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

Tlieory. ari/ing from multiplying the verfed fine of half the arch
v ' by the cube of its cofine, and dividing this product by the
difference between the cube of the radius, and the cube of
the cofine ; or, to change the expre (lions, the thicknefs of
the roadway above the keystone, when the roadway is a
Jlraight line, is equal to the quotient arifing from multi¬
plying the height of the arch, by the cube of the differ¬
ence between the radius of the arch and its height, and
dividing this produci by the difference between the cube
of the radius, and the cube of the difference between the
radius and the heigh t of the arch.
331. When the arch is a femicircle R—y vaniflies,
and m becomes equal o, fo that the femicircular arch is
evidently inadmiflible. But when the arch is lefs than
a femicircle, the value of m will be finite. Thus, if
the arches are refpectively
Arch.
6o°, we have the fpan,
90°, we have m—~ of the fpan, or
iio°, we have m~T\ of the fpan nearly.
The two firft arches of 60® and 90°, manifeftly give
too great a thicknefs to the part BU or m. In the
third arch of iio°, the thicknefs of BD is nearly what
is given to it by good archite&s, and is therefore the
bell in praftice ; for if the arch were made greater
than no0, the thicknefs of BU or m would be too
fmall. It is obvious, however, that an arch of no0 is
not an arch of perfeB equilibration, for this can be the
cafe only when the roadway has the form Uar. When
the roadway, therefore, is horizontal, as U r, there is
an unbalanced preffure on both fides of the keyflone,
produced by the weight of the materials in the mixti-
linear fpace r%U. It is indeed very fmall, and might
be countera£led, by making the materials below" Z
lighter than thofe below U : but the unbalanced pref¬
fure is fo trifling, that it may be fafely negle&ed. We
MECHANICS.
may, therefore, conclude, that when the arch is to be
circular with a horizontal roadway, an arch of no de¬
grees approaches nearefi to an arch of equilibration.
332. When the arch is elliptical, it will be found, Elliptical.
as in the circle, that mz
_ y X R—y|3
"R3—R—if
arches fu-
An elliptical Peno[to
^ nrril n
_yi ' circular
arch, however, has the advantage of a circular one, when their
when the tranfverfe axis is horizontal j for as it is tranfverfe
much flatter, the point of contrary flexure in the extra- ax*s ^
dos is thrown at a greater diftance, and therefore itzon a ’
will, with lefs inconvenience, admit of a horizontal
roadway. Elliptical arches have alfo the advantage of
being more elegant, and likewife require lefs labour and
materials.
333. The cycloidal arch is likewife fuperior to a cir¬
cular one, but inferior to thofe which are elliptical.
Parabolic, hyperbolic, and catenarian arches, may be
employed when the bridge has only one arch, and is
to rife high ; but in other cafes they are inadmiflible.
The method of determining the roadway for all thefe
forms of arches Avill be found in Dr Hutton’s excellent
work on the Principles of Bridges, p. 3. See alfo
Emerfon’s Mifcellanies, p. 156.3 and his work on
Fluxions, publiihed in 1742.
334. When the form of the roadway is given, the On the me-
fliape of the intrados for an arch of equilibration may chan'cai
be determined. As the inveftigation is very difficult, cuUf-,ot
unlefs when the roadway is a horizontal line, we (hall turn.1 ^
merely give the formula, which will enable any perfon
to conftrudt the curve. In all other curves the equi¬
librium of the arch is imperfeft 3 but the curve de-
fcribed by the following formula is an arch of perfect
equilibration, and has been called the mechanical curve
of equilibration.
ED=AFx
Hyperbol. log. BU x BD-f-BD*
Hyperbol. log. BU+BF-kA BU x BF+BF-
From this formula, which correfponds with figure 11.
Dr Hutton has computed the following table, contaim
ing the values of c U and cE, for an arch whofe fpan
AC is 100, whofe height BF is 40, and whofe thick¬
nefs at the crown or BU is 6. The table will anfwer
BU
for any other arch whofe fpan and thicknefs are as the
numbers 100, 40, 6 5 only the values of cU and cE
muff be increafed or diminiftied in the fame ratio as
thefe numbers.
1 ABLE for confiruciing the Curve of Equilibration, when the fpan, height, and thicknefs at the crown, are as the
numbers 100, 40, and 6.
Value of
rU.
Value of
cE.
o
2
4
6
8
10
12
*3
6.000
6- °35
6.144
6.324
6.580
6.914
7- 33°
I'.Sl1
7-834
Value of
cU.
15
16
17
18
!9
20
21
22
23
Value of
cE.
8.120
8.430
8.766
9.168
9-S'l
9-934
10.381
10.858
11.368
Value of
cU.
24
25
26
27
28
29
30
31
32
Value of
cE.
11.911
12.489
13.106
13- 76i
14- 457
15.196
15.980
16.811
17.693
Vol. XIII. Part I.
Value of
cU.
33
34
35
36
37
38
39
40
41
Value of
cE.
18.627
19.617
20.665
2I-774
22.948
24.190
25-5°5
26.894
28.364
Value of
c\J.
42
43
44
45
46
47
48
49
50
Value of
c E.
29-9I9
31-563
33-299
35-I35
37-075
39.126
4I-293
43-58i
46.000
N
335* The