THE AMERICAN APICULTURIST. 



53 



view. Hence, not a few writers, 

 even to the present day, maintain 

 that we have here a typical case of 

 "instinct" in the old acceptation of 

 the word — that is, of blind uncon- 

 scious, untaught action, producing 

 results which man can only reach 

 by dint of highly cultivated reason. 



So many of these so-called in- 

 stincts have lately found a scien- 

 tific explanation that naturalists of 

 the old school have recognized the 

 cell of the bee as one of their last 

 entrenchments. 



It is therefore very satisfactory 

 that Herr K. Mullenhoff has found 

 a quite simple and satisfactory 

 solution of the question, which 

 neither admits of any mysterious 

 instinct nor, on the other hand, 

 credits the bee with the knowledge 

 of the differential calculus. 



Taking first a preliminary view 

 of the case, we find that Pappus 

 fifteen hundred years ago, noted 

 that bees constructed their cells in 

 the form of regular six-sided col- 

 umns, and proved mathematically 

 the superiority of this shape to 

 any other. 



In the last century Maraldi, and 

 after him Reaumur, examined the 

 form of the middle plate of the 

 entire comb, ^.e., the bottoms of 

 the cells, formed each of three 

 rhombs. 



At the instigation of Reaumur 

 the mathematician Konig, in 1739, 

 found hy calculation the most suit- 

 able, i.e., the most economical, of 

 all possible forms for the middle 

 plate, and ascertained that it must 

 consist of pyramids formed by three 

 rhombs having at the apex the 



angle 109° 28'. Maraldi had found 

 this as the very angle actually em- 

 ployed by the bees. 



These investigators show that ' 

 each cell represents a six-sided 

 column bounded at the middle 

 plate of the comb by a three-sided 

 pja'amid. The edges meeting at the 

 deepest point of the cell form angles 

 of 109° 28' ; other angles of the 

 same magnitude are enclosed by 

 the short side of the hexagonal 

 column and the two adjacent sides 

 of the rhombs. In the terminal 

 points of the long sides of the prism 

 there meet therefore, four edges at 

 angles of 70° 32'. 



The arrangement of the wax 

 plates which compose the entire 

 comb may therefore be formulated 

 as follows : 



1. On one edge there intersect 

 each other each time, three films, 

 and these form with each other 

 equal angles of 120°. 



2. At the terminal points of the 

 short sides of the prism meet in 

 each case four edges at angles of 

 109° 28'. 



3. In the terminal points of the 

 long sides of the prism four edges 

 cut each other at 70° 32'. 



These properties correspond al- 

 most exactly with the laws which 

 Plateau has discovered for his 

 equilibrium figures, namely : at a 

 liquid edge there intersect each 

 other never more than three films, 

 and these form with each other 

 equal angles,and when liquid edges 

 intersect each other in the interior 

 of the figure they are always four 

 in number, and form with each 

 other equal angles. 



