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. Fig. 84. Obtuse-Angle Plan. Fig. 85. Acute-Angle Plan. Fig. 86. Semicircular Plan.

In practical work it will be required to find the correct goemetrical 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 porjections 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. Joint Fig. 87. Illustrating Plane Inclined in One Direction Only. Fig. 88. Plan Line of Rail Projected into Oblique Plane Inclined to One Side Only.

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. 89. Finding Angle between Tangents. Fig. 90. Two Tangents Equally-Inclined. Fig. 91. Plan Lines Projected into Oblique Plane Inclined to Two Sides.

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 // 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. Fig. 92. Finding Angle between Tangents.

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. Fig. 93. Laying Out Line of Wreath to Start from Level-Landing Rail, Wind around Well-Hole, and Connect at Landing with Flight to Upper Story.

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 to 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. Tangents Unfolded to Find Their Inclination.

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' to". 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 FIRST-FLOOR PLAN OF RESIDENCE FOR MR. HANS HOFFMAN, MILWAUKEE, WIS.