As it is necessary the student should be acquainted with the methods of development of the angle of tangents which give shape and joints of the face moulds directly from the pitch lines, a couple of examples are herewith illustrated. Fig. 12 shows a straight pitch in which both tangents are of equal length, while Fig. 13 shows the tangents of unequal lengths and different pitches, and I advise the student to thoroughly master these two problems by frequently reproducing them, as these two examples are the very foundation of the system we are about to submit.
Stair Parts Fig. 12
A tangent is a line touching a circle at right angles to the radius as shown at Fig. 14, and is readily constructed and as easily understood.
To construct Stair Parts Fig. 12, from center 0 with the radius OA, describe a quarter circle, APC; draw tangents AB and CB, join AC; through the point B draw a straight line parallel to AC; with center B, with radius BA, describe the arcs AD and CE; at the point E erect the perpendicular EF at right angles to DE to any desired height (in laying out a handrail this height will be the same as the height of the number of risers contained in the wreath); let F be the given height (this being one pitch); join FD, extend OB to G; from G draw GH at right angles to FD; make GH equal to BI. With the center H and radius DG describe arcs, cutting DF at K and L; draw HK, which are
Stair Parts Fig. 13
Tangents on the pitch, and which, when placed in position, would stand plumb over ABC
Stair Parts Fig. 14
To construct Stair Parts Fig. 13, proceed in the same manner as in Fig. 12, until the height is located. It will be noticed that in this example BG is lifted higher, making the pitch-lines and tangents FG and DG of unequal lengths. To obtain, the angle continue BG to H, making BH equal to EF; from H draw the line HJ to any distance at right angles to DG. With the center G and radius GF describe an arc cutting the line HJ at S; join SG and SD and the angle is completed.
An easy way to prove the correctness of these problems is to draw them on common thick paper or cardboard on a larger scale than shown in these diagrams; then take a knife and cut out the angle DEF, place It perpendicularly over ABC, bringing D over A and 2 over C; then cut out the angle HKL, and if drawn, correctly it will lie on the pitch-lines and fit the sides exactly.
To draw the curve line in the most practical- way take B as a center, and with radius BP describe an arc touching the curve APC in the angle ABC; from H as a center, with the same radius, describe an arc cutting HG at M; then take a thin flexible strip' of wood of an even thickness, bend it until it touches the points KLM; mark around it with a pencil, and the curve is completed, and near enough to absolute accuracy for all practical purposes. The curve so obtained in its perfection should be a portion of an ellipse, which it will be if correctly drawn,
Let- us now go back to Stair Parts Fig. 1 and describe the method for obtaining the face moulds and bevels of turnout and wreath pieces for that style of a stair.
To build these stairs correctly and with an easy, Guild carved Stair parts, baluster, spindles and newel posts graceful handrail, two or three things must be carefully observed in taking dimensions and laying down the plan. Measure the height from top of first, to top of second floor; set the rod you measure with plumb at the trimmer where the stairs land, and be sure that the lower end is level from where the stairs start. Measure the width of opening from studding to face of trimmer, also the depth of joist, that the cylinder may curve round and meet the face board level; plumb down from the header at landing, and measure back the amount of run where the stairs start; divide the height into the necessary number of risers, space off the run, making one less than in dividing the height, and also make allowance for the cylinder, landing and swell of the turnout steps. Where it is practicable make the rise seven inches or as near to it as possible, and make the tread, or step, ten inches or as near as can be, as this combination makes a very easy stair for dwellings, but of course the height of riser and width of tread will be dependent to a great extent on the surrounding conditions.