The first thing like a principle that we meet with is in the assertion of the eminent Dr Hooke, that the figure into which a heavy chain or rope arranges itself, when suspended at the two extremities, being the curve com monly called the catenaria, is, when inverted, 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 general ly adopted, and affords some useful hints, it may he well worth while to examine it. Let A, B, Plate LXXX, Fig. 2, be a string or festoon of heavy bodies, hanging by the points A, B, and so connected, that they cannot separate although flexible. These bodies having arranged them selves in the catenaria ACB, conceive this to be turned exactly upside down. The bodies A and B being firm 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 between the dotted lines, the stability will be more considerable. And as the thrust from each spherule to its neighbour is in a direction parallel to the tangent of the arch at the point ofj unction, it is obvious, that the joints of our cubi cal pieces must be perpendicular to that, so as to pre vent 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 catena rian curve : but, in our opinion, this is at present unne cessary. 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 a bridge, which, independent of the varying loads that pass over it, must be filled up at the haunches, so as to form a convenient roadway. In this case, some
farther modification becomes necessary. The haunch E of the arch ACB, (Plate LXXX, Fig. 3,) bearing a much greater depth of stuff than the crown, it must be so contrived as to resist this additional pressure. Eve ry variation of the line FGH, or extrados, will require a new modification of the curve ACB, or intrados, and the contrary. Accordingly, M. de la Hire has suggested a good popular mode of investigating this subject. Let it be required to determine the form of an arch of the span AB, and height CD, proper for carrying a roadway of the form FGH. Mark off, upon a vertical wall, the points A, B, C', inverting 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 AB into any number of equal parts, and drawing the perpendiculars ab, cd, Eke. suspend from the intersec tions e, f, bits of chain eb,fd, Sec. so trimmed, that their ends may fall on the line of roadway ; and it may be ob served, 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, AC'B, the same ratio which the stuff to be filled into the haunches has to the whole weight of the archstones; the depth of which must of course he previously determined. 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 proportion. This being equally divided among the small chains, will give a roadway very nearly parallel to the former. The curve will evi dently be a perfect curve of equilibration, and extreme ly near the one wanted. And this whole process is so easy, that it may be gone through in a short time by any intelligent mason.
But although this mechanical way of forming an equi librated arch be founded upon principles sufficiently just, and be perhaps the simplest and best way in which the practical builder could form the original design of such an arch, yet as it affords no general rules that may be applied to the construction of arches, we pro ceed to consider the same subject in a mathematical point of view.