THE PRINCIPLES OF BRIDGES.

TRACT I.

34 .

number of equal parts, answering to a breadth of 5 degreeseach, or such that the angle formed by every two adjacentjoints, when produced, shall be an angle of 5 degrees. Letus then draw a line through the middle point of every oneof these breadths, bisecting them, and in a direction perpen-dicular to the curve at every point. Then, by setting- oil',upon these lines, from the curve upwards, by a proper scale,lengths which shall have the same ratios to each other as theweights of the corresponding wedges through which theselines pass, or proportional to the numbers in the last columnof the foregoing table; then will the lengths of these linesbe the extent of the several voussoirs nearly, and therefore,their upper extremities or points being connected, by draw-ing short lines from one to another, they will limit or formthe extrados, or the upper curve or side of the arch, whenbuilt of uniform materials, so as to be very nearly in equi-librio.

As it is manifest that the theorems and the, calculationhave no peculiar restricted reference to any particular curvefor the intrados, or under side of the arch, we are thereforeat liberty to assume that curve of any form at pleasure;therefore the form of it being so assumed, by then applyingthe numbers of the foregoing table to it, in the mannerabove mentioned, we shall have a balanced arch as required.And thus by assuming any different shapes of curve for theintrados, the same numbers in the table will give as manybalanced arches as we please. Assuming then, for the innercurve, a semicircle, as in the next fig. having its span or dia-meter LM 84 feet, consequently its pitch or height oa 42 feet.We shall also assume ab the thickness of the crown or key-piece, equal to 6 feet, or the 14th part of the span, beingnearly the proportion employed by good engineers. Divid-ing each half arc al, am, into 9 equal parts, of 10 degreeseach, which will be sufficiently small to show the nature andform of the extrados, containing each an extent of twowedges or voussoirs; then from the centre o drawing radiithrough all the points of division, these, when continued,