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Vault, Cylindro-cylindric, that which is formed of thesurfaces of two unequal cylinders. When their axes cut eachother, they are denominated Welsh groins.

Vault, Groined , a compound vault, which rises to thesame height in its surfaces as that of two equal cylinders,or a cylinder with a cylindroid.

In the Temple of Peace, at Rome , the middle aisle isgroined, all the arches are elliptic, their chord, or springingline, being that of the lesser axis: the small groins springfrom mere points. The main vault of this edifice was sup-ported by columns, which have long been removed ; theentablatures, however, and the springing of the groins, stillremain. The side vaulting is cylindrical, and coffered inoctagons and squares. The elliptic passages of the Coliseumare groined. Groins are to be found in many Roman build-ings still remaining. See Adams’s Ruins of Spalalro ; alsothe Ruins of Balbec and Palmyra, by Wood. The domeof the Pantheon is coffered ; that of the Temple of Bacchusis plain.

The groins of Gothic edifices are numerous, and form adifferent class, which it would be difficult to define. The.most beautiful specimens are to be found in England ; andamong 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 isthe same as that of the arch, which has been given under thearticle Stone Bridge. The four walls of a building arestrong ties to every kind of vaulting ; and more particularlythat of groins, where the horizontal pressure is directedagainst the angles, and, consequently, the four walls in thiscase will become ties, which ought, therefore, to be firmlybuilt or bound together with a cast-iron bar, which will beentirely concealed. Besides what has been shown under thearticle Stone Bridge, the reader should also consult thearticle Dome, and the following quotation from Hutton’sMathematics, vol. iii., which is necessary to be understood byevery engineer.

“ In the practice of engineering, with respect to the erec-tion of powder-magazines, the exterior shape is usually madelike the roof of a house, having two sloping sides, formingtwo inclined planes, to throw off the rain, and meeting in anangle, or ridge, at the top ; while the interior represents avault, more or less extended as the occasion may require ;and the shape, or transverse section, in the form of somearch, both for strength and commodious room, for placingthe powder-barrels. It has been usual to make this interiorcurve a semicircle. But, against this shape, for such a pur-pose, I must enter my decided protest; as it is an arch thefarthest of any from being in equilibrium in itself, and theweakest of any, by being unavoidably much thinner in onepart than in others. Besides, it is constantly found, that afterthe centering of semicircular arches is struck, and removed,they settle at the crown, and rise up at the flanks, even witha straight horizontal form at top, and still much more so inpowder-magazines with a sloping roof; which effects areexactly what might be expected from a contemplation of thetrue theory of arches. Now, this shrinking of the archesmust be attended with other additional bad effects, by break-ing the texture of the cement, after it has been in somedegree dried, and also by opening the joints of the voussoirsat one end. Instead of the circular arch, therefore, we shallin this place give an investigation, founded on the true prin-ciples of equilibrium, of the only just form of the interiorwhich is properly adapted to the usual sloped roof.

“ For this purpose, put a = d k the thickness of the archat the top, x = any absciss, dp, of the required arch ad cm,

V A U

u = k R the corresponding absciss of the g V en exterior linek i, and y = PC = r i their equal ordinates. Then by theprinciples of arches, in my tracts on that subject,; s found

that c I or w = « + a: — #=QX

y x — x y

, or=

supposing y a constant quantity, and where Q is some ctain quantity to be determined hereafter. But KB ° ris = ty, if t be put to denote the tangent of the given ang.^of elevation kib, to radius 1 ; and then the equation

is

w = a 4- x — ty = —

y

“ Now, the fluxion of the equation w = a + x W 1io = x — ty , and the second fluxion is w — x; thereto 1- >

q ii> i hence

the foregoing general equation becomes w = —p ; ana

Q W W

w w == -—, the fluent of which gives w 2

y 1

__ . but at V

_ r'

the value of w is = a, and w = 0, the curve at

p

■2

Q

parallel to k i ; therefore the correct fluent is w 2 — a ~~ ijQ w

Hence, then, y 2

or y

W Q . . t he correc 1

w 2 — a" ” f (w 2 — afluent of which gives y zz *J Q X hyp. log. ofw + f (w 2 — a 2 )

a ' that

“ Now, to determine the value of Q, we are to consider nwhen the vertical line C i is in the position ALor 51N ! cw — c I becomes = A l or m n = the given q u; ' ntl ^j o!!suppose, and y = AQorQM = i suppose, in which p° 8the last equation becomes b — q x hyp. log.

c + f (c 2 —a 2 )

; and hence it is found that the value

of t )]e

constant quantity f q, is

. wh lC ^

h. 1. c + ^ (c* — a 2 ) '

being substituted for it, in the above general value of# tl)8 ‘

value becomes

log. of

w + f (w 2 — a s )

b x

log. of

o + V (c 2

b x

log. of tv + f (w 2 — a 2 ) — log. o .

log. of c + f (c 2 — a 2 ) — log. a ' lv?a y 9

from which equation the value of the ordinate P C m ybe found, to every given value of the vertical C !• wb lC ^

“ But if, on the other hand, PC be given, to find. fobwill be the more convenient way, it may be found in

_i-x m -0

lowing manner: Put a = log. of a, and C — ^

0 + f (c - a ' the above equation gives oy

+

the

nuW

bet

log. of w -f- hf O 2 — a 2 ); again, put n ^

whose log. is c y + a ; then n = w + f { w

. an 0

a 2 + n 2

hence w = —~- = C I.

2 n

e tin 9

real ca8 ® p 0 v crt*'

“ Now, for an example in numbers, in a an svers enature, let the foregoing figure represent a r