By Prof. E.J. HALLOCK.
A very simple apparatus for describing an oval or ellipse may be constructed by any apprentice or school boy as follows: Procure a straight piece of wood about ¼ inch wide by 1/8 inch thick and 13 inches long. Beginning ½ inch from the end, bore a row of small holes only large enough for a darning needle to pass through and half an inch apart. Mark the first one (at A) 0, the third 1, the fifth 2, and so on to 12, so that the numbers represent the distance from O in inches. A small slit may be made in the end of the ruler or strip of wood near A, but a better plan is to attach a small clip on one side.
Next procure a strong piece of linen thread about four feet long; pass it through the eye of a coarse needle, wax and twist it until it forms a single cord. Pass the needle upward through the hole marked 0, and tie a knot in the end of the thread to prevent its slipping through. The apparatus is now ready for immediate use. It only remains to set it to the size of the oval desired.
Suppose it is required to describe an ellipse the longer diameter of which is 8 inches, and the distance between the foci 5 inches. Insert a pin or small tack loosely in the hole between 6 and 7, which is distant 6-½ inches from O. Pass the needle through hole 5, allowing the thread to pass around the tack or pin; draw it tightly and fasten it in the slit or clip at the end. Lay the apparatus on a smooth sheet of paper, place the point of a pencil at E, and keeping the string tight pass it around and describe the curve, just in the same manner as when the two ends of the string are fastened to the paper at the foci. The chief advantage claimed over the usual method is that it may be applied to metal and stone, where it is difficult to attach a string. On drawings it avoids the necessity of perforating the paper with pins.
As the pencil point is liable to slip out of the loop formed by the string, it should have a nick cut or filed in one side, like a crochet needle.
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.