In most cases strength depends upon form, in some cases strength has nothing whatever to do with form; if we cut an iron bar in two its cross-section of say one square inch may be round, oblong, or of other contour, while the effort required to work the dividing shears will in any case be the same. But shearing stresses, such as those here in play, are not so common or important as the tension which tugs the wires of Brooklyn Bridge, or the compression which comes upon a pillar beneath the dome of the national capitol. When we place a lintel over a door or a window, we are concerned that it shall not sag and let down the wall above it in ruin: we ensure safety from disaster by giving the lintel a suitable shape. When we build a bridge we wish its roadway to remain as level as possible while a load passes, so that no hills and hollows may waste tractive power: levelness is secured by a design which is rigid as well as strong. If a railroad has weak, yielding rails, a great deal of energy is uselessly exerted in bending the metal as the wheels pass by. A stiff rail, giving way but little, avoids this waste. To create forms which in use will firmly keep their shape is accordingly one of the chief tasks of the engineer and the architect. [Illustration: Rubber strip suspended plank-wise, and joist-wise.] [Illustration: Board doubled breadthwise through small semi-circle AB, then edgewise through large semi-circle CD.] Plank and Joist. Forms of this kind, well exemplified in the steel columns and girders of to-day, have been arrived at by pursuing a path opened long ago by some shrewd observer. This man noticed that a plank laid flatwise bent much beneath a load, but that when the plank rested on its narrow edge, joist fashion, it curved much less, or hardly at all. Thus simply by changing the position of his plank he in effect altered its form with reference to the strain to be borne, securing a decided gain in rigidity. Let us repeat his experiment, using material much more yielding than wood. We take a piece of rubber eight inches long, one inch wide and one quarter of an inch thick. Placing it flatwise on supports close to its ends we find that its own weight causes a decided sag. We next place it edgewise, taking care to keep it perpendicular throughout its length, when it sags very little. Why? Because now the rubber has to bend through an arc four times greater in radius than in the first experiment. Suppose we had a large board yielding enough to be bent double, we can see that there would be much more work in doubling it edgewise than flatwise. The rule for joists is that breadth for breadth their stiffness varies as the square of their depth, because the circle through which the bending takes place varies in area as the square of its radius. In our experiment with the rubber strip by increasing depth four-fold, we accordingly increased stiffness sixteen-fold; but the breadth of our rubber when laid as a joist is only one-fourth of its