Before we give a more rigid determination, we should wish to shew the practical builder, that the solution of this problem may be easily approximated to, by the help of the trigonometrical tables. For we may suppose the matter of the archstones to be the same in specific gra vity with that which lies above it ; and as there can be no impropriety in considering the arch as polygonal, from joint to joint, our mean versed sine is only half the sum of those at the two joints. The supposition is not strictly accurate, but it is sufficiently near : greater strictness would only serve to render the calculation more complicated, without making it more useful.
The following table exhibits, in the first line, the sup posed sections of the polygonal vault, taken 5° asunder. The second line is the angles of abutment, or inclination of each joint to the vertical. The third, the ratio of weight in each section, taking as the standard 874 or the length of the arch of 5° in 1000ths of radius ; of course this line is merely the differences of the natural tangents taken at every 5°.
The use of this will be understood by the following example. Suppose the thickness at crown 4- of the ra dius of the circle, or .142857, which answers very nearly to the 1 1 th key. We see by this last table, that the flanks of the arch over that key, or at 50° on each side of the arch, must be filled solid. Between that and the crown they must be lightened, by using lighter matter, or making vacant spaces in the spandrel: and at a greater distance from the crown, the flank, although solid, will be too light for a crown of 4-; so that we must expand or increase the breadth of the arch, in order to preserve Accordingly, the density beginning at about the crown, must be about 4 at 30° from it, and thence gra dually increase till about 60°, where it must be 1, or equal to that of the arch. After that, if no denser material can be employed, the arch must be expanded in breadth, hav ing already arrived at the limit of density.
If we make the thickness at the crown of radius, the densities will just be 2 of the above numbers ; the point of solidity will be removed a little farther from the crown ; and indeed whatever be the thickness, the densities will be proportional to the above numbers, and may easily be had from them.
The above is for a horizontal roadway. There will be some alteration requisite if the road be made to slope up the arch : the quantity of pressure that is thus lost, must be corrected by increasing the density of the spandrel ; and this will be more necessary towards the springing.
It will not be difficult for the practical builder to form an the equilibration. Every different thickness of crown will require a different arrangement in this respect. Without therefore prosecuting this farther, let us as sume the thickness equal to A of the radius, or -4 of the span in a semicircle ; a proportion not unusual, and of easy calculation ; from thence, to find the density of the matter in the spandrel, take the numbers of line 7, di vide by those of line 9, and multiply by the given thick ness that is, divide by 10; we have for the density in the spandrel, in that case, idea of its effect. Take the section at any part, say 30° from the crown, where the horizontal distance is 2 of ra dius; suppose the road to slope 1 in 10, for example, which is great, the fall will have become of radius, or .05; the versed sine is .1339, accordingly the height in the spandrel is reduced to .0839, and the density being in creased, inversely as the height, we have .552 in this case at 30° instead of .346, other things being the same. Yet this density is too great ; for the solid matter in the road way will be increased, being lengthened by sloping. At the same time it admits of doubt, whether it may not be made thinner, in the same proportion; for its oblique po sition gives a greater vertical thickness. This will pre serve the density at .552, and the whole series will he found, by deducting of the sine from the versed sine in col. 8, and proceeding with the remainders as with col. 8, as follows: Where we find the density in no case less than 1, which is about 30° from the crown, and it increases both ways, about i at 45° and at 19°, and solid about 53° and 162°, the first 10' are marked negative ; for we should observe, that when we keep the thickness at the crown the parallel to the roadway cuts the curve of arch stones. We ought in fact to make the roadway of a pro per thickness where the arch approaches nearest to it, and relieve the crown by rounding the two inclined planes into each other. This will also tend to diminish the density necessary in the spandrel ; for the height will be a little increased, while at the same time a greater pressure is derived from the solid roadway. But we choose to allow the example to remain in this way, that the reader may see that every necessary information can be got, even in this way of considering it.