This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

The principles employed in finding the lines in stairs are nearly allied to those required to find the bevels for splayed work - such as hoppers, bread-trays, etc. A method by which these may be obtained will, therefore, here be shown. In Fig. 178, a b c is the angle at which the work is splayed, and b d, on the upper edge of the board, is at right angles to a b; make the angle f g j equal to a b c, and from f draw f h parallel to e a; from b draw b o at right angles to a b; through o draw ie parallel to c b, and join e and d; then the angle a e d will be the proper bevil for the ends from the inside, or k d e from the outside. If a mitre-joint is required, set fg, the thickness of the stuff on the level, from e to m, and join m and d; then kdm will be the proper bevil for a mitre-joint. If the upper edge of the splayed work is to be bevelled, so as to be horizontal when the work is placed in its proper position, then fgj, the same as a b c, will be the proper bevel for that purpose. Suppose, therefore, that a piece indicated by the lines k g, g f, and f h were taken off; then a line drawn upon the bevelled surface from d at right angles to k d would show the true position of the joint, because it would be in the direction of the board for the other side; but a line so drawn would pass through the point 0, thus proving the principle correct. So, if a line were drawn upon the bevelled surface from d at an angle of 45 degrees to k d, it would pass through the point n.

Fig. 178.

View In The Alhambra.

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