BRIDGE. 



given by Emerfon in iiis fluxions, publidied in the year 1742, 

 and which lias been fo ably and judicioufly handled by Dr. 

 Hutton in particular, is confident with natiirt and with 

 truth. This theory ellablilhes an equilibrium among all the 

 vertical prelTures of the whole fabric contained between the 

 foffit, or under-fide of the arch, and the road way over all. 

 It is now very generally adopted by the moll (Ivilful engi- 

 neers and architefls, as the only true one ; becaufe it le- 

 cures a balance in the whole of the poudtratiug matter, by 

 makinff an equality at every point of the curve, between all 

 the adjacent preflures when reduced to the taii;5ential direc- 

 tions, or perpendicular to the joints, which are fuppofed to 



2, 'J, Sec. and draw vertical lines, cuttinjj the chain in the 

 correfponding points 1, 2, 3, Sec. Now take pieces of an- 

 other chain, wiiofe links arc eafily feparated, and hanij 

 them on at all the points I, 2, 3, Sec. of the chain A (^G : 

 trim thcfe pieces of chain, by taking off links at fome 

 placen, and hanging on at others, till their lower ends all 

 coincide with tlie inverted road-way b c c. The greater 

 lengths hung on in the vicinity of A and G, will pull do«ii 

 thofe points of the chain, and catife the middle point <if 

 which is lefs loaded, to rife a little, and bring it near to its 

 proper height. It is obvious this is an arch of equilibration 

 for a bridge fo loaded, that the weight of the arch-'lones 



be at right angles to the curve of tiie arch in every part, as is to that of the fuperincumbent matter between the arcli 

 fuch ftruetures naturally require them to be : for, if the and road-way, as the weight of the chain A ^' G, is to the 



fum of the weights of all the little bits of chain, very near- 

 ly. But this proportion is r.ot known before-hand ; wc 

 mufl, therefore, proceed thus : adapt to the curve produced 

 in this way fuch a thicknefs of tlie arch-ilones as may be 

 thought fufEcient to enfure (lability ; then compute the 

 weight of the arch-Hones, and the weight of the rubbUs, 

 or other materials with which the haunches are to be filled 

 up to the road-way. If the proportion of thefe two 

 weights be nearly the fame with the proportion of the 

 P/.XXXVIII. ylrchttelhire, f.g. I. kj:g.2. PI. VI. (referred weights of the chain, we may rell fatisfied with the curve 

 to in the article Arch) completely inverted. Let this rcprc- now found: but if it be much different, we may foon fnid 

 fent a flexible chord or chain, void of gravity, hanging from how much Ihould be added to, Or taken from, the appended 

 the points A and G, which are fixed ; at the points B, C, bits of chain, in order to make the two proportions equal. 

 D, E, F, fuppofe weights to be fufpendcd, (acling in the We fhall then have a curve infinitely near to the inverfion of 

 direftions BH, CI, DK, &c.) proportional to the feveral the curve wanted. This method we can fafely recommend. 



joints be perpendicular to the curve, there will arife a lateral 

 prefiure, whofe diredlion is not along the tangent ; which, 

 wanting a force to fuftain it, will dLllroy the equilibrium, 

 and fome of the ftones will endeavour to fly out. 



When fpeaking of the principle advanced by Dr. Ilooke, 

 we obferved, that by means of peculiar modifications, it 

 was capable of univerfal application to cafes occurring in 

 praiflice, and was at the fame time confillent with the theory 

 we have afTumed. This we fliall now p-oceed to (liew. In 



lines B »', Cm, D /;, E j, and Yy. Tlien the cafe now be- 

 fore us will be the complete inverfion of that which is firft 

 ftated in Prob. i. article Arch, the drawing forces in this 

 inftance being refpeftively equal and oppofite to the feveral 

 prefling forces in that : therefore, every thing proved there, 

 by means of the compofition and refolution of forces, will 

 be found to obtain here, only in a contrary direftion. Con- 

 fequently, if the number of weights hanging fjom the chord 

 ADG be indefinitely increafcd, it will alTume a curvilinear 

 (hape, fimilar in its nature to the arch of equilibration, only 

 in an inverted pofition ; and the various theorems which re- 

 late to the weights and preiTures of the Handing arch, apply 

 with equal facility and accuracy to the weights hanging 



as we know it to have been frequently ufcd with facility aiid 

 fuccefs. 



It would draw us far beyond the limits we are obh'ged to 

 aflign ourlelves, were we to give a complete view of the 

 theory in all its branches : thofe who are defirous of obtain- 

 ing a moi-e intimate acquaintance with the fubjeA are, 

 therefore, referred to Dr. Hutton's ingenious treatife on 

 bridges ; for our own parts, we muft content ourfelves with 

 jull touching upon a few of the moll important particulars. 

 Under the article Arch, and the correfponding plate, we 

 have given figures of the extrados, of a circular and ellipti- 

 cal arch of equilibration ; from which it may be feen how 

 far the extrados extends from the vertex of the curve, before 



from the fnfpended arch. Whether, therefore, v.-e confider it becomes unfit for a road-v>-ay by reafon ot its bending up. 

 the Handing or the hanging arch, it is equally true, that in wards : in this refpedl, it appears, that the flat ellipfis has 

 the cale of juft equilibration, the column cither prefling or the advantage of the circular arch ; but the cycloidal arch of 

 drawing at any point of the arch is reciprocally as the radius equilibration, though fimilar to thefe, has the advantage of 

 of curvature and the cube of the fine of the angle, in which both, becaufe the extrados runs farther on, nearly parallel 

 the vertical line cuts the curve in that point (Cor. 2. pr. i. to the arch before it comes to the point of inflcftion. We 

 Arch) ; or, fince the cofecant vaiiesa.s the fine inverfely, the (hould obferve, however, that in many cafes, even of circu- 

 column above-mentioned is reciprocally as the radius of cur- lar or elliptical arches, the evil arifing from the infletlion of 

 vature, and direftly as the cube of the fecant of the curve's the extrados may be thrown off to a greater dillance, by a very 

 inclination to the horizon, in the given point. fimple expedient: for, in an arch of equilibration, as NBH, 

 But the analogy between the Handing and the hanging ^f^.J. /"/. XXXVIII. of y/;-f/j;VfiS;//r, whofe extradosis EIK 

 arch has been traced out, not fo much for the purpofe of SF, fince the points at in, n, 0, &c. are kept in equilibrio by- 

 corroborating the true theory of equilibration, as for the the heights of the wall I m, K «, L 0, &c. if the lines I ;«, 

 fake of deducing from it a very popular and general mode Kn, Lo, &c. be divided in a given ratio, in i,!:, I, &c. the 

 of conftruftion ; Hriftly accurate in its principle, and yet fo fmaller mafs, under the new extrados e, i, i,/,/, will Hill fe- 

 fimple in its application, that the moH illiterate artiH may cure the equilibrium. Now it is obvious, that the lower extra- 

 fafely praftife it. Suppofe it were required to afcertain the dos runs much farther from the crown than the upper one, 

 form of an arch which fhall have the fpan AG (fg-2. PL before it has a point of inflexion : and hence appears one great 

 XXXVIII. Architeaure) and the height D 8, and which Hiall advantage arifing from the ufe of iron in bridges inHead of 

 have a road-way of the form B EC above it. Let the outline Hone. Suppofe, for inftance, that an arch was to be con- 

 figure ABECG be inverted, fo as to form a figure A ^^^ G. ftruiled, having the fpan AD, and height CB, and that 

 Sufpend a fine chain of uniform thicknefs from the points A the neceffary thicknefs of a ftone arch at the crown was 

 and G, and of fuch a length, that its lower point will hang BS ; here it is plain, that if the road way were made, having 

 a little below d, coiTefponding to D. Divide AG into a a pradicable Hope as SK«, it would fall far below the re- 

 number of equal parts (the more the better) in the points i, quired extrados at KIE^ and confequently, the arch, for 



E. r 2 want 



