VII 



OF THE BEE'S CELL 



535 



But AC and FT do not vaiy, while BS varies with the position of E. 

 Accordingly, AC{PT + RS), or PE. AC-BE. BC, is a minimum when RS 

 vanishes: that is to say, when E coincides with L. 



"Therefore A LCI is the Rhombus of the most advantageous Form in respect 

 of Frugality, when BL is to PL as BC to .46'." 



Again, since OB = BC, and OP = PB, BC^ = 4PB\ and PC^ = 3PB^, and 

 AC = 2PC = 2V3.PB. 



Therefore 

 and, by hypothesis. 



and 



Therefore 



BC:AC::2PB:2VS.PB:: 1 : Vs, 



"That is, the angle CLP is that whose Tangent is to the Radius as V2 

 is to 1, or as 1-4142135 to 1-0000000; and therefore is of 54° 44' 08", and 

 consequently the Angle of the Rhombus of the Best Form is that of 

 109° 28' 16".". 



When we have thus ascertained that the characteristic angles of 

 the rhombs are 109° 28' 16" and its supplement 70° 31' 44", the 

 cosine of which latter angle is 1/3, the construction of a model is 

 of the easiest. 



Fig. 207. Construction of a model of the bee's cell 



On AD make AB = BC = CD. Let AF = AD meet the perpen- 

 dicular BE in F. Then the angle BAF (whose cosine = 1/3, or 

 whose tangent = 2\/2) = 70° 31' 44". Complete the rhomb ADGF, 

 and repeat three times as indicated. Make a developed hexagonal 

 prism with sides ab, be, = BF . Cut away angles bb'a, bb'c, etc., 

 = BAF. Fold, and attach together. 



