THE PRINCIPLES OF BRIDGES.

TRACT I.

Lo

And this property comes to the very same thing as theproperties in the foregoing propositions, because the anglesof elevation of the curve at every point, or of the directionof the tangents there, or of the curve itself, are equal to theangles in this proposition, which the joints form with the ver-tical direction. So that, all the three theories in these fourpropositions arc all one and the same in effect, amounting tothe very same thing, and yielding the same conclusions.And therefore, whatever consequences may further be drawnfrom any one of them, may be understood as deduced fromthe whole.

Scholium .—In the practice of bridge-building, the heypiece, or wedge at the crown, is a solid, having its magni-tude and weight half on each side of the middle vertical line ;whereas, in this proposition, it has been supposed that thiswedge is divided and actually separated in two by that lineab: this however will cause no difference in the theory, noryet in the practice; for, in any calculations that may be re-quired, it is only necessary to suppose the key piece dividedexactly in the middle, then taking half its weight for theweight of the piece ad, and computing all the other weightsand angles from the middle line AB.

It has also been supposed, in all the three theories thathave been contemplated, that the constituent parts areformed of materials perfectly smooth and polished, and puttogether without cement, and without all kinds of ties orbars, so as to leave them quite at liberty to slide over eachother, the parts being kept in a perfect balance by means oftheir shape, weight, and disposition only. This, it must beacknowledged, is not the case in real practice; ashore allthe materials are quite rough, which very much preventsthem from sliding by each other, even when their abuttingsurfaces are laid at a considerable slope or angle. But thiscircumstance however, so far from being a disadvantage, bythus deviating from the theory, is on that very account ofgreat use and benefit. For, the equilibrium among the con-