DUBLIN NATURAL HISTORY SOCIETY. 131 



hexagon species. Now it is thus evident that they construct these by 

 a certain geometrical foresight; for they consider it fit that all the 

 figures should cohere together and have common sides, lest anything, 

 falling into the intervening spaces, should spoil and corrupt their work. 



"Hence, three rectilinear and ordinate figures can effect what is 

 proposed — I mean ordinate figures which are equilateral and equiangu- 

 lar, for ordinate and dissimilar figures did not please the bees them- 

 selves. JSTow, equilateral triangles, and squares, and hexagons (ne- 

 glecting other dissimilar figures filling space), may be placed next each 

 other, so as to have common sides — other ordinate figures cannot ; for 

 the space about the same point is filled, either by six equilateral tri- 

 angles, or by four squares, or by three hexagons — but three pentagons 

 are less than sufiicient, and four are more than sufficient to fill the 

 space round a point — neither can three heptagons be established, so as 

 to fill the space round a point.*' 



" The same reasoning will apply much more to figures having a 

 greater number of sides. There being, then, three figures, which, of 

 themselves, can fill up the space round a point, viz., the triangle, the 

 square, and the hexagon; the bees have wisely selected for their struc- 

 ture, that which contains most angles, suspecting, indeed, that it could 

 hold more honey than either of the others. 



" The bees, forsooth, know only what is useful to themselves, viz., 

 that the hexagon is greater than the square or triangle, and can hold 

 more honey, an equal quantity of material being employed in the con- 

 struction of each; but we, who profess to have more wisdom than the 

 bees, will investigate something even more remarkable, viz., that of plane 

 figures which are equilateral and equiangular and have equal peri- 

 meters, that is always the greatest which consists of most angles, and 

 the circle is the greatest of all, provided it be included in a perimeter 

 equal to theirs." — Pappus. 



In 1712, Maraldi published in the Memoir es de VAcademie des Sci- 

 ences, Paris, 1712, page 299, a remarkable paper, in which is investi- 

 gated, for the first time, the terminal planes of the bees' cell, which 

 are now well known to be formed of the faces of the rhombic dodecahe- 

 dron. He appears to have believed, that the object of having lozenges 

 of the same form, as terminating planes, was to enable the bees to 

 carry in their mind the idea of one geometrical form only, in addition 

 to their original idea of the hexagon. The angles of the lozenge are 

 found by him to be 110"^ and 70°, by observation; and 109° 28' and 

 70° 32', by calculation. He gives, also, the following mean measure- 

 ments of the cells : — In a foot long of comb, there are from 60 to 66 cells, 

 about two lines for each cell, and the depth of the ceU is five lines. 



Eeaumur appears to have been the first who introduced the fantastic 

 idea of economy of wax, as the motive cause of the peculiar shape of the 

 terminating planes, and, not being a geometer, he obtained the assistance 



* The proofs of these asaertipns are omitted in this translatipn. 



