Fig. 117. Upper Tangent Inclined, Lower Tangent Level, Over Acute-Angle Plan.

## Fourth Case

In Fig. 113 is shown how to find the bevels for a wreath when the upper tangent inclines less than the bottom tangent. This example is the reverse of the preceding one; it is the condition of tangents found in the bottom piece of wreath shown in Fig. 95. To find the bevel, continue the upper tangent b" to the ground line, as shown at n; connect n to a, which will be the horizontal trace of the plane. From o, draw a line parallel to n a, as shown from o to d; upon d, erect a perpendicular line to cut the continued portion of the upper tangent b" in m; from m, draw the line m u o" across as shown. Now place the dividers on u; extend to touch the upper tangent, and turn over to 1, connect 1 to o"; the bevel at 1 will be the one to apply to the tangent b" at h, where the two wreaths are shown connected in Fig. 95. Again place the dividers on u; extend to touch the line c; turn over to 2; connect 2 to o"; the bevel at 2 is to be applied to the bottom tangent a" at the joint where it is shown to connect with the rail of the flight.

## Fifth Case

In this case we have two equally inclined tangents over an obtuse-angle plan. In Fig. 102 is shown a plan of this kind; and in Fig. 103, the development of the face-mould.

In Fig. 114 is shown how to find the bevel. From a, draw a line to a', square to the ground line. Place the dividers on a'; extend to touch the pitch of tangents, and turn over as shown to m; connect m to a. The bevel at m will be the only one required for this wreath, but it will have to be applied to both ends, owing to the two tangents being inclined.

Fig. 118. Finding Bevels for Wreath of Plan, Fig. 117.

## Sixth Case

In this case we have one tangent inclining and one tangent level, over an acute-angle plan.

In Fig. 115 is shown the same plan as in Fig. 114; but in this case the bottom tangent a" is to be a level tangent. Probably this condition is the most commonly met with in wreath construction at the present time. A small curve is considered to add to the appearance of the stair and rail; and consequently it has become almost a "fad" to have a little curve or stretch-out at the bottom of the stairway, and in most cases the rail is ramped to intersect the newel at right angles instead of at the pitch of the flight. In such a case, the bottom tangent a" will have to be a level tangent, as shown at a" in Fig. 115, the pitch of the flight being over the plan tangent b only.

Fig. 119. Laying Out Curves on Face-Mould with Pins and String.

To find the bevels when tangent b" inclines and tangent a" is level, make a c in Fig. 116 equal to a c in Fig. 115. This line will be the base of the two bevels. Upon a, erect the line a w m at right angles to a c; make a w equal to o w in Fig. 115; connect w and c; the bevel at w will be the one to apply to tangent b" at n where the wreath is joined to the rail of the flight. Again, make a m in Fig. 116 equal the distance shown in Fig. 115 between w and m, which is the full height over which tangent b" is inclined; connect m to c in Fig. 116, and at m is the bevel to be applied to the level tangent a".

## Seventh Case

In this case, illustrated in Fig. 117, the upper tangent b" is shown to incline, and the bottom tangent a" to be level, over an acute - angle plan. The plan here is the same as that in Fig. 100, where a curve is shown to stretch out from the line of the straight stringer at the bottom of a flight to a newel, and is large enough to contain five treads, which are gracefully rounded to cut the curve of the central line of rail in 1, 2, 3, 4. This curve also may be used to connect a landing rail to a flight, either at top or bottom, when the plan is acute-angled, as will be shown further on.

Fig. 120. Simple Method of Drawing Curves on Face-Mould.

Fig. 121. Tangents, Bevels, Mould-Curves, etc., from Bottom Wreath of Fig. 95, in which Upper Tangent Inclines Less than Lower One.

RESIDENCE FOR MR. HANS HOFFMAN, MILWAUKEE, WIS.

Fernekes & Cramer, Architects.

To find the bevels - for there will be two bevels necessary for this wreath, owing to one tangent b" being inclined and the other tangent a" being level - make a c, Fig. 118, equal to a c in Fig. 117, which is a line drawn square to the ground line from the newel and shown in all preceding figures to have been used for the base of a triangle containing the bevel. Make a w in Fig. 118 equal to w o in Fig. 117, which is a line drawn square to the inclined tangent b" from w; connect w and c in Fig. 118. The bevel shown at w will be the one to be applied to the joint 5 on tangent b", Fig. 117. Again, make am in Fig. 118 equal to the distance shown in Fig. 117 between the line representing the level tangent and the line m' 5, which is the height that tangent b" is shown to rise; connect m to c in Fig. 118; the bevel shown at m is to be applied to the end that intersects with the newel as shown at m in Fig. 117.

Fig. 122. Developed Section of Plane Inclining Unequally in Two Directions.

Fig 123. Arranging Risers around Well-Hole on Level-Landing Stair, with Radius of Central Line of Rail One-Half Width of Tread.

The wreath is shown developed in Fig. 101 for this case; so that, with Fig. 100 for plan, Fig. 101 for the development of the wreath, and Figs. 117 and 118 for finding the bevels, the method of handling any similar case in practical work can be found.