This section is from the book "The Tinman's Manual And Builder's And Mechanic's Handbook", by Isaac Ridler Butt. Also available from Amazon: The Tinman's Manual And Builder's And Mechanic's Handbook.
[Drawn for this work by L. W. Truesdell, Tinman, Owego, N. Y.]
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 DE perpendicular to the line AB, and from C, the point of intersection, as a centre, describe a circle FG, having the diameter equal to the length of the ellipse; from the same centre C describe a circle HJ equal to the width; then describe the end circles LK' and LK, as much less than the width as the width is less than the length; then draw the lines MN and MN tangent to the circles K'L, HJ and KL; from the middle of the line MN at 0 erect a perpendicular produced until it intersects the indefinite line DE; from the point of intersection P as a centre, describe the arc K'HK, and with the same sweep of the dividers mark the point R on the line DE; from the point R draw the lines RU and RV through the points K' and K where the arc K'HK touches the end circles K'L and KL; then place one foot of the dividers on the point R and span them to the point H, and describe the arc Q'HQ, which will be equal in length to the arc K'HK; from the same centre R describe the arc UWV the width of the pattern; then span the dividers the diameter of the end circle KL; place one foot of the dividers on the line RV, at point Q, and the other at Y as a centre, describe the arc QT the length of the curve line KG, and with the same sweep of the dividers describe the arc T'Q' from the centre Y' on the line RU; then span the dividers from Y' to U, and from Y' as a centre, describe the arc TJX, and from Y as a centre, describe the arc VX, 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 H, you must move the centre Y towards, or from Q, or which would be the same as spanning the dividers less, or greater, than the diameter of the end circle KL.