these in terms of the sines and tangents of the successive angles of inclination ; but in reducing these to numbers, he has been led, by 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 approximation.
The weight of the section C may be determined in the same way as the foregoing. But surely more simply thus : From c draw c s parallel to w z, that is, at right angles to K o, and make it equal to w z b; draw s c at right angles to a o, meeting the vertical c e in c, then c c repre sents the weight of C. From n draw n T parallel and equal to s c, draw T d perpendicular to n o, meeting the vertical D d in d, D d is the weight of D, and so on successively.
" Nay, instead of drawing n T parallel to s c, and T d per pendicular to n o, we may at once draw from s, s perpen dicular to D o, which will cut oil' for us e = D d, the weight of the section D. It is of no consequence, although the lines of abutment do not all run to the same centre o.
" And thus we obtain a general construction for all the sections, which turns out abundantly simple ; for upon any vertical line, 6' e, Figure 9, it c 6 be taken to represent the given weight of any section, 0, and c T be drawn at right angles to c o, and b T at right angles to B o, meeting the other in T; then T b represents the pressure against the abutment o n, and T c the pressure against o c; and by drawing T d at right angles to n o, T e to E o, &c., we have the weights of the successive sections represented by c d, d e, &c., and the pressure on their lower abutments represented by T d, T 0, &C.
" We may carry the same mode of determination to the other side of C, and pass the vertex of the arch. The divi sions representing the weights of the sections will run upwards along the indefinite line c The pressures on the abutments will be determined as before. Should the two sides of a section be parallel, the perpendiculars through T upon them will coincide ; such a section, therefore, should have no weight. But should the two lines of abutment diverge towards the lower side, the line expressing the weight of that section will return upon the vertical, showing that such a section requires the reverse of weight, viz., a support
from below. The line T v drawn horizontally through T, exhibits the horizontal pressure, which is uniform through the same equilibrated arch. But it is evidently greater, the less b T and c T are inclined to each other, the weight 6 c being constant, that is, the smaller the angle of the wedges, or sections. It also increases directly as the weight of the section &c. The line v e expresses the weight of the semi arch, or perpendicular pressure on each pier ; being the sum of the weights of alt the sections in the semi-arch.
"Again, it is obvious that the angles 6 T c, or c T d, &c. are equal to the angles of the sections B o c, c o D, &c. therefore, the weight of any section, E, be given=d c, and the requisite angle of that section be required, everything else being known, we have only to join 1. e, and the line E o, being drawn perpendicular to T e, will exhibit the inclination of the lower abutment of the section; d T e is the angle of that section. And here it matters not where the point E be, that is, how great. the base of the section be, provided the weight is equal to d c. We also see, that while the angles remain the same, and the weights proportional, it is of no consequence what the curve passing through the lower edges of the sections, or through their upper edges, may be, they may even be straight lines. According to this principle, the architect is not confined to given forms of intrados or 4tra dos; he may take whatever curve appears most beautiful or useful : and, what is more, by the proper adjustment of the joints, he may cast the ultimate pressure in any direc tion which he thinks most conducive to the strength of the edifice.
" We now proceed to show the application of this investi gation to some practical eases ; and the first we shall con is that known by the common, though awkward name of the flat arch ; one with which every mason i.; perfectly fluniliar, though it be seldom noticed by writers one quill bration.