F; join B F, which being produced will meet D a in a. Then a b is equal to A B, the distance required. 18. To measure the distance between two objects, both being inaccessible. From any point c draw any line c c, and bisect it in D; take any point E in the prolongation of A C, and draw the line E e, making D e equal to D E; in like manner take any point F in the prolongation of в C, and make D ƒ equal to F D. Produce A D and e c till they meet in a, and also B D and fc till they meet in b; then a bis A e F B E a equal to A B, or the distance between the objects as required. 19. A round piece of timber being given, out of which to cut a beam of strongest section. Divide into three equal parts any D diameter in the circle, as A d, e c; from d or e, erect a perpendicular meeting the circumference of the circle, as d B; draw A B and B C, also A D equal to в C, and D C equal to A B, and the rectangle will be a section of the beam as required. A 20. To find the proper position for an eccentric, in relation to the crank in a steam engine, the angle of eccentric rod and travel of the valve being given. Draw the right line A B, as the situation of the crank at commencement of the stroke; draw also the line c d, as the proper given angle of eccentric rod with the crank; then from cas centre, describe a circle equal to the travel of the valve; draw the line e ƒ at right angles to the line c d; draw also the lines 1 1, and 2 2, parallel to the line ef; and at a distance from ef on each side, equal to the lap and lead of the valve, draw the angular lines c 1, c 2, which are the angles of eccentric with the crank, for forward or backward motion, as may be required. 21. The throw of an eccentric, and the travel of the valve in a steam engine, also the length of one lever for communicating motion to the valve, being given, to determine the proper length for the other. On any right line, as A B, describe a circle A D, equal to the throw of eccentric and travel of valve; then from c as a centre, with a radius equal to the length of lever given, cut the line A B, as at d, on which describe a circle, equal to the throw of eccentric or travel of valve, as may be required; draw the tangents Ba, Ba, cutting each other in the line ▲ B, and d в is the length of the lever as required. Note. The throw of an eccentric is equal to the sum of twice the distance between the centres of formation and revolution, as a b, or to the degree of eccentricity it is made to describe, as c d. And The travel of a valve is equal to the sum of the widths of the two steam openings, and a b c d the valve's excess of length more than just sufficient to cover the openings. 22. To inscribe any regular polygon in a given circle. Divide any diameter, as A B, into so many equal parts as the polygon is required to have sides; from A and B as centres, with a radius equal to the diameter, describe arcs cutting each other in c; draw the line C D through e, the second point of division on the diameter, and the line D B is one side of the polygon required. e A B 23. To construct a square upon a given right line. From A and B as centres, with the radius A B, describe the arcs A c b, в c d, and from c with an equal radius describe the circle or portion of a circle eda Bbc; from b, d, cut the circle at e and c; draw the lines A e, B c, also the line s t, d b A which completes the square as required. 24. To form a square equal in area to a given triangle. Let A B C be the given triangle: let fall the perpendicular в d, and make a e half the height dв; bisect e c, t n B e A d and describe the semicircle e n c; erect the perpendicular A s, or side of the square, then A s t x is the square of equal area as required. 25. To form a square equal in area to a given rectangle. D A C e B Let the line A B equal the length and breadth of the given rectangle: bisect the line in e, and describe the semicircle A D B; then from A with the breadth, or from в with the length of the rectangle, cut the line A B at C, and erect the perpendicular c D, meeting the curve at D, and C D is equal to a side of the square required. 26. To find the length for a rectangle whose area shall be equal to that of a given square, the breadth of the rectangle being also given. parallel to D F, cutting D C produced in g, through which draw g d parallel to D E, meeting E F produced in d. EDgd is the rectangle required. 27. To bisect any given triangle. Suppose A B C the given triangle: bisect one of its sides, as A B in e, from which C describe the semicircle a ↑ B; bisect the same in r, and from B with the distance в r, cut the diameter A B in v; draw the line v y parallel to a C, which will bisect the triangle as required. 28. To describe a circle of greatest diameter in a given triangle. Bisect the angles a and B, and draw the intersecting lines A D, B D, cutting each other in D; then from D as centre, with the distance D c, which is drawn perpendicular to A B, describe the circle, c e ƒ, as required. 29. To form a rectangle of greatest surface, in a given triangle. Let A B C be the given triangle: bisect any two of its sides, as a B, B, C, in e and d; draw the line e d also at right angles with the line e d, draw the lines e P, d and epp P, is the rectangle required. p C Р d d A e B 30. To make a rectangle equal to a given triangle. Suppose A B C the given е с triangle: bisect A в in d, and A พ d a B |