The analytical writers have assumed one leading prin ciple, that the arch is in every point kept in equilibration solely by the gravity of the superincumbent column of matter. Now, it is even doubtful whether this prin ciple be true. At any rate, they do not consider the numerous modifications which it receives, from the cohe sion of that matter among itself; from the mutual cohe sion and friction of the archstones ; from the position of their joints ; from the different specific gravity which the arch and superincumbent matter have, or which they may be made to have ; from the lateral, and in some cases hydrostatic pressure, propagated to the masonry throughout that matter ; and, in fine, from a number of other causes, which, if not singly, are, when combined, at least of as much importance as the gravity of the ver tical column of matter alone.
Let us turn, therefore, to another mode of considering this subject, which has been adopted by De la Hire, Parent, Belidor, and many others on the continent, and in our own country by the ingenious Mr Atwood.
The latter has, from the known properties of the wedge, and the elementary laws of mechanics, exhibited to us a geometrical construction for adjusting the equi libration of arches of every form. The mathematical reader, who has not lost his relish for the ancient ge ometry, will find there an elegant specimen of its appli cation ; for he completes his geometrical construction without once having recourse to any other than the principles of elementary geometry and trigonometry. It had been well, indeed, if he had adhered longer to that mode of investigation ; for, by applying the analytic form too early, he has been led unawares to consider that only as an approximation to the values of the quantities sought after, which, in fact, is the expression for the values of these quantities themselves. Nevertheless we owe much to Atwood : he has shewn, that the advan tages of equilibration are not confined to any particular curve ; that the drift or horizontal thrust of an arch may be easily found; and that an arch may have all the ad vantages of equilibration, whatever its figure may be, merely by adjusting the joints of the arch-stones.
The stones, or sections of an arch, being of a wedge like form, have their tendency to descend opposed by the pressure which their sides sustain from the similar tendency of the adjoining sections. Should this pres sure be too small, the stone will descend ; should the pressure be too great, the stone will be forced upwards.
These pressures act in directions perpendicular to the touching surfaces ; for, if the original direction of any pressure should be oblique, it may be resolved into two forces, of which, while one is perpendicular to the surface, the other is parallel to it, and,of course, neither increases nor diminishes the perpendicular pressure.
The wedge A, Plate LXXX. Fig. 9, if unimpeded, would descend in the direction vo, but is prevented by the re-action of B and B', acting in the directions rq and perpendicular to the sides AG, Qn; and it is known, from the properties of the wedge, that if pq, or be to the weight of the wedge A, as no is to nG, the wedge A will remain at rest. If also the wedge A be only at li berty to slide down GA, considered as a fixed abutment, then the force rq alone will keep it in equilibrio. The force Pct being perpendicular to no, has no tendency to make A slide either up or down on that line, but pro duce it towards N, making Ntit equal to rq ; then this force acting obliquely at x, may be reduced to two others, viz. 1%111 perpendicular to AG, expressing the per
pendicular pressure on the abutment of A, and RN ex pressing the force or tendency it has to make A slide upwards along AG. Again, take the vertical line A a, expressing the weight of A, and draw a it at right an gles to AG ; it is very evident, that An expresses the tendency of A by its weight to slide clown GA. All is opposite, and is equal to Na.
For, draw the perpendiculars D d and A p, then the triangles A a u, DG d are evidently similar ; and also the triangles on d, (KIN, MNR, as they have always a common angle besides the right angle. Now the force rq, that is, MN is to the weight of A, that is AC, as OD to DG by supposition.
And A a : All :: AG : AP DG : D d Therefore, MN : AH OD : 1) d :: DfN : Nit.
Or mx has the same ratio to AH, that it has to that is, Au and sit are equal, or the tendency of A to slide downwards by its weight, is balanced by the ten dency of MN to make it slide upwards : wherefore the section A remains at rest in equilihrio.
Considering the whole arch as completed, with its parts mutually balancing each other, the force rq, which is necessary for sustaining the wedge A, will be sup plied by the reaction of the adjacent wedge B. Now, let it be required to ascertain the weight of B in propor tion to A, so that they, being adjusted to equipoise, may continue to be in equilibrio, when left free to slide along KB. Since MR is the pressure produced by rq in a direction perpendicular to AG, we must add to this it a, which is derived from the wedge A ; therefore make m n equal to H a, produce MR to Y, take yz equal to R draw zw at right angles to NB ; YW is the force tending to make B slide up BE : take therefore au' equal to YW, draw the perpendicular b meeting the vertical B b in b ; B b will represent the necessary weight of the wedge B; and the whole is so evident from the composition of pressures, as to require no further demonstration. Such is Atwood's construction ; he has rendered the demon stration much more prolix, by the unnecessary introduc tion of trigonometry ; and after shewing how the weight of the sections C, D, &c. may be found in the same way, he goes on to reduce these weights and pressures to analytical and numerical values. He finds these in terms of the sines and tangents of the successive angles of inclination ; but in reducing these to numbers, he has been led to the accumulation of small errors in that very operose way of proceeding, to give erroneous results ; and into the singular mistake of conceiving, that the real expression of these values was only an approxima tion. Had he recalculated the whole by more extended trigonometrical tables, they would have quickly unde ceived him ; and they would have shown him, that what he was thus searching so deeply for, was all the while lying exposed at the surface ; that the apparent difficul ties were entirely of his own creation, and his imagined accuracy was error. This should teach mathematicians to beware of thinking calculation the surest mode of eli citing truth. It should be the last thing employed. Nothing is so simple, so perspicuous, as the diagram. Its geometrical properties should be pursued as far as possible. They are not only clear, they are palpable. And in such applications of mathematical theory, the whole being a creature of the mind, it seldom admits of an approximating value in any part.