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water when highest. Where the arches rise above one- third of thespan, the piers may be somewhat lower. The thickness of piersdepends very much on the height and form of the arch, and fromthese circumstances will vary from one-fourth to one-eighth of thespan of the arch. Under low water, the piers must increase inbreadth about three inches in every foot of height,' which maybe taken off in steps at the height of every course, observingnever to make large scarcements or intakes at once, as it wouldthereby come within the solid of the stretchers in the undercourse of stones. All the piers should be built solid with squarestones, having headers over the centre of every stretcher, andall the sides of the stones squared the whole way back.
The size and form of the arches will very much depend onthe height of the banks on each side of the river. The conveni-ence of the road requires the bridge to rise as little in the mid-dle as possible, and never above one in twenty-four. Where the
. banks are low, therefore, circular arches, having the versed sine
I or rise not above one-fourth of the span, will suit best; and, as
| the materials of the arch are the most expensive, such arches
having the shortest length of curve will be the most economical.The lateral pressure of arches is greater in proportion as they areless than the semicircle, and therefore the strength of the abut-ments must increase with the flatness of the arch. In caseswhere the piers are high on account of the depth of water, the| abutments will become expensive for arches with at. J all rise;and where the line of the road above will not admit of their be-ing raised, arches of a semielliptic form may be introduced withadvantage, as they press upon the piers and abutments in thesame manner as those of a semicircle. The depth of the archstones must be in proportion to the radius of the curve, which isa circumstance to be particularly attended to, as the size of thearch stones materially affects the expense. The hardness or soft-ness of the stones will require some consideration in fixing theproper depth ; but in common it will be found, that the depththe arch should be one-twelfth of the length of the radius :thus a semicircular arch of 100 feet span, whose radius is 50’ eet , requires the arch stones to be 4 feet 2 inches deep, and so°n of others.
All the beds or joints of the arch stones must be in the line of“c radius.
, Mathematicians have made numerous and intricate calcula-tions of the equilibrium of arches; but as most of these have notincluded the parts joined to the arch, and necessary to form abnd ge, they have not been of much use in the practical opera-tions. The great variety of constructions now existing, withfailure and destruction of many, under a variety of circum-stances, have enabled engineers to form correct principles suitedto the construction of bridges in all the variety of situations,voi.. u, 7