A piece of hand-railing intended for the curved part of a stairs, when properly shaped, has a twisted form, deviating widely from plane surfaces. If laid upon a table it may easily be rocked to and fro, and can be made to coincide with the surface of the table in only three points. And yet it is usual to cut such twisted pieces from ordinary parallel-faced plank; and to cut the plank in form according to a face-mould, previously formed from given dimensions obtained from the plan of the stairs. The shape of the finished wreath differs so widely from the piece when first cut from the plank as to make it appear to a novice a matter of exceeding difficulty, if not an impossibility, to design a face-mould which shall cover accurately the form of the completed wreath. But he will find, as he progresses in a study of the subject, that it is not only a possibility, but that the science has been reduced to such a system that all necessary moulds may be obtained with great facility. To attain to this proficiency, however, requires close attention and continued persistent study, yet no more than this important science deserves. The young carpenter may entertain a less worthy ambition than that of desiring to be able to form from planks of black-walnut or mahogany those pieces of hand-railing which, when secured together with rail-screws, shall, on applying them over the stairs for which they are intended, be found to fit their places exactly, and to form graceful curves at the cylinders. That railing which requires to be placed upon the stairs before cutting the joints, or which requires the curves or butt-joints to be refitted after leaving the shop, is discreditable to the workman who makes it. No true mechanic will be content until he shall be proved able to form the curves and cut the joints in the shop, and so accurately that no alteration shall be needed when the railing is brought to its place on the stairs. The science of hand-railing requires some knowledge of descriptive geometry - that branch of geometry which has for its object the solution of problems involving three dimensions by means of intersecting planes. The method of obtaining the lengths and bevils of hip and valley rafters, etc., as in Art. 233, is a practical example of descriptive geometry. The lines and angles to be developed in problems of hand-railing are to be obtained by methods dependent upon like principles.

264. - Hand-Railing: Definitions; Planes and Solids. - Preliminary to an exposition of the method for drawing the face-moulds of a hand-rail wreath, certain terms used in descriptive geometry need to be defined. Among the tools used by a carpenter are those well-known implements called planes, such as the jack-plane, fore-plane, smoothing-plane, etc. These enable the workman to straighten and smooth the faces of boards and plank, and to dress them out of wind, or so that their surfaces shall be true and unwinding. The term plane, as used in descriptive geometry, however, refers not to the implement aforesaid, but to the unwinding surface formed by these implements. A plane in geometry is defined to be such a surface that if any two points in it be joined by a straight line, this line will be in contact with the surface at every point in its length. With like results lines may be drawn in all possible directions upon such a surface. This can be done only upon an unwinding surface; therefore, a plane is an unwinding surface. Planes are understood to be unlimited in their extent, and to pass freely through other planes encountered.

The science of stair-building has to do with prisms and cylinders, examples of which are shown in Figs. 136, 137, and 138. A right prism (Figs. 136 and 137) is a solid standing upon a horizontal plane, and with faces each of which is a plane. Two of these faces - top and bottom - are horizontal and are equal polygons, having their corresponding sides parallel.

The other faces of the prism are parallelograms, each of which is a vertical plane. When the vertical sides of a prism are of equal width, and in number increased indefinitely, the two polygonal faces of the prism do not differ essentially from circles, and thence the prism becomes a cylinder. Thus a right cylinder may be defined to be a prism, with circles for the horizontal faces (Fig. 138).

Fig. 136.

Fig. 137.

Fig. 138.

265. - Hand - Railing: Preliminary Considerations. - If within the well-hole, or stair-opening, of a circular stairs a solid cylinder be constructed of such diameter as shall fill the well-hole completely, touching the hand-railing at all points, and then if the top of this Cylinder be cut off on a line with the top of the hand-railing, the upper end of the cylinder would present a winding surface. But if, instead of cutting the cylinder as suggested, it be cut by several planes, each of which shall extend so as to cover only one of the wreaths of the railing, and be so inclined as to touch its top in three points, then the form of each of these planes, at its intersection with the vertical sides of the cylinder, would present the shape of the concave edge of the face-mould for that particular piece of hand - railing covered by the plane. Again, if a hollow cylinder be constructed so as to be in contact with the outer edge of the hand-railing throughout its length, and this cylinder be also cut by the aforesaid planes, then each of said planes at its intersection with this latter cylinder would present the form of the convex edge of the said face-mould. A plank of proper thickness may now have marked upon it the shape of this face-mould, and the piece covered by the face-mould, when cut from the plank, will evidently contain a wreath like that over which the face-mould was formed, and which, by cutting away the surplus material above and below, may be gradually wrought into the graceful form of the required wreath.

By the considerations here presented some general idea may be had of the method pursued, by which the form of a face-mould for hand-railing is obtained. A little reflection upon what has been advanced will show that the problem to be solved is to pass a plane obliquely through a cylinder at certain given points, and find its shape at its intersection with the vertical surface of the cylinder. Peter Nicholson was the first to show how this might be done, and for the invention was rewarded, by a scientific society of London, with a gold medal. Other writers have suggested some slight improvements on Nicholson's methods. The method to which preference is now given, for its simplicity of working and certainty of results, is that which deals with the tangents to the curves, instead of with the curves themselves; so we do not pass a plane through a cylinder, but through a prism the vertical sides of which are tangent to the cylinder, and contain the controlling tangents of the face-moulds. The task, therefore, is confined principally to finding the tangents upon the face-mould. This accomplished, the rest is easy, as will be seen.

The method by which is found the form of the top of a prism cut by an oblique plane will now be shown.