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within the range of his hypothesis. I would, however, submit that the 
bee finding the square root of two to four or five places of decimals is as 
marvellous as anything Mr. Darwin seeks to explain away. But though I 
do not suppose that the bee is acquainted with our decimal system of notation 
or can count the measures of its cell, I will concede that the structure of the 
lozenge of the bee’s cell gives not approximately, but most accurately, the 
square root of two and the square root of three, not numerically, but geo- 
metrically. If you take the side of one of the lozenges terminating the 
bee’s cell for a unit, you will find that half the smaller diagonal of the lozenge 
is accurately the square root of two, and half the larger diagonal is the square 
root of three. But conceding all this, supposing the bee knows the exact 
radius to choose in excavating its hemispheres on both sides of a wall of 
wax, Mr. Darwin sets the bee to the solution of a most difficult problem in 
geometry ; one, the difficulty of which he will only appreciate if he en- 
deavours to solve it with the aid of good mathematical instruments. The 
problem is this : — Given a wall ; to find a point on one side of the wall, sup- 
posing the wall of the same uniform thickness and bounded by parallel planes 
equidistant from three equidistant points on the other side. Let any of my 
auditory attempt the solution of this problem, even with line and compass, 
and they will learn to marvel only the more, if the bee does take this method of 
constructing its cell. But where are the bee’s compasses, where its accurate 
rulers ? I remember hearing the architect of the British Museum discoursing 
on the marvellous structure of the bee’s cell before the most distinguished 
architects in London. Some one talked about the bee excavating a hemi- 
spherical cell as the first step of the process. The learned lecturer asked 
where were the instruments which would enable the bee to make such a 
structure. Nay, he asked how many of his auditory could make an accurate 
circle without a pair of compasses, much less excavate a hemispherical cell 
out of a mass of wax. I bring this to your consideration to show the per- 
plexities and mistakes learned naturalists, such as Darwin, undoubtedly can 
fall into. Because here mathematical considerations enable me to demon- 
strate the incredibility of such explanations of the bee’s marvellous instinct. 
Mr. Warington has told us a good deal about the transmission of acquired 
habits by animals. I do not think, however, that he demonstrated such 
habits to be transmissible. But let us assume that they are. Let us assume, 
too, that some mathematical bee solved the problem of the perfect cell. How 
was this acquired habit transmitted ? Not by that bee, assuredly. The hive- 
making bees are females indeed, but they are imperfect, sterile females, 
incapable of propagating their species. Transmission of acquired habits, 
therefore, could not have anything to do with the perfection of the bee s cell. 
Tried here by this fact alone, the crucial instance selected by Mr. Darwin 
fails. It becomes utterly incredible, even by the laws of his own hypothesis. 
Again, why does the bee, showing such economy in the construction of 
ordinary cells, ignore this economy altogether when forming the cell of the 
future queen ? — a bee that will never excavate a cell, and yet be the parent of 
a whole hive of architects ? Again, we may ask how does Mr. Darwin’s hypo- 
