a very short time the plank would begin to bend and buckle in the center. In order to increase the strength of this primitive bridge, we could place another plank beside it, making the bridge twenty-four inches wide, and, if the passing load were made to bear upon the entire width of this bridge, of course the bridge would bear just twice as much as one plank; but, in order to double the strength of the bridge, we have also doubled the amount of material necessary in its construction, and therefore have not in any way increased the economy.
This system might be carried on to infinity, and almost any amount of required strength be obtained by placing a sufficient number of planks one beside the other. But, returning again to the two planks, instead of placing them one beside the other, suppose we place one plank on top of the other, and nail them together firmly, so that they shall act as one plank (Fig. 2). We
then have a bridge eighteen feet long, twelve inches wide, and six inches deep. In this bridge we have exactly the same amount of material we had when the two planks were placed side by side, but we have four times as strong a bridge instead of only twice; that is, we have doubled the amount of material, but we have multiplied the strength by four.
If one plank would hold up one hundred pounds on the center, then the two planks placed side by side would hold up two hundred pounds; while, placing the planks one on top of the other, and nailing them firmly together, they would hold up four hundred pounds. In this way we see that, in order to increase the strength of the bridge or beam faster than we increase the amount of material, the increased amount of material should go into the depth of the beam and not into the width of it. This is one of the first principles in the resistance of material, that the strength of a beam varies directly as the width—that is, if we make the beam twice as wide, it will hold twice as much; and that the strength varies as the square of the depth that is, if we make it twice as deep, it will hold up four times as much. If we make it three times as deep, it will hold up nine times as much of a load. So that you can readily understand that, in order to increase the strength of the bridge or beam without increasing the material in the same proportion, the increased amount of material should be put into the depth and not into the width.
We now have a bridge twelve inches wide and six inches deep, which will hold up four times as much as our original bridge, twelve inches wide and three inches deep, and the amount of material is simply doubled. To advance one step beyond this,
