draw also the line ns D; bisect s D with a line at right angles, and meeting the line CD in g; draw the line gq, make CP equal to Ck, and draw the line g Pi; then from g as a centre with the radius g D, describe the arc s Di, and from k and P as centres, with the radius Ak, describe the arcs As and Bi, which completes the arch as required. Or, A B 14. Bisect the chord A B, and fix at right angles any straight guide, as be; prepare of any suitable material a rod or staff, equal to half the chord's length, as def; from the end of the staff, equal to the height of the arch, fix a pin e, and at the extremity a tracer f; move the staff, keeping its end d e to the guide and the fixed pin to the chord, and the tracer will describe one-half the arc required. 15. To describe a parabola, the dimensions being given. Let A B equal the length, and CD the breadth of the required parabola; divide CA, CB into any number of equal parts, also divide the perpendiculars A a and Bb into the same number of equal parts; then from a and b draw lines meeting each division on the line AC B, and a curve line drawn through each intersection will form the parabola required. 16. To obtain by measurement the length of any direct line, though intercepted by some material object. Suppose the distance between A and B is re A B quired, but the right line is intercepted by the object C. On the point d, with any convenient radius, describe the arc cc, make the arc twice the radius in length, through which draw the line dce, and on e describe another arc equal in length to once the radius, as eff; draw the line efr equal to efd; on describe the arc jj, in length twice the radius; continue the line through rj, which will be a right line, and de, or er, equal the distance between dr, by which the distance between A and B is obtained as required. 17. To ascertain the distance geometrically, of any inaccessible object on an equal plane. Let it be required to find the distance between A and BA, being inaccessible; produce the line in the direction of AB to any point, as D; draw the line Dd at any angle to the line A B D 18. Otherwise. Prolong A B to any point D, making BC equal to CD; draw the line Da at any angle with D A, and the line Cb similar to Bc; draw also the line DEF, which intersects the line Da, then ab equal BA, or the distance required. B E C a 19. To measure the distance between two objects, both being inaccessible. From any point C draw any line Cc, and bisect it in D; take any point E in the prolongation of A C, and draw the line E e, making De equal to DE; in like manner take any point F in the prolongation of B C, and make Dƒ equal to FD. Produce A D and ec till they meet in a, and also BD and fe till they meet in b; then ab equal A e F b B E a A B, or the distance between the objects as required. 20. A round piece of timber being given, out of which to cut a beam of strongest section. Divide into three equal parts any diameter in the circle, as A d, eC; from d or e, erect a perpendicular meeting the circumference of the circle, as dB; draw A B and B C, also A D equal to BC, and DC equal to A B, and the rectangle will be a section of the beam as required. 21. 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 Cd, as the proper given angle of eccentric rod with the crank; then from C as centre, describe a circle A e B d equal to the travel of the valve; draw the line ef at right angles to the line Cd, draw also the lines 1 1, and 22, 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 C1, C2, which are the angles of eccentric with the crank, for forward or backward motion, as may be required. 22. 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 & travel of valve; then from C as a centre, with a radius equal to the length of lever given, cut the line AB, 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 A B, and d B 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 the sum of the widths of the two steam openings, and the c d valve's excess of length more than just sufficient to cover the openings. 23. 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 CD A through the second point of division on the diameter e, and the line DB is one side of the polygon required. d e 24. To construct a square upon a given right line. From A and B as centres, with the radius A B, describe the arcs A cb, Bc d, and from c with an equal radius describe the circle or portion of a circle ed, AB, bc; from bd cut the circle at e and c; draw the lines Ae, Bc, also the line s t, which completes the square as required. b B B |