Everybody knows that the cells
of the honey-comb are 6-sided, and I presume most people know
why they are 6-sided. If they were square, the young bee would
have a much more uncomfortable cradle in which to grow up, and
it would take a much greater space to accommodate a given number
of bees. This last would, of itself, be a fatal objection; for
to have the greatest benefit of the accumulated animal heat of
the brood, they must be closely packed together. This is not
only the case with the unhatched bees, but with the bees of a
whole colony in winter; when each bee is snugly ensconced in
a cell, they occupy less room than they could by any other arrangement.
 If
the cells were round, they could be grouped together much in
the same way as they are now; viz., one in the center, and 6
all around it, equally distant from the central one, and from
each other, like the cut, in the figure A; but even then, the
circles will leave much waste room in the corners, that the bees
would have to fill with wax. |
At B, we see the cells are nearly as comfortable for the young
bee as a round one would beof course. I mean from our point
of view, for it is quite likely that the bees know just what
they need a great deal better than we doand, at the same
time, they come together in such a way that no space is
left to be filled up at all. The bees, therefore, can make the
walls of their cells so thin that they are little more than a
silky covering, as it were, that separates each one from it's
neighbor. It must also be remembered that a bee, when in his
cell, is squeezed up, if we may so term it, so as to occupy much
less space than he otherwise would; and this is why the combined
animal heat of the cluster is so much better economized in winter,
when the bees have a small circle of empty cells to cluster in,
with sealed stores all around them.
But, my friends, this is not half of the ingenuity displayed
about the cell of the bee. These hexagonal cells must have some
kind of a wall or partition between the inmates of one series
of cells, and those in the cells on the opposite side. If we
had a plain partition running across the cell at right angles
with the sides, the cells would have flat bottoms which would
not fit the rounded body of the bee, besides leaving useless
corners, just as there would have been if the cells had been
made round or square. Well, this problem was solved in much the
same way, by making the bottom of the cell of three little lozenge-shaped
plates. In the figure below we give one of these little plates,
and also show the manner in which three of them are put together
to form the bottom of the cell.
 If
the cells were round, they could be grouped together much in
the same way as they are now; viz., one in the center, and 6
all around it, equally distant from the central one, and from
each other, like the cut, in the figure A; but even then, the
circles will leave much waste room in the corners, that the bees
would have to fill with wax. |
 Now,
if the little lozenge plates were square, we should have much
the same arrangement, but the bottom would be too sharp-pointed,
as it were, to use wax with the best economy, or to best accommodate
the body of the infantile bee. Should we, on the contrary, make
the lozenge a little longer, we should have the bottom of the
cell too nearly flat, to use wax with most economy, or for the
comfort of the young bee. Either extreme is bad, and there is
an exact point, or rather a precise proportion that the width
of this lozenge should bear to the length. This proportion has
been long ago decided to be such that, if the width of the lozenge
is equal to the side of a square, the length should be exactly
equal to the diagonal of this same square. This has been proven
by quite an intricate geometrical problem; but a short time ago,
while getting out our machine for making the foundation, I discovered
a much shorter way of working this beautiful problem. |
In the figure above, let A B C D represent the lozenge at the
bottom of the cell, and A C, the width, while B D is the length
of said lozenge. Now, the point I wish to prove is, that A C
bears the same proportion to B D that the side of a square does
to the diagonal of the same square.
THE MATHEMATICS OF THE HONEY-COMB.
 Suppose
we have a cubical block, E B C G F, and that we pile small blocks
on its sides as shown, so as to raise pyramids of such an inclination
that a line from any apex to the next, as from A to D, will just
touch the edge of the cube, B C. Now A C D B is the geometric
lozenge we are seeking. Its width, B C, is equal to one side
of the square, E B F H, for it is one side of the cube. Now,
to prove that A D is equal to the diagonal E F, we will use the
diagram below. |
 Let
E B F H represent the cube, and the dotted lines the pyramids.
If the pyramids are so made that the line A D is a straight,
continuous one, it is evident, by a little reflection, that the
angles A and D will be right angles. If this is so, A D is exactly
equal to E F, the point we were to prove. Now, referring to the
former figure, if we should go on building these pyramids on
all sides of the cube, we will have the beautiful geometrical
figure called the rhombic dodecahedron; it is so called, because
it is a solid figure having 12 equal sides, and each side is
a rhomb, or lozenge, such as we have described. Where the obtuse
angles of three of these rhombs meet, as at C, we shall have
the exact figure of the bottom of a honey-comb cell. A picture
of the geometrical solid we have mentioned is given below. |
 How
does it come that the bees have solved so exactly this intricate
problem, and know in just what form and shape their precious
wax can be used, so as to hold the most honey, with the very
least expenditure of labor and material? Some are content with
saying that they do it by instinct, and let it drop there; but
I believe God has given us something farther to do than to invent
names for things, and then let them drop. By carefully studying
the different hives in a large apiary, we see that not all of
them build comb precisely alike, and not all colonies are equally
skilled in working wax down to this wonderful thinness. Some
bees will waste their precious momentsand waxin making
great, awkard lumps of wax; coarse, irregular cells; crooked,
uneven comb, etc., with very bad economy either for the production
of brood or for the storing of honey; while others will have
all their work so even and true, and so little wax will be wasted,
that it is wonderful to contemplate the regularity and system
with which the little fellows have labored. Now, it does not
require any great amount of wisdom to predict that the latter
would, in a state of nature, stand a far better chance of wintering
than the ones that were wasteful and irregular in their ways
of doing things. If this be the case, those queens whose progeny
were best laborers, most skillful wax-workers, as well as most
energetic honey-gatherers, would be most sure to perpetuate themselves,
while the others would, sooner or later, become extinct. I have
found more of a tendency in bees to sport, or to show queer peculiarities,
than in any other department of the animal or vegetable kingdom.
They vary in color, in shape, in size, in disposition, in energy;
and almost every colony, if studied closely, will be found to
have some little fashion or way of doing things, different from
all the rest in the apiary. Now, when we take into account tha
fact that many generations can be reared in a single summer,
we see how rapidly, by fostering and encouraging any desirable
trait or disposition, the bees may be molded to our will. The
egg that is laid by a queen to-day may, by proper care, be made
to produce a queen laying eggs of the same kind herself, in the
short time of only 25 days, as I have explained heretofore. Well,
if we should pick out a queen whose progeny made the thinnest
comb, and rear others from her, doing the same thing for several
generations, we should probably get bees whose combs would break
down by the weight of the honey. In a state of nature this extreme
would correct itself, as well as the other; but the point I wish
you to see is right here: Geometrical accuracy in the shape
of the cells can never be overdone, and can be reached only by
absolute perfection; and this absolute perfection, the bees have
been constantly aiming at through endless ages. Is it any
thing strange, my friends, that the bees have got the honey-comb
pretty nearly right by this time? I will give you a little story,
and one which has been very interesting to me, from page 150,
Vol. II., American Bee Journal. |
If a single cell
be isolated, it will be seen that the sides rise from the outer
edges 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 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 degrees 28', and the smaller 70
degress 32', the two together making 180 degress, 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 in the figure below.
Some time afterward, Reaumur, thinking that this remarkable uniformity
of angle might have some connection with the wonderful economy
of space which is observed in the bee-comb, hit upon a very ingenious
plan. Without mentioning his reasons for the question, he asked
Koenig, the mathematician, to make the following calculation:
Given a hexagonal vessel terminated by three lozenge-shaped plates,
what are the angles which would give the greatest amount of space
with the amount of material?
Koenig made his calculations, and found that the angles were
109 degrees 26' and 70 degrees 34', almost precisely 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 negative in the actual construction of so small
an object as the bee-cell.
 Mathematicians were naturally
delighted with the results of the investigation, for it showed
how beautifully practical science could be aided by theoretial
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 production.
It is, therefore, essential that the quantity of wax employed
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 instruments could measure the angles for
himself, and the calculations of a mathematician like Koenig
would hardly be questioned. However, Maclaurin, the well-known
Scotch mathematician, was not satisfied. The two results very
nearly tallied with each other, but not quite, and he felt that,
in a mathematical question, precision was a necessity. So he
tried the whole question himself, and found Maraldi's measurement
correctnamely, 109 degrees 28', and 70 degrees 32'.
He then set to work at the problem which was worked out by Koenig,
and found that the true theoretical angles were 109 degrees 28'
and 70 degrees 32', precisely corresponding with the actual measurement
of the bee-cell.
Another question now arose. How did this discrepancy occur? On
investigation, it was found that no blame attached to Koenig,
but that the error lay in the book of Logarithms which he used.
Thus a mistake in a mathematical work was accidentally discovered
by measuring the angles of a bee-cella mistake sufficiently
great to have caused the loss of a ship whose captain happened
to use a copy of the same Logarithmic tables for calculating
his longitudes. |
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