Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

JOI
Theory.
M E C H
are generally built for ornament j and we fee that there
is great room for indulging our fancy in the choice.
All curves which are concave outwards will give
domes of great lirmnefs : they are alfo beautiful. The
Gothic dome, wrhofe outline is an undulated curve, may
be made abundantly firm, efpecially if the upper part
be convex and the lower concave outwards.
“ The chief difficulty in the cafe of this analyfis
arifes from the neceflity of exprefling the weight of
the incumbent part, or j'dy^JThis requires
the mcafurement of the conoidal furface, which, in
moft cafes, can be had only by approximation by means
of infinite feriefes.
“ The furface of any circular portion of a fphere is
very eafily had, being equal to the circle deferibed
with a radius equal to the chord of half the arch, ihis
radius is evidently —s/x'+if.
“ In order to difcover what portion of a hemifphere
may be employed (for it is evident we cannot employ
the whole) when the thicknefs of the vaulting is uni¬
form, we may recur to the equation or formula
dy x \/X1 -f-y1. Let a be the ra-
x
t • a y y
dius of the hemifphere. We have x ———
s/a'—y'
• • a* v%
and ar— ■ ■—. Subftituting thefe values in the
a'—y'fi
/ a* v y
formula, we obtain the equation y*v —y2—
We eafily obtain the fluent of the fecond member
rra3—a's/a1—if, and y—a'J—4 + ^ Therefore if
the radius of the fphere be i, the half breadth of the
dome mufl not exceed —4-x v'l, or 0.786, and the
height will be .618. The arch from the vertex is
about 510 49'. Much more of the hemifphere can¬
not ftand, even though aided by the cement, and by
the fri&ion of the. courfing joints. This laft circum-
ftance, by giving connexion to the upper parts, caufes
the whole to prefs more vertically on the courfe be¬
low, and thus diminifhes the outward thruft 5 but it at
the fame time diminifhes the mutual abutment of the
vertical joints, which is a great caufe of-firmnefs in the
vaulting. A Gothic dome, of which the upper part is
a portion of a fphere not exceeding 450 from the ver¬
tex, and the lower part is concave outwards, will be
very flrong, and not ungraceful.
“341. Perfuaded that what has been faid on the fub-
je£I convinces the reader that a vaulting perfectly equi¬
librated throughout is by no means the bell form, pro¬
vided that the bafe is fecured from feparating, we
think it unnecefiary to give the inveftigation of that
form, which lias a confiderable intricacy, and fhall
merely give its dimenfions. The thicknefs is fuppofed
uniform. The numbers in the firft column of the table
exprefs the portion of the axis counted from the vertex,
and thofe of the fecond column are the length of the
•rdinates.
“ The curve formed according to thefe dimenfions
will not appear very graceful, becaufe there is an
abrupt change in its curvature at a fmall diitance from
its vertex 5 if, however, the middle be occupied by a
lantern of equal or of fmaller weight than the part
whofe place it fupplies, the whole will be elegant, and.
free from this defect.
“ The connexion of the parts arifing from cement and
from fridtion has a great effect on dome-vaulting. In
the fame way as in common arches and cylindrical
vaulting, it enables an overload on one place to break
the dome in a diftant place. But the refiftance to this
effedl is much greater in dome-vaulting, becaufe it ope¬
rates all round the overloaded part. Hence it hap¬
pens that domes are much lefs fhattered by partial
violence, fuch as the falling of a bomb, or the like.
Large holes may be broken in them without much
affefting the reft 5 but, on the other hand, it greatly
diminifhes the ftrength which fhould be derived from
the mutual preffure in the vertical joints.- Fridlion
prevents the Hiding in of the arch-ftones which pro¬
duces this mutual preffure in the vertical joints, ex¬
cept in the very higheft courfes, and even there it
greatly diminifhes it. Thefe caufes make a great
change in the form which gives the greateft ftrength \
and as their laws of aftion are but very imperfedtly
underftood as yet, it is perhaps impoflible, in the pre-
fent ft ate of our knowledge, to determine this form
with tolerable precifion. We fee plainly, however,
that it allows a greater deviation from the beft form
than the other kind of vaulting ; and domes may be
made to rife perpendicular to the horizon at the bafe,
although of no great thicknefs ; a thing which muft
not be attempted in a plane arch. The immenfe ad¬
dition of ftrength which may be derived from hooping
largely compenfates for all defects 5 and there is hard¬
ly any bounds to the extent to which a very thin dome¬
vaulting may be carried, when it is hooped or framed
in the diredtion of the horizontal courfes. The roof
of the Halle du Bled at Paris is but a foot thick, and
its diameter is more than 200, yet it appears to have
abundant ftrength.”
Scholium.
342. The fedtion of the dome of St Paul’s cathedral
is part of an ellipfe whofe conjugate diameter is parallel
to the horizon. It is built of wood, and confined by
ftrong iron chains ; and is fupported by carpentry refting
on a cone of brick work.
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