the angle ADF; then apply one extreme point of the inner edge of the face-mould to the top of the plank, and bring it to that point of the arris* where the line intersects, and bring the other extreme point to the same arris; then, the under side of the face-mould coinciding with the face of the plank, draw a line by each edge of the mould. Apply the mould to the under side of the plank in the same manner: then cut away the superfluous wood on the outside of the lines drawn on the plank; which being done, apply the falling-mould to the convex side of the piece thus formed.

It is not easy to give such directions as to make a workman perfectly understand at once the application of the face-mould and falling-mould; but, by a little practice, a perfect knowledge of the terms, and ruminating upon the subject, the directions here given will conduct him through every difficulty; and, unless a person collects his ideas into a proper train, except on selfevident subjects, the plainest directions that can be written may be mistaken.

HAND-RAILS OF ELLIPTICAL STAIRS.

To find the Falling and Face-Moulds for the Hand-Rail of an Elliptical Stair, where the Steps next to the Well-hole are equally divided.

204. Let ABCD (pl. LXXXIII, No. 1) be the plan of the external side of the rail, and a bed that of the internal side; these two curve lines comprehending between them the breadth of the rail. In order to cut the rail out to the greatest advantage, or, so as to waste the least wood, it is made in three lengths AB, BC, CD; the joints being represented by Bb, Cc, Dd, the scroll and the adjoining part being formed of a separate piece.

205. The first thing to be done, as in all other cases of this kind, is, to find the falling-mould. For this purpose, let AB, No. 2, be the stretch-out of the line AB on the plan; and make AB, BE, in the proportion of any number of steps to their height, and draw the line AE. Then draw ae, at such a distance from AE, that AE, ae, may comprehend between them the thickness of the rail. In like manner, draw BC, No. 3, equal to the stretch-out of BC on the plan, and make BC to CF as the tread of any number of steps next to the well-hole is to their height, and draw bf parallel to BF, so that BF and bƒ may comprehend between them the entire thickness of wood for the rail.

Bisect AB, No. 2, in G, and draw GH perpendicular to AB, cutting AE in the point H. Bisect BC, No. 3, in the point I, and draw IJ perpendicular to BC, cutting BF in J. Divide the length of the curve AB, No. 1, into two equal parts in the point K; also divide the curve BC into two equal parts in the point L, and divide the curve CD into two equal parts in the point M; then the points a, K, B, are the three resting points for the portion of the rail over AKB; and the three resting points of the rail, over BLC, are b, L, C; and the three resting points of the rail, over CMD, are c, M, D. Now, according to the falling-mould, laid down at No. 2 and 3, the height at A, No. 2, is nothing; therefore, the height upon A, No. 1, is nothing: the height of the rail upon the point K, No. 1, is equal to the line GH, No. 2; and the height upon the point B, No. 1, is equal to the straight line BE, No. 2. Again, since in the falling-mould, No. 3, the height upon B is nothing, so the height upon B, for the rail over its plan BLC, is nothing; the height upon L is the line IJ, and the height upon C is the straight line CF. The heights upon the points c, M, D, of the rail over CMD are the same as the heights upon the points a, K, B, of the rail over AKB.

• The explanation of Arris, and other terms, will be found in the Glossary of Technical Terms at the end.

Therefore, with the heights GH, BE, No. 2, find the intersections BN, No. 1, of a plane that would pass through the point a, and through the upper extremities of the lines erected upon K and B: next find CO, the intersection of a plane that would pass through the point b, and through the upper extremities of the lines standing upon L and C, and find DP, the intersection of a plane passing through the point c, and through the lines standing upon M and D.

Draw any line, QN, in No. 1, perpendicular to BN. Draw any line, OS, perpendicular to CO, and draw any line, PU, perpendicular to DP. Draw AR perpendicular to QN, cutting QN in Q; draw BT perpendicular to OS, cutting OS in S; and draw CV perpendicular to PU, cutting PU in U. Make QR equal to BE in No. 2, ST equal to CF in No. 3, and UV equal to BE in No. 2. Join NR, OT, and PV. The moulds, No. 4, 5, and 6, being traced in the usual manner, are applied to the plank, as at No. 7, where the edge and the two adjacent sides of the plank are stretched out so as to be seen at once. The moulds are usually traced, as at No.

8; but this position will evidently give the same thing as the method here taken, exhibited in No. 4, 5, and 6.

206. To find the Face and Falling-Moulds of the Scroll of the Hand-rail.

Place the edge DH, plate LXXXIV. of the pitch-board DHI, upon the side DQ of the shank of the scroll, as exhibited in No. 1 and No. 2.

In DH take any number of points, e, f, g, &c.; and through these points draw lines, perpendicular to DH, cutting the convex edge of the scroll in h, i, j, and the concave edge in the points k, l, m, &c. Produce the lines ke, lf, mg, &c., to meet the other edge, HI, of the pitch-board in o, p, q. Draw the straight lines orv, psw, qtx, perpendicular to HI, and make the distances ov, pw, qx, &c., respectively equal to ek, fl, gm, &c.; then, through all the points, u, v, w,x, &c., draw a curve.

Again, make or, ps, qt, &c., respectively equal to eh,fi, gj, &c., and through the points n, r, s, t, &c., draw a curve, and the two curved parts of the face-mould will be completed. Draw cE, CH, and Dn, perpendicular to HD, meeting the edge HI of the pitch-board in the points E and n. Draw HG, EF, nu, perpendicular to HI. Make HG equal to HC, and draw GF parallel to HI; also make nu equal to Dd, and draw u K parallel to HI; then the whole of the face-mould will be completed. The part In u K is termed the shank of the face-mould, the same as the part PdDQ is termed the shank of the plan of the scroll. The part Ccd PQDC is got out of the plank by means of the face-mould, by drawing the pitch-line on the edge of the stuff, then applying the mould, No. 2, to both sides of the plank, so that the same point of the facemould may agree with the pitch-line, while the shank is parallel to the edge of the plank; then, the stuff being cut away, the piece, thus formed, being set up in its due position, will range to the plan.

207. The Falling Moulds for the concave and convex Sides of the Rail, are to be found as follows:

Suppose now that the rail is to have a continued fall to the point A, see the plan, No. 1. Stretch out the curve ABCD, No. 1, and draw the straight line AD, No. 3, and make AD, No. 3, equal to the curve line ABCD, No. 1. Apply the angular point of the pitch-board to some convenient point, B, in the line AD, No. 3, so that the edge may be on the line AD; then the other edge, LB, will form an angle, LBA, with AD. Draw AS perpendicular to AD, and make AS equal to the thickness of the scroll. Draw SR parallel to AD, cutting BL in R: ease off the angle LRS, by means of a parabolic curve, as shown in art. 190; and this being done, will form the upper edge of the falling-mould, for the convex side of the rail: then, through the point A, draw a curve line parallel to the upper curve; and thus the whole of the outside falling-mould will be completed.