and therefore the joint of the stones there must be perpendicular to AB, and so also must be its projection on the horizontal plane.
Proceeding along the line of the curve to the point R, we observe that the pressure there must be in the direction of a tangent to the curve, wherefore the surface of a joint at R must be perpendicular to that tangent, and the exposed face of the stone must be right-angled. Now, the projection upon a horizontal surface of a right angle placed obliquely is not necessarily right ; in this case it cannot be right, and therefore the course of a line of joints represented in plan must bend away from being perpendicular to the side wall towards being parallel to the line of the abutment. Thus a continuous course of joints beginning at I must be shown in plan by sonic curved line such as IPp.
In many of the skewed bridges actually built, the outline of the arch is divided into equal parts, as seen on the ends of the vault ; the curved joint-lines IPp thus become portions of screws drawn on an oblique cylinder, and, although the arch-stone at the crown be rectangular, those on the slope cease to be so. The bearing surface is thus inclined to the direction of the pressure, and the tendency is to thrust out the arch-stones at the acute corners F and G. The fault is exactly the same as if, in ordinary building, the mason were to bed the stones off the level. The consequence is that skewed stone-bridges have not given satisfaction, the fault being attributed to the principle of the skew, whereas it should have been assigned to the unskilfulness of the design.
Let figure 9 be an elevation projected on a vertical plane parallel to AB, EIG, FSH, being the outlines of the ends of the arch, and the sections taken at equal intervals along the crown line being also shown; then, since the projection of a right angle upon a plane parallel to one of its sides is always right, the joint at R, as seen on this elevation, must be perpendicular to the curve at 11, and thus the curve IPp, representing one of the joint-courses, must cross each of the vertical sections perpendicularly. In this way each of the four-sided curvilinear spaces into which this elevation is divided must be right-angled at its four corners. This law is general, and enables us to determine the details of any proposed oblique arch.
If we draw, as in figure 9, the end elevation of the vault as intersected by numerous parallel planes, and lead a curved line crossing all these intersections perpendicularly, we obtain the end elevation of one of the joint-lines, and are able from it to prepare any other of its projections. The form and character of this end elevation IPp depends entirely on the nature of the curve BIG, hut is the same whatever may be the angle of the skew. In order to examine its general character, let us take in the crown line two closely contiguous points I, K, and from these lead the joint-lines IP, KQ, of equal length, then the straight line PQ is equal and parallel to IK, on any of the projections.
If in the end elevation, figure 9, we continue the joint IP to meet the vertical section OQ in p, we may regard PQp as a small rectilineal triangle, right angled at p, while PQp is the inclination to the horizon. Now, PQ Qp : : It : cos PQp, while PQ is equal to KI, the breadth of the arch-stone at the crown, wherefore the breadth of the course at the crown is to the breadth of. the same course at any other place as radius is to the cosine of the inclination there. -Hence it follows, as is shown in the end elevation, figure 10, that the arch-stones gradually diminish in breadth from still narrower, and an infinity of them would be needed to reach the abutment of a semicircular or semi-elliptic arch, because the cosine of the inclination there is zero. In no properly-built skewed bridge can the arch-stones show equal divisions ; and it is impossible to continue the arch to the complete half circle or half ellipse.
Passing from the end elevation, figure 9, to the plan, figure 8, we observe that Qp on the plan is less than the actual Qp 