118 Mr. J. W. L. Glaisher on the Form of the Cells of Bees. 



part of the walls adjoining, the six small triangles, which are 

 formed by a line drawn parallel to the base through the points 

 where the rhomboidal plates cut the walls. This is manifest 

 upon inspection ; and I have tried it by weighing equal parts in 

 superficial extent, as far as it was possible, of the base and of 

 the sides, and uniformly found the latter sensibly lighter. It 

 did not seem that the proportion was always the same ; but I 

 never found the difference less than in the proportion of 3 to 2. 

 The thickness of the walls varies much more than that of the 

 base in different combs. But any considerable difference be- 

 tween the two portions at once destroys the argument of M. 

 L'Huillier. If it is as 3 to 2, then the saving is nearly an 

 eighth upon the thicker part, and consequently about -^ instead 

 of -jfc of the whole/'' 



The ratio of one ninth is found by comparing the bottoms of 

 the cells, the bottom of the actual cell being taken as the three 

 rhombs and six triangles, and the bottom of the hypothetical 

 prismatic cell as the hexagonal floor and the portions of the sides 

 below a horizontal plane through highest points of the rhombs 

 (supposing the cell placed with its axis vertical and mouth upwards); 

 and the ratio of -^ follows by considering the ratio of the saving 

 to the whole amount of wax, the bottom (defined as above) being 

 supposed half as thick again as the rest of the cell. 



The above quotations contain, I venture to think, as striking 

 instances of bad reasoning as are often met with in writings 

 relating to mathematical subjects. The comparison is between 

 two kinds of cells ; the one requires an amount n of wax, the 

 other an amount n + ifon; and the amount saved is -^n, that 

 is to say, if they built fiat- bottomed cells, the bees would have 

 to manufacture ^ more wax than in reality they do manufac- 

 ture. If a house with a V-roof costs £1000, and with a flat 

 roof £1020, the builder who prefers the former saves £20 out 

 of an expenditure of £1000, or -J of what he spends*; his 

 saving is not a fifth of his expenditure because the walls cost 

 £900, and are the same for both. Lord Brougham says, "Why 

 is the wax of the wall to be imported into the calculation, with 

 which it has nothing to do ? " and the reply is, " Why is the wax 

 of the wall to be left out of the calculation ? it is manufactured 

 by the bees in exactly the same way as is the wax for the rhombs 

 and triangles, and is part of the cell built by the bees/'' In 

 effect, Lord Brougham replaces "the bees must have walls for their 



* Or, to state the same thing differently, if m be the wax required for 

 the hypothetical cell, m—^m represents the amount used in the real cell, 

 so -that the saving is T \ of the amount that would have been needed for 

 the hypothetical cell ; it is because Lhuillier so regarded it that he writes 

 w \ and not T V • 



