1499h. "In analysing a dome, it will be found that it is nothing more than rib-vaulting carried to its maximum, that it consists of as many ribs as there are vertical sections to be made in the dome, or is composed wholly of ribs abutting against each other, in direct opposition, by which the force of each is destroyed. In the ceilings of King's College Chapel, Cambridge, and Henry VII.'s Chapel, London, this most admirable invention is exemplified. The author ventures an hypothesis, that, in an equilibrated dome, the thickness of the vaulting will decrease from the vertex to the springing, and assigns the following reason theoretically, and the Gothic vaulting practically, in confirmation. 1499i. "The parts of a circular wall compose a horizontal arch; but the whole gravity of each part is resisted by the bed on which it rests, therefore the parts cannot be in mutual opposition; and, although the parts are posited like those of an arch, a circular wall has not the properties of one. In a semi-spherical dome the first course answers this description, no part gravitating in the direction of its radius. When the beds are oblique on which the parts of the wall rest, each course may then be called an oblique arch, as it then assumes the property of an arch, by having a double action, the one at right angles to, or on the bed, and the other in the direction of the radius; and if this arch be of equal thickness throughout, and has an equal inclination to the horizon, it will be an arch of equilibration. All the courses in a dome are oblique arches of equilibration, of various inclinations, between the horizontal line at the springing, and the perpendicular at its vertex. 1499j. "A dome is comprised of as many vertical arches as there are diameters, and as many oblique arches as there are chords. The actions of the parts of a vertical arch are eccentric, an oblique arch concentric; "consequently they will be in opposition, and the greater force will lose power equal to that of the less. An oblique arch bears the same relation to a dome as a voussoir does to an arch; when the vertical arches are not in equilibration, the action is upon the whole oblique arch, not upon the voussoirs separately; although a whole course or oblique arch (which must be the case, or no part of it, admitting that each course in itself is similar and equal throughout) be thrust outwards by the inequilibration of the vertical arches; the incumbent oblique arches will descend perpendicularly, keeping the same congruity of their own parts. 1499k. "As the voussoirs of each oblique arch are in equilibration, no one can approach nearer to the centre of the dome than another, unless the other voussoirs squeeze or crush, which, in investigations of subjects of this nature, are always assumed perfectly rigid ; therefore, in their position in the dome, they have obtained their concentration. Hence we obtain the essential distinction between an arch and a dome, that no part of the latter can fall inwardly. Since no part of a dome can fall inwards, it resembles an arch resting on the centre on which it has been constructed, and the resistance which the vertical arch meets with from that centre is similar to the opposition of the oblique arches to the vertical arches. If this deduction be just, the mechanician will be able to describe the extrados of equilibration to a dome and its abutment wall, with the same facility as he may to an arch and its abutment piers." 14997. Pasley has likewise stated that "as soon as any course is completed all round, the stones or bricks composing it form a circular arch like that of a cone, which cannot by any means fall inwards. Hence there is an important difference between the dome and the common arch, which latter cannot stand at all without its centering, unless the whole curve be completed, and when finished, the crown or upper segment tends to overset the haunches or lower segments. The dome, on the contrary, is perfectly strong, and is a complete arch without its upper segment; and thus, as the pressure acts differently, there is less strain upon the haunches and abutments of a dome, than on those of a common arch of the same curve. Hence a sufficient dome may be constructed with much thinner materials than would be proper for a common arch of the same section. The dome of St. Paul's Cathedral offers a fine specimen of this kind of work." It has been described in par. 472. 1499m. The Pantheon, at Paris, has a dome formed of three portions. The first, or interior one, is a regular hemisphere of about 66 ft. 9 in. span, with a circular opening at top of about 31 ft. 44 in. in diameter. It is built of cut stones, varying from 18 in. thick at bottom, to 10 in. at top. Thus the thickness is only about 3rd part of the span. The intermediate dome is a catenarian curve having a span of about 70 ft. with a rise of 50 ft.; and it has to support considerable weight at top. It has four large openings in its sides to give light, about 37 ft. high by 31 ft. wide, arched at top in a somewhat parabolic form. The outer dome has an external diameter of 78 ft. Its height is not stated, but it appears to be a moderately pointed Gothic arch had it been continued, without forming an opening at top for the sides of a lantern, which it was intended to support. The thickness of the stone at bottom is about 28 in. and 14 in. at top. A great part of the surface is only half the above thickness, as the dome is laid out internally in piers, supporting three tiers of arched recesses, or niches, of less substance, and showing like the panels in joiners' work. (See figs. 177 and 178.) 1499n. Partington, in the British Cyclopædia, 1835, expresses the opinion that "the weight of the dome may force out its lower parts, if it rises in a direction too nearly verCC tical; and supposing its form to be spherical, and its thickness equal, it will require to be confined by a hoop or chain as soon as the span becomes eleven fourteenths of the whole diameter. But if the thickness of the dome be diminished as it rises, it will not require to be bound so high. Thus, if the increase of thickness in descending begins at about 30° from the summit, and be continued until at about 60°, the dome becomes little more than twice as thick as at first, the equilibrium will be so far secure. At this distance it would be proper to employ either a chain or some external pressure to prove the stability, since the weight itself would require to be increased without limit, if it were the only source of pressure on the lower parts. The dome of the Pantheon, at Rome, is nearly circular, and its lower parts are so much thicker than its upper parts, as to afford a sufficient resistance to their pressure; they are supported by walls of great thickness, and furnished with many projections, which answer the purpose of abutments and buttresses." 14999. Keeping to the theory of the dome, we must avoid noticing its history, beyond pointing out the papers which have of late years treated on the subject. These are published in the Transactions of the Royal Institute of British Architects. The first was by J. Fergusson, On the Architectural Splendour of the City of Beejapore, November 1854; the discussion in December following, when J. W. Papworth detail d his interesting and novel theory, to be presently noticed; and two papers by T. H. Lewis, Some Remarks on Domes, June 1857; and On the Construction of Domes, May 1859, in which, however, great care must be taken by the reader to separate the arch from the dome constructions, as in our opinion they are treated therein as of one principle. The question of a Gothic dome was much discussed without a solution in the journals of the period named. Domes and pendentives are illustrated in Fergusson's Handbook of Architecture. The very interesting paper On the Mathematical Theory of Domes. by E. B. Denison, Q C., read at the Institute on 6th February, 1871, should be consulted by all students on this difficult subject; as well as the papers by E. W. Tarn, M. A., printed in the Civil Engineer and Architect's Journal of March 1868 and February 1870. 1499p. On the occasion referred to, Mr. Papworth asserted that a dome was not an arch, and that domes were not governed by the same laws as vaults. He then entered into calculations on the causes of the stability of domes, showing that in domes of great thicknes the upper half of each gore was only about one-third in weight of the lower half, and aaduced the possibility of loading the crown to a certain extent. Ho produced a series of drawings of domes, constructed upon principles which ought theoretically, if they were arches, to lead to their failure, but which had nevertheless proved perfectly sound; his views being fortified by Mr. Fergusson's concurrence as to the absence of examples of failure where the bases were stable. He then alluded to the following arguments of others, and explained his reasons for not agreeing with them. Such as, that the dome of the Pantheon, at Rome, had been built on the principle of a bridge, 'i.e. of an arch; that it was impossible to plan a large dome without great thickness of walls, i.e. greater than sufficient to bear the weight and its consequences; that it was necessary for the exterior of a dome to stand flush with the wall of the building to which it belonged; that it was desirable to append heavy corbelling to the inside of the wall to counteract the thrust of the K M N 0 E Figs. 590b and 590c. D B A A dome, with special reference to some circular tambours, of which he exhibit d sketches; to the supposed unnecessarily great weight on the top of some examples: and to the supposed beauty of principle exhibited in the dome of Sta. Maria, at Florence, which he characterised as a piece of octagonal vaulting and not a dome. He also explained that domes which had failed had not been supported on a stable foundation; that he saw great beauty in the idea of forming an eye in so large a dome as that of the Gol Goomuz, at Beejapore, where the centre of the curve on each side of the section was in the edge of the eye; that the outer face of the springing of the dome might be within the inside of the square enclosing wall of the building; that if the principles of vaulting were applied, the wagon-headed section of the Gol Goomuz dome would not be expected, theoretically, to stand; and concluded by some observations in expla nation of his illustrations, as to the requisite thickness of domes. All writers, so far as he had seen, considered the dome as a case of vaulting on principles deduced from their experiments on arches, which was a mode repudiated by him. 14999. The causes of the stability of domes, as thus put forward for the first time, by Mr. Papworth, are the following :—Let the plan (fig. 590b.), of a semicircular dome be divided, say, into twelve or more equal parts, and the section (fig. 590c.), say, into nine or more. Give a thickness by an inner line for stone or brick work. Then it will be at once perceived that the lower block K has to support a mass L of less dimensions as to horizontal length; that the block L supports a still less mass M; that M supports a much less mass N; and that N supports a mass of but a small length in comparison with K, whilst in breadth it dimi. aishes from a few feet to nothing at the apex. If the dimensions of a dome were worked out, say of 50 fr. internal diameter, and of 4 ft. in thickness, it would be found that the block K would be about 4133 ft. cube; L 3684 ft. cube; M 274 ft. cube; N 1463 ft. cube; and the half block O 223 ft. cube. The fact has to be remembered, that all domes are built in courses of stones which are bonded one into the other, forming circular rings; and that even if a dome be cut down into four quarters, each quarter will stand of itself. D У 1499r. Rankine, Applied Mechanics, 1858, points out that the tendency of a dome to spread at its base is resisted by the stability of a cylindrical wall, or of a series of buttresses surrounding the base of the domes, or by the tenacity of a metal hoop encircling the base of the dome. The conditions of stability of a dome are ascertained by him in the following manner. Let fig. 592d. represent a vertical section of a dome springing from a cylindrical wall BB. The shell of the dome is supposed to be thin as compared with its external and internal dimensions. Let the centre of the crown of the dome, O, be taken as origin of coordinates; let be the depth of any circular joint in the shell, such as CC; and y the radius of that jo'nt Let i be the angle of inclination of the shell at C to the horizon, and ds the length of an elementary are of the vertical section of the dome, such as CD, whose vertical height is dr, and the difference of its lower and upper radii dy; so that =cotan i; =cosec i. Let Pr be the weight of the part of the dome above the circular joint CC. Then the total thrust in the direction of a set of tangents to the dome, radiating obliquely downwards all round the joint CC, is Pr - Pr 'cosec i ; dy dr ds dr dx ds dx Fig. 500d. B and the total horizontal component of that radiating thrust is PP cotan i. Let Py denote the intensity of that horizontal radiating thrust, per unit of periphery of the joint CC; then because the periphery of that joint is 2 y ( = 6·2832 y), we have py P. cotan i = dPy 1 = d Pa cotan i (Pr cotan i.) 1499s. If there be an inward radiating pressure upon a ring, of a given intensity per unit of arc, there is a thrust exerted all round that ring, whose amount is the product of that intensity into the radius of the ring. The same proposition is true, substituting an outward for an inward radiating pressure, and a tension all round the ring for a thrust. If, therefore, the horizontal radiating pressure of the dome at the joint CC be resisted by the tenacity of a hoop, the tension at each point of that hoop, being denoted by Py, is given by the equation Py =yPy Now conceive the hoop to be removed to the circular joint DD, distant by the arc ds from CC, and let its tension in this new position be Py -dPy. The difference, dPy, when the tension of the hoop at CC is the greater, represents a thrust which must be exerted all round the ring of brickwork CC DD, and whose intensity per unit of length of the arc CD is Pz 14994. Every ring of brickwork for which pz is either nothing or positive, is stable, independently of the tenacity of cement; for in each such ring there is no tension in any direction. When Pz becomes negative, that is, when Py has passed its maximum and begins to diminish, there is tension horizontally round each ring of brickwork, which, in order to secure the stability of the dome, must be resisted by the tenacity of cement, or of external hoops, or by the assistance of abutments. Such is the condition of the stability of a dome. The inclination to the horizon of the surface of the dome at the joint where l'2 =0, and below which that quantity becomes negative, is the angle of rupture of the done; and the horizontal component of its thrust at that joint, is its total horizontal thrust against the abutment, hoop or hoops, by which it is prevented from spreading. A dome may have a circular opening in its crown. Oval-arched openings may also be made at lower points, provided at such points there is no tension; and the ratio of the horizontal to the inclined axis of any such opening should be fixed by the equation A building in Rankine concludes with examples of "spherical," and "truncated conical," domes. 1499u. Cones. These are used in tile-kilns, glass-houses, and such like. the shape of a hollow cone forms everywhere a species of circular arch, which may be constructed without centering or support, provided the joints be made to radiate towards the centre. The courses should be laid perpendicular to the sides of the proposed cone. A rod of variable length, turning on a pivot, must be stretched all round from time to time, upon a moveable centre, rising as the work proceeds, in order to regulate the internal outline. Such is the strength of this form that the highest kilns are seldom built more than one brick thick, although this dimension would be altogether insufficient for a common wall of the same height. It is, probably, tais principle which has conduced to the existence of the Round Towers of Ireland. That of Kilkenny, for example, 100 ft. in height, was built on, or very near. the surface, for at 2 ft. below it, wood coffins with skeletons were found partly under the walls, thus affording an unstable foundation. POINTED ARCH VAULTING. 1499v. We now proceed to enter into a view of the general forms of groining in pointed architecture, observing, by the way, that the groins at the arrises, up to the twelfth century, were seldom moulded with more than a simple torus or some fillets. In the twelfth century, however, the torus is doubled, and the doubling parted by a fillet. Towards the end of the twelfth century, three tori often occur; and at the beginning of the thirteenth, the moulded arrises become similar to the moulded archivolts of the arches, both in their form and arrangement. In France, until the middle of the fifteenth century, the arrises of the groins only were moulded; but in this country the practice took place much earlier, for, instead of simple groining, the introduction of a number of subdivisions in the soffits of arches had become common. In fig. 590e. is given a plan of the soffit of a vault of this kind, in which A is an arc doubleau (by which is understood an arc supposited below another at certain intervals, and concentric with the latter); B is an upper arch, called by the French antiquaries formeret; C, the wall arch, or formeret du mur; D is a diagonal rib, or croisée d'ogire; E, intermediate rib or tierceron; FF, summit ribs or liernes; G, the key or boss, clef de voute. Willis has used the French terms here given, and as we have no simple terms to express them in English, it may be convenient to adopt the practice. 1499w. The ribs formed by the intersections of the groins perform the office of supporting Fig. 5906. Mr. the vaulting which lies upon them, they in their turn being borne by the pillars. Thus, in the simple groin (fig. 590f.), the arches AA, and diagonal rib C, carry the vaulting BB, a rebate being formed at the lower part of the ribs on which the vaulting lies. This figure exhibits the simplest form of groining in any species of vaulting, the intersecting arches being of equal height. The contrivance in its earliest state was ingenious, and the study attractive, and we cannot be surprised at Dr. Robison observing, in respect of the artists of the thirteenth and two following centuries, that "an art so multifarious, and so much out of the road of ordinary thought, could not but become an object of fond study to the architects most eminent for ingenuity and invention: becoming thus the dupes of their own ingenuity, they were fond of displaying it where not necessary." This observation would be fully verified had we room for showing the reader the infinite number of devices that ingenuity has created: he will, however, from the few elementary ones that we do give, be enabled to see the germs of countless others. 1499г. Ware, in his Tracts on l'aults and Bridges, 1822-a work which, notwithstanding the quaint method in which the subject is treated, contains extremely valuable matter, -has made some remarks which we must introduce at length, or justice would not be done to them. "In the vaulting," he says, "of the aisles of Durham and Canterbury cathedrals are to be observed the arcs doubleaux and groined ribs in round-headed vaults. In the naves of the same buildings is the same character of vaulting, except that the arch of the vault is pointed. Some vaults of this kind are to be distinguished from others by the Fig. 590f. positing of the stones of the vault between the ribs, which, instead of being parallel to cach side of the plan, as in Roman groined vaults, take a mean direction between the groined rib and the ribs of the arches over the sides; whence they meet at the vertex at an acute angle, and are received by stones running along the vertex, cut in the form of a ratchet. The advantage of this method consists in requiring less centering, and originates in the position of the ribs at the springing." "From these beginnings vaulting began to assume those practical advantages which the joint adaptation of the pointed arch and ribs was calculted to produce." "The second step differed from the first, inasmuch as at the vertex of the vault a continued keystone or ridge projects below the surface of the vault, and forms a feature similar to the ribs. But here it was necessary that the ridge should be a stone ot great length, or having artificially that property, because its suspension by a thinner vault than itself would be unsafe, unless assisted by the rib arches over the diagonals and side, a distance equal to half the width of the vault. To obviate this objection, other ribs were introduced at intervals, which may be conceived to be groined ribs over various oblongs, one side continually decreasing. This practice had a further advantage, as the panels or vaults between the ribs might become proportionally thinner as the principal supports increased. It is now that the apparent magic hardiness of pointed vaulting and the high embowered roof began to display itself; from slender columns to stretch shades as broad as those of the oak's thick branches, and, in the levity of the panel to the rib, to imitate that of the leaf to the branch." On comparing rib-pointed vaulting with Roman vaulting, it will be invariably found that the rib itself is thinner than the uniform thickness of the Roman vault under similar circumstances; and that the panel, which is the principal part of the vault in superficial quantity, sometimes does not exceed one ninth part of the rib in thickness. The Gothic architects, it has been expressively said, have given to stone an apparent flexibility equal to the most ductile metals, and have made it forget its nature, weaning it from its fondness to descend to the centre." 1499y. In the second example (fig. 590g.), another rib, a b, is introduced, which on plan produces the form of a star of four points. The forms of these thus inserted ribs result from curves of the lines on the plan in the space to be vaulted. As many radii are drawn |