INSTINCT AND FINAL TENDENCY. 25 



correspond to the smaller interior //, G, F, without 

 being shortened (i. e. a b and cd are equal iolm and 

 z" k respectively) , the second S-curve must necessarily 

 lie in a more horizontal position. 



This is one part of our little builder's problem. 

 The other consists in the suitableness of the chosen 

 curve to the formation of a funnel. Supposing that 

 the beetle wished to construct from the birch leaf the 

 largest and strongest funnel possible, and that, too, 

 in the shortest time and with the expenditure of the 

 least amount of its limited strength, it could really 

 not choose a more suitable curve. 



The funnel may be considered as a surface coiii- 

 cally evolvable which, when spread out upon a plane, 

 coincides with it in all its points. Now such a sur- 

 face can be rolled up in two ways, so that the lines 

 of convolution meet either in one point or in a row of 

 points, lying in a straight or curved line (Fig. 3). 

 To have them meet in one point is, in our case, alto- 

 gether out of question. For apart from the fact that 

 the central rib would most probably tear in the course 

 of the operation, it would exceed the strength of our 

 little beetle to handle the whole surface o a at once. 

 Therefore, the second manner of convolution had to be 

 chosen. Yet here again it would not do to have the 

 upper margin in a straight line, for in rolling up the 

 leaf, the upper and lower openings would have to be- 

 come either equal in their respective diameters, and 

 we would have no funnel, but only a useless cylinder; 

 or they would be unequal, the larger opening being 

 either above or below. If above, the funnel, be- 

 cause reversed, evidently would not serve its purpose ; 



