530 THE FORMS OF TISSUES [ch. 



the old geometry and requiring the methods of Newton and Leibniz. 

 Whereupon Fontenelle, as Secretaire Perpetuel, summed up the 

 case in a famous judgment, in which he denied intelhgence to the 

 bees but nevertheless found them blindly using the highest mathe- 

 matics by divine guidance and command*. 



When CoHn Maclaurin studied the honeycomb in Edinburgh, a 

 few years after Maraldi in Paris, he proceeded to solve the problem 

 without using "any higher Geometry than was known to the 

 Antients," and he began his account by saying: "These bases are 

 fornled from Three equal Rhombus's, the obtuse angles of which 

 are found to be the doubles of an Angle that often offers itself to 

 mathematicians in Questions relating to Maxima and Minima. f" It 

 was an angle of 109° 28' 16", with its supplement of 70° 31' 44". 

 And this angle of the bee's cell determined by Maraldi, Koenig and 

 Maclaurin in their several ways, this angle which has for its cosine 1/3 

 and is double of the angle which has for its tangent V2, is on the 

 one hand an angle of the rhombic dodecahedron, and on the other 

 is that very angle of simple tetrahedral symmetry which the soap- 

 films within the tetrahedral cage spontaneously assume, and whose 

 frequent appearance and wide importance we havo already touched 

 upon J. 



That "the true theoretical angles were 109° 28' and 70° 32', 

 precisely corresponding ivith the actual measurement of the bee's cell,'' 

 and that the bees had been "proved to be right and the mathema- 

 ticians wrong," was long believed by many. Lord Brougham 



* La grande merveille est que la determination de ces angles passe de beaucoup 

 les forces de la Geometrie commune, et n'appartient qu'aux nouvelles Methodes 

 fondees sur la Theorie de I'lnfini. Mais a la fin les Abeilles en S9auraient trop, 

 et I'exces de leur gloire en est la ruine. II faut remonter jusqu'a une Intelligence 

 infinie, qui les fait agir aveuglemerit sous ses ordres, sans leur accorder de ces 

 lumieres capables de s'accroitre et de se fortifier par elles-memes, qui font I'honneur 

 de notre Raison." Histolre de VAcademie Royale, 1739, p. 3.j. 



t Colin Maclaurin, On the bases of the cells wherein the bees deposit their 

 honey, Phil. Trans, xlii, pp. 561-571, 1743; also in the Abridgement, viii, pp. 709- 

 713, 1809; it was characteristic of Maclaurin to use geometrical methods for 

 wellnigh everything, even in his book on Fluxions, or in his famous essay on the 

 equilibrium of spinning planets. Cf. also Lhuiller, Memoire sur le minimum du 

 cire des alveoles des Abeilles, et en particulier sur un minimum minimorum relatif a 

 cette matiere, Nouv. Mem. de VAcad. de Berlin, 1781 (1783), pp. 277-300. Cf. 

 Castillon, ibid, (commenting on Lhuiller); also Ettore Carruccio, Notizie storiche 

 eulla geometria delle api, Periodica di Mathematiche, (4) xvi, pp. 35-54, 1936. 



X Supra, p. 497. The faces of a regular octahedron meet at the same angle. 



