When the breadth and length are the same the strength obviously increases with the depth, but not in the same proportion. The increase of strength is found to be much greater in proportion than the increase of depth. By the theory of Galileo, a double or treble thickness ought to increase the strength in a four-fold or nine-fold proportion, and experiments in most cases do not materially vary from this rule. If while the breadth and depth remain the same, the length of the beam, or rather, the distance between the points of support, vary, the strength will vary accordingly, decreasing in the same proportion as the length increases. From these observations it appears, that the transverse strength of a beam depends more on its thickness than its breadth. Hence we find that a broad thin board is much stronger when its edge is presented upwards. On this principle the joists or rafters of floors and roofs are constructed. If two beams be in all respects similar, their strengths will be in the proportion of the squares of their lengths. Let the length, breadth, and depth of the one be respectively double the length, breadth, and depth of the other. By the double breadth the beam doubles its strength, but by doubling the length half this strength is lost. Thus the increase of length and breadth counteract each other’s effects, and as far as they are concerned the strength of the beam is not changed. But by doubling the thickness the strength is increased in a four-fold proportion, that is, as the square of the length. In the same manner it may be shown, that when all the dimensions are trebled, the strength is increased in a nine-fold proportion, and so on. (334.) In all structures the materials have to support their own weight, and therefore their available strength is to be estimated by the excess of their absolute strength above that degree of strength which is just sufficient to support their own weight. This consideration leads to some conclusions, of which numerous and striking illustrations are presented in the works of nature and art. We have seen that the absolute strength with which a lateral strain is resisted is in the proportion of the square of the linear dimensions of similar parts of a structure, and therefore the amount of this strength increases rapidly with every increase of the dimensions of a body. But at the same time the weight of the body increases in a still more rapid proportion. Thus, if the several dimensions be doubled, the strength will be increased in a four-fold but the weight in an eight-fold proportion. If the dimensions be trebled, the strength will be multiplied nine times, but the weight twenty-seven times. Again, if the dimensions be multiplied four times, the strength will be multiplied sixteen times, and the weight sixty-four times, and so on.