198 The Book of Bugs. 



and half a mind that they have no sense at all, but are 

 pretty little automata wound up and set a-going at an 

 angle of sixty degrees. If you will sit tight now and 

 hold fast, I will try to show you how far advanced they 

 would have to be in plane and solid geometry to lay off 

 one of those three diamond-shaped facets that close the 

 end of the six-sided cell. If we are going to be mathe- 

 matical we might as well be hanged for an old sheep as 

 killed for a young lamb, so we will call the diamond- 

 shaped facet a rhomb, an equilateral parallelogram hav- 

 ing oblique angles. Now, assuming that the comb 

 equals its ideal form, to close up a cell each of the three 

 rhombs must have its wide angle I beg your pardon, 

 obtuse, I should have said must have its obtuse angle 

 of such size that half of it has for its tangent the square 

 root of 2. That quite clear to you? Some day, when 

 you are not otherwise employed, you may spend a pleas- 

 ant afternoon extracting the square root of 2. I will 

 put it another way. The diagonals of each rhomb must 

 be to each other as the side and the diagonal of a 

 square. One can imagine the first bee that landed 

 ciphering out this problem on the blackboard, thumb- 

 ing over its book of logarithms till it finally set down: 

 " Tan. square root of 2 equals 54 deg. 44 min. 8 sec."; 

 multiplying that by 2 to get the whole obtuse angle, 

 109 deg. 28 min. 16 sec.; subtracting that from 180 

 deg. to get the acute angle, which is 70 deg. 31 min. 44 

 sec., and then dusting the chalk off its clothes with a 

 happy sigh and hunting up the scissors to cut out a pat- 

 tern to fit a hexagon whose diameter is one-fifth of an 

 inch. 



But wait. There's more. Maraldi measured the 

 angles of these rhombs and then gave the problem to 

 Koenig to solve. Koenig's calculations for the pyramid 



