a succession of grooves 14 inches apart, and parallel with the lines for the risers on the face. These grooves must be tut along the whole length of the plank, and deep enough to admit of the, plank's bending around the curve, a b c d. Then construct a drum, or cylinder, of any common kind of stuff, and made to fit a curve having a radius the thickness of the string less than o a; upon this the string must be bent, and the grooves filled with strips of wood, called keys, which must be very nicely fitted and glued in. After it has dried, a board thin enough to bend around on the outside of the string, must be glued on from one end to the other and nailed with clout nails. In doing this, be careful not to nail into any place where a riser or step is to enter on the face.

After the string has been on the drum a sufficient time for the glue to set, take it off, and cut the mortices for the steps and risers on the face at the lines previously made; which may be done by boring with a centre-bit half through the string, and nicely chiseling to the line. The drum need not be made so large as the whole space occupied by the stairs, but merely large enough to receive one piece of the wall-string at once-for it is evident that more than one will be required. The front string may be constructed in the same manner; taking e l instead of a k for the tread of the pitch-board, dadoing it with a smaller dado plane, and bending it on a drum of the proper size.

471.---To find the shape and position of the timbers necessary to support a winding stairs. The dotted lines in Fig. 303 show the proper position of the timbers as regards the plan : the shape of each is obtained as follows. In Fig. 304, the line, 1 a, is equal to a riser, less the thickness of the floor, and the ines, 2 m, 3 n, 4 o, 5 p and 6 q, are each equal to one riser. The

line, a 2, is equal to a m in Fig. 303, the line, m 3 to m n in that figure, &c. In drawing this figure, commence at a, and make

the lines, a 1 and a 2, of the length above specified, and draw them at right angles to each other; draw 2 m, at right angles to a 2, and m 3, at right angles to m 2, and make 2 m and m 3 of the lengths as above specified; and so proceed to the end. Then, through the points, 1, 2, 3, 4, 5 and 6, trace the line, 1 b; upon the points, 1, 2, 3, 4, &c., with the size of the timber for radius, describe arcs as shown in the figure, and by these the lower line may be traced narallel to the upper. This will give the proper shape for the timber, a b, in Fig. 303 ; and that of the others may be found in the same manner. In ordinary cases, the shape of one face of the timber will be sufficient, for a good workman can easily hew it to its proper level by that; but where great accuracy is desirable, a pattern for the other side may be found in the same manner as for the first.

472.-To find the falling-mould for the rail of a winding stairs. In Fig. 305, a cb represents the plan of a rail around half the cylinder, A the cap of the newel, and 1, 2, 3, &c., the face of the risers in the order they ascend. Find the stretch-out, ef, of a c b, according to Art. 92; from o, through the point of the mitre at the newel-cap, draw o s; obtain on the tangent, e d, the position of the points, s and h3⁄4,* as at t and ƒ2; from e t f2 and f, draw e x, tu, fg and f h, all at right angles to e d; make e g equal to one rise and f2 g2 equal to 12, as this line is drawn from the 12th riser; from g, through g2, drawg i; make g x equal to about three-fourths of a rise, (the top of the newel, r, should be 3 feet from the floor;) draw x u, at right angles to ex, and ease off the angle at u; at a distance equal to the thickness of

In the above, the references, a2, b2, &c., are introduced for the first time. During the time taken to refer to the figure, the memory of the form of these may pass from the mind, while that of the sound alone remains; they may then be mistaken for a 2, b 2, &c. This Dan be avoided in reading by giving them a so. nd corresponding to their meaning, which I second a second b, &c. or a second, b second.

the rail, draw v w y, parallel to rui; from the centre of the plan, o, draw o l, at right angles to e d; bisect hn in p, and through p, at right angles to g i, draw a line for the joint; in the same manner, draw the joint at k; then x y will be the falling-mould for that part of the rail which extends from s to b on the plan.

473. To find the face-mould for the rail of a winding-stairs. From the extremities of the joints in the falling-mould, as k, z and y, (Fig. 305,) draw k a2, z b2 and y d, at right angles to e d; make be equal to f d. Then, to obtain the direction of the joint, a c2, or b'd, proceed as at Fig 306, at which the parts are

shown at half their full size. A is the plan of the rail, and B is the falling-mould; in which k z is the direction of the butt-joint. From k, draw k b, parallel to lo, and k e, at right angles to k b: from b, draw bf, tending to the centre of the plan, and from ƒ, draw f e, parallel to bk; from l, through e draw li, and from i, drawi d, parallel to ef; join d and b, and db will be the proper direction

for the joint or the plan. The direction of the joint on the other side, a c, can be found by transferring the distancës, x b and o d to xa and o c. (See Art. 477.)

Having obtained the direction of the joint, make s r d b, (Fig. 307,) equal to s r d b' in Fig. 305; through r and d, draw ta, throughs and from d, draw t u and d e, at right angles to ta; make tu and d e equal to t u and b2 m, respectively, in Fig. 305; from u, through e, draw u o; through b, from r, and from as many other points in the line, t a, as is thought necessary, as f, h and j draw the ordinates, r c, fg, hi, j k and a o; from u, c, g, i, k, e and o, draw the ordinates, u 1, c 2, g 3, i 4, k 5, e 6 and o 7, at right angles to u o; make u 1 equal to t s, c 2 equal to r 2, g 3 equal to f 3, &c., and trace the curve, 17, through the points thus found; find the curve, c e, in the same manner, by transferring the distances between the line, t a, and the arc, r d; join 1 and c, also e and 7; then, 1 c e 7 will be the face-mould required for that part of the rail which is denoted by the letters, s r ď2 12, on the plan at Fig. 305.

To ascertain the mould for the next quarter, make a c je, (Fig