Chapter III: Part I: Stair-Building (2)

There is also a framed spandrel which helps materially to carry the weight, makes a sound job, and adds greatly to the appearance. This spandrel may be made of 1¼-inch material, with panels and mouldings on the front side, as shown in Fig. 36. The joint between the top and bottom rails of the spandrel at the angle, should be made as shown in Fig. 42 with a cross-tongue, and glued and fastened with long screws. Fig. 43 is simply one of the panels showing the miters on the moulding and the shape of the sections. As there is a convenient space under the landing, it is commonly used for a closet.

In setting out stairs, not only the proportions of treads and risers must be considered, but also the material available. As this material runs, as a rule, in certain sizes, it is best to work so as to conform to it as nearly as possible. In ordinary stairs, 11 by 1-inch common stock is used for strings and treads, and 7-inch by ¾-inch stock for risers; in stairs of a better class, wider and thicker material may be used. The rails are set at various heights; 2 feet 8 inches may be taken as an average height on the stairs, and 3 feet 1 inch on landings, with two balusters to each step.

In Fig. 36, all the newels and balusters are shown square; but it is much better, and is the more common practice, to have them turned, as this gives the stairs a much more artistic appearance. The spandrel under the string of the stairway shows a style in which many stairs are finished in hallways and other similar places. Plaster is sometimes used instead of the panel work, but is not nearly so good as woodwork. The door under the landing may open into a closet, or may lead to a cellarway, or through to some other room.

In stairs with winders, the width of a winder should, if possible, be nearly the width of the regular tread, at a distance of 14 inches from the narrow end, so that the length of the step in walking up or down the stairs may not be interrupted; and for this reason and several others, it is always best to have three winders only in each quarter-turn. Above all, avoid a four-winder turn, as this makes a breakneck stair, which is more difficult to construct and inconvenient to use.

_Bullnose Tread._ No other stair, perhaps, looks so well at the starting point as one having a _bullnose_ step. In Fig. 44 are shown a plan and elevation of a flight of stairs having a bullnose tread. The method of obtaining the lines and setting out the body of the stairs, is the same as has already been explained for other stairs, with the exception of the first two steps, which are made with circular ends, as shown in the plan. These circular ends are worked out as hereafter described, and are attached to the newel and string as shown. The example shows an open, cut string with brackets. The spandrel under the string contains short panels, and makes a very handsome finish. The newels and balusters in this case are turned, and the latter have cutwork panels between them.

Bullnose steps are usually built up with a three-piece block, as shown in Fig. 45, which is a section through the step indicating the blocks, tread, and riser.

Fig. 46 is a plan showing how the veneer of the riser is prepared before being bent into position. The block _A_ indicates a wedge which is glued and driven home after the veneer is put in place. This tightens up the work and makes it sound and clear. Figs. 47 and 48 show other methods of forming bullnose steps.

Fig. 49 is the side elevation of an open-string stair with bullnose steps at the bottom; while Fig. 50 is a view showing the lower end of the string, and the manner in which it is prepared for fixing to the blocks of the step. Fig. 51 is a section through the string, showing the bracket, cove, and projection of tread over same.

Figs. 52 and 53 show respectively a plan and vertical section of the bottom part of the stair. The blocks are shown at the ends of the steps (Fig. 53), with the veneered parts of the risers going round them; also the position where the string is fixed to the blocks (Fig. 52); and the tenon of the newel is marked on the upper step. The section (Fig. 53) shows the manner in which the blocks are built up and the newel tenoned into them.

The newel, Fig. 49, is rather an elaborate affair, being carved at the base and on the body, and having a carved rosette planted in a small, sunken panel on three sides, the rail butting against the fourth side.

=Open-Newel Stairs.= Before leaving the subject of straight and dog-legged stairs, the student should be made familiar with at least one example of an open-newel stair. As the same principles of construction govern all styles of open-newel stairs, a single example will be sufficient. The student must, of course, understand that he himself is the greatest factor in planning stairs of this type; that the setting out and designing will generally devolve on him. By exercising a little thought and foresight, he can so arrange his plan that a minimum of both labor and material will be required.

Fig. 54 shows a plan of an open-newel stair having two landings and closed strings, shown in elevation in Fig. 55. The dotted lines show the carriage timbers and trimmers, also the lines of risers; while the treads are shown by full lines. It will be noticed that the strings and trimmers at the first landing are framed into the shank of the second newel post, which runs down to the floor; while the top newel drops below the fascia, and has a turned and carved drop. This drop hangs below both the fascia and the string. The lines of treads and risers are shown by dotted lines and crosshatched sections. The position of the carriage timbers is shown both in the landings and in the runs of the stairs, the projecting ends of these timbers being supposed to be resting on the wall. A scale of the plan and elevation is attached to the plan. In Fig. 55, a story rod is shown at the right, with the number of risers spaced off thereon. The design of the newels, spandrel, framing, and paneling is shown.

Only the central carriage timbers are shown in Fig. 54; but in a stair of this width, there ought to be two other timbers, not so heavy, perhaps, as the central one, yet strong enough to be of service in lending additional strength to the stairway, and also to help carry the laths and plaster or the paneling which may be necessary in completing the under side or soffit. The strings being closed, the butts of their balusters must rest on a subrail which caps the upper edge of the outer string.

The first newel should pass through the lower floor, and, to insure solidity, should be secured by bolts to a joist, as shown in the elevation. The rail is attached to the newels in the usual manner, with handrail bolts or other suitable device. The upper newel should be made fast to the joists as shown, either by bolts or in some other efficient manner. The intermediate newels are left square on the shank below the stairs, and may be fastened in the floor below either by mortise and tenon or by making use of joint bolts.

Everything about a stair should be made solid and sound; and every joint should set firmly and closely; or a shaky, rickety, squeaky stair will be the result, which is an abomination.

=Stairs with Curved Turns.= Sufficient examples of stairs having angles of greater or less degree at the turn or change of direction, to enable the student to build any stair of this class, have now been given. There are, however, other types of stairs in common use, whose turns are curved, and in which newels are employed only at the foot, and sometimes at the finish of the flight. These curved turns may be any part of a circle, according to the requirements of the case, but turns of a quarter-circle or half-circle are the more common. The string forming the curve is called a _cylinder_, or part of a cylinder, as the case may be. The radius of this circle or cylinder may be any length, according to the space assigned for the stair. The opening around which the stair winds is called the _well-hole_.

Fig. 56 shows a portion of a stairway having a well-hole with a 7-inch radius. This stair is rather peculiar, as it shows a quarter-space landing, and a quarter-space having three winders. The reason for this is the fact that the landing is on a level with the floor of another room, into which a door opens from the landing. This is a problem very often met with in practical work, where the main stair is often made to do the work of two flights because of one floor being so much lower than another.

A curved stair, sometimes called a _geometrical stair_, is shown in Fig. 57, containing seven winders in the cylinder or well-hole, the first and last aligning with the diameter.

In Fig. 58 is shown another example of this kind of stair, containing nine winders in the well-hole, with a circular wall-string. It is not often that stairs are built in this fashion, as most stairs having a circular well-hole finish against the wall in a manner similar to that shown in Fig. 57.

Sometimes, however, the workman will be confronted with a plan such as shown in Fig. 58; and he should know how to lay out the wall-string. In the elevation, Fig. 58, the string is shown to be straight, similar to the string of a common straight flight. This results from having an equal width in the winders along the wall-string, and, as we have of necessity an equal width in the risers, the development of the string is merely a straight piece of board, as in an ordinary straight flight. In laying out the string, all we have to do is to make a common pitch-board, and, with it as a templet, mark the lines of the treads and risers on a straight piece of board, as shown at 1, 2, 3, 4, etc.

If you can manage to bend the string without kerfing (grooving), it will be all the better; if not, the kerfs (grooves) must be parallel to the rise. You can set out with a straight edge, full size, on a rough platform, just as shown in the diagram; and when the string is bent and set in place, the risers and winders will have their correct positions.

To bend these strings or otherwise prepare them for fastening against the wall, perhaps the easiest way is to saw the string with a fine saw, across the face, making parallel grooves. This method of bending is called _kerfing_, above referred to. The kerfs or grooves must be cut parallel to the lines of the risers, so as to be vertical when the string is in place. This method, however—handy though it may be—is not a good one, inasmuch as the saw groove will show more or less in the finished work.

Another method is to build up or _stave_ the string. There are several ways of doing this. In one, comparatively narrow pieces are cut to the required curve or to portions of it, and are fastened together, edge to edge, with glue and screws, until the necessary width is obtained (see Fig. 59). The heading joints may be either butted or beveled, the latter being stronger, and should be cross-tongued.

Fig. 60 shows a method that may be followed when a wide string is required, or a piece curved in the direction of its width is needed for any purpose. The pieces are stepped over each other to suit the desired curve; and though shown square-edged in the figure, they are usually cut beveled, as then, by reversing them, two may be cut out of a batten.

Panels and quick sweeps for similar purposes are obtained in the manner shown in Fig. 61, by joining up narrow boards edge to edge at a suitable bevel to give the desired curve. The internal curve is frequently worked approximately, before gluing up. The numerous joints incidental to these methods limit their uses to painted or unimportant work.

In Fig. 62 is shown a wreath-piece or curved portion of the outside string rising around the cylinder at the half-space. This is formed by reducing a short piece of string to a veneer between the springings; bending it upon a cylinder made to fit the plan; then, when it is secured in position, filling up the back of the veneer with staves glued across it; and, finally, gluing a piece of canvas over the whole. The appearance of the wreath-piece after it has been built up and removed from the cylinder is indicated in Fig. 63. The canvas back has been omitted to show the staving; and the counter-wedge key used for connecting the wreath-piece with the string is shown. The wreath-piece is, at this stage, ready for marking the outlines of the steps.

Fig. 62 also shows the drum or shape around which strings may be bent, whether the strings are formed of veneers, staved, or kerfed. Another drum or shape is shown in Fig. 64. In this, a portion of a cylinder is formed in the manner clearly indicated; and the string, being set out on a veneer board sufficiently thin to bend easily, is laid down round the curve, such a number of pieces of like thickness being then added as will make the required thickness of the string. In working this method, glue is introduced between the veneers, which are then quickly strained down to the curved piece with hand screws. A string of almost any length can be formed in this way, by gluing a few feet at a time, and when that dries, removing the cylindrical curve and gluing down more, until the whole is completed. Several other methods will suggest themselves to the workman, of building up good, solid, circular strings.

One method of laying out the treads and risers around a cylinder or drum, is shown in Fig. 65. The line _D_ shows the curve of the rail. The lines showing treads and risers may be marked off on the cylinder, or they may be marked off after the veneer is bent around the drum or cylinder.

There are various methods of making inside cylinders or wells, and of fastening same to strings. One method is shown in Fig. 66. This gives a strong joint when properly made. It will be noticed that the cylinder is notched out on the back; the two blocks shown at the back of the offsets are wedges driven in to secure the cylinder in place, and to drive it up tight to the strings. Fig. 67 shows an 8-inch well-hole with cylinder complete; also the method of trimming and finishing same. The cylinder, too, is shown in such a manner that its construction will be readily understood.

Stairs having a cylindrical or circular opening always require a weight support underneath them. This support, which is generally made of rough lumber, is called the _carriage_, because it is supposed to carry any reasonable load that may be placed upon the stairway. Fig. 68 shows the under side of a half-space stair having a carriage beneath it. The timbers marked _S_ are of rough stuff, and may be 2-inch by 6-inch or of greater dimensions. If they are cut to fit the risers and treads, they will require to be at least 2-inch by 8-inch.

In preparing the rough carriage for the winders, it will be best to let the back edge of the tread project beyond the back of the riser so that it forms a ledge as shown under _C_ in Fig. 69. Then fix the cross-carriage pieces under the winders, with the back edge about flush with the backs of risers, securing one end to the well with screws, and the other to the wall string or the wall. Now cut short pieces, marked _O O_ (Fig. 68), and fix them tightly in between the cross-carriage and the back of the riser as at _B B_ in the section, Fig. 69. These carriages should be of 3-inch by 2-inch material. Now get a piece of wood, 1-inch by 3-inch, and cut pieces _C C_ to fit tightly between the top back edge of the winders (or the ledge) and the pieces marked _B B_ in section. This method makes a very sound and strong job of the winders; and if the stuff is roughly planed, and blocks are glued on each side of the short cross-pieces _O O O_, it is next to impossible for the winders ever to spring or squeak. When the weight is carried in this manner, the plasterer will have very little trouble in lathing so that a graceful soffit will be made under the stairs.

The manner of placing the main stringers of the carriage _S S_, is shown at _A_, Fig. 69. Fig. 68 shows a complete half-space stair; one-half of this, finished as shown, will answer well for a quarter-space stair.

Another method of forming a carriage for a stair is shown in Fig. 70. This is a peculiar but very handsome stair, inasmuch as the first and the last four steps are parallel, but the remainder _balance_ or _dance_. The treads are numbered in this illustration; and the plan of the handrail is shown extending from the scroll at the bottom of the stairs to the landing on the second story. The trimmer _T_ at the top of the stairs is also shown; and the rough strings or carriages, _R S_, _R S_, _R S_, are represented by dotted lines.

This plan represents a stair with a curtail step, and a scroll handrail resting over the curve of the curtail step. This type of stair is not now much in vogue in this country, though it is adopted occasionally in some of the larger cities. The use of heavy newel posts instead of curtail steps, is the prevailing style at present.

In laying out geometrical stairs, the steps are arranged on principles already described. The well-hole in the center is first laid down and the steps arranged around it. In circular stairs with an open well-hole, the handrail being on the inner side, the width of tread for the steps should be set off at about 18 inches from the handrail, this giving an approximately uniform rate of progress for anyone ascending or descending the stairway. In stairs with the rail on the outside, as sometimes occurs, it will be sufficient if the treads have the proper width at the middle point of their length.

Where a flight of stairs will likely be subject to great stress and wear, the carriages should be made much heavier than indicated in the foregoing figures; and there may be cases when it will be necessary to use iron bolts in the sides of the rough strings in order to give them greater strength. This necessity, however, will arise only in the case of stairs built in public buildings, churches, halls, factories, warehouses, or other buildings of a similar kind. Sometimes, even in house stairs it may be wise to strengthen the treads and risers by spiking pieces of board to the rough string, ends up, fitting them snugly against the under side of the tread and the back of the riser. The method of doing this is shown in Fig. 71, in which the letter _O_ shows the pieces nailed to the string.

=Types of Stairs in Common Use.= In order to make the student familiar with types of stairs in general use at the present day, plans of a few of those most likely to be met with will now be given.

Fig. 72 is a plan of a straight stair, with an ordinary cylinder at the top provided for a return rail on the landing. It also shows a stretch-out stringer at the starting.

Fig. 73 is a plan of a stair with a landing and return steps.

Fig. 74 is a plan of a stair with an acute angular landing and cylinder.

Fig. 75 illustrates the same kind of stair as Fig. 74, the angle, however, being obtuse.

Fig. 76 exhibits a stair having a half-turn with two risers on landings.

Fig. 77 is a plan of a quarter-space stair with four winders.

Fig. 78 shows a stair similar to Fig. 77, but with six winders.

Fig. 79 shows a stair having five dancing winders.

Fig. 80 is a plan of a half-space stair having five dancing winders and a quarter-space landing.

Fig. 81 shows a half-space stair with dancing winders all around the cylinder.

Fig. 82 shows a geometrical stair having winders all around the cylinder.

Fig. 83 shows the plan and elevation of stairs which turn around a central post. This kind of stair is frequently used in large stores and in clubhouses and other similar places, and has a very graceful appearance. It is not very difficult to build if properly planned.

The only form of stair not shown which the student may be called upon to build, would very likely be one having an elliptical plan; but, as this form is so seldom used—being found, in fact, only in public buildings or great mansions—it rarely falls to the lot of the ordinary workman to be called upon to design or construct a stairway of this type.

GEOMETRICAL STAIRWAYS AND HAND-RAILING

The term _geometrical_ is applied to stairways having any kind of curve for a plan.

The rails over the steps are made continuous from one story to another. The resulting winding or twisting pieces are called _wreaths_.

=Wreaths.= The construction of wreaths is based on a few geometrical problems—namely, the projection of straight and curved lines into an oblique plane; and the finding of the angle of inclination of the plane into which the lines and curves are projected. This angle is called the _bevel_, and by its use the wreath is made to twist.

In Fig. 84 is shown an obtuse-angle plan; in Fig. 85, an acute-angle plan; and in Fig. 86, a semicircle enclosed within straight lines.

=Projection.= A knowledge of how to project the lines and curves in each of these plans into an oblique plane, and to find the angle of inclination of the plane, will enable the student to construct any and all kinds of wreaths.

The straight lines _a_, _b_, _c_, _d_ in the plan, Fig. 86, are known as _tangents_; and the curve, the _central line_ of the plan wreath.

The straight line across from _n_ to _n_ is the _diameter_; and the perpendicular line from it to the lines _c_ and _b_ is the _radius_.

A _tangent_ line may be defined as a line touching a curve without cutting it, and is made use of in handrailing to square the joints of the wreaths.

=Tangent System.= The _tangent system_ of handrailing takes its name from the use made of the tangents for this purpose.

In Fig. 86, it is shown that the joints connecting the central line of rail with the plan rails _w_ of the straight flights, are placed right at the springing; that is, they are in line with the diameter of the semicircle, and square to the side tangents _a_ and _d_.

The center joint of the crown tangents is shown to be square to tangents _b_ and _c_. When these lines are projected into an oblique plane, the joints of the wreaths can be made to butt square by applying the bevel to them.

All handrail wreaths are assumed to rest on an oblique plane while ascending around a well-hole, either in connecting two flights or in connecting one flight to a landing, as the case may be.

In the simplest cases of construction, the wreath rests on an inclined plane that inclines in one direction only, to either side of the well-hole; while in other cases it rests on a plane that inclines to two sides.

Fig. 87 illustrates what is meant by a plane inclining in one direction. It will be noticed that the lower part of the figure is a reproduction of the quadrant enclosed by the tangents _a_ and _b_ in Fig. 86. The quadrant, Fig. 87, represents a central line of a wreath that is to ascend from the joint on the plan tangent _a_ the height of _h_ above the tangent _b_.

In Fig. 88, a view of Fig. 87 is given in which the tangents _a_ and _b_ are shown in plan, and also the quadrant representing the plan central line of a wreath. The curved line extending from _a_ to _h_ in this figure represents the development of the central line of the plan wreath, and, as shown, it rests on an oblique plane inclining to one side only—namely, to the side of the plan tangent _a_. The joints are made square to the developed tangents _a_ and _m_ of the inclined plane; it is for this purpose only that tangents are made use of in wreath construction. They are shown in the figure to consist of two lines, _a_ and _m_, which are two adjoining sides of a developed section (in this case, of a square prism), the section being the assumed inclined plane whereon the wreath rests in its ascent from _a_ to _h_. The joint at _h_, if made square to the tangent _m_, will be a true, square butt-joint; so also will be the joint at _a_, if made square to the tangent _a_.

In practical work it will be required to find the correct geometrical angle between the two developed tangents _a_ and _m_; and here, again, it may be observed that the finding of the correct angle between the two developed tangents is the essential purpose of every tangent system of handrailing.

In Fig. 89 is shown the geometrical solution—the one necessary to find the angle between the tangents as required on the face-mould to square the joints of the wreath. The figure is shown to be similar to Fig. 87, except that it has an additional portion marked “Section.” This section is the true shape of the oblique plane whereon the wreath ascends, a view of which is given in Fig. 88. It will be observed that one side of it is the developed tangent _m_; another side, the developed tangent _a″_ (= _a_). The angle between the two as here presented is the one required on the face-mould to square the joints.

In this example, Fig. 89, owing to the plane being oblique in one direction only, the shape of the section is found by merely drawing the tangent _a″_ at right angles to the tangent _m_, making it equal in length to the level tangent _a_ in the plan. By drawing lines parallel to _a″_ and _m_ respectively, the form of the section will be found, its outlines being the projections of the plan lines; and the angle between the two tangents, as already said, is the angle required on the face-mould to square the joints of the wreath.

The solution here presented will enable the student to find the correct direction of the tangents as required on the face-mould to square joints, in all cases of practical work where one tangent of a wreath is level and the other tangent is inclined, a condition usually met with in level-landing stairways.

Fig. 90 exhibits a condition of tangents where the two are equally inclined. The plan here also is taken from Fig. 86. The inclination of the tangents is made equal to the inclination of tangent _b_ in Fig. 86, as shown at _m_ in Figs. 87, 88, and 89.

In Fig. 91, a view of Fig. 90 is given, showing clearly the inclination of the tangents _c″_ and _d″_ over and above the plan tangents _c_ and _d_. The central line of the wreath is shown extending along the sectional plane, over and above its plan lines, from one joint to the other, and, at the joints, made square to the inclined tangents _c″_ and _d″_. It is evident from the view here given, that the condition necessary to square the joint at each end would be to find the true angle between the tangents _c″_ and _d″_, which would give the correct direction to each tangent.

In Fig. 92 is shown how to find this angle correctly as required on the face-mould to square the joints. In this figure is shown the same plan as in Figs. 90 and 91, and the same inclination to the tangents as in Fig. 90, so that, except for the portion marked “Section,” it would be similar to Fig. 90.

To find the correct angle for the tangents of the face-mould, draw the line _m_ from _d_, square to the inclined line of the tangents _c′_ _d″_; revolve the bottom inclined tangent _c′_ to cut line _m_ in _n_, where the joint is shown fixed; and from this point draw the line _c″_ to _w_. The intersection of this line with the upper tangent _d″_ forms the correct angle as required on the face-mould. By drawing the joints square to these two lines, they will butt square with the rail that is to connect with them, or to the joint of another wreath that may belong to the cylinder or well-hole.

Fig. 93 is another view of these tangents in position placed over and above the plan tangents of the well-hole. It will be observed that this figure is made up of Figs. 88 and 91 combined. Fig. 88, as here presented, is shown to connect with a level-landing rail at _a_. The joint having been made square to the level tangent, _a_ will butt square to a square end of the level rail. The joint at _h_ is shown to connect the two wreaths and is made square to the inclined tangent _m_ of the lower wreath, and also square to the inclined tangent _c″_ of the upper wreath; the two tangents, aligning, guarantee a square butt-joint. The upper joint is made square to the tangent _d″_, which is here shown to align with the rail of the connecting flight; the joint will consequently butt square to the end of the rail of the flight above.

The view given in this diagram is that of a wreath starting from a level landing, and winding around a well-hole, connecting the landing with a flight of stairs leading to a second story. It is presented to elucidate the use made of tangents to square the joints in wreath construction. The wreath is shown to be in two sections, one extending from the level-landing rail at _a_ to a joint in the center of the well-hole at _h_, this section having one level tangent _a_ and one inclined tangent _m_; the other section is shown to extend from _h_ to _n_, where it is butt-jointed to the rail of the flight above.

This figure clearly shows that the joint at _a_ of the bottom wreath—owing to the tangent _a_ being level and therefore aligning with the level rail of the landing—will be a true butt-joint; and that the joint at _h_, which connects the two wreaths, will also be a true butt-joint, owing to it being made square to the tangent _m_ of the bottom wreath and to the tangent _c″_ of the upper wreath, both tangents having the same inclination; also the joint at _n_ will butt square to the rail of the flight above, owing to it being made square to the tangent _d″_, which is shown to have the same inclination as the rail of the flight adjoining.

As previously stated, the use made of tangents is to square the joints of the wreaths; and in this diagram it is clearly shown that the way they can be made of use is by giving each tangent its true direction. How to find the true direction, or the angle between the tangents _a_ and _m_ shown in this diagram, was demonstrated in Fig. 89; and how to find the direction of the tangents _c″_ and _d″_ was shown in Fig. 92.

Fig. 94 is presented to help further toward an understanding of the tangents. In this diagram they are unfolded; that is, they are stretched out for the purpose of finding the inclination of each one over and above the plan tangents. The side plan tangent _a_ is shown stretched out to the floor line, and its elevation _a′_ is a level line. The side plan tangent _d_ is also stretched out to the floor line, as shown by the arc _n′ m′_. By this process the plan tangents are now in one straight line on the floor line, as shown from _w_ to _m′_. Upon each one, erect a perpendicular line as shown, and from _m′_ measure to _n_, the height the wreath is to ascend around the well-hole. In practice, the number of risers in the well-hole will determine this height.

Now, from point _n_, draw a few treads and risers as shown; and along the nosing of the steps, draw the pitch-line; continue this line over the tangents _d″_, _c″_, and _m_, down to where it connects with the bottom level tangent, as shown. This gives the pitch or inclination to the tangents over and above the well-hole. The same line is shown in Fig. 93, folded around the well-hole, from _n_, where it connects with the flight at the upper end of the well-hole, to _a_, where it connects with the level-landing rail at the bottom of the well-hole. It will be observed that the upper portion, from joint _n_ to joint _h_, over the tangents _c″_ and _d″_, coincides with the pitch-line of the same tangents as presented in Fig. 92, where they are used to find the true angle between the tangents as it is required on the face-mould to square the joints of the wreath at _h_.

In Fig. 89 the same pitch is shown given to tangent _m_ as in Fig. 94; and in both figures the pitch is shown to be the same as that over and above the upper connecting tangents _c″_ and _d″_, which is a necessary condition where a joint, as shown at _h_ in Figs. 93 and 94, is to connect two pieces of wreath as in this example.

In Fig. 94 are shown the two face-moulds for the wreaths, placed upon the pitch-line of the tangents over the well-hole. The angles between the tangents of the face-moulds have been found in this figure by the same method as in Figs. 89 and 92, which, if compared with the present figure, will be found to correspond, excepting only the curves of the face-moulds in Fig. 94.

The foregoing explanation of the tangents will give the student a fairly good idea of the use made of tangents in wreath construction. The treatment, however, would not be complete if left off at this point, as it shows how to handle tangents under only two conditions—namely, first, when one tangent inclines and the other is level, as at _a_ and _m_; second, when both tangents incline, as shown at _c″_ and _d″_.

In Fig. 95 is shown a well-hole connecting two flights, where two portions of unequal pitch occur in both pieces of wreath. The first piece over the tangents _a_ and _b_ is shown to extend from the square end of the straight rail of the bottom flight, to the joint in the center of the well-hole, the bottom tangent _a″_ in this wreath inclining more than the upper tangent _b″_. The other piece of wreath is shown to connect with the bottom one at the joint _h″_ in the center of the well-hole, and to extend over tangents _c″_ and _d″_ to connect with the rail of the upper flight. The relative inclination of the two tangents in this wreath, is the reverse of that of the two tangents of the lower wreath. In the lower piece, the bottom tangent _a″_, as previously stated, inclines considerably more than does the upper tangent _b″_; while in the upper piece, the bottom tangent _c″_ inclines considerably less than the upper tangent _d″_.

The question may arise: What causes this? Is it for variation in the inclination of the tangents over the well-hole? It is simply owing to the tangents being used in handrailing to square the joints.

The inclination of the bottom tangent _a″_ of the bottom wreath is clearly shown in the diagram to be determined by the inclination of the bottom flight. The joint at _a″_ is made square to both the straight rail of the flight and to the bottom tangent of the wreath; the rail and tangent, therefore, must be equally inclined, otherwise the joint will not be a true butt-joint. The same remarks apply to the joint at 5, where the upper wreath is shown jointed to the straight rail of the upper flight. In this case, tangent _d″_ must be fixed to incline conformably to the inclination of the upper rail; otherwise the joint at 5 will not be a true butt-joint.

The same principle is applied in determining the pitch or inclination over the crown tangents _b″_ and _c″_. Owing to the necessity of jointing the two wreaths, as shown at _h_, these two tangents must have the same inclination, and therefore must be fixed, as shown from 2 to 4, over the crown of the well-hole.

The tangents as here presented are those of the elevation, not of the face-mould. Tangent _a″_ is the elevation of the side plan tangent _a_; tangents _b″_ and _c″_ are shown to be the elevations of the plan tangents _b_ and _c_; so, also, is the tangent _d″_ the elevation of the side plan tangent _d_.

If this diagram were folded, as Fig. 94 was shown to be in Fig. 93, the tangents of the elevation—namely, _a″_, _b″_, _c″_, _d″_—would stand over and above the plan tangents _a_, _b_, _c_, _d_ of the well-hole. In practical work, this diagram must be drawn full size. It gives the correct length to each tangent as required on the face-mould, and furnishes also the data for the layout of the mould.

Fig. 96 shows how to find the angle between the tangents of the face-mould for the bottom wreath, which, as shown in Fig. 95, is to span over the first plan quadrant _a b_. The elevation tangents _a″_ and _b″_, as shown, will be the tangents of the mould. To find the angle between the tangents, draw the line _a h_ in Fig. 96; and from _a_, measure to 2 the length of the bottom tangent _a″_ in Fig. 95; the length from 2 to _h_, Fig. 96, will equal the length of the upper tangent _b″_, Fig. 95.

From 2 to 1, measure a distance equal to 2-1 in Fig. 95, the latter being found by dropping a perpendicular from _w_ to meet the tangent _b″_ extended. Upon 1, erect a perpendicular line; and placing the dividers on 2, extend to _a_; turn over to the perpendicular at _a″_; connect this point with 2, and the line will be the bottom tangent as required on the face-mould. The upper tangent will be the line 2-_h_, and the angle between the two lines is shown at 2. Make the joint at _h_ square to 2-_h_, and at _a″_ square to _a″_-2.

The mould as it appears in Fig. 96 is complete, except the curve, which is comparatively a small matter to put on, as will be shown further on. The main thing is to find the angle between the tangents, which is shown at 2, to give them the direction to square the joints.

In Fig. 97 is shown how to find the angle between the tangents _c″_ and _d″_ shown in Fig. 95, as required on the face-mould. On the line _h_-5, make _h_-4 equal to the length of the bottom tangent of the wreath, as shown at _h″_-4 in Fig. 95; and 4-5 equal to the length of the upper tangent _d″_. Measure from 4 the distance shown at 4-6 in Fig. 95, and place it from 4 to 6 as shown in Fig. 97; upon 6 erect a perpendicular line. Now place the dividers on 4; extend to _h_; turn over to cut the perpendicular in _h″_; connect this point with 4, and the angle shown at 4 will be the angle required to square the joints of the wreath as shown at _h″_ and 5, where the joint at 5 is shown drawn square to the line 4-5, and the joint at _h″_ square to the line 4 _h″_.

Fig. 98 is a diagram of tangents and face-mould for a stairway having a well-hole at the top landing. The tangents in this example will be two equally inclined tangents for the bottom wreath; and for the top wreath, one inclined and one level, the latter aligning with the level rail of the landing.

The face-mould, as here presented, will further help toward an understanding of the layout of face-moulds as shown in Figs. 96 and 97. It will be observed that the pitch of the bottom rail is continued from _a″_ to _b″_, a condition caused by the necessity of jointing the wreath to the end of the straight rail at _a″_, the joint being made square to both the straight rail and the bottom tangent _a″_. From _b″_ a line is drawn to _d″_, which is a fixed point determined by the number of risers in the well-hole. From point _d″_, the level tangent _d″_ 5 is drawn in line with the level rail of the landing; thus the pitch-line of the tangents over the well-hole is found, and, as was shown in the explanation of Fig. 95, the tangents as here presented will be those required on the face-mould to square the joints of the wreath.

In Fig. 98 the tangents of the face-mould for the bottom wreath are shown to be _a″_ and _b″_. To place tangent _a″_ in position on the face-mould, it is revolved, as shown by the arc, to _m_, cutting a line previously drawn from _w_ square to the tangent _b″_ extended. Then, by connecting _m_ to _b″_, the bottom tangent is placed in position on the face-mould. The joint at _m_ is to be made square to it; and the joint at _c_, the other end of the mould, is to be made square to the tangent _b″_.

The upper piece of wreath in this example is shown to have tangent _c″_ inclining, the inclination being the same as that of the upper tangent _b″_ of the bottom wreath, so that the joint at _c″_, when made square to both tangents, will butt square when put together. The tangent _d″_ is shown to be level, so that the joint at 5, when squared with it, will butt square with the square end of the level-landing rail. The level tangent is shown revolved to its position on the face-mould, as from 5 to 2. In this last position, it will be observed that its angle with the inclined tangent _c″_ is a right angle; and it should be remembered that in every similar case where one tangent inclines and one is level over a square-angle plan tangent, the angle between the two tangents will be a right angle on the face-mould. A knowledge of this principle will enable the student to draw the mould for this wreath, as shown in Fig. 99, by merely drawing two lines perpendicular to each other, as _d″_ 5 and _d″ c″_, equal respectively to the level tangent _d″_ 5 and the inclined tangent _c″_ in Fig. 98. The joint at 5 is to be made square to _d″_ 5; and that at _c″_, to _d″ c″_. Comparing this figure with the face-mould as shown for the upper wreath in Fig. 98, it will be observed that both are alike.

In practical work the stair-builder is often called upon to deal with cases in which the conditions of tangents differ from all the examples thus far given. An instance of this sort is shown in Fig. 100, in which the angles between the tangents on the plan are acute. In all the preceding examples, the tangents on the plan were at right angles; that is, they were square to one another.

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.

=Bevels to Square the Wreaths.= The next process in the construction of a wreath that the handrailer will be called upon to perform, is to find the bevels that will, by being applied to each end of it, give the correct angle to _square_ or _twist_ it when winding around the well-hole from one flight to another flight, or from a flight to a landing, as the case may be.

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.

A method of finding the bevels for _all wreaths_ (which is considered rather difficult) will now be explained:

_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.