22 THE HUMANIZING OF THE BRUTE. 



timid a nature, exhibits such an interest and fervor 

 that, as I myself more than once have observed, it 

 does not desist from its ingenious work once begun, 

 even though taken into the observer's hand. 



Now in what does the real problem of the beetle 

 consist, and what has it to do with the conservation of 

 its species? 



Unrolling the leaf and spreading it on a plain sur- 

 face (Fig 2), we shall find that the exterior margin 

 of the leaf and the S-curve cut by the beetle are in 

 the same relation to each other as the two curves of 

 higher mathematics, the involute and evolute, i. e., 

 v w, t u, r s, p g, I m are almost perpendicular to 

 the exterior margin w u s q m, and are equal to 

 the corresponding curves v y g, t y g, r y g, p y g, 

 lyg, respectively. In other words, our little mathe- 

 matician cuts its S-curve so that the length of the cut 

 made and the distance from the exterior margin always 

 remain the same. This problem coincides with the task 

 of higher mathematics, from a given involute to con- 

 struct the corresponding evolute, and consequently in- 

 volves a most complicate combination of differential 

 calculus and geometry. 



But to what kind of curve does the evolute of Rh. 

 betula belong? As Prof. Heis first discovered, the 

 evolute in this case is nothing else than an unfinished 

 circle, which has its terminals in the joints g and 

 y. According to the same authority, the more 

 horizontal curve of the second half of the leaf is to 

 be considered as a very appropiate flattening of the 

 first curve, which has a more perpendicular position. 

 For, since the broader exterior windings A, B, C> 



