332 A E C H at P tend towards 0, so that the draughtsman, while mak ing 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 of that string along the straight line HEF, P would always move towards the then position of the point O, 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 Tractory. 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 cases 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 Fig. 12. the arch. The computations have been made ior equal motions of the point O, corresponding, therefore, to equal dis tances measured along the crown-line of the arch. The head ings 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 iO of figure 10 is clearly v sin s, and as the result of the integration, we obtain v sin s XT . /AKO , t - = Nap. log tan (45 + %i) , 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 448 19, the above equation becomes . v . sin s = log tan (45 + |i) , from which it is quite easy to tabulate the values of i cor responding to equidifferent values of v, because the constant factor M . sin s ~~r 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 Lo able to lay off the lines. I i sin i COS t f-sin i o 00 ooooo 1-00000 ooooo 1 5 43 09967 99502 00033 2 11 23 19738 98033 00262 3 10 56 29131 95663 00869 4 22 20 37995 92501 02005 5 27 31 46212 88682 03783 6 32 29 53705 84355 06295 7 37 11 60437 79670 09563 8 41 37 66404 74770 13596 9 45 45 71630 69779 1S370 1-0 49 36 76159 64805 23S41 11 53 11 80050 59933 29950 1-2 56 29 83365 55229 36635 1-3 59 31 86172 50738 43828 1-4 62 18 "88535 46492 51465 1-5 64 51 90515 42510 59485 1-6 67 10 92167 38793 57833 17 69 18 93541 35357 76459 1-8 71 14 94681 32180 85319 1-9 72 59 95624 29259 94376 2-0 74 35 96403 26580 1-03597 2-1 76 02 97045 24129 1-12955 2-2 77 21 97574 21892 1-22426 2-3 78 33 98010 19852 1-31990 2-4 79 38 98367 17995 1-41633 2-5 80 37 98661 16307 1-51339 2-6 81 30 98903 14773 1-61097 27 82 19 99101 13381 170899 2-8 83 02 99263 12117 1-80737 2-9 83 42 99396 10971 1-90604 3-0 84 18 99505 09933 2-00495 10 iro OT TP if The only other kind of skewed arch likely to possess any interest is the elliptic. In right arches the semi-ellipse is somatimes 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, B6rfD, inclined to suit the breadth of the course, but in the skewed bridge the corre sponding 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 Fig. 13. edges may easily be computed when its intended position is known, and thus the degree of twist on each of its faces may be ascertained, and the lines may then be marked off on the ends of the stone. The theory of the skewed arch was given for the first time in the Transactions of the Royal Scottish Society of Arts for 1833; from which it was copied into the Civil Engineer and Architect s Journal for July 1840, which see. (For the history and various forms of the arch see ABCHI-
TECTURE.) ( E . S.)