150 



THE AMERICAN BEE JOURNAL AND GAZETTE. 



of the hive-bee from that of other insects, is the 

 manner in which the cells are arranged in a 

 double scries. The combs of the wasp or the 

 hornet are single, and are arranged horizon- 

 tall}', so that their cells are vertical, with the 

 mouths downward, and the bases upward, the 

 united bases forming a floor on which the nurse 

 WMsps can walk while feeding the young in- 

 closed in the row of cells immediately above 

 them. 



Such, however, is not the case with the hive- 

 bee. As every one knows who has seen a bee- 

 comb, the cells are laid nearly horizontally, and 

 in a double series, just as if a couple of thimbles 

 were laid on the table, with the points touching 

 each other, and their mouths pointing in oppo- 

 site directions. Increase the number of 

 thimbles, and there will be a tolerable imitation 

 of a bee-comb. 



There is another point which must now be 

 examined. If the bases of the cells were to be 

 rounded like those of the thimbles, it is clear 

 that they would hive but little adhesion to each 

 other, and that a large amount of space would 

 be wasted. The simplest plan of obviating 

 these defects is evidently to square off the 

 rounded bases, and to fill up the ends of each 

 cell with a hexagonal flat plate, which is actu- 

 ally done by the wasp. 



If, however, wc look at a piece of bee-comb, 

 we shall find that no such arrangement is em- 

 ployed, but that the bottom of each cell is formed 

 into a kind of three-sided cup. Now, if we 

 break away the walls of the cell so as onl}' to 

 leave the bases, we shall see that each cup con- 

 sists of three lozenge-shaped plates of wax, all 

 the lozenges being exactly alike. 



These lozenge-shaped plates contain the key 

 to the bee-cell, and their properties will there- 

 fore be explained at length. Before doing so, 

 I must acknowledge my thanks to the Rev. 

 Walter Mitchell, vicar and hospitaler at St. 

 Bartholomew's Hospital, who has long exercised 

 his well-known mathematical powers on this 

 subject, and has kindly supplied me with the 

 outline of the present history. 



If a single cell be isolated, it will be seen that 

 the sides rise from the outer eilges of the three 

 lozenges above mentioned, so that there are of 

 course six sides, the transverse section of which 

 gives a perfect hexagon. Many years ago, 

 Maraldi, being struck with the fact that the 



A 70O32' 



F 



109O28' 



70O32' 



109^28' 



70O32' 



Fig. 1. 



lozenge-shaped plates always had the same 

 angles, took the trouble to measure them, and 



found that in each lozenge the large angles 

 measured 109■-28^ and the smaller 70-32^ the 

 two together making 180°, the equivalent of 

 two right angles. He also noted the fact that 

 the apex of the three-sided cup was formed by 

 the union of three of the greater angles. The 

 three united lozenges are seen at Fig. 1. 



Sometime aflerward, Reaumur, thinking that 

 this remarkable uniformity of angle might have 

 some connection with the wonderful economy 

 of space which is observable in the bee-comb, 

 hit upon a very ingenious plan. Without men- 

 tioning his reasons for the question, he asked 

 Koenig, the mathematician, to make the follow- 

 ing calculation : Given a hexagonal vessel 

 terminated by three lozenge-shaped plates, what 

 arc the angles which would give the greatest 

 amount of space with the least amount of ma- 

 terial ? 



Koenig made his calculations, and found that 

 the angles were 129^26^ and 70^^34^, almost pre- 

 cisely agreeing with the measurements of 

 Maraldi. The reader is requested to remember 

 these angles. 



Reaumur, on receiving the answer, concluded 

 that the bee had very nearly solved the difficult 

 mathematical problem, the difference between 

 the measurement and the calculation being so 

 small as to be practically negatived in the 

 actual construction of so small an object as the 

 bee-cell. 



Mathematicians were naturally delighted with 

 the result of the investigation, for it showed 

 how beautifully practical science could be aided 

 by theoretical knowledge; and the construction 

 of the bee-cell became a famous problem in the 

 economy of nature. In comparison with the 

 honey which the cell is intended to contain, the 

 wax is a rare and costly substance, secreted in 

 very small quantities, and requiring much time 

 and a large expenditure of honey for its pro- 

 duction. It is therefore essential that the quan- 

 tity of wax emploj'ed in making the comb 

 should be as little, and that of the honey which 

 could be stored in it as great, as possible. 



For a long time these statements remained 

 uncontroverted. Any one with the proper in- 

 struments could measure the angles for himself, 

 and the calculations of a mathematician like 

 Ktenig would hardly be questioned. However, 

 Maclaurin, the well-known Scotch mathema- 

 tician, was not satisfied. The two results very 

 nearly tallied with each other, but not quite, 

 and he felt that in a mathematical question pre- 

 cision was a necessity. So he tried the whole 

 question himself, and found Muraldi's measure- 

 ment correct— namely, 109^28', and 70-'32'. 



He then set to work at the problem which 

 was worked out by Kcenig, and found that the 

 true theoretical angles were 109'^'28' and 70^32', 

 precisely corresponding with the actual mea- 

 surement of the bee-cell. 



Another question now arose. How did 

 this discrepancy occur ? On investigation, it 

 was found that no blame attached to Ka?nig, 

 but that tlie error lay in the book ,of Logarithms 

 wliich he used. Tlius a mistake in a mathema- 

 ticial work was accidentally discovered by mea- 

 suring the angles of a bee-cell — a mistake suffl- 

 i cicntly great to have caused the lo«f of a skip 



