Fig. 140. - To Divide a Given Straight Line into Any Number of Equal Parts

11. To Divide a Straight Line Into Any Number of Equal Parts by Means of a Scale. - It may be more convenient to transfer the length of a given line to a slip of paper, and by laying the paper across a scale, as shown in lug. 141, mark the required dimensions upon it, and afterward transfer them to a given line, than to divide the line itself by one of the methods explained for that purpose. It may also occur that it is desirable to divide lines of different lengths into the same number of equal parts, or the same lengths of lines into different numbers of equal parts. Such a scale as is shown in Fig. 142 is adapted to all of these purposes. The scale may be ruled upon a piece of paper or upon a sheet of metal, as is preferred. The lines may be all of one color, or two or more colors may he alternated, in order to facilitate counting the lines or following them by the eye across the sheet. In size, the scale should be adapted to the special purposes for which it is intended to be used. By the contrast of two colors in ruling the lines, one scale may be adapted to both coarse and fine work. For instance, if the lines are ruled a quarter of an inch apart, in colors alternating red and blue, in fine work all the lines in a given space may be used, while in large work, in which the dimensions are not required to be so small, either all the red or all the blue lines may be used, to the exclusion of those of the other color. Let it be required to divide the line A B in Fig. 141 into thirty equal parts. Transfer the length A B to one edge of a slip of paper, as shown by A' B', and placing A1 against the first line of the scale, carry B1 to the thirtieth line. Then mark divisions upon the edge of the strip of paper opposite each of the several lines it crosses, as shown. Let it be required to divide the same length A B into fifteen equal parts by the scale. Transfer the length A B to a straight strip of paper, as before. Place A' against the first line and carry B2 against the fifteenth line, as shown. Then mark divisions upon the edge of the paper opposite each line of the scale, as shown.

Fig. 141. - To Divide a Straight Line into Any Number of Equal Parts by Means of a Scale.

12. To Divide a Given Angle into Two Equal Parts. - In Fig. 142, let A C B represent any angle which it. is required to bisect. From the vertex, or point C, as center, with any convenient radius, strike the arc D E, cutting the two sides of the angle. From D and E as centers, with any radius greater than one-half the length of the are D E, strike short arcs intersecting at G, as shown. Through the point of intersection, G, draw a line to the vertex of the angle, as shown by F C. Then F C will divide the angle into two equal parts.

Fig. 142. - To Bisect a Given Angle.

13. To Trisect an Angle. - No strictly geometrical method of solving this problem has ever been discovered. The following method, partly geometrical and partly mechanical, is, however, perfectly accurate and can be used to advantage whenever it becomes necessary to find an exact one-third or two-thirds of an angle:

Let ABC, Fig. 143, be the angle of which it is required to find one-third. Extend one of its sides beyond the vertex indefinitely, as shown by B E, and upon this line from B as center with any convenient radius describe a semicircle A C D, cutting both sides of the angle. Place a straight edge firmly against the extended side as at F, and a pin at the point C. On another straight edge (G) having a perfect corner at E, set off from one end a distance equal to the radius of the semicircle as shown by point x; and placing this straight edge, with the end upon which the radius was set off, against the other straight edge (F) and its edge near the other end, against the pin at the point C, all as shown, slide it along until the mark x comes to the semicircle establishing the point D. Draw the line D B, then the angle D E B will be one-third of the angle ABC, and C D B will he two-thirds of it.

Fig. 143. - To Trisect a Given Angle.

14. To Find the Center from which a Given Arc is Struck. - In fig. 144, let A B C represent the given are, the center from which it was struck being unknown and to be found. From any point near the middle of the arc, as B, with any convenient radius, strike the arc F G, as shown. Then from the points A and C, with the same radius, strike the intersecting arcs I H and E D. Through the points of intersection draw the lines K M and L M, which will meet in M. Then M is the center from which the given are was struck. Instead of the points A and C being taken at the extremities of the arc, which would be quite inconvenient in the case of a long arc, these points may he located in any part of the arc which is most convenient. The greater the distance between A and B and B and C, the greater will be the accu-racy of succeeding operations. The essential feature of this rule is to strike an arc from the middle one of the points, and then strike intersecting ares from the other two points, using the same radius. It is not necessary that the distance from A to B and from B to C shall be exactly the same.

Fig. 144. - To Find the Center from which a Given Arc is Struck.