164 BEES AND BEE-KEEPING. 



sufficiently large, I have found extremely useful for 

 lecturing purposes. 



Supposing that comb equals its ideal or theoretical 

 form, Cramer's' 55 ' very elegant geometrical demonstra- 

 tion shows that the angles of the rhomb must be 

 such that their two diagonals [ef, gh, E, Fig. 36) 

 are to each other in the ratio of the side and diagonal 

 of a square ; or, to use Cramer's less popular, though 

 equivalent, form, the obtuse angle of the rhomb must 

 be such that its half has for its tangent ^J 2. This 

 is only true of the angle 54° 44' 8". The two angles 

 of the rhomb are, therefore, double the foregoing, 

 viz., 109 28' 16", and its supplement, 70 31' 44 // . 

 Thence, as geometric sequences, the angles at which 

 the sides of the prism [s, D) are cut at the base, in 

 order to fit on to the rhombs, is precisely equal to 

 those of the rhombs themselves ; and, further, the solid 

 angle formed at the apex of the pyramid, by the 

 meeting of the three obtuse angles (0, 0, o, C) of 

 the rhombs will be equal to the solid angles formed 

 by the meeting of one obtuse angle (0) of the 

 rhomb, and the two similar obtuse angles (o' t ', C) 

 of the sides. It is also true, that no other angles 

 give these equalities, which every geometrician will 

 recognise as affording the nearest approach to the 

 form of the larva possible to the number of plane 

 surfaces composing the cell. 



It has sometimes been thought that these angles 

 gave greater space than any other; but this is an 

 error, as F will show ; for here the actual inclination 



* Hutton's "Mathematical Recreations," or Huber's " Nouvelles 

 Observations," vol. ii., page 475. 



