To construct a square that shall be a Multiple of any given square.
—Let A B C D ([Fig. 16]) be the given square, and let it be required to construct a square that shall contain 2, 3, 4, &c., times its surface. Draw the diagonal B C, then the square described on B C will be double the square A B C D. Lay off D E, equal to B C, and draw C E; then the square described on C E will be three times the square A B C D. In the same manner lay off D F, equal to C E, and the square described on C F will be four times the square A B C D; and so for any multiple of the square A B C D.
Fig. 17.
To construct a square that shall be equal to 1⁄2, 1⁄4, &c., of any given square.
—Let A B C D ([Fig. 17]) be the given square. On A B, as a diameter, describe the semicircle A G B, and erect at the centre E the perpendicular E G. Draw G B, which will be the side of a square equal to one-half of A B C D. Lay off B F, equal to one-fourth of A B, and erect the perpendicular F H; then the square described upon H B will be equal to one-fourth of A B C D. In the same manner a square may be constructed equal to any part of A B C D.
Fig. 18.
