While the method of “cut and try” had brought success in the models, and was perhaps the only method by which they could have been successfully developed, it was thought that, with these models as a basis of design, much time would be saved by making an analytical study of them as engineering structures, and from the data thus obtained the proper proportions for the parts of the larger machine could be calculated. Such an analytical study, however, revealed very little from which to make calculations as to the strength necessary for the various parts of the large machine, but it did show very clearly that most of the parts were working under stresses generally far above the elastic limit of the materials, and in many cases the ultimate breaking strength was closely approached. Such a condition was the natural outcome of the method by which these models had been developed—all the various parts having been built at first of the least possible weight and, when they proved too weak, strengthened until they would withstand the stresses imposed on them. It is extremely doubtful if previous calculations as to the strength necessary would have been of any assistance, in fact it is probable that it would have been a distinct disadvantage and would have resulted in the machines being entirely too heavy for flight. The exact strength which had been incorporated in the frames of the models was as unknown as was the exact amount of the stresses which they has been made to withstand. Their static strength was easily determined by calculation, but the stresses due to the live loads were incapable of exact determination from the available data, for stresses produce strains, which in turn generally cause distortions accompanied by greatly increased stresses. While exact data were, therefore, lacking as to stresses and strengths in many of the important parts, yet the models furnished most important illustrations of unusual strength for minimum weight, and a careful study of them showed many ways in which increased strength could be obtained with decreased weight which could hardly have been devised without these concrete examples. It was, however, by no means possible to build the large aerodrome within the permissible limits of weight by simply increasing the various parts of the models according to some predetermined function of the size of the whole. The fundamental difficulty is that inevitably, by the laws of geometry, which are mere expressions of the properties of space, if a solid of any form is magnified, the weight increases as the cube, while the surface increases only as the square, of the linear dimensions. Successive generations of physicists and mathematicians pointed out that while this “law of the cube” is of advantage in the construction of balloons, yet it is a stumbling block that will prevent man [p130] from ever building a dynamic flying machine sufficiently large to carry even one human being.[38]