WHEN up on the heights, among the imposing wilderness of rocks, crags and pines, the mountaineer is struck by the roaring sound of a storm, he may observe clearly that the weather-beaten trees of a mountain forest, like other organic beings, have to defend themselves against the external attacks of nature. In other words, they have to make provisions to grow in spite of precarious circumstances and to resist many violent disturbances. The adaptability of organic beings to surrounding conditions and the existence of special means of resistance against inner and outer enemies are well known biological facts. That nature in its domain of activity, however, also makes extended use of such principles to which the engineer is accustomed in carrying out his projects, seems to be less generally known. In many instances nature is far in advance of the best human efforts in regard to rational construction. To the eye of the attentive observer, nature even may show pictures which in a beautiful manner reveal definite geometric configurations and relations.
It is not surprising that this should be the case, and might be expected. The axioms of geometry are abstract statements of primitive experiences in space. In fact, according to Picard, geometry may be called the theory of space and, as such, has its origin in experience. Geometric configurations as exhibited by nature are therefore necessarily in accord with the results deduced from the geometric premises. Conversely, within the space of our experience the theorems deduced in ordinary geometry are not contradicted by nature. This statement does, of course, not exclude the possibility of other consistent theories of space, as, for instance, established in the so-called non-Euclidean geometries. The tremendous advantage of the ordinary, or Euclidean geometry, lies in the relative simplicity and adequacy of its application to physical space. As Painlevé states, the science of mechanics, in the philosophical aspect of its foundations, does not differ from that of geometry. Its axioms also are derived from primitive experiences. No science can be created by purely formalistic logic.
Returning to the innumerable objects of natural growth, I shall confine myself to a description of the architectural and mechanical features of a few most conspicuous examples.
The contour-lines of a column or tower, all of whose horizontal
- "De la méthode dans les sciences," pp. 1-30, Paris, 1909.
- "De la méthode dans les sciences," pp. 336-407.