124 A R C H I T 
is formed for the correfponding values of x andy, by which 
She curve is conftruCted for any particular occafion. Thus, 
fuppofing a or DK=6, k or A05=50, and r or D<i=40; 
then the correfponding values of K 1 and IC, or horizon¬ 
tal and vertical lines, will be as in this Table: 
Table for conjlruEling the Curve of Equilibration. 
Valueof 
KI. 
Valueof 
IC. 
IValueol 
1 KI. 
Value of 
IC. 
Valueof 
KI. 
Valueof 
IC. 
O 
6 -ooo 
21 
10-381 
36 
21-774 
2 
6 '°35 
22 
10-858 
37 
22-948 
4 
6-144 
23 
11-368 
38 
24‘190 
6 
6-324 
24 
11*911 
39 
25-505 
8 
6-580 
2 5 
12-489 
40 
26-894 
IO 
6'9>4 
26 
13-106 
4* 
28-364 
j 12 
7-330 
27 
13-761 
42 
2 9"9 I 9 
13 
7-57i 
28 
I 4-457 
43 
3 1 "5^3 
14 
7-834 
29 
15-196 
44 
33-299 
15 
8 'i 20 
30 
15-980 
45 
35'i35 
16 
8-430 
3i 
i6’8i 1 
46 
37-075 
17 
8-766 
32 
17-693 
47 
39-126 
18 
9-168 
33 
18-627 
48 
41-293 
19 
9 "5 1 7 
34 
19*617 
49 
43 -58 * 
20 
9‘934 
35 
20665 
5° 
46-000 
Among all the arches there is no one, except the me¬ 
chanical curve of the arch of equilibration, that can ad¬ 
mit of a horizontal line at top : yet this arch is of a form 
both graceful and convenient, and it may be made higher 
or lower at pleafure, with the fame fpan or opening. All 
other arches require extrados that are curved, more or 
lefs, either upwards or downwards. Of thefe, the ellip¬ 
tical arch approaches the neared to that of equilibration 
for equality of drength and convenience ; and it is alfo the 
bed form for mod bridges, as it can be made of any height 
to the fame fpan, its handles being at the fame time diffi¬ 
dently elevated above the water, even when it is very flat 
at top : elliptical arches alfo look bolder and lighter, are 
more uniformly drong, and much cheaper than mod others, 
as they require lefs materials and labour. Of the other 
curves, the cycloidal arch is next in quality to the ellip¬ 
tical one, for thofe properties, and, ladly, the circle. As 
to the others, the parabola, hyperbola, and catenary, they 
are quite inadmiffible in bridges that confid of feveral 
arches ; but may, in fome cafes, be employed for abridge 
of one Angle arch which may be intended to rife very high, 
as in fuch cafes they are not much loaded at the handles.” 
The lize of the piers for bridges is to be determined by 
that of the arch; and, according to Palladio, they ffiould 
never be lefs than one-fixth part of its width, nor more 
than one-fourth. In the plans of piers for bridges they 
are generally drawn of an hexagonal figure, having two 
long fides parallel to each other, and at the ends are pla¬ 
ced two fliort ones facing the courfe of the river at right 
angles to each other. Palladio fays, they are fometimes 
made femicircular facing the ftream in order to divide the 
the water, that thofe things which are impetuoufly brought 
down the river, when they ftrike again!! them, may be 
thrown from the piers, and pafsthrough the middle of the 
arch. With refpeCt to the dimenfions of the key-ftones, 
or archivolts, he gives to their height a feventeenth part 
of the width of the arch, This proportion is fomewhat 
1 
E C T U R E. 
fmaller than that propofed by M. Gautier, an experienced 
engineer, who makes the length of the arch-ftones, of an 
arch twenty-four feet wide, two feet; and to arches forty- 
five, fixty, feventy-five, or ninety, feet wide, he gives 
three, four, five, and fix, feet refpeClively, and more than 
this when the done is of a loft nature. But Mr. Belidore 
fays, the length of the key-ftones ought always to be one- 
twenty-fourth part of the width of the arch, whether the 
ftone be hard or foft, becatife, if they are foft, their weight 
is proportionably lefs. It appears fomewhat ftrange, that 
fcientific men fliould differ fo widely about the fize of a 
key-ftone. However, as a juft and proper medium be¬ 
tween the two extremes, we may fafely venture to recom¬ 
mend the judicious proportion adopted by Palladio. Of a 
bridge which that celebrated architect defigned at the re- 
queft of fome gentlemen, he mentions the following pro¬ 
portions : The river was 180 feet wide, which he divided 
into three arches, giving fixty feet to the centre arch, and 
to the other two forty-eight feet each. The piers, whioh 
he terms pilajlers, were twelve feet thick, or one-fifth of 
the width of the middle arch, and a fourth of the fmaller 
ones. The arches were a fmall portion lefs than a femi- 
circle; and their archivolt one-feventeenth part of the 
aperture of the centre arch, and a fourteenth part of the 
other two. According to this proportion, the key-ftone 
of the centre arch was three feet fix inches long, but, ac¬ 
cording to M. Gautier, it fliould have been four feet, and 
agreeably to Mr. Belidore only two feet ten inches. In an 
arch of twenty-four feet, Palladio’s rule makes the length 
of the key-ftone a fmall part more than fixteen inches, 
which is a much more eligible fize than either the one or 
the two feet key-ftone of the former gentlemen. 
The width commonly allowed to bridges is thirty feet: 
but, in large ones near great towns, a banquette is gene¬ 
rally added on each fide for foot paflengers of fix to nine 
feet each, raifed about a foot above the common road; 
the parapet-walls are about eighteen inches thick, and four 
feet high; they generally project from the bridge with a 
cornice underneath: fometimes balluftrades of ftone or 
iron are placed upon the parapet, as at Weftminfter; but 
this is only praCtifed where a bridge is of great length, 
and built on a magnificent plan. The ends of bridges 
open from the middle of the two large arches with two 
wings, making an angle of forty-five degrees with the reft, 
in order to make their entrance more free : thefe wings 
are fupported by a continuation of the arches; that im¬ 
mediately under each wing being fmaller than the reft. 
But the wings of bridges are generally fupported by the 
folid butment alone. 
The moft proper feafon for beginning the work, and lay¬ 
ing the foundations of a bridge, is in autumn, when the 
waters are loweft, and the weather moft fuitable for aqua¬ 
tic undertakings. The ancients ufed feveral methods to 
obtain a lading foundation for their ftone bridges. Some 
of thefe it will be proper to defcribe, in order that the 
reader may form his own judgment, by drawing a compa- 
rifon between what they have done fo long before our time, 
and what architects are doing now. 
Alberti, who is reckoned among the firft who wrote on 
bridges, gives us the following directions for laying the 
foundation of a pier : “ Firft raife an cnclofure to keep off 
the water, by driving a double row of flakes very clofe 
and thick fet, with their heads above the top of the water 
like a trench. Then put hurdles within this double 
row of flakes. Clofe that fide of the row which is next 
to the intended pier, and fill up the hollow between the 
two rows with rulhes and mud, ramming them together fo 
hard, that no water can poffibly get through. Then what¬ 
ever you find within this inclofure, water, mud, fand, or 
whatever elfe is a hindrance to you, throw ®ut, and dig 
till you come at a folid foundation ; or, if you find it ne- 
ceffiiry, make a foundation of wooden piles burnt at the 
ends, and driven in as clofe together as poffible. And 
here I have obferved, that the bell architects ufed to make 
a continued foundation of the whole length of the bridge. 
