254 Geometrical Construction of the Bees Cell. [Nov. 26, 



A side of the lozenge is thus equal to three times the difference 

 between the two parallel edges of the trapezium. If a line equal to 

 the side of the hexagon be bisected by an equal line at right angles, 

 and the extremities of both joined, half the resulting line will be 

 equal to x, so that the side of one of the lozenges is three times the half 

 side of a square whose diagonal equals the side of the hexagon. It is 

 moreover manifest that 



which is positive for x — <^ y-g anc ^ therefore the surface u is the least 



possible. As the angle at top of the trapezium has for its tangent 



h ' 



-=2V2, this angle is 70° 31' 44", the supplement of 109° 28' 16" for 



the obtuse angle of the lozenge, found by Maclaurin. 



It is manifest that E = /in/ 3 and e— v^2E, or the longer diagonal, is 

 equal to the diagonal of a square whose side is the shorter diagonal 

 of the lozenge. Also the acute angle of the lozenge is equal to the 

 acute angle of the trapezium, for the tangent of half the former 



is — -.* 

 ^2 



From these results a model of the bee's cell can be easily con- 

 structed, with the aid of a pair of compasses, as follows : — 



1. Inscribe an equilateral triangle in a hexagon ; a side of this 

 triangle is the long diagonal of the lozenge; bisect this, and the 

 diagonal of a square erected on the half is the shorter diagonal of the 

 lozenge. 



2. Draw six parallel lines at distances equal to the side of the 

 hexagon, and a straight line perpendicular to them from the second 

 of the parallel lines ; inflect a straight line equal to a side of the 

 lozenge above constructed, and repeat this process until six tra- 

 peziums are finished. 



3. On folding these trapeziums a hexagonal prism is formed into 

 which three lozenges equal to that constructed will accurately fit, and 

 the entire structure will be completed. Models in cardboard have 

 been easily made in this way, and I had one afterwards made in 

 glass. 



* As tan 6= 2tan ^ if tan 10=4=, tan v = 2Vo. 

 l-tan 2 |e *2 



