490 



BRIDGE. 



Theory. 



A new 



theory of 

 arched pro> 

 posed. 



Common 

 theory of 

 quili bra- 

 tion. 



Catenarian 



curve 



proposed 



byDr 



Hooke. 



PLATE 

 LXXX. 



INS. .'. 



in doubting of the justice of such theories, at least 

 until they are more consonant to the approved prac- 

 tice. 



It is our intention, in the present article, to point 

 out a new mode of considering this subject, to which, 

 with great diffidence, we request the attention of the 

 intelligent practitioner. It may indeed still be defi- 

 cient, if not in some respects erroneous. But it will, 

 we think, have this merit, that of being readily ap- 

 prehended, and easily applied, without requiring much 

 previous scientific information. Indeed though we 

 highly value the sublime geometry, we are inclined 

 to think that the unnecessary parade of calculus in the 

 application of science to the arts, has been one of the 

 chief causes of the dislike, which many able practi- 

 cal men of our country have shewn to analytical 

 investigation. 



Nevertheless, as many of our readers are well qua- 

 lified to comprehend, and will naturally expect that 

 we should point out, the modes of investigation, 

 usually pursued in this interesting subject ; we shall 

 previously, and in as succinct a manner as possible, 

 endeavour to lay before them the commonly received 

 t'neory of equilibration. From which, having clear- 

 ed away the useless rubbish, if we can extract 

 any proper materials, we may, like ceconomieal build- 

 ers, make good use of them in our future structure. 



The first thing like a principle that we meet with is 

 in the assertion of the eminent Dr Hooke, that the fi- 

 gure iuto which a heavy chain or rope arranges itself, 

 when suspended at the two extremities, being the 

 curve commonly called the calenaria, is, when invert- 

 ed, the proper form for an arch ; the stones of which 

 are all of equal size and weight. 



Now, as this idea, strictly just, has been very gene- 

 rally adopted, and affords some useful hints, it may be 

 well worth while to examine it. Let A, B, Fig. 2, be a 

 string or festoon of heavy bodies, hanging by the 

 points A, B, and so connected, that they cannot sepa- 

 rate although flexible. These bodies having arranged 

 themselves in the catenaria ACB, conceive this to be 

 turned exactly upside down. The bodies A and B 

 being firmly fixed, then each body in the arch ADB, 

 being acted on by gravity, and the push of its two 

 neighbours with forces exactly equal and opposite to 

 the former, must still retain its relative position, and 

 the whole will form an arch of equilibration. 



This arch, however, would support only itself ; 

 nay, a mere breath will derange it, and the whole will 

 fall down. But if we suppose each spherule to be 

 altered into a cubical form, occupying all the space be- 

 tween the dotted lines, the stability will be more con- 

 siderable. And as the thrust from each spherule to 

 its neighbour is in a direction parallel to the tangent 

 of the arch at the point of junction, it is obvious, 

 that the joints of our cubical pieces must be perpen- 

 dicular to that, so as to prevent any possibility of 

 sliding. 



Our arch is now composed of a series of truncated 

 wedges, arranged in the curve of the catenaria, which 

 passes through their centres ; and we are disposed, 

 with David Gregory, to infer, that when other arches 

 are supported, it is only because in their thickness 

 some catenaria is included. 



We might pursue this subject a great deal farther, 



by investigating all the useful properties of the cate- Tlteor 

 wtrian curve : but, in our opinion, this is at present v- 

 unnecessary. This curve is, indeed, the only one 

 proper for an arch consisting of stones of an equal 

 weight, and touching in single points, but is not at all 

 adapted to the arch of abridge, which, independent of 

 the varying loads that pass over it, must be filled up at p 

 the haunches, so as to form a convenient road-way. I.XXX 

 In this case, some farther modification becomes ne- Fig. 3. 

 cessary. The haunch E of the arch ACB, bearing a 

 much greater depth of stuff than the crown, it must 

 be so contrived as to resist this additional pressure. 

 Every variation of the line FGH, or extrudes, will re- 

 quire a new modification of the curve ACB, or infra- 

 dos, and the contrary. Accordingly, M. de la Hire 

 has suggested a good popular mode of investigating n "^^ 

 this subject. Let it be required to determine the jj e j a H 

 form of an arch of the span AB, and height CD, 

 proper for carrying a road-way of the form FGH. 

 Mark off, upon a vertical wall, the points A,B,C', in- 

 verting the required figure : Suspend from A, B, a 

 uniform chain or rope, so that its middle may hang 

 a little below the point C', and dividing the span A B 

 into any number of equal parts, and drawing the per- 

 pendiculars a b, c d, &c. suspend from the intersec- 

 tions cubits of chain eb,fd, &c. so trimmed, that 

 their ends may fall on the line of road-way 5 and it 

 may be observed, that as those pieces, which hang 

 near the haunch, will bring it down, the crown C will 

 thereby be raised into its proper position. 



All will now do, provided that the sum of the 

 small pieces of chain has to the large one, A C'B, the 

 same ratio which the stuff to be filled into the haunch- 

 es has to the whole weight of the archstones ; the 

 depth of which must of course be previously deter- 

 mined. But, if this is not the case, it will be easy to 

 calculate how much must be added to, or subtracted 

 from, the small chains, in order to obtain this propor- 

 tion. This being equally divided among the small 

 chains, will give a road-way very nearly parallel to 

 the former. The curve will evidently be a perfect 

 curve of equilibration, and extremely near the one 

 wanted. And this whole process is so easy, that it 

 may be gone through in a short time by any intelli- 

 gent mason. 



But although this mechanical way of forming an Mathe 

 equilibrated arch be founded upon principles suffi- tical th 

 ciently just, and be perhaps the simplest and best f C <J' 

 way in which the practical builder could form the Dratl l 

 original design of such an arch, yet as it affords no 

 general rules that may be applied to the construction 

 of arches, we proceed to consider the same subject in 

 a mathematical point of view. 



And first, then, in the semicircular polygon, as it is 

 called, Fig. 4, where weights are hung on the thread F'g- 4 - 

 AC'CC"B, which bring it into the position ACB, we 

 have at each angle three forces hi equilibrio. Where- 

 fore, by the principles of statics, they are to one ano- 

 ther as the sines of the opposite angles ; that is, the 

 tension rC is to the tension / C, as sine / C W is to sine 

 r CW, but the tension from C to / is the same as from 

 C' to r. Also sine I CW' is the same as sine r' C'W', 

 since these angles are supplementary, CW, C'W' be- 

 ing parallel ; therefore the tension rC is to the ten- 

 sion r' C', as sine r' C'W' to sine r C W. Or,thc tens'mn 



