"Nevertheless, as many of our readers are well qualified 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 com monly received theory of equilibration. From which, having cleared away the useless rubbish, if we can extract any proper materials, we may, like economical builders, make good use of them in our future structure.
"The first thing like a principle that we meet with is the assertion of the eminent Dr. Hook, that the figure into which a heavy chain, or rope, arranges itself, when suspended at the two extremities, being the curve commonly 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 generally adopted, and affords some useful hints, it may be well wolth while to examine it.
"Figure 1.—Let A B be a string or festoon of heavy bodies. hanging by the points A, B, and so connected, that they can not separate although flexible. These bodies having arranged themselves in the calenaria A c B. conceive this to be turned exactly upside down. The bodies A and 13 being firmly fixed, then each body in the arch A B n, 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 posi tion, 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 of junction, it is obvious, that the joints of our cubical pieces must be perpendicular 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 be cause in their thickness some catenaria is included.
"This curve is, indeed, the only one proper for an arch consisting of stones of 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 road way. In this case, some further modification becomes
necessary.
" Figure 2.—The haunch, E, of the arch A c n, bearing a much greater depth of stuff than the crown, it must be so contrived as to resist this additional pressure. Every varia tion of the line F G n, or extrados, will require a new mod i fieation of the curve A C B, 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 A 13, and height c n, proper for carrying a road-way of the form F G it. Mark off upon a vertical wall, the points A, a', n, inverting the required figure: suspend from A Ban uniform chain or rope, so that its middle may hang a little below the point c', and dividing the span, t B, into any number of equal parts, and drawing the perpendiculars a b, c d, &e., suspend from the intersec tions, e, f, bits of chain, e b, f d, &c., so trimmed, that their ends may fall on the line of the road-way ; 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 in to its proper position.
" But although this mechanical way of forming an equili brated arch be founded upon principles sufficiently just, and be perhaps the simplest and best way in which the practical builder could farm the original design of such an arch ; yet as it affords no general rules that may be applied to the con struction of arches, we proceed to consider the same subject ,• in a mathematical point of view.
"Figure 3.—And first, then, in the semicircular polygon, as it is called, where weights arc hung on the thread A c' c c" 13, which bring it into the position A C 13, we have at each angle three forces in equilibria. Wherefore, by the princi ples of statics, they are to one another as the sines of the opposite angles; that is, the tension r c is to the tension lc, as sine 1 c w is to sine r c w, but the tension from c to 1 is the same as from c' to r. Also, sine 1 c w is the same as sine r' c' w', since these angles are supplementary, c w, c' uv' being parallel ; therefore the tension r c is to the tension r' c' as sine r' c' w' to sine r c w. Or, the tension in each part of the chord is inversely as the sine of its inclination to the vertical.