In the chapter upon terms and definitions under the word degree (def. 68) and in some of those immediately following the dimensions of the circle are described and their use explained; and in the chapter upon Drawing Tools and Materials (on page 21) the triangles or set-squares in common use are described and illustrated. As all regular polygons depend, for their construction, upon the equal division of the circle, some explanation of the application of the foregoing will serve to fix a few facts in the mind of the student and thus prepare him for the use of the set-square.

A well-known and easily demonstrated geometrical principle is that the sum of the three interior angles of a triangle is equal to two right angles, or in other words, as a right angle is one of 90 degrees, if the three angles of any triangle be added together their sum will equal 180 degrees. Hence, if one of the angles of a set-square be fixed at 90 degrees (which is done for convenience in drawing perpendicular lines) the sum of the two remaining angles must also be 90 degrees, and, if then the two other angles be made equal, each will be 45 degrees, which is the half of 90 degrees. If, however, one of the other angles is fixed at 30 (one-third of 90 degrees), the remaining angle must be 60 degrees, as 30 + 60 = 90.

By means, then, of the 45-degree and the 30 X 60-degree triangles, the draftsman has at his command the means of drawing lines at angles of 90, 60, 45 and 30 degrees, and by combination 75 degrees (45 + 30) and 15 degrees (90 - 75). With the 45-degree angle he can bisect a right angle, and with the 30 and 60-degree angles he can trisect it.

Fig. 177.   Circle Divided into Four Equal Parts by the Use of a Triangle or Set Square.

Fig. 177. - Circle Divided into Four Equal Parts by the Use of a Triangle or Set-Square.

Fig. 178.   Circle Divided into Eight Equal Parts by the Use of a 45 degree Triangle.

Fig. 178. - Circle Divided into Eight Equal Parts by the Use of a 45-degree Triangle.

Fig. 179   Circle Divided into Twelve Equal Parts by the Use of a 80 x 60 degree Triangle.

Fig. 179 - Circle Divided into Twelve Equal Parts by the Use of a 80 x 60-degree Triangle.

The pattern draftsman sometimes finds it convenient to have a set-square in which the sharpest angle is one of 22 1/2 degrees (one-half of 45) for use in drawing the octagon in a certain position which will be referred to later.

In Figs. 177, 178, 179 and 180 are illustrated the application of the foregoing, in which the circle is divided, by the use of the triangles above described, into four, eight and twelve equal parts. In Fig. 177 the horizontal division A B of the circle is drawn by means of the T-square placed against the side of the drawing board, after which one of the shorter sides of the 45-degree triangle, as A E, is placed against the blade of the T-square, and the vertical division of the circle is drawn along the other short side C E.

In Fig. 178 the vertical and horizontal divisions of the circle, A B and C D, are drawn as before, after which one of the shorter sides of the 45-degree triangle is placed against the T-square, and the long or oblique side E F is brought to the center of the circle and another division G I is drawn. By reversing the position of the triangle the last division H K is drawn, thus dividing the circle into eight equal parts.

In Fig. 179, after drawing the divisions A B and C D as before, the 30 x 60-degree triangle is placed in the position shown at A E F, and the division E N is drawn along its hypothenuse or oblique side. By reversing the position of the triangle, still keeping the side A F against the blade of the T-square, the division J K may be drawn. Changing the position of the triangle now so that its shortest side comes against the blade of the T-square, as shown dotted at G H F, the division G M is drawn, and again reversing its position, still keeping its shortest side against the T-square, the last division I L may be drawn, thus dividing the circle into twelve equal parts.

In Fig. 180 the circle is divided into eight equal parts, but differing from that shown above in this respect that, while in Fig. 178 two of the divisions lie parallel with the sides of the drawing board, in the latter case none of the divisions are parallel with the sides of the board or can be drawn with the T-square; but if this method is used in drawing an octagon, as shown dotted in Fig. 180, four of the sides of the oc tagon can be drawn with the T-square and the other four with the 45-degree triangle. The position of the 22 1/2 x 67 1/2-degree triangle in drawing the divisions of the circle is shown at A B C and DE C, while the position of the 45-degree triangle in drawing the oblique sides of an octagon figure is shown at F. It will thus be seen that the 22 1/2 X 67 1/2-degree triangle is available in drawing accurately the miter line for all octagon miters.

As a triangle in whatever form it may he constructed is intended to he used by sliding it against the blade of the T-square, all the angles above mentioned are calculated with reference to the lines drawn by the T-square. In practical use it will be found inconvenient in drawing such lines to actually bring the point of a set-square to the center of a circle. A better method, and one which makes use of the same principles, is shown in Fig. 181. The blade of the T-square is placed tangent to or near the circle, as shown by A B. One side of a 45-degree triangle is placed against it, as shown, its side C F being brought against the center. The line C F is then drawn. By reversing the triangle, as shown by the dotted lines, the line E D is drawn at right angles to C F, thus dividing the circle into quarters.