Though the geometrical construction we have just given is so simple, that it appears likely to answer every practical purpose ; yet it may be proper to express ana lytically, or lather arithmetically, the values of the seve ral quantities concerned in the investigation. This is attended with no difficulty, as T v e being a right angled triangle, it is obvisus that the weight v e ol the same arch is the tangent of v T e, or of the inclination of the lower abutment, when TV the horizontal force is radius ; at the same time also, the pressure T C on the abutment is the secant of the same angle ; and the weight c d of any section is the difference of the tangents of the incli nations of its upper and lower abutments. In like man ner a v, the weight of half the key-stone, is to TV the ho rizontal force as the tangent ol half the angle of that sec tion is to the radius; or, as radius is to the cotangent of the same angle.
We now proceed to spew the application of this inves tigation to some practical cases, and the. first we shall consider, is that known by the common, though awk ward name of the flat arch ; Plate LXXXI. Fig. I. one with which every mason is perfectly laminar, though it be seldom noticed by writers on equilibration. AB b a is a structure of this kind, adjusted to this equilibrium, and resting on the abutments A a, B b. Its construction is exceedingly simple ; nothing more is necessary than to craw all the joints in M, /L, &c. to one centre C ; and the reason is obvious ; for DK, KL, &c. are the differ ences of the natural tangents of the inclinations of the abutments, the perpendicular CD being radius ; and the same thing is true in the line d a, and in every other pa rallel section. The surface therefore A m, M 1, that is, the bulks or weights of the stones, are in the same ra tio, and it is that which is required by the above princi ples. Also, if we assume the line of its base to repre sent the weight of any stone in the arch, for example, KD for half the keystone ; then the perpendicular CD is the horizontal thrust, drift, or shoot of the arch. By increas ing DC, or diminishing it, that is, by drawing the joints to a lower, or a higher centre, we may alter this thrust at pleasure. What if we should take C up to D ? Some curious ideas occur here, but being chiefly speculative, we shall not now pursue them. They serve to connect this case very neatly with the lintel and the Egyptian arch, (or that formed by flat courses of stones gradually overlapping each other, until the opening be covered), in each of which the horizontal thrust vanishes. We ought also to observe, that whatever weight of stuff lies on an arch of this kind, there is no change of design re quisite, so long as the upper surface or roadway is hori zontal. For being every where of the same height, the mass incumbent on any stone will be proportional to its base, viz. the back of that stone ; since we must conceive the stuff to press vertically. It is therefore the same as
if the whole arch had undergone a change of specific gravity ; every pressure will be increased in the same proportion.
The design of an equilibrated horizontal arch, or plat band, being thus easily formed, it will not be difficult to extend it to a curve of any form, Plate LXXXI. Fig. I. is an arch of this kind. It is a circular segment from the centre C, to which the joints of the horizontal arch were directed ; the two key stones have the same weight and obliquity of abutment ; consequently the horizontal thrusts arc the same. The other arch stones being previously intended to have the same weight with those of the flat arch, it is only necessary to draw the lines 1 1, 2 2, 3 3, parallel to Kk, L/, INIm, and so as to produce this equality. This being merely a simple pro blem in mensuration, we shall not occupy the reader's attention with the solution of it. In the figure referred to, we have divided the soffit AB of the flat arch into equal parts ; all the stones therefore of that as well as the curvilineal form, are of equal magnitude and weight, the angles of the arch stones only varying. We might make a table of these angles, to any given form of key stone, but it is really unnecessary ; for we have only to take the tangent of half the angle of the key stone, or more correctly, of the angle of inclination to the vertical of one abutment to the key-stone, from a table of natural tangents, and by adding to it twice the same number successively, we have the natural tangents of the incli nations of all the other abutments. We believe, how ever, that the practical builder will prefer a geometrical construction to this, and lay off his joints by means of the common bevel.
Before we take leave of the straight or flat arch, there is another of its properties we would wish parti cularly to be noticed. The reader must have already observed, that when CD expresses the horizontal thrust, or pressure of the vertex, CK, CL, CM, &c. express the perpendicular pressures on the successive joints Kk, L/, Mm, &c. Now, it is obvious, that Kk, L/, &c. are proportional to CK, CL ; for AD, ad, are parallel. Therefore the vertical sides of the arch being parallel, the pressure on each joint of the flat arch is always propor tional to the surface of that joint, and the pressure on each square inch of joint throughout the arch is always the same. It may readily be found too, by dividing the horizontal thrust by the area of the vertical section Dd. This is a most valuable property, for it secures unifor mity of action in every part of the structure. But it is not to be found in the arch abd ; for there, the joints being nearly equal, the pressure on each increases as we descend from the vertex, and may, at the lower sec tions, be eventually so great as to overcome the cohesion of the materials.