B; also through the points A and B, draw A F, and B G, parallel to D E, cutting F G, in F and G.

2. Divide A E and E B into a like number of equal parts, as four; likewise B G, and A F, into the same number of equal parts.

3. From the point D, through the points 1, 2, 3, in A F, and B G, draw 1 D, 2 D, 3 D.

4. From the point C, through the points 1, 2, 3, in A B, draw C a, Cb, Cc, cutting the lines 1 D, 2 D, 3 D, in a, b, c, they will be in the periphery of the ellipsis; a curve being traced through these points, will form the ellipsis required.

But if the curve is very large, as in practical works; the best way is to put in nails or pins at the points a, b, c, &c, bend a slip round them, and draw a curve by it, it will appear quite regular.