Chap. XXV.

ON STONE BRIDGES.

1497

ness of the abutments, and Bernoulli gave a solution of the problem. In the hypothesisof the perfectly polished joints, the arches where the tangent at the springing is not verticalare those only which can have a thrust; but experience proving that other arches alsohave one, we must recur to a supposition to determine its value. In a memoir publishedin 1712, in the History of the Academy of Sciences , Lahire supposes that a semicirculararch tends to disjoint at D d, and that the upper part DErf acts as a wedge to overthrowthe inferior parts, sliding down on the joints of rup-ture, and making the lower parts turn on K£.

According to this hypothesis, we can easily deter-mine the effort produced by the weight of thesuperior part of the arch, following a direction D Vperpendicular to the joint of rupture; and in ex-pressing that this force is in equilibrio round thepoint K with that which results from the stabilityof the inferior part of the arch, we obtain an equa-tion which gives K B, or the thickness which thepier should have to resist the thrust. It was sup-posed in the calculation that the arc B D was equalto half the quarter of the circle, because it was thought that the rupture of semicirculararches usually took place at an angle of 45° ; but it has since been observed that thecalculation of Lahire makes the thicknesses of the piers greater as they approached thejoints of rupture of the summit F of the vault.

This method of determining the thickness of the abutments of arches has been admittedby all engineers who have paid attention to this subject. Belidor has applied it to archesof different kinds, and tables have been calculated, which have appeared in the “ Coursd’Architecture de Blondel,” and it was supposed that in the arches described with three arcsof a circle, the joints of rupture would be at the junction of the arcs, in the “ Memoiresde 1’Academie des Sciences” for the year 1729. Couplet adopted the same theory, andresolved, always upon the hypothesis of the joints being perfectly, polished, the principalquestions relative to arches : he examined also the pressure which they exercised on theircentres, a subject which Pitot had already treated upon in 1726. In a second memoirprinted in the year following, Couplet endeavoured to establish a theory on arches, on anentirely contrary hypothesis to that which he had previously adopted : supposing that in thefall of an arch the rubbing of the joints of the voussoirs was such that they could never slideon their beds, and that to separate themselves from each other it was necessary that theirjoints should open and turn round upon the edge, he made moreover an abstraction for theresistance which the adhesion of the mortar might oppose to this movement. Althoughthis supposition, as well as the preceding, is not conformable to truth, it conducted theauthor to a theory approaching it: the observations and experiments which have beenmade since the epoch at which he wrote have taught us that, in almost all cases, the consi-deration of the friction was nothing in the nature of the movements of the arch, and that inthese different movements, the voussoirs turned in certain places round their edges, whilstthe joints tended to open and shut; but the expressions to which Couplet arrived are socomplicated, and of so little value in practice, that no one has used them ; the suppositionon which they were established being also false, no confidence could be given to them.

A little time after the publication of the two memoirs of Couplet, Danisy repeatedbefore the Academy at Montpellier experiments on models of arches in plaster divided intoa certain number of voussoirs, and pointed out the effect of the weights with which heloaded them, and the manner in which they produced the rupture : these experiments areindicated in the “ Coupe des Pierres,” by Frazier, which also contains a rule the authordeduced from them, and in which he attended less to exactness, than to the convenienceof the workman, as, in determining the thickness of the abutment, he neglected to take intoconsideration the height of the piers, which nevertheless materially influences it. Theseexperiments are the first indications of the true theory of arches; but they were made upontoo small a scale, and the calculations do not appear to have been properly applied to

them. v

We find in the “Recueil des Savants Etrangers,” for the year 1773, a memoir by Coulomb,where there is a question on the equilibrium of arches : this celebrated natural philosopherconsiders them successively in the hypothesis of the joints perfectly polished, and in thatwhere the friction and adherence produced by the mortar would oppose this disjunction ofthe voussoirs • in the latter ease, he regards first the half DE of the upper part of the archas a body carried on an inclined plane, and, having regard to the effect of the friction andof the cohesion, gives the limits between which are found comprised the value of a hori-zontal force acting ill the sense E i, which opposes itself to the sliding of the body, andsupports it in the inclined plane D F. He afterwards observes that the result of the twoforces which act on the portion of the arch DE must necessarily meet DI‘, between the

Fig. 2736.