A smaller degree of slope, as 1 in 20, or 1 in 40, will tend to diminish still farther the density necessary in the spandrel, and approximate it to those found for the horizontal line. We might calculate the densities as well for these useful slopes, as for other thicknesses of the crown or proportions between the key-stones and superincumbent roadway, which, in the preceding en quiry, is taken at equality ; but we forbear doing so, being satisfied with giving the intelligent practitioner clear ideas of the subject. Ile already knows there are pretty wide limits to his practice ; and, if the case be any way delicate, we should think any person deserving the name of architect may, after what we have said, go over the necessary calculations for himself.
The mathematical reader will perhaps say, that we have taken a very awkward and unscientific mode of resolving this problem ; we are not, however, inclined to admit that opinion. Our object has not been to give a specimen of the application of calculus; but to show the practical builder how a good conception may be formed of the relative pressures in different parts of his arch, and this by a process purely arithmetical, and which is level to every capacity. We conceive that this is the way to make our speculations really useful, and perhaps it were well if scientific men had this oftener in view. Neither have we carried our results to many figures, like some authors, who give five or six places of decimals ; for we have considered that no common modes of measuring either distances, angles, or weights, can proceed to any thing near that nicety. Yet, that we may not rest satisfied with an approximation without sheaving what degree of accuracy can be obtained, and especially that we may render this mode of conceiving the subject more useful by a more complete solution of the problem, we proceed to the following analytical in vest igation.
We have already shewn that the weights of the sec tions must he proportional to the differences of the tan gents of the successive angles of abutment.
This is to be provided for, 1st, By the weight of the arch-stones ; here taken as constant.
2d, By the weight of matter forming the roadway, &c. ; here taken as of uniform thickness, and varying in effect only as the difference of the sines of the distan ces from the crown.
3d, By the matter in the spandrel ; which may be made to vary in density, and is equal in the longitudinal SCLaiUn Lii the versed sine multiplied into the difference of the sines.
Take z, the angular distance from the vertex, ti the density in the spandrel, a the thickness of the arch or keystone, r the thickness of the road, &c. at the crown.
Which may be thus expressed : In an arch of uniform thickness, with a horizontal road way, given the thickness of the arch and roadway ; re quired the density in every part of the spandrel, so that the whole may be preserved in equilibrio.
To twice the log. tangent of the angular dis nee front the vertex, add the log. secant, and subtract the log. versed sine ; take the corresponding number, and multiPly by the thickness of crown ; and add to this the secant mul tiplied by the thickness of roadway. These being e.rpress ed in terms of the radius, the resulting number gives ac density in the spandrel, or proportion which the solid matter in measuring transversely across the arch bears to the whole breadth at the crown.
Accordingly, we have constructed the following short table front this formula. The first line shews the mul tiplier for the thickness at the vertex. The second shews that of the roadway ; and is merely the table of natural secants.
And if these densities be compared with those of line 1 lth, the reader will satisfy himself as to the value of the approximation which is there employed.
It would not be difficult, upon principles similar to the above, to establish a theorem for the elliptic, para bolic, and other curves, similar to that we have now given for the circle. But this is of less general use ; and the limits assigned to an article of this kind, pre vent us from entering at present upon an investigation, through which, perhaps, few of our readers would be inclined to follow as Another opportunity may be found of offering this to the public notice.
But, in the meantime, the reader must at once see, that by this mode of expressing the density in the span drel, the solution we have given applies to any form of roadway. All that is necessary, is to compare the den sities in the table above given for a circular arch, with the relative height between the back of the archstone and the bottom of the roadway in the given design ; and this will, we are sure, be more readily, and more satis factorily done by the common builder with his sector and compasses, than by giving him equations for any number of figures of extrados.