Let us now return to the geometrical construction.
The weight of the section C, may be determined in the same way as the foregoing. But surely more sim ply thus : From c draw cs parallel to wz, that is at right angles to NO, and make it equal to wz-He b ; draw sc at right angles to Lo, meeting the vertical c c in c, then c represents the weight of C. From D, draw DT pa rallel and equal to s c, draw T d perpendicular to DO, meeting the vertical D d in d, D d is the weight of D, and so on successively.
Nay, instead of drawing DT parallel to s c, and T d perpendicular to Do, we may at once draw from s, sd' perpendicular to DO, which will cut off for us c a'=n d, the weight of the section I). 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, Fig. 10 : for, upon any vertical line b' e, if c b be taken to rep e sent the given weight of any section C, and c T be drawn at right angles to co, and b T at right angles to BO, meet ing the other in T: then T b represents the pressure against the abutment OB, and ire the pressure against o c, and drawing i d at right angles to DO, •C to 110, &c. we have the weights of the successive sections re presented by c d, d e, &c. and the pressure on their low er abutments represented by T d, T C, &c.
We may carry the same mode of determination to the other side of C, and pass the vertex of the arch. The divisions representing the weights of the sections will run upwards along the indefinite line c The pres sures on the abutments will be determined as before. Should the two sides of a section be parallel, the per pendiculars 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 re turn upon the vertical, shewing that such a section re quires the reverse of weight, viz. a support from below.
The line TV drawn horizontally through 'r 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 b 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, &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 all the sec tions in the semi-arch.
Again, it is obvious that the angles b T c, or C T d, &c. arc equal to the angles of the sections noc, con, &c. If therefore the weight of any section E be given =d c, and the requisite angle of that section be required, every thing else being known, we have only to join 'r e, and the line E0 being drawn perpendicular to T e, will exhibit the inclination of the lower abutment of the section ;dTe 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 e. We also see that while the angles remain the same, and the weights pro portional, 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 he straight lines. According to this principle, the architect is not confined to given forms of intrados or extrados ; lie 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 direction which he thinks most conducive to the strength of the edifice.
The reader will easily perceive, that the segments of the vertical line rapidly increase, as the perpendiculars to the line of abutment approach the vertical ; that is, as the abutments approach the horizontal line ; and in that 'Position, the last segment becoming infinite, it is impos sible by mere weight alone to effect the adjustment of the sections.