Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

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Arch, suspended from the points A and C. If this figure be in-
^ verted, preserving the same points of contact, they will
remain in equilibrio. It will indeed be that kind of equi¬
librium which will admit of no disturbance, and which
may be called a tottering equilibrium. If the form be al¬
tered in the smallest degree, by varying the points of con¬
tact (which indeed are points in the figure of equilibration),
the magnets will no more recover their former position
than a needle, which we had made to stand on its point,
will regain its perpendicular position after it has been
disturbed.
But if we suppose planes hi, &c. drawn through
the points of mutual contact a, b, c, each bisecting the angle
formed by the lines that unite the adjoining contacts (fig,
for example, bisecting the angle formed by ab,b c), and if
we suppose that the pieces are changed for others of the
same weights, but having flat sides, which meet in the
planes de, fg, hi, &c., it is evident that we shall have an
arch of equilibration, and that the arch will have some sta¬
bility, or will bear a little change of form without tumbling
down: for it is plain that the equilibrium of the original
festoon obtained only in the points a, b, c, of contact,
where the pressures were perpendicular to the touching
surfaces; therefore, if the curve a, b, c, still passes through
the touching surfaces perpendicularly, the conditions that
are required for equilibrium still obtain. The case is
quite similar to that of the stability of a body resting on
a horizontal plane. If the perpendicular through the
centre of gravity falls within the base of the body, it will
not only stand, but it will require some force to push it
over. In the original festoon, if a small weight be added
in any part, it will change the form of the curve of equi¬
libration a little, by changing the points of mutual con¬
tact. This new curve will gradually separate from the
former curve as it recedes from A or C. In like manner,
when the festoon is set up as an arch, if a small weight be
laid on any part of it, it will bring the whole to the
ground, because the shifting of the points of contact will
be just the contrary to what it should be to suit the new
curve of equilibration ; but if the same weight be laid on
the same part of the arch now constructed with flat joints,
it will be sustained if the new curve of equilibration still
passes through the touching surfaces.
17. These conclusions, which are very obviously de-
ducible from the principle of the festoon, show us, without
any further discussion, that the longer the joints are, the
greater will be the stability of the arch, or that it will re¬
quire a greater force to break it down. Therefore it is of
the greatest importance to have the arch-stones as long as
economy will permit; and this was the great use of the
ribs and other apparent ornaments in the Gothic architec¬
ture. The great projections of those ribs augmented their
stiffness, and enabled them to support the unadorned com¬
partments of the roof, composed of very small stones, sel¬
dom above six inches thick. Many old bridges are still
remaining, which are strengthened in the same way by
ribs.
Having thus explained, in a very familiar manner, the
stability of an arch, we proceed to give the same popular
account of the general application of the principle.
ap- 18. Suppose it be required to ascertain the form of an
tHl' arch which shall have the span AB (fig. 7), and the height
1* 8, and which shall have a road-way of the dimensions
CDE above it. Let the figure ACDEB be inverted, so
as to form a figure AcrfeB. Let a chain of uniform
thickness be suspended from the points A and B, and let
it be of such a length that its lower point will hang at, or
rather a little below, fi corresponding to F. Divide AB
into a number of equal parts, in the points 1, 2, 3, &c.
and draw vertical lines, cutting the chain in the corre¬
sponding points 1, 2, 3, &c. Now take pieces of another Arch.
chain, and hang them on at the points 1, 2, 3, &c. of the
chain A/B. This will alter the form of the curve. Cut
or trim these pieces of chain, till their lower ends all co¬
incide with the inverted road-way cde. The greater
lengths that are hung on in the vicinity of A and B will
pull down these points of the chain, and cause the middle
point f (which is less loaded) to rise a little, and will
bring it near to its proper height.
It is plain that this process will produce an arch of per¬
fect equilibration; but some further considerations are ne¬
cessary for making it exactly suit our purpose. It is an
arch of equilibration for a bridge that is so loaded that
the weight of the arch-stones is to the weight of the mat¬
ter with which the haunches and crown are loaded, as the
weight of the chain A/B is to the sum of the weights of
all the little bits of chain very nearly. But this propor¬
tion is not known beforehand ; we must therefore proceed
in the following manner:—Adapt to the curve produced
in this way a thickness of the arch-stones as great as are
thought sufficient to insure stability; then compute the
weight of the arch-stones, and the weight of the gravel or
rubbish with which the haunches are to be filled up to the
road-way. If the proportion of these two weights be the
same with the proportion of the weights of chain, we may
rest satisfied with the curve now found; but if different,
we can easily calculate how much must be added equally
to or taken from each appended bit of chain, in order to
make the two proportions equal. Having altered the ap¬
pended pieces accordingly, we shall get a new curve,
which may perhaps require a very small trimming of the
bits of chain to make them fit the road-way. This curve
will be very near to the curve wanted.
We have practised this method for an arch of 60 feet
span and 21 feet high, the arch-stones of which were only
two feet nine inches long. It was to be loaded with gra¬
vel and shivers. We made a previous computation, on the
supposition that the arch was to be nearly elliptical. The
distance between the points 1, 2, 3, &c. were adjusted, so
as to determine the proportion of the weights of chain
agreeable to the supposition. The curve differed consi¬
derably from an ellipse, making a considerable angle with
the verticals at the spring of the arch. The real propor¬
tion of the weights of chain, when all was trimmed so as
to suit the road-way, was considerably different from what
was expected. It was adjusted. The adjustment made
very little change in the curve. It would not have changed
it two inches in any part of the real arch. When the pro¬
cess was completed, we constructed the curve mathema¬
tically. It did not differ sensibly from this mechanical
construction. This was very agreeable information; for
it showed us that the first curve, formed by about two
hours’ labour, on a supposition considerably different from
the truth, would have been sufficiently exact for the pur¬
pose, being in no place three inches from the accurate
curve, and therefore far within the joints of the intended
arch-stones. Therefore this process, which any intelli¬
gent mason, though ignorant of mathematical science,
may go through with little trouble, will give a very pro¬
per form for an arch subject to any conditions.
19. The chief defect of the curve found in this way is
a want of elegance, because it does not spring at right
angles to the horizontal line; but this is the case with all
curves of equilibration, as we shall see by and by. It is
not material; for, in the very neighbourhood of the piers,
we may give it any form we please, because the masonry
is solid in that place ; nay, we apprehend that a deviation
from the curve of equilibration is proper. Ihe construc¬
tion of that curve supposes that the pressure on every
part of the arch is vertical; but gravel, earth, and rub-