* The demonstration of this theorem will be found in the sixth book of Euclid and the sixth proposition.

Take the standard from which it is desired to grade a set on the geometric method, and trace its outline with a fine line on a sheet 01 paper as Fig. 150, where A, B, C, D is the pattern. Make a line from the counter-height E, towards the toe of the standard A, continuing it a sufficient length beyond the pattern to mark thereon the number of sizes that are required in grade. A central radial point is now to be found, and in principle it matters little where the point is placed; but for convenience it is best to make it some fractional part of the line AE, and the point selected should also be convenient to cut as many of the curves of the pattern as sharply as possible. This will usually be attained by selecting the point one-third of the length AE from E (see G, Fig. 150). A shoe size (if English measures, then 1/3 in.) is to be divided into the same proportion as the toe-counter line, and the subdivisions placed outside of E and A, repeating them for the other sizes required. In Fig. 150 the amount placed to the credit of E, for each size is 1/9 in., and at A 2/9 in., making in. for the whole increase. A radial tool should now be constructed to compare results with what has been done. This radial tool is made as follows: A line AE (Fig. 151) is drawn on a separate piece of paper, equal in length to the line from the toe to the counter of the standard for which it is intended. At one end, say A, a line Aa is drawn at right angles to AE. From A, in the direction of a, mark as many shoe sizes as are required, 1, 2, 3, etc. Connect these points with E, as illustrated, and when the paper has been cut to the lines 3E, EA, and Aa, it is ready for use. To use, put the end E at the radial centre G, in Fig. 150, and lay the edge AE of the tool level with the line AE. Where the toe, A, crosses at H (Fig. 151), Hh gives the grade from A (Fig. 150).

Fig. 150.

Fig. 151.

Lines radiating from G will be drawn through the principal points desired to be graded (see Fig. 150, GB, GC, GD, etc.). If continuing to use the radial tool, the end E of it should be placed upon the radial centre G, and where the base-line crosses the radial-line at the portion of the pattern to be graded, the transverse distance of the tool will give the amount of grade.

If the pattern be prepared as described above, and the amount of proportional grade be marked from the points AE, as shown in Fig. 150, the remaining points may be found without using or making the radial tool. After the radiating lines have been drawn from the centre G to the various points required, a set-square may "be used to obtain the grade at B, C, D, etc. Place one side of the set-square to two points* of the pattern, such as AB (Fig. 150), and while in this relation put a straight-edge to one other convenient side of the set square, as shown in Fig. 150. Firmly hold the straight-edge to the paper, and slide the set-square - keeping it to the straight-edge - until one of the graded divisions at A is reached. Then, where the set-square crosses the radial line from B, the new amount of grade is found.

Instead of the set-square a parallel rule may be employed. One edge of the rule should be laid to the points, such as C and E in Fig. 150, and the fingers put firmly on the bottom portion of the rule to keep it from moving; the top half of the rule to be carefully moved to the point already found, and the required grade will be given.

A sector will give the same results exactly as the parallel rule, the set-square, or the radial tool, but it is not so convenient to use.

The proportional compass, illustrated by Fig. 152, is a very useful instrument to the pattern cutter, and can be applied here to grade the amounts for the method illustrated in Fig. 150. If a compass of 6 in. be used, it will be necessary to divide the line AE into two equal parts, instead of the usual three. But if a pair of compasses of larger size be used, it will not be necessary to depart from the rule laid down. Take the compass and loosen the nut C, and by trial remove it until the distance between the legs AB is equal to the length of the standard AE (Fig. 150), and at the same time the legs ab shall span the number of sizes to be graded. The nut should then be fastened rather tightly. The legs AB are then made to span any radial line, say as GA (Fig. 150), and by using the other legs ab, the required grade is found.

* One of the points used each time must have the grade previously marked or found, otherwise any two points may be used.

Fig. 152.