BRIDGE. 



49.3 



iod of 

 Hirc, 

 nt, 13?- 

 ,ar,d 

 30d. 



principle, that the arch is in every point kept in equi- 

 libration solely by the gravity of the superincumbent 

 column of mutter. Now, it is even doubtful v> 1 



incipla be true. At any rate, tlu-y do not consi- 

 der the numerous modifications which it receives, from 

 the cohesion ot that matter among itself; from the 

 mutual cohesion and friction of the :,, ; from 



the position of their joints ; from the different specific 

 gravity which the arch and superincumbent matter 

 have, or which they may be made to have ; from the 

 lateral, and in some cases hydrostatic pressure, pro- 

 1 to the masonry throughout that matter ; 

 and, in line, from a numbiT of other causes, which, 

 if not singly, are, when combined, at least of as much 

 importance as the gravity of the vertical column of 

 matter alone. , 



Let us turn, therefore, to another mode of consi- 

 this subject, which has been adopted by De 

 la Hire, Parent, Belidor, and many others on the 

 continent, and in our own country by the ingenious 

 Mr Atwood. 



The latter has, from the known properties of the 

 , and the elementary laws of mechanics, exhi- 

 bited to us a geometrical construction for adjusting 

 the equilibration of arches of every form. The ma- 

 thematical reader, who has not lost his relish for the 

 ancient geometry, will find there an elegant speci- 

 men of its application ; for he completes his geome- 

 trical construction without once having recourse to 

 any other than the principles of elementary geometry 

 and trigonometry. It had been well, indeed, if he 

 had adhered longer to that mode of investigation ; 

 for, hy applying the analytic form too early, he has 

 been led unawares to consider that only as an ap- 

 proximation to the values of the quantities sought 

 after, which, in fact, is the expression for the values 

 of these quantities themselves. Nevertheless we owe 

 much to Atwood : he has shewn, that the advantages 

 of equilibration are not confined to any particular 

 curve ; that the drift or horizontal thrust of an arch 

 m?.v be easily found ; and that an arch may have all 

 the advantages of equilibration, whatever its figure 

 may be, merely by adjusting the joints of the arch- 



The stones, or sections of an arch, being of a 

 wedge-like form, have their tendency to descend op- 

 posed by the pressure which their sides sustain from 

 the similar tendency of the adjoining sections. Should 

 this pressure be too small, the stone will descend ; 

 should the pressure be too great, the stone will be 

 forced upwards. 



These pressures act in directions perpendicular to 

 the touching surfaces ; for, if the original direction 

 of any -pressure should be oblique, it may be resol- 

 ved into two forces, of which, while one is perpen- 

 dicular to the surface, the other is parallel to it, and, 

 .: .e, neither increases nor diminishes the per- 

 pendicular pressure. 



The wv'.lgt- A, Fig. 9, if unimpeded, would de- 

 in the direction vo, but is prevented by the re- 

 action of B and I}', acting in the directions pg and 

 K'I, perpendicular to the sides AO, QD ; and it is 

 ki.oun. trom the properties of the wedge, that if PQ, 

 or K'I be to the weight of the v > is to 



BO, the wedge A will remain at rest. If also the 

 wedge A be only at liberty to slide down GA, con- 



sidered as a fixed abutment, then the force PQ alone Theory, 

 will keep it in equilibrio. The forte i-o ')< -lug per- 

 pendicular to no, ha. no tendency to make A dlidc 

 eitlu T up or down on that line, but produce it to- 

 wards \, making NM equal to i>g; then this force act- 

 ing obliquely at N, may be reduced to two others, 

 viz. Mil perpendicular to Ad, expressing the perpen , 

 dicular pressure on the abutment of A, and UN ex- 

 pressing the force or tendency it has to make A slide 

 upwards along AC;. Again, take the vertical line A a, 

 expressing the weight 'of A, and draw a si at right 

 angles to AO j it is very evident, that AH expresses 

 the tendency of A by its weight to slide down UA. 

 All is opposite, and is equal to Mt 



For, draw the perpendiculars ot/and A p, then 

 the triangles A an, M,p, norfare evidently similar; 

 and also the triangles OD </, oox, ,>INR, a-, they have- 

 always a common angle besides the right angle. Now 

 the force PQ, that is, MN is to the weight of A, that is 

 AH, as OD to DG by supposition. 



And A : AH :: AG : A/) :: DO : oil 

 Therefore, M.V : AH :: OD : od :: MX : NR. 



Or MN has the same ratio to AH, that it has to XR ; 

 that is, AH and NU are equal, or the tendency of A 

 to slide downwards by its weight, is balanced by the 

 tendency of MN to make it slide upwards : wherefore 

 the section A remains at rest in equilibrio. 



Considering the whole arch as completed, with its 

 parts mutually balancing each other, the force PQ, 

 which is H.-essary for sustaining the wedge A, will 

 be supplied by the reaction of the adjacent wedge B. 

 Now, let it be required to ascertain the weight of B 

 in proportion to A, so that they, being adjusted U> 

 equipoise, may continue to be in equilibria, when 

 left free to slide along KB. Since MR is the pressure- 

 produced by PQ in a direction perpendicular to AO, 

 we must add to this n a, which is derived from the 

 wedge A ; therefore make M n equal to H a, produce 

 MR to Y, take \x equal to n /;, draw y.\\ a: 

 angles to KB j YW is the force tending to make ii 

 slide up UK : take therefore BH' equal to YW, draw 

 the perpendicular n' b meeting the vertical fli in It ; 

 B b will represent the necessary weight of the wedge 

 B ; and the whole is so evident from the composition 

 of pressures, as to require no further demonstration. 

 Such is Atwood's construction ; he has rend:red the 

 demonstration much more prolix, by the unnecessary 

 introduction of trigonometry ; and after slievvi.ig how 

 the weight of the sections C, D, &c. may be found 

 in the same way, he goes on to reduce these weights 

 and pressures to analytical and numerical values. He 

 iinds these in terms of the sines and tangents of the 

 successive angles ot inclination; but in reducing these 

 to numbers, he has been led to the accumulation of 

 small errors in that very operose way of proceeding, 

 to give erroneous results ; and into the singular mis- 

 take of conceiving, that the real ex; these 

 values was only an approximation. Had lie recalcu- 

 lated the whole by more extended trigonous 

 tables, they would have quickly undeceived him ; 

 and they would have shewn him, thai what In- 

 thus searching so deeply for, was all the while lying 

 exposed at the surface; that the app .. ulties 

 were entirely of his own creation, and his imagined 

 accuracy was error. This should teach mathemati- 



