As the mechanic frequently wants to make an oval having a given width and length, but does not know what the distance between the foci must be to produce this effect, a few directions on this point may be useful:
It is a fact well known to mathematicians that if the distance between the foci and the shorter diameter of an ellipse be made the sides of a right angled triangle, its hypothenuse will equal the greater diameter. Hence in order to find the distance between the foci, when the length and width of the ellipse are known, these two are squared and the lesser square subtracted from the greater, when the square root of the difference will be the quantity sought. For example, if it be required to describe an ellipse that shall have a length of 5 inches and a width of 3 inches, the distance between the foci will be found as follows:
(5 x 5) - (3 x 3) = (4 x 4)
or __
25 - 9 = 16 and \/16 = 4.
In the shop this distance may be found experimentally by laying a foot rule on a square so that one end of the former will touch the figure marking the lesser diameter on the latter, and then bringing the figure on the rule that represents the greater diameter to the edge of the square; the figure on the square at this point is the distance sought. Unfortunately they rarely represent whole numbers. We present herewith a table giving the width to the eighth of an inch for several different ovals when the length and distance between foci are given.
Length. Distance between foci. Width.
Inches. Inches. Inches.
2 1 1¾
2 1½ 1¼
2½ 1 2¼
2½ 1½ 2
2½ 2 1½
3 1 1½
3 1½ 2-7/8
3 2 2-5/8
3 2½ 2¼
3½ 1 3-3/8
3½ 1½ 3-1/8
3½ 2 2-7/8
3½ 2½ 2½
3½ 3 1¾
4 2 3½
4 2½ 3-1/8
4 3 2-5/8
4 3½ 2
5 3 4
5 4 3
For larger ovals multiples of these numbers may be taken; thus for 7 and 4, take from the table twice the width corresponding to 3½ and 2, which is twice 2-7/8, or 5¾. It will be noticed also that columns 2 and 3 are interchangeable.
To use the apparatus in connection with the table: Find the length of the desired oval in the first column of the table, and the width most nearly corresponding to that desired in the third column. The corresponding number in the middle column tells which hole the needle must be passed through. The tack D, around which the string must pass, is so placed that the total length of the string AD + DC, or its equal AE + EC, shall equal the greater diameter of the ellipse. In the figure it is placed 6½ inches from A, and 1½ inches from C, making the total length of string 8 inches. The oval described will then be 8 inches long and 6¼ inches wide.
The above table will be found equally useful in describing ovals by fastening the ends of the string to the drawing as is recommended in all the text books on the subject. On the other hand, the instrument may be set "by guess" when no particular accuracy is required.