of the elevation in the ratio of the cosine of the inclination to radius, and, therefore, on the plan, the breadth at the crown is to the apparent breadth of the course at any other place as the square of the radius is to the square of the cosine of the inclination there; so that, at the inclination of GO° the apparent breadth will be quarter of that at the crown.
Again, in figure 11, which is the side elevation of the vault, or its projection on a vertical plane perpendicular to the road, the apparent distance Qp is to the actual distance Qp of figure 9 as the sine of the inclination is to radius, wherefore, the apparent breadth Qp on this projection is proportional to the product of the sine by the cosine of the inclination, that is, to half the sine of twice the inclination. The width on this projection is therefore greatest at an inclination of 45°, being there just one-half of the actual breadth at the crown of the arch.
This reasoning is founded on the supposition that the distance IK is excessively small, and the resulting conclusions are strictly true only of an infinitely narrow course of arch-stones; they are, indeed, differential equations which must be integrated in order to be applied to actual practice. Thus we have seen that the curved line IP, figure 9, crosses the section NP perpendicularly at P, but then it does not continue in this direction for any perceptible distance. The draughtsman may attempt to trace it by making the sections very numerous, and by drawing perpendiculars across the successive intervals; but however numerous he may make these sections, he can thus only effect an approximation to the true curve. We must integrate, that is, we must obtain the aggregate of an infinite number of infinitely small portions in order to reach an absolutely true result.
These conclusions hold good whatever may be the outline of the arch. The most common, and therefore the most interesting case, is when the longitudinal section is circular, the cross section taken perpendicularly to the abutment being then an ellipse with its shorter diameter placed horizontally, the vault being an oblique cylinder. Figure 9 is actually drawn for the circular arch. If then 0 be the centre of the circular arc NP, the curve IP must at P tend towards 0, so that the draughtsman, while making the step across one of the intervals, has only to keep his straight edge up to the corresponding place of the centre. If we place the paper horizontally, fix a small heavy round body at P to the end of a thread OP, and then draw the end 0 of that string along the straight line HEF, P would always more towards the then position of the point 0, and would trace out the curve of which we are in search. The projection, then, of the joint of an oblique circular arch upon a vertical plane parallel to the road, is always the curve known by the name of the Tractorg. All tractories have the same shape, the size merely is regulated by the length of the thread OP, that is, by the radius of curvature of the circular arch. Hence, if the delineation of it have been accurately made in one case, the curve for another case may be obtained by mere enlargement or reduction ; or, still better, in all eases it may be traced by help of a table of co-ordinates, such as that subjoined, which shows the dimensions of the tractory as represented in figure 12, in decimal parts of the radius of curvature of
the arch. The computations have been made for equal motions of the point 0, corresponding, therefore, to equal distances measured along the crown-line of the arch. The headings of the columns sufficiently explain their contents. By help of these the form of the tractory may easily be obtained, and with a piece of veneer or of thin metal cut to this shape, the architect may obtain all the details of the intended structure, first working out the said elevation, figure 9, and transferring the several points therefrom to the other projections.
If we put s for the angle of the skew, v for the distance IN measured along the crown of the vault, and i for the inclination at 