I IAN D, a measure of' four inches.
aND-I rtoxs. Sec END-IRONS.
Hasu-1:a11., of a stair, a rail raised upon slender posts, called balusters, intended to assist persons in ascending and descending, and to protect them from falling down the well. hole.
I I the art of making hand rails by moulds, according to geometrical principles.
The art of forming hand-rails was never, before the pub lication of tne Carpenter's Guide, subjected to any certain geometrical principle. The best method then was that of tracing it, like an angle-bracket, from the rise and tread of as many steps as the rail was supposed to occupy in winders, and making the fitcc-mould of a parallel after obtaining the middle of the concave side. This is manifestly false ; and the magnitude of the error will be greater as the circumference of the arc at the plan of the rail is greater than its chord. A rail, or any portion of a rail, formed upon this principle, could never stand vertically over its plan. The method here shown is founded upon the fol lowing principles : that if a cylinder be cut in any direction except parallel to the axis or base, the section will be an ellipsis; if cut parallel to the axis, its section is a rectangle ; and if parallel to the base, its section is a circle.
Let us suppose a hollow cylinder made to a given plan, the interior will be concave, and the exterior convex ; and let this cylinder be cut by any inclined or oblique plane, the section formed will be bounded by two concentric similar ellipses ; consequently, the section will be at its greatest bread that each extremity of the greater axis, and at its least breadth at each extremity of the lesser axis. Therefore, in any quarter of the ellipsis, there will be a continued increase of breadth from the extremity of the lesser axis to that of the greater. Now a cylinder can be cut by a plane through any three points ; therefore suppose we have the height of the rail at any three points in the cylinder, and cut the cylinder through these points, the section will be a figure equal and similar to the face-mould of the rail ; and if the cylinder be cut by another plane, parallel to the section, at such a distance from it as to contain the thickness of the rail, this portion of the cylinder will represent a part of the rail with its vertical surfaces already wrought ; and if the back and lower surface of this cylindric portion be squared to vertical lines, either on the convex or concave side, through two certain parallel lines, drawn by a thin piece of wood bent upon that side ; the portion of the cylinder thus formed will represent the part of the rail intended to be made.
The principles upon which this art depends are those of cutting a right prism through any three given points in space, and of forming a development of any portion of the surface of the prism.
Thus, let the interior surface of the surrounding wall he that of an entire cylinder, the breadth of the steps divided into the frustums of equal and similar sectors, and the heights all equal, as is universally the case ; then, if an inte rior cylinder surface be erected concentric with the wall, and the ends of the steps or surfaces to be trodden upon, and the planes of the risers tending to the axis be supposed to meet the interior cylindric surface, it is evident that if the por tion of the intercepted surface contained between the indented line formed by the ends of the steps, and the circumferent line at the base be developed, or stretched out, all the points of the indented line formed by the outward or salient angles, will be In the same straight line. and all the points formed by the inward or re-entrant angles will be in another straight line. It is equally evident, that this will not only be the case with evlinders, but with cylindroids, and every other description of prisms : that is, the points of the development of the indented line will always have such a position, that two straight lines parallel to each other may be drawn through the whole number of them.