Fig. 4.

To describe an ellipse of any length and width, and by it to describe a pattern for the sides of a vessel of any flare.

First draw an indefinite line D E perpendicular to the line A B, and from C, the point of intersection, as a centre, describe a circle F G, having the diameter equal to the length of the ellipse; from the same centre C describe a circle H J equal to the width; then describe the end circles L K' and L K, as much less than the width as the width is less than the length ; then draw the lines M N and M N tangent to the circles K'L, H J and K L ; from the middle of the line M N at 0 erect a perpendicular produced until it intersects the indefinite line D E; from the point of intersection P as a centre, describe the arc K' H K, and with the same sweep of the dividers mark the point R on the line D E; from the point R draw the lines R U and R V through the points K' and K where the arc K' H K touches the end circles K'L and K L; then place one foot of the dividers on the point R and span them to the point H, and describe the arc Q' H Q, which will be equal in length to the arc K' H K; from the same centre R describe the arc U W V the width of the pattern; then span the dividers the diameter of the end circle K L; place one foot of the dividers on the line R V, at point Q, and the other at Y as a centre, describe the arc Q T the length of the curve line K G, and with the same sweep of the dividers describe the arc T' Q' from the centre Y' on the line R U; then span the dividers from Y' to U, and from Y' as a centre, describe the arc U X, and from Y as a centre, describe the arc V X, which completes the description of the pattern.

The more flare you wish the pattern to have, the nearer the centre point R must be to H ; and the less flare, the further the centre point R must be from H ; in the same proportion as you move the centre R towards, or from II, you must move the centre Y towards, or from Q, or which would be the same as spanning the dividers lees, or greater, than the diameter of the end circle K L.