**ABC—XYZ**

498 [jOINEKY. spiral being given to describe the curve, let AB be the total height, and AC the intended height of the eye, and let the spiral be required to make two revolutions. Divide BC into four times as many parts as there are revolutions required (4x2 = 8), because there are four quadrants in every revolution. Draw any line DE equal to the height of the spiral. Set down from D half the number of the parts and one more (4 + 1 = 5); this is the top of the eye. Set down half AC at O, and describe the eye ; then at set up half a part to F, and make FG, FH = OF; then, as in fig 105, draw OG, OH, GI, and from draw a line parallel to GH, and divide the same into as many parts as there are to be revolutions. Fig. 105 is for one, and fig. 106 for two revolutions. Bisect Ol at X ; make 21 = 2X, and join XI ; and through 1 draw MN parallel to OF to meet OG and XI. Draw the quarter circles, as in the diagram, HD being the first opening of the compasses, GP the next, and H, G, I, K, L, M, and N being the centres. To describe the scroll let AB, as in fig. 107, be the width across, usually about 10 or 12 inches ; let EB be the intended diameter of the eye; and let the scroll be required to make one revolution and a half, or six quadrants (these are shown at greater size by the side of fig. 108), then proceed as last directed, and complete the scroll, also dot in the lines of the nosings and risers. For the curtail step transfer the lines of nosings a, and the lines of the risers b, to another place, as in fig. 108, and set out the thick- ness of the veneer within the line of nos- 107, 108 illustrate the formation of Scro11 and CurtaU Step " ing, the part within this represents the solid block of the curtail. The places of the balusters are shown in fig. 107. It is obvious that in every geometrical staircase, the half of a cylinder placed upright in the well-hole would touch the wreathed string K in all parts, another a little less would touch all parts of the hand-rail. Let us suppose ACB, as in fig. 109, to be the plan of half a cylinder so set up right in the well- hole, and let us sup pose A E to be the height of the same, Divide the curved line ACB into any convenient number Fia ^-Construction of Hand-Bail. of parts, and set the same off by compasses on the straight line from C to A and C to B . Or, in case ACB is a semicircle, divide the line AB, draw the diameter CD, making aD equal to three-fourths of the radius, and draw DA, DB , and the rest of the lines through the points of division, as shown in the diagram. Then A B is the stretch out or length of the circumference ACB unrolled, But A E is said to be the whole height. From E set down the respective heights of the winders, step by step, as shown. Now let G be the representation of the cylinder, with tho different lines squared up and across, these will give a representation of the curve at which the winders must ascend, and which, of course, must regulate the hand-rail. The other faint lines show the edge of the covering. Fig. 112 represents the plan of a staircase, beginning with a scroll, and having steps winding round the circular part of the well-hole. In the first place, let the end of the steps be developed according to the method we have just given (fig. 110 shows this development). Now the hand-rail ought to follow the inclination of a line drawn to touch the nosings of the steps, except where there is an abrupt Fig. .110. Fig. 111. Fig. 112. FIGS. 110-112. Development of Circular Hand-Rail. transition from the rake of the winding to that of the other steps ; at such places it must be curved, the curve may be drawn by the help of intersecting lines, as in fig. 111. The part which is shaded in fig. 109 represents the hand rail and ends of the steps when spread out, and the hand rail is only drawn close to the steps for convenience, as it would require too much space to raise it to its proper position. This development of the rail is called the falling- mould. We will now refer to fig. 113, and will suppose the inner semicircle of ACB to be the plan of the well- hole, and <?A, B, the width of the rail ; then the outer shaded part ACB will be the plan of the rail on the level ; ADEB is the cy linder referred to before ADE being the angle at which the stairs ascend. Now since by the principles of Conic Sections the ob lique section of a circular cylinder is an ellipse, if the cylinder be circular the lines may then be found FlG. 113. Tracing the Face-Mould of Rail. by a trammel. Be it of what section it may, the delineation of a cylinder cut at any angle ADE may be found by dividing it into equal parts, and setting up the ordinates ol, &2, &c., as shown. This delineation is a plan " on the oblique," or the face- mould of the rail, to be cut " on the plumb." The wood used for hand-rails being of an expensive kind, it becomes of some importance to consider how the plank may be cut so as to require the least quantity of

material for the curved part of the rail. Now, if we were