_To construct a square that shall be in any Proportion to a given square._--Let A B C D (Fig. 18) be the given square. It is required to construct a square which shall be to A B C D as 2 is to 5. Upon the side A B as a diameter describe the semicircle A F B, and divide the line A B into five equal parts. At the second point of division erect the perpendicular E F and join A F; the square described upon A F will be to the given square A B C D as 2 is to 5.
_To construct, upon a given base, a Rectangle, which shall be similar to a given rectangle._--Let A E F G (Fig. 19) be the given rectangle. It is required to construct upon the base A B, one that shall be similar to A E F G. Produce A E and lay off the given base from A to B; draw the diagonal A G and produce it indefinitely. Erect a perpendicular to A B at B, and from the point D where it intersects the diagonal produced, draw D C perpendicular to A F produced. Then A B C D will be similar to A E F G. All rectangles having their diagonals in the same line are similar.
_To describe a regular Pentagon on a given line._--Let A B (Fig. 20) be the given line. Bisect A B at C, draw C F perpendicular to A B, and make C D equal to A B. Draw A D and produce it indefinitely; make D E equal to half A B. From A as a centre, with A E as a radius, describe an arc cutting the perpendicular C D in F; and from A F and B as centres, with radius A B, describe arcs cutting each other in G and H; join A G, B H, F G and F H; then A G F H B will be the pentagon required.
_To describe a regular Hexagon._--With a radius equal to the length of one side of the required hexagon, describe a circle (Fig. 21), and set off the same radius round the circumference of the circle, which will be thus divided into six equal parts. Join the points thus found, and the required hexagon will be completed as A B C D E F.
_To draw a Parabola, the base and height being given._--Let C A (Fig. 22) equal half the base, and C D the height. From the point D draw D E parallel and equal to A C, and from the point A draw A E parallel and equal to C D. Divide D E and A E similarly, making the end E of A E correspond to the end D of E D. Through 1, 2, &c., in DE draw 1, 1; 2, 2, &c., parallel to D C. Join D to the several points 1′, 2′, &c., in A E. The parabola will pass through the points of intersections of these lines with the verticals drawn from D E to C A.