simply the angles have been changed. Now, suppose we divide this rectangle, by means of a brace or tie, into two triangles. Then not one of these timbers can be moved, or the form of the rectangle changed in any way, without lengthening or shortening the diagonal which divides it into triangles, and, therefore, the rectangle with the brace and tie forms a perfectly rigid figure (Fig. 9).
In other words, the triangle is the only figure the form of which can not be changed without changing the length of one of the sides; and thus any truss, to be perfectly braced and able to withstand any strains that come upon it, must be framed so as to be divided into a series of triangles.
Returning to our original beam thrown across an opening, we will suppose that we have a beam long enough and strong enough with the required load to cross an opening eighteen feet wide, and that we have an opening thirty-six feet wide which we wish to cross. That could be done by building a pier in the center of the opening and dividing it into two openings, each eighteen feet, as shown in Fig. 10; but, in the case of a bridge over a road or over a small river, it would not be advisable to block up the way by this pier, and some other method must be found to support the two inner ends of the beams. The simplest plan of doing it is shown in Fig. 11. Taking two beams that are each slightly
longer than eighteen feet, we throw them across the opening, as shown in Fig. 11. These two beams meet at the angle, the apex, A, of which is up, and, if the two lower ends are kept from sliding apart, will stand in that position. Now, if from the angle where these two beams meet we let down a rope or iron rod, run out the floor beams eighteen feet long, and connect the inner end of each to this rope or rod, we have a bridge covering an opening thirty-six feet long—that is, one end of each floor beam rests upon the ground, the other end is sustained by the rope or
