VAULT, Groined, a compound vault, which rises to the same height in its surfaces as that of two equal cylinders, or a cylinder with a eylindroid.
In the Temple of Peace, at Rome, the middle aisle is groined, all the arches are elliptic, their chord, or springing line, being that of the lesser axis : the small groins spring from mere points. The main vault of this edifice was sup ported by columns, which have long been removed ; the entablatures, however, and the springing of the groins, still remain. The side vaulting is cylindrical, and coffered in octagons and squares. The elliptic passages of the Coliseum are groined. Groins are to be found in many Roman build ings still remaining. See Adams's Ruins of Spolatro ; also the Ruins of Bulbcc and Palmyra, by Wood. The dome of the Pantheon is coffered; that of the Temple of Bacchus is plain.
The groins of Gothic edifices are numerous, and form a different class, which it would be difficult to define. The most beautiful specimens are to be found in England; and among the numerous variety, we might mention King's Col lege, Cambridge, and King Henry the Seventh's Chapel, Westminster. See GOTHIC ARCHITECTURE.
A vault is an extended arch, and therefore the theory is the same as that of the arch, which has been given under the article STONE BRIDGE. The four walls of a building are strong ties to every kind of vaulting; and more particularly that of groins, where the horizontal pressure is directed against the angles, and, consequently, the four walls in this case will become ties, which ought, therefore, to be firmly built or bound together with a cast-iron bar, which will be entirely concealed. Besides what has been shown under the article STONE BRIDGE, the reader should also consult the article DOME, and the following quotation from Hutton's Mathematics, vol. iii., which is necessary to be understood by every engineer.
" In the practice of engineering, with respect to the erec tion of powder-magazines, the exterior shape is usually made like the roof of a house, having two sloping sides, forming two inclined planes, to throw off the rain, and meeting in an angle, or ridge, at tho top ; while the interior represents a vault, more or less extended as the occasion may require; and the shape, or transverse section, in the form of some arch, both for strength and commodious room, for placing the powder-barrels. It has been usual to make this interior curve a semicircle. But, against this shape, for such a pur pose, I must enter my decided protest ; as it is an arch the farthest of any from being in equilibrium in itself, and the weakest of any, by being unavoidably much thinner in one part than in others. Besides, it is constantly found, that after the centering of semicircular arches is struck, and removed, they settle at the crown, and rise up at the flanks, even with a straight horizontal form at top, and still much more so in powder-magazines with a sloping roof; which effects are exactly what might be expected from a contemplation of the true theory of arches. Now, this shrinking of the arches
must be attended with other additional had effects, by break ing the texture of the cement, after it has been in some degree dried, and also by opening the joints of the voussoirs at one end. Instead of the circular arch, therefore, we shall in this place give an investigation, founded on the true prin ciples of equilibrium, of the only just form of the interior which is properly adapted to the usual sloped roof.
" For this purpose, put a = D x the thickness of the arch at the t1 -= any abseiss, D r, of the required arch A D C u = x It the corresponding absciss of the given exterior line and y=pc =at their equal ordinates. Then by the principles of arches, in my tracts on that subject, it is found y that o or to =a+x u= yx , Or= Q X x suppossing y a constant quantity, and where Q is some cer tain quantity to be determined hereafter. But r. R or ee is = ty, if t be put to denote the tangent of the given angle of elevation s t R, to radius 1 ; and then the equation is Q X 20 = a + x ty = .. "Now, the fluxion of the equation w = Is + x ty, is w = x ty, and the second fluxion is w = x; therefore, 0 2 the foregoing general equation becomes w = and hence Q 2/2 2 Q /0 TO 20 = , the fluent of which gives w but at D y the value of w is = a, and w = 0, the curve at n being parallel to K I therefore the correct fluent is a2 Q s Hence, then, = 111 or y the correct 20U fluent of which gives y= voX hyp. log. of w + a') a cal section of a magazine arch irdanced in all its parts, in which the spur or width A M is 20 feet, the pitch or height D Q is 10 feet, thickness at the crown n K = 7 feet, and the angle of the ridge 1.1: N 37', or the half of it L s v =-. lb-}' the complement of which, or the elevation K I R, is 33° the tangent of which is = which will therefore be the value. of I in the foregoing investigation. The values of the other letters will be as follows, viz. D K = a = 7 ; AQ=b=10; DQ=1a=10; AL=c= 101 =!;.;A= 1 log. of 7 = .S4509S0 ; c = b X ofc a log. of 31 ± 520 21 = log. of 2.56207 = .0408591 ; c y + A = .0408591 y + .8450980 = log. of a. From the a2 n a2 general equation, then, viz. c 2 = w = n, n by assuming y successively equal to 1, 2, 3, 4, &c., thence finding the corresponding values of c y A or .0408591 y + .8450980, and to these, as common logs. taking out the corresponding natural numbers, which will be the values of ; then the above theorem will give the several values of w or c r, as they are here arranged in the annexed table, from which the figure of. the curve is to be constructed, by thus finding so many points in it.