1870.] 



THE AMERICAN BEE JOURNAL. 



29 



and equiangular, for ordinate and dissimilar 

 figures did not please the bees themselves." 



"Now equilateral triangles, and squares, and 

 hexagons, ( neglecting other dissimilar figures 

 filling spaces, ) 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 triangles, 

 or by four squares, or by three hexagons ; but 

 three pentagons are less than sufficient, and 

 four are more tlian sufficient, to fill tlie space 

 around the point ; neither can three heptagons 

 be established so as to fill the space around 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 till up the space around a point, viz.: the 

 triangle, the square, and tlie hexagon, the bees 

 have wisely selected for their structure that 

 which contains most angles, suspecting indeed 

 that it could hold more honey than eitlier of the 

 others. 



The bees, forsooth, know only what is usefiil 

 to themselves, viz. : that the hexagon is greater 

 than the square or triangle, and can hold more 

 honey, an equal quantity of material being em- 

 ployed in the construct on of each. But we, 

 who profess to have more wisdom than the bees, 

 will investigate sometl ling even more reniarkabli', 

 viz : of three plane figures which are equilateral 

 and equiangular, and have equal perimeters, 

 that is always tlie greatest of all provided it be 

 included in a perimeter eqiuil to theirs. 



In 1712, Maraldi published in the Memoirks 

 DE li'AcADEMiE DES SciENCi':s, Paris 1712, page 

 299, a remarkable paper, in which is investigated 

 for the first time, the tern iual planes of the bee's 

 cell, which are now well known to be formed of 

 the faces of the rhombic dodecahedion He 

 appears to have believed the object of having 

 lozenges of the same form, as terminating planes, 

 was to enable bees to carry in their mind tlie 

 idea of one geometrical form only, in addition to 

 their original idea of the hexagon. The angles 

 of the lozenge are found by observation to be 

 110' and 70' ; and 10!)' 28' and 70' .32' by calcula- 

 tion. He gives also the following mean measure- 

 ments of tlie cells :— in a foot long of comb there 

 are 60 to 60 cells, about two lines for each cell, 

 and the depth of the cell is five lines. 



Reaumur appears to have been tiie first who 

 introduces 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 of Konig to calculate 

 the angle of the lozenge which should give the 

 least surface with a given volume. Konig 

 determined this angle at 109' 26', agreeing with 

 Maraldi within two minutes. 



Maclaurin published in the Philosophical 

 Transactions, 17-43, page .■)'i5, an elaborate geo- 

 metrical paper on the subject, in which he 

 proves that the tangent of the angle in question 

 is the square root of 2, and that it is the efore 

 equal to 109 ' 28' 16" ; and he compntes the saving 

 of wax as almost one-fourth part of the pains 

 and expense of wax they bestow, above what 

 was necessary for completing the parallelogram 

 side of the cells. 



L"Hullier, in 1781, published iu the Berlin 

 ^Memoirs page 277, an elaborate discussion of 

 the entire problem, in which he arrived at the 

 following results, alreadj^ found by Maclaurin" s 

 geometrical method. 



a, That the economy of wax is less than one- 

 fifth of what would make a flat base. 



b, That the economy of wax referred to the 

 total expenditure is rl-^^fit, so that the bees can 

 make 51 cells instead of 50, by the adoption of 

 the rhombic dodecahedron. 



He does not share in the enthusiasm of the 

 naturalist, but maintains and proves that the 

 mathematician could make cells of the same 

 form as those of the bees, wliich instead of using 

 only a miiiimiuii of wax, would use a minimum- 

 minimorum, so that five cells could be made of 

 less wax than that which now makes only four, 

 instead of fifty-one out of fiftJ^ 



Notwithstanding this conclusive decision in 

 favor of the mathematicians, the advocates of 

 final cause, and those who maintain that economy 

 of wax can create a new species, have both per- 

 sisted in using the bee's cell in illustration of 

 their respective theories, with a pertinacity tliiit 

 proves the persistent vitality of exploded theory. 



In fact the whole question of the economy of 

 wax, like other such questions, requires a 

 thorough sifting. To my mind it is evident that 

 economy of wax has nothing to do with the 

 making of the bee's cells ; but that this and other 

 l^roperties necessarily reside in the bee's cell, 

 because they are inherent i^roperties of the 

 rhombic dodecahedron. Tlie true cause of that 

 shape is, the crowding together of the bees at 

 work, jostling and elbowing each other, as was 

 first sliown by Buft'on. From this crowding 

 together they cannot help making cells with 

 diliedial angles of 120" of the rhombic dodeca- 

 hedron ; and the economy of wax has nothing 

 to do with the origen of the cells, but is a 

 geometrical projierty of the figure named — 

 Annals of Natural History, Third 8 ries, Vol. XI. 



[For tlie American Bee .Tournal.] 



Th^ Sting of the Honey Be^. 



]\Ir. Editor : — In the summer of 1868, at sev- 

 eral dift'erent times, when stung by a bee, I 

 noticed, that in extracting the sting a portion of 

 it would remain in the wound ; tliat the part 

 remaining fixed in the fiesli was much finer in 

 size and sometimes fully as lontr as the portion 

 removed, audit appeared also to be a tube pulled 

 out of the main sting, much in the manner of 

 the working of a telescope, and which I thought 

 to be the form of its construction, particu- 

 larly as I had read in several places that such 

 was its working. This led me to give the bee 

 sting a thorough examination recently, with a 

 powerful microscope then in my possession. 

 The result has jjroved to me several interesting 

 facts, which I have never seen published any 

 where, and thinking they might be interesting to 

 the readers of your valuable journal, I have 

 taken the liberty of detailing them and forward- 

 ing them to you. 



The bee sting, in the first place, is not a per- 



