SECT. 2.

OP THE ARCHES.

33

From this calculation, as well as from the theorems bywhich it is made, it is manifest how greatly the weight andthe pressure of the semiarch increase towards the bottom orthe extremity, where the position of the joint approachestowards the horizontal direction, or the angle it makes withthe vertical approaches towards a right-angle; and whenthat angle actually becomes a right-angle, or the joint quitehorizontal, then the weight and pressure become equal andinfinite, which must naturally be expected, both because thetangent and secant of the angle'(being a right one) are theninfinite, and also because it. must require an infinite weightor pressure to balance there the constant given horizontalthrust, which is perpendicular to the former.

We may here, by the way, stop to examine a little inwhat manner the preceding calculation of the weights of thevoussoirs may be employed to give a familiar and easy me-chanical construction, that may approach very near to atrue balanced arch. In order to this, we are to consider,that since the bases, or extents of the under sides, of all thevoussoirs, are equal, it will thence happen that their weightswill have to each other nearly the same ratios as their lengths,from the under to the upper side of them, or taken in thedirection of the radius, that is perpendicular to the undercurve or intrados, at least when the breadth or angle of thesewedges is very small, which is the case in real practice, theapproach to equality being the nearer indeed as their breadthis the smaller. And though the angle of 5 degrees, em-ployed in the preceding calculation, be not such a smallbreadth as to render the equality and the construction per-fect, it will yet serve to show the manner of proceeding insuch a way of forming the arch, and will besides approachtolerably near to the truth.

As it is most proper that the joints between the wedges,in the arch of a bridge, should be in directions perpendicu-lar to the under curve of the arch, we shall only exemplifythe method in cases of that sort. For this purpose then, letus suppose the intrados or under curve to be divided into a

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