712 PHILOSOPHICAL TRANSACTIONS. [aNNO J 743. 



base flat, and not of the pyramidal form described above, then, besides com- 

 pleting the parallelograms cgnk and bmnk, the surface of the base had been 

 3cB X AK ; what they really do form amounts in surface to the same parallelo- 

 grams, and 3CB X ah : the savings therefore amount to 3cb x ak — ah 



= 3cB X AH X ^ — , which is almost a 4th part of the pains and expence 



of wax, they bestow above what was necessary for completing the parallelogram 

 sides of the cells : and at the same time they seem also to have other advan- 

 tages from this form, besides the saving of their wax ; such as a greater strength 

 of the work, and more convenience for moving in these larger solid angles. 



It remains that we should show, that the plane angles clb, cln, and bln, are 

 equal to each other. We before found, that kl : al :: kc : BC :: ka (= -J-kc): 

 AC ; consequently kl : ka :: al : ac, and the triangles lka, l^c, are similar: 

 therefore lk : al :: al : lc :: kc : bc :: 1 : v^3, and lc = 3lk. With the 

 centre l and radius lc, describe, in the plane cgnk, fig. 3, the semicircle dcp, 

 meeting the line kn, in d and p ; join cp and cd, and let La be perpendicular 

 to CP in a, then will the angle cdk be equal to qlp, and we shall have pq : La :: 

 PC : DC :: -/PK : \^dk :: 'V^lc -|- lk : ■/lc — lk :: v'4 : \/2 :: t/2 : 1 :: ac : 

 AL. Consequently the angle aLP = alc, and clp =: clb, or the obtuse angle 

 of the Rhombus clbI is equal to clp, the obtuse angle of the trapezium ; and 

 consequently, the 3 plane angles that form the solid angle at l, or the apex at 

 1, are equal to each other : from which it is obvious, that the 4 acute plane 

 angles, which form the solid angle at c or b, are likewise equal among them- 

 selves. 



Though M. Maraldi had found, by his mensuration, these obtuse angles to 

 be of about 1 10 degrees; the small difference between this and the 109° 28' 16", 

 just found by calculation, seems to have been either accidental, or owing to the 

 difficulty of measuring such angles with exactness : besides that he seems to 

 admit the real equality of the several plane angles, that form as well the apex, 

 as the other solid ones we have been treating of. And, as to the small differ- 

 ence between our angle and that determined by Mr. Koenig, who first con - 

 sidered this problem, but has not yet published his demonstration of it, that 

 can only be owing to his not carrying on his computation so far, and would 

 scarcely have been worth the mentioning, were it not yet in favour of the 

 practice of these industrious little insects ; and did it not therefore give us 

 ground to conclude, that in general, and when the particular form and circum- 

 stance of the honey-comb does not require a variation from their rule, the bees 

 do truly construct their cells of the best figure, and that not only nearly, but 

 with exactness ; and that their proceeding could not have been more perfect 



