330 



THE FORMS OF TISSUES 



[CH. 



appears to have thought of the impossibility of measuring such a 

 thing as the end of a bee's cell to the nearest minute." At any 

 rate, it now occurred to Reaumur (as curiously enough, it had not 

 done to Maraldi) that, just as the closely packed hexagons gave 

 the minimal extent of boundary in a plane, so the actual solid 

 figure, as determined by Maraldi, might be that which, for a given 

 solid content, gives the minimum of surface : or which, in other 

 words, would hold the most honey for the least wax. He set this 

 problem before Koenig, and the geometer confirmed his conjecture, 

 the result of his calculations agreeing within two minutes (109° 26' 

 and 70° 34') with Maraldi's determination. But again, Maclaurin* 

 and LhuiUert, by different methods, obtained a result identical 

 with Maraldi's ; and were able to shew that the discrepancy of 

 2' was due to an error in^ Koenig's calculation (of tan 6 = \/2), 

 y — that is to say to the imper- 



fection of his logarithmic tables, — 

 not (as the books say J) "to a 

 mistake on the part of the Bee." 

 "Not to a mistake on the part of 

 Maraldi" is, of course, all that we 

 are entitled to say. 



The theorem may be proved as 

 follows : 



ABCDEF, abcdef, is a right 

 prism upon a regular hexagonal base. 

 The corners BDF are cut ofE by 

 planes through the lines AC, CE, 

 EA, meeting in a point V on the 

 axis VN of the prism, and intersect- 

 ing Bh, Dd, Ff, at X, Y, Z. It is 

 evident that the volume of the figure 

 thus formed is the same as that of 



the original prism with hexagonal 

 Fig. 132. & r & 



ends. For, if the axis cut the 



hexagon ABCDEF in N, the volumes ACVN, ACBX are equal. 



* Phil. Trans, xlu, 1743, pp. 5(55-571. t ^^em. de VAcad. de Beilin, 1781. 



J Cf. Gregory, Examples, p. 106, Wood's Homes without Hands, 1865, p. 428, 

 Mach, Science of Mechanics, 1902, p. 453, etc., etc. 



