so

THE PRINCIPLES OP BRIDGES.

TRACT I.

the breadth of each wedge (5 degrees), to the said de-grees; by which it will be found that the angles at the cen-tre, formed with tiie vertical, by the said lower edges of thearch pieces, in order after the key, will be as follows, viz,that of the 2d wedge degrees; that of the 3d, ]2f de-grees; that of the 4th, IT*-degrees; and so on to the 17thor last on each side the key, which will have its lower edgemaking an angle of 87*- degrees with the vertical direction:all which angles, of inclination to the vertical, arc ranged inthe 2d column of the following tablet, the first, or half themiddle wedge, making an angle of 2 } degrees. We shallalso suppose the weight of the middle wedge at the crownto be a certain given quantity, represented by unity or 1,and express the several other weights and pressures, as inthe other columns of the said tablet, in terms of that uniteso that all these proportional numbers for the other weightsand pressures, will require to be multiplied by any otherweight of middle wedge which may happen to occur in anyother case.

Now, in regard to the rule for computing all the otherweights and pressures, according to the conclusions fromthe preceding theory, it is very easv and simple indeed, viz,that the weight of any part of the arch, counted from thevertex or crown downward, is always proportional to thetangent of the angle of inclination of the lower wedge to thevertical, while the oblique push or pressure, in direction ofthe curve, is proportional to the secant of the same angle,and the constant horizontal thrust is proportional to the ra-dius. For which reason it is, as formerly observed, that theconstant horizontal thrust is a proper radical measuring unit,by means of which to compute the two other circumstances,namely, the weight of the arch, and the oblique push orpressure in the direction of the curve: for, the horizontalthrust being taken for radius, then the weight of the semi-arch will be the tangent of the angle with the vertex, andthe oblique pressure the secant of the same angle, to thatradius. Consequently, if the constant horizontal push becalled A, then the weight of the semiarch will be h x if, or A