ng 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 the point P, r being the radius of the arch, the distance IN or i0 of figure 10 is clearly v sin s, and as the result of the integration, we obtain - v sin s - Nap. log tan (450 + , by help of which equation we can readily determine i when v is known, or v when i is given. The table of Napierian logarithmic tangents being very scarce, it is convenient to convert these into denary or common logarithms. Putting, as is usual, M for the modulus of denary logarithms, that is, for '43429 44819, the above equation becomes - - r . v . sin s = log tan (45° + from which it is quite easy to tabulate the values of i corresponding to equidifferent values of v, because the constant factor - has to be only once computed; i, that is, the number of degrees in the arc NP being thus computed for each of the successive sections of the vault, we have only to divide a tape-line so as to show degrees and minutes of the actual circle in order to be able at once to mark the course of the joints upon the centering of the arch; or better still, instead of the degrees, we may write upon the tape the successive values of NP, and then the commonest workman will Le able to lay off the lines.
The only other kind of skewed arch likely to possess any interest is the elliptic. In right arches the semi•ellipse is sometimes used on account of the grace of its form, but this reason for its adoption disappears in the case of the skew, because then we can only use a portion of the semi-ellipse. The end elevation of a joint in an elliptic skewed arch is a modified form of the tractory, and the general features of the arrangement are analogous to those of the circular arch.
The arch-stone of a common bridge is wedge-shaped, having two flat faces AacC, BbdD, inclined to suit the breadth of the course, but in the skewed bridge the corresponding faces are twisted, Cc not being parallel to Aa, and thus the dressing of them requires both skill and care. The dimensions of the stone and the inclinations of its four edges may easily be computed when its intended