s across corners of the polygon is that of the distance from O
along O B to the horizontal line, giving the length of the side of the
polygon, and the width across corners for a given length of one side of
the square, is measured by the length of the lines A, B, C, etc. Thus,
dotted line 2 shows the length of the side of a nine-sided figure, of
2-inch radius, the radius across corners of the figure being 2 inches.

[Illustration: Fig. 183.]

The dotted line 2-1/2 shows the length of the side of a nine-sided
polygon, having a radius across corners of 2-1/2 inches. The dotted line
1 shows the diameter, across corners, of a square whose sides measure an
inch, and so on.

[Illustration: Fig. 184.]

This scale lacks, however, one element, in that the diameter across the
flats of a regular polygon being given, it will not give the diameter
across the corners. This, however, we may obtain by a somewhat similar
construction. Thus, in Figure 184, draw the line O B, and divide it into
inches and parts of an inch. From these points of division draw
horizontal lines; from the point O draw the following lines and at the
following angles from the horizontal line O P.

[Illustration: Fig. 185.]

A line at 75 deg. for polygons having 12 sides.
" 72 deg. " " 10 "
" 67-1/2 deg. " " 8 "
" 60 deg. " " 6 "

From the point O to the numerals denoting the radius of the polygon is
the radius across the flats, while from point O to the horizontal line
drawn from those numerals is the radius across corners of the polygon.

[Illustration: Fig. 186.]

A hexagon measures two inches across the flats: what is its diameter
measured across the corners? Now from point O to the horizontal line
marked 1 inch, measured along the line of 60 degrees, is 1 5-32nds
inches: hence the hexagon measures twice that, or 2 5-16ths inches
across corners. The proof of the construction is shown in the figure,
the hexagon and other polygons being marked simply for clearness of
illustration.

[Illustration: Fig. 187.]

[Illustration: Fig. 188.]

Let it be required to draw the stud shown in Figure 185, and the
construction would be, for the pencil lines, as shown in Figure 186; line
1 is the centre line, arcs, 2 and 3 give the large, and arcs 4 and 5
the small diameter, to touch which lines 6, 7, 8, and 9 may be drawn.
Lines 10, 11, and 12 are then drawn for the lengths, and it remai