450 THE POPULAR SCIENCE MONTHLY 



MATHEMATICS AND ENGINEERING IN NATUEE 



By Professor ARNOLD EMCH 



UNIVERSITY OF ILLINOIS 



« "TTTHEN up on the heights, among the imposing wilderness of 

 V V 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 them- 

 selves 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, 1 geometry may be 

 called the theory of space and, as such, has its origin in experience. 

 Geometric configurations as exhibited by nature are therefore neces- 

 sarily 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 Painleve 2 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 



1 " De la methode dans les sciences, ' ' pp. 1-30, Paris, 1909. 

 2 " De la methode dans les sciences, ' ' pp. 336-407. 



