Fig. 100 is a plan of a few curved steps placed at the bottom of a stairway with a curved stringer, which is struck from a center o. The plan tangents a and b are shown to form an acute angle with each other. The rail above a plan of this design is usually ramped at the bottom end, where it intersects the newel post, and, when so treated, the bottom tangent a will have to be level.

In Fig. 101 is shown how to find the angle between the tangents on the face-mould that gives them the correct direction for squaring the joints of the wreath when it is determined to have it ramped. This figure must be drawn full size. Usually an ordinary drawing-board will answer the purpose. Upon the board, reproduce the plan of the tangents and curve of the center line of rail as shown in Fig. 100. Measure the height of 5 risers, as shown in Fig. 101, from the floor line to 5; and draw the pitch of the flight adjoining the wreath, from 5 to the floor line. From the newel, draw the dotted line to w, square to the floor line; from w, draw the line w m, square to the pitch-line b″. Now take the length of the bottom level tangent on a trammel, or on dividers if large enough, and extend it from n to m, cutting the line drawn previously from w, at m. Connect m to n as shown by the line a″. The intersection of this line with b″ determines the angle between the two tangents a″ and b″ of the face-mould, which gives them the correct direction as required on the face-mould for squaring the joints. The joint at m is made square to tangent a″; and the joint at 5, to tangent b″.

In Fig. 102 is presented an example of a few steps at the bottom of a stairway in which the tangents of the plan form an obtuse angle with each other. The curve of the central line of the rail in this case will be less than a quadrant, and, as shown, is struck from the center o, the curve covering the three first steps from the newel to the springing.

In Fig. 103 is shown how to develop the tangents of the face-mould. Reproduce the tangents and curve of the plan in full size. Fix point 3 at a height equal to 3 risers from the floor line; at this point place the pitch-board of the flight to determine the pitch over the curve as shown from 3 through b″ to the floor line. From the newel, draw a line to w, square to the floor line; and from w, square to the pitch-line b″, draw the line w m; connect m to n. This last line is the development of the bottom plan tangent a; and the line b″ is the development of the plan tangent b; and the angle between the two lines a″ and b″ will give each line its true direction as required on the face-mould for squaring the joints of the wreath, as shown at m to connect square with the newel, and at 3 to connect square to the rail of the connecting flight.

The wreath in this example follows the nosing line of the steps without being ramped as it was in the examples shown in Figs. 100 and 101. In those figures the bottom tangent a was level, while in Fig. 103 it inclines equal to the pitch of the upper tangent b″ and of the flight adjoining. In other words, the method shown in Fig. 101 is applied to a construction in which the wreath is ramped; while in Fig. 103 the method is applicable to a wreath following the nosing line all along the curve to the newel.

The stair-builder is supposed to know how to construct a wreath under both conditions, as the conditions are usually determined by the Architect.

The foregoing examples cover all conditions of tangents that are likely to turn up in practice, and, if clearly understood, will enable the student to lay out the face-moulds for all kinds of curves.

The wreath is first cut from the plank square to its surface as shown in Fig. 104. After the application of the bevels, it is twisted, as shown in Fig. 105, ready to be moulded; and when in position, ascending from one end of the curve to the other end, over the inclined plane of the section around the well-hole, its sides will be plumb, as shown in Fig. 106 at b. In this figure, as also in Fig. 105, the wreath a lies in a horizontal position in which its sides appear to be out of plumb as much as the bevels are out of plumb. In the upper part of the figure, the wreath b is shown placed in its position upon the plane of the section, where its sides are seen to be plumb. It is evident, as shown in the relative position of the wreath in this figure, that, if the bevel is the correct angle of the plane of the section whereon the wreath b rests in its ascent over the well-hole, the wreath will in that case have its sides plumb all along when in position. It is for this purpose that the bevels are needed.

First Case. In Fig. 107 is shown a case where the bottom tangent of a wreath is inclining, and the top one level, similar to the top wreath shown in Fig. 98. It has already been noted that the plane of the section for this kind of wreath inclines to one side only; therefore one bevel only will be required to square it, which is shown at d, Fig. 107. A view of this plane is given in Fig. 108; and the bevel d, as there shown, indicates the angle of the inclination, which also is the bevel required to square the end d of the wreath. The bevel is shown applied to the end of the landing rail in exactly the same manner in which it is to be applied to the end of the wreath. The true bevel for this wreath is found at the upper angle of the pitch-board. At the end a, as already stated, no bevel is required, owing to the plane inclining in one direction only. Fig. 109 shows a face-mould and bevel for a wreath with the bottom tangent level and the top tangent inclining, such as the piece at the bottom connecting with the landing rail in Fig. 94.

Second Case. It may be required to find the bevels for a wreath having two equally inclined tangents. An example of this kind also is shown in Fig. 94, where both the tangents c″ and d″ of the upper wreath incline equally. Two bevels are required in this case, because the plane of the section is inclined in two directions; but, owing to the inclinations being alike, it follows that the two will be the same. They are to be applied to both ends of the wreath, and, as shown in Fig. 105, in the same direction—namely, toward the inside of the wreath for the bottom end, and toward the outside for the upper end.

In Fig. 110 the method of finding the bevels is shown. A line is drawn from w to c″, square to the pitch of the tangents, and turned over to the ground line at h, which point is connected to a as shown. The bevel is at h. To show that equal tangents have equal bevels, the line m is drawn, having the same inclination as the bottom tangent c″, but in another direction. Place the dividers on o′, and turn to touch the lines d″ and m, as shown by the semicircle. The line from o′ to n is equal to the side plan tangent w a, and both the bevels here shown are equal to the one already found. They represent the angle of inclination of the plane whereon the wreath ascends, a view of which is given in Fig. 111, where the plane is shown to incline equally in two directions. At both ends is shown a section of a rail; and the bevels are applied to show how, by means of them, the wreath is squared or twisted when winding around the well-hole and ascending upon the plane of the section. The view given in this figure will enable the student to understand the nature of the bevels found in Fig. 110 for a wreath having two equally inclined tangents; also for all other wreaths of equally inclined tangents, in that every wreath in such case is assumed to rest upon an inclined plane in its ascent over the well-hole, the bevel in every case being the angle of the inclined plane.

Third Case. In this example, two unequal tangents are given, the upper tangent inclining more than the bottom one. The method shown in Fig. 110 to find the bevels for a wreath with two equal tangents, is applicable to all conditions of variation in the inclination of the tangents. In Fig. 112 is shown a case where the upper tangent d″ inclines more than the bottom one c″. The method in all cases is to continue the line of the upper tangent d″, Fig. 112, to the ground line as shown at n; from n, draw a line to a, which will be the horizontal trace of the plane. Now, from o, draw a line parallel to a n, as shown from o to d, upon d, erect a perpendicular line to cut the tangent d″, as shown, at m; and draw the line m u o″. Make u o″ equal to the length of the plan tangent as shown by the arc from o. Put one leg of the dividers on u; extend to touch the upper tangent d″, and turn over to 1; connect 1 to o″; the bevel at 1 is to be applied to tangent d″. Again place the dividers on u; extend to the line h, and turn over to 2 as shown; connect 2 to o″, and the bevel shown at 2 will be the one to apply to the bottom tangent c″. It will be observed that the line h represents the bottom tangent. It is the same length and has the same inclination. An example of this kind of wreath was shown in Fig. 95, where the upper tangent d″ is shown to incline more than the bottom tangent c″ in the top piece extending from h″ to 5. Bevel 1, found in Fig. 112, is the real bevel for the end 5; and bevel 2, for the end h″ of the wreath shown from h″ to 5 in Fig. 95.

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.

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.

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.

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 a m 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.

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.

It has been shown how to find the angle between the tangents of the face-mould, and that the angle is for the purpose of squaring the joints at the ends of the wreath. In Fig. 119 is shown how to lay out the curves by means of pins and a string—a very common practice among stair-builders. In this example the face-mould has equal tangents as shown at c″ and d″. The angle between the two tangents is shown at m as it will be required on the face-mould. In this figure a line is drawn from m parallel to the line drawn from h, which is marked in the diagram as “Directing Ordinate of Section.” The line drawn from m will contain the minor axes; and a line drawn through the corner of the section at 3 will contain the major axes of the ellipses that will constitute the curves of the mould.

The major is to be drawn square to the minor, as shown. Place, from point 3, the circle shown on the minor, at the same distance as the circle in the plan is fixed from the point o. The diameter of this circle indicates the width of the curve at this point The width at each end is determined by the bevels. The distance a b, as shown upon the long edge of the bevel, is equal to ½ the width of the mould, and is the hypotenuse of a right-angled triangle whose base is ½ the width of the rail. By placing this dimension on each side of n, as shown at b and b, and on each side of h″ on the other end of the mould, as shown also at b and b, we obtain the points b 2 b on the inside of the curve, and the points b 1 b on the outside. It will now be necessary to find the elliptical curves that will contain these points; and before this can be done, the exact length of the minor and major axes respectively must be determined. The length of the minor axis for the inside curve will be the distance shown from 3 to 2; and its length for the outside will be the distance shown from 3 to 1.

To find the length of the major axis for the inside, take the length of half the minor for the inside on the dividers: place one leg on b, extend to cut the major in z, continue to the minor as shown at k. The distance from b to k will be the length of the semi-major axis for the inside curve.

To draw the curve, the points or foci where the pins are to be fixed must be found on the major axis. To find these points, take the length of b k (which is, as previously found, the exact length of the semi-major for the inside curve) on the dividers; fix one leg at 2, and describe the arc Y, cutting the major where the pins are shown fixed, at o and o. Now take a piece of string long enough to form a loop around the two and extending, when tight, to 2, where the pencil is placed; and, keeping the string tight, sweep the curve from b to b.

To draw the curves of the mould according to this method, which is a scientific one, may seem a complicated problem; but once it is understood, it becomes very simple. A simpler way to draw them, however, is shown in Fig. 120.

The width on the minor and at each end will have to be determined by the method just explained in connection with Fig. 119. In Fig. 120, the points b at the ends, and the points in which the circumference of the circle cuts the minor axis, will be points contained in the curves, as already explained. Now take a flexible lath; bend it to touch b, z, and b for the inside curve, and b, w, and b for the outside curve. This method is handy where the curve is comparatively flat, as in the example here shown; but where the mould has a sharp curvature, as in case of the one shown in Fig. 101, the method shown in Fig. 119 must be adhered to.

The mould shown in these two diagrams, Figs. 119 and 120, is for the upper wreath, extending from h to n in Fig. 94. A practical handrailer would draw only what is shown in Fig. 120. He would take the lengths of tangents from Fig. 94, and place them as shown at h m and m n. By comparing Fig. 120 with the tangents of the upper wreath in Fig. 94, it will be easy for the student to understand the remaining lines shown in Fig. 120. The bevels are shown applied to the mould in Fig. 105, to give it the twist. In Fig. 106, is shown how, after the rail is twisted and placed in position over and above the quadrant c d in Fig. 94, its sides will be plumb.

In Fig. 121 are shown the tangents taken from the bottom wreath in Fig. 95. It was shown how to develop the section and find the angle for the tangents in the face-mould, in Fig. 113. The method shown in Fig. 119 for putting on the curves, would be the most suitable.

Fig. 121 is presented more for the purposes of study than as a method of construction. It contains all the lines made use of to find the developed section of a plane inclining unequally in two different directions, as shown in Fig. 122.

In level-landing stairways, the easiest example is the one shown in Fig. 123, in which the radius of the central line of rail is made equal to one-half the width of a tread. In the diagram the radius is shown to be 5 inches, and the treads 10 inches. The risers are placed in the springing, as at a and a. The elevation of the tangents by this arrangement will be, as shown, one level and one inclined, for each piece of wreath. When in this position, there is no trouble in finding the angle of the tangent as required on the face-mould, owing to that angle, as in every such case, being a right angle, as shown at w; also no special bevel will have to be found, because the upper bevel of the pitch-board contains the angle required.

The same results are obtained in the example shown in Fig. 124, in which the radius of the well-hole is larger than half the width of a tread, by placing the riser a at a distance from c equal to half the width of a tread, instead of at the springing as in the preceding example.

In Fig. 125 is shown a case where the risers are placed at a distance from c equal to a full tread, the effect in respect to the tangents of the face-mould and bevel being the same as in the two preceding examples. In Fig. 126 is shown the plan of Fig. 123; in Fig. 127, the plan of Fig. 124; and in Fig. 128, the plan of Fig. 125. For the wreaths shown in all these figures, there will be no necessity of springing the plank, which is a term used in handrailing to denote the twisting of the wreath; and no other bevel than the one at the upper end of the pitch-board will be required. This type of wreath, also, is the one that is required at the top of a landing when the rail of the flight intersects with a level-landing rail.

In Fig. 129 is shown a very simple method of drawing the face-mould for this wreath from the pitch-board. Make a c equal to the radius of the plan central line of rail as shown at the curve in Fig. 130. From where line c c″ cuts the long side of the pitch-board, the line c″ a″ is drawn at right angles to the long edge, and is made equal to the length of the plan tangent a c, Fig. 130. The curve is drawn by means of pins and string or a trammel.

In Fig. 131 is shown a quarter-turn between two flights. The correct method of placing the risers in and around the curve, is to put the last one in the first flight and the first one in the second flight one-half a step from the intersection of the crown tangents. By this arrangement, as shown in Fig. 132, the pitch-line of the tangents will equal the pitch of the connecting flight, thus securing the second easiest condition of tangents for the face-mould—namely, as shown, two equal tangents. For this wreath, only one bevel will be needed, and it is made up of the radius of the plan central line of the rail o c, Fig. 131, for base, and the line 1-2, Fig. 132, for altitude, as shown in Fig. 133.

The bevel shown in this figure has been previously explained in Figs. 105 and 106. It is to be applied to both ends of the wreath.

The example shown in Fig. 134 is of a well-hole having a riser in the center. If the radius of the plan central line of rail is made equal to one-half a tread, the pitch of tangents will be the same as of the flights adjoining, thus securing two equal tangents for the two sections of wreath. In this figure the tangents of the face-mould are developed, and also the central line of the rail, as shown over and above each quadrant and upon the pitch-line of tangents.

The same method may be employed in stairways having obtuse-angle and acute-angle plans, as shown in Fig. 135, in which two flights are placed at an obtuse angle to each other. If the risers shown at a and a are placed one-half a tread from c, this will produce in the elevation a pitch-line over the tangents equal to that over the flights adjoining, as shown in Fig. 136, in which also is shown the face-mould for the wreath that will span over the curve from one flight to another.

In Fig. 137 is shown a flight having the same curve at a landing. The same arrangement is adhered to respecting the placing of the risers, as shown at a and a. In Fig. 138 is shown how to develop the face-moulds.