The solution of this difficulty will be better considered by means of a simple and more familiar case. The intersecting vault in its most normal form is plain enough in its application to a square compartment, but becomes difficult when applied to a space longer one way than the other; yet oblong spaces continually present themselves as requiring to be vaulted.

Mathematically this is readily met, and that with perfect accuracy, by making one of the intersecting vaults elliptical instead of circular in its curvature; making, for instance, the narrower arch a semi-ellipse with its longer semi-diameter vertical. This, however, is an unsightly form, and was always rejected, though the natural mode of effecting the object, and though it would give intersecting curves which would be complete and in vertical planes.

The Roman builders solved the problem at the sacrifice of mathematical accuracy, by what is called stilting the narrower arch; that is, raising its springing till its crown becomes level with that of the wider arch. This is a practical solution of the difficulty, but is not a very pleasing one, inasmuch as the line of intersection is most uncouthly twisted, and, in point of fact, begins at considerable height above the springing of the vault (Figs. 324, 325.)

To go back, however, to our previous case of the apsidal termination of a vaulted space, it affords a very fair solution of the difficulty by which we were before encountered; for it is clear that the arches on the sides of the octagon may be lifted up till their crowns become level with that of the main vault; and, as the intersecting angles of a polygonal groined vault coincide with its transverse ribs, we have nothing to do but to raise from every angle a transverse rib similar, or very nearly similar, to those of the main vault, and to make the smaller vaults of the octagon to intersect upon them (Fig. 326). There will be a little geometrical inaccuracy in the forms of these cells of vaulting; but, as the angle ribs would assume correct lines, these inaccuracies would not seriously offend the eye.

There is, however, another method of meeting the difficulty; but before describing it, I will say a few words on the treatment of other difficulties resulting from the irregularities in form of spaces which have to be vaulted.

Let us, as an example, suppose an aisle or corridor passing round such a polygonal figure as we have been considering. It is manifest that its compartments will have a form enclosed by unequal sides, or, to say the least, one side will differ greatly in width from that opposite to it.

The stilting system before-mentioned is the most obvious method of getting over the difficulty. It may be, that three of the arches surrounding such a compartment may be about equal, and no great difficulty would occur as to their intersection; but the fourth, being far narrower, would have to be stilted to raise its crown to the level of the others, and its lines of intersection will consequently be more or less disturbed.

The difficulty is, in early specimens, increased through the apse being usually round instead of polygonal; though this does not very materially alter the case. We have in London two excellent examples of this apsidal aisle; that in the chapel of the Tower of London[43] (Fig. 327), and that in St. Bartholomew’s Church in Smithfield (Figs. 328, 329),—the former of an early, and the latter of a later type.

In both, the narrow arches are greatly stilted; and at first sight the two may appear to be similarly treated; but on closer examination there will be found to be much difference between them. In the Tower Chapel the transverse ribs are made to increase prodigiously in width towards the outer wall, so as to reduce the want of parallelism of the groined compartments, a very unsightly expedient; and the capitals of the columns are square, which makes the backs of the arches they support nearly double the width they present in front: while at St. Bartholomew’s, the ribs are of uniform width, and the capitals, instead of being square, have their sides radiating from the centre of the apse, so as to share with their arches the spreading of their outer side. The difficulty is really increased in the later work, but is met by more skilful workmanship. Somewhat similar to the case of the aisle round a semicircular apse, is the case of vaulting a circular building with a central pillar. In each, the main surrounding vault, if uncut by others, would assume the form of a portion of an annulus or ring. In the aisle such a ring would be wide in the opening it surrounds, but in the circular building its opening would be reduced to the diameter of the central column or its capital.

This annulus, or curved vault, would become divided in plan into triangular portions by the transverse ribs which would meet on the central pillar and the cross vaults, proceeding from the surrounding arches, would intersect with only the outer portion of the vault, the inner portion which rests on the pillars being uncut by them, and assume the form of a concave conoid, something like the flower of the convolvulus.

This form of vault is well seen in two instances in the Cathedral at Worcester. The best known of these is the Chapter-house (Fig. 330), a circular building, between 50 ft. and 60 ft. in diameter, whose circumference is divided into ten parts, from which small ribs run across to the central pillar. The intersecting cells of groining are at present pointed, possibly the result of a subsequent alteration, and simply intersect with the surrounding vault. In this case the central conoid is broken into a polygonal form to give piquancy to its otherwise too unbroken surface. This may be considered the father of our beautiful polygonal chapter-houses, of which I shall have more to say as I proceed.

The other instance I have alluded to at Worcester is in the crypt (Figs. 331, 332). In this, the case in question occurs not in a distinct form, but in combination with an apsidal aisle on the one side, and a vaulted span, with a central range of pillars, on the other; the last pillar forming the central point of the semicircular apse, is exactly parallel in position, and forms very similar groining to that of the Chapter-house.

The same problem, when applied to a polygon instead of a circle, is open to two different modes of solution. In the one, the main vault is always supposed to run from each side towards the central pillar; in the other, from each angle towards the pillar. I shall, however, have to go more minutely into this when I come to pointed-arch vaulting, to which the last-named system more especially applies.[44]

Having now briefly touched upon the most prominent forms of round-arched vaulting in its more normal form, as resulting from the barrel vault and its intersections, I will digress for a short time to consider some of the conditions which relate to what I in my last lecture stated to be the other most simple kind of vault—the dome. I do so, however, not with any idea of treating at large on a form which should be made the subject of a separate lecture, but merely to facilitate the explanation of certain indirect influences which it exercised upon ordinary vaulting.

A dome in its most typical form stands upon a circular wall; this, however, is by no means a necessary condition. It may in reality cover a square or polygonal space just as well; for, suppose a square or a polygon inscribed within the base of a hemisphere, it is clear, from the properties of a sphere, that vertical planes erected on the sides of such square or polygon will cut the hemisphere in semicircles of the diameter of those sides (Fig. 333). It follows, therefore, that the walls of a square or polygonal building would intersect with a dome in the form of semicircular arches standing on each of its sides; and, consequently, that such a square or polygon will carry a hemispherical dome, or rather the remainder of it left after cutting the base into a square or polygon.

For our immediate purpose we will limit the case to that in which the inscribed figure is a square.

Now, a dome cut in this manner by four planes is not a very sightly form, and needs some embellishment (Fig. 334); but if a horizontal circle be drawn within it by means of a cornice resting on the crowns of the supporting arches, it assumes at once an agreeable form, and one which has been largely used both in Byzantine and in modern architecture (Fig. 335). My present purpose, however, suggests another mode of giving sightliness to the squared dome. The lines drawn on its surface may lie in vertical as easily as in horizontal planes, and by making such lines pass through the angles of the square, touching the dome throughout their length, and intersecting one another at its apex, we obtain a form not wholly unlike a square groined vault; the great differences being that the intersecting diagonals of a groined vault assume elliptical curves, whereas these are semicircles; that in the one they represent an actual angle, while in the other they are arbitrarily drawn on an unbroken surface; and that the ridges or crowns of the vault in one case are horizontal, while in the other they are raised and circular. This mode of vaulting, though frequent in some parts of France, is seldom found in this country.

There is, however, an instance of it in the vaulting beneath the tower of Goring Church, Oxfordshire. (Fig. 336).

Though this is not really groining, but a disguised dome, there is a ready process by which it may be, and continually was, converted into genuine groining.

I have defined the barrel vault as the prolongation of an arch in a direct line at right angles to its plane.[45] But an arch may be prolonged in other than a straight line. Let us, in the previous figure, suppose the arches which rise from the sides of the square to be prolonged, not horizontally, but in a curve rising as it proceeds, and so regulated that the semicircle as it moves forward retains its vertical position, and is guided in its motion by the diagonal lines drawn in the dome. This process at once generates a new form of vault (Fig. 337). For each of the triangular gores of the dome we now substitute a vault, of which every vertical section parallel to the side of the square is a portion of a circle of the same diameter with those raised on the sides, while the angles of the intersection of these newly generated vaults are themselves semicircles. It is a perfectly accurate geometrical figure, none of whose salient lines are other than portions of circles, though the ridge or crown lines now become elliptical. It is a most useful development, as being much stronger than the ordinary groined vault. Oddly enough, it has—so far as I am aware—no suitable name. It is usual to speak of such vaults as being “domed up,” but this is a very rough description. When adapted to the pointed arch, it has been called by Mr. Petit the Angevine vault. I know no better way of describing it than as round-arched vaulting with a raised ridge (Fig. 338).

Now, though less obvious at first sight, the very same processes are applicable either to an oblong, to a tapering four-sided figure, such as the bay of the aisle of an apse, or even to one of the triangular compartments of the apse itself, or of a circle.

For, in either case, we have only to cut out the required slice from a hemispherical dome, to draw the diagonal lines from the angles of such form to the apex, and then to substitute for the gores of the dome the vault generated by the motion of the semicircle, produced by the plane of the sides of the figure parallel to itself, and rising under the guidance of the diagonal lines (Fig. 339). This process, it will at once be seen, is capable of solving all the problems of irregular figures which I have enumerated at an earlier stage in my lecture, without the aid of stilting, and without giving intersecting curves, which deviate from the vertical plane, while it avoids the use of the ellipse for any prominent line (Fig. 340).

The last case I have named—that of the triangular gore of an apse or circle (Fig. 341)—also solves the difficult case I mentioned at the beginning of this lecture as arising in the groining of a polygon, owing to the excessive lowness of the arch formed by the intersecting angles. These are now raised to the full height of a semicircle, while if half of such a polygonal vault be used for an apse, it agrees in height with the main vault without the use of stilting.

It may, however, be mentioned that, as stilting is sometimes most useful in making room for windows, it was not superseded by this invention; the two systems continuing to be used at pleasure, and sometimes a union of the two, which, however, is so arbitrary as to defy definition. The form last described for a vaulted circle is often used as a variety of the dome by raising numerous small arches round its circumference, and giving a sort of fluted or shell-like surface to the dome.

I think I have now described the principal varieties of round-arched vaulting with two exceptions. The one is that in which the side vaults of oblong compartments cut the higher and main vault at a level lower than its crown. This is vulgarly known as “Welsh” groining, and though not quite pleasing in effect, it is a very legitimate mode of covering an oblong compartment. It is customary to obviate the unpleasing coal-scuttle shape of the true line of intersection (Fig. 342) (such as may be seen in St. Martin’s Church), by making them take the lines given by vertical planes, and throwing the irregular geometrical curve into the surfaces of the cells where it does not strike the eye, or perhaps generating them by the motion forward of the side arch (Fig. 343). This has been done in the Sistine Chapel, and Mr. Smirke has, I think, done the same in our Great Exhibition Room. In a ceiling to be decorated with painting, this form of vaulting possesses the great advantage of leaving the central range unbroken by diagonal lines.

The other form I have omitted is the square or polygonal dome, or that generated by the intersection of vaults running parallel to the sides of the base, instead of, as in the groined vault, running at right angles to them (Figs. 344, 345).

The square dome is, in fact, the exact correlation or complement to the square groined vault. Like it, it is generated by the intersection of two barrel vaults of the diameter of the sides of the square; but the parts of such vaults which are retained in the one, are precisely those which are omitted in the other. The angular lines are the same, though in the one case the angles project, and in the other they recede; and while the groined vault is reduced in its bearing to four points in the corners, the square dome demands for its support the whole line of the walls, which, however, it reduces in height to the level of the springing line; while the other allows them to rise in their centres to the full height of the vault. In some cases, as in the vaulting beneath the tower of Grantham Church, “Welsh” groins are united with the polygonal dome, a form quite applicable to the vaulting of an apse (Fig. 346).

There is another peculiar feature in the square or other straight-sided dome, viz., that it may be cut by vertical planes, as is the case with the spherical dome. Thus, if we inscribe within the base of a square or triangular dome, another square or triangle whose corners bisect the sides of the original base, and erect upon the sides of this newly-formed figure vertical planes, these will intersect the dome in arched forms, and the parts left will give a new form of vaulting, rising from the angles of the figure, and terminating in an unaltered portion of the original dome. This form was not unfrequently used, especially in vaulting triangular spaces (Figs. 347, 348).

I have now gone through all the normal varieties (of which I can think) of the round-arched vault, and it is time that I should allude to a great step which, after perhaps the first quarter of the twelfth century was introduced into their construction. I allude to the addition of a moulded rib beneath their intersecting angles. It is clear that these angular lines are both in reality and in effect, the weak points of plain groining. I have mentioned that the Romans fortified them in construction, by using in them stronger material than in the rest of the vault; and the early Norman builders made a feeble attempt to take off from the dulness of the intersection, where it approaches its apex, by artificially sharpening the edge in plastering it; for, without this, it becomes (in a vault where the courses of stone or brick are concealed) almost invisible.

The great step in advance which I have now to mention provided both the constructive and the artistic strength which the line required.

It is also clear that any irregularity of form may render these lines shapeless and unpleasing, and it is an obvious gain from an artistic point of view, to adopt a system which will at once render them pronounced and regular. While, then, the introduction of the angular rib was in many cases a departure from geometrical accuracy, it was a vast gain both in strength and beauty.

In that form of vaulting, which I have defined as that with the raised ridges, no geometrical inaccuracy would arise, the angles of intersection being semicircular, and in vertical planes; but in the more ordinary form of vaulting, where these lines are elliptical, that curve being unpleasing, two courses offered themselves for choice: the use of segments of circles for the diagonal ribs, or the bringing down the springing to a lower level than that of the vault. In either case the true geometrical figure has to be departed from, and the error has to be thrown into the vaulting-surfaces—a course which subsequently became so thoroughly adopted as a principle, that it may be received as an axiom that in ribbed vaulting, where the ridges are not raised, the ribs are made of such forms as will satisfy the eye, and the vaulting surfaces made to fit themselves to them as best they may, apart from geometrical accuracy,—a principle which, though it may at first sight offend the mathematical mind, has proved in practice so wonderfully useful, and to offer so many facilities, as to be a sort of Magna Charta to the art of vaulting.

This step once taken, round-arched vaulting seems to have completed its work. Square and oblong spaces were vaulted either with mathematical accuracy on the raised-ridge principle, or with deliberate departure from such accuracy on the level-ridge principle. Irregular spaces were covered over by expedients which satisfied the eye, and met practical conditions tolerably well, and many beautiful works were the results. The diagonal ribs, too, became a new source of decoration, not only by means of their own mouldings or enrichments, and through the bosses now sometimes placed at their point of intersection, but also because they were suggestive of additional colonnettes, and thus added more richness and intricacy to the piers; and sometimes they were carried upon sculptured corbels, as in the cathedral at Oxford. Among the richest specimens of this vaulting may be mentioned the gateway and the Chapter-house of Bristol Cathedral, the chancel of St. Peter’s Church at Oxford, etc.

We have now arrived at a stage of our investigation when we must pause for the sake of asking ourselves what need or requirement yet remained unsatisfied which was essential to the perfecting of our arcuated developments, and what means remained—hitherto unused—by which such need might be met.

We have followed out our arched construction, and the process by which it was rendered at once susceptible and productive of artistic beauty, till we might fancy it to need nothing but the gradual additions of refined art to render it a perfect style; and it would be both an interesting and a profitable field of speculation to take up the style at such a point, and to study how best to clothe it with the charms of the highest art, irrespective of our knowledge of its historical destiny; how, in fact, to perfect our round-arched style to the highest and most refined artistic status; and I feel that any one who could fulfil such a task would be a benefactor to our art.

The semicircle is unquestionably the typical form for an arch, and one well suited to the majority of purposes and positions. I therefore wish well to him who will push a style which claims it as its leading element to its highest possible pitch of perfection. I should rejoice to see a round-arched style rendered as perfect, and its accompanying art as noble, as the Greeks did their trabeated architecture and its ever-glorious sister arts; nor do I see why such an end should not be attained, and God speed the man who worthily attempts it!

This task was, in fact, nobly though unconsciously approached by the artists of the twelfth century; nor can any one examine their works, particularly from the close of the first quarter of that century, without being filled with the warmest admiration at their determined strivings after refinement; their earnest aim to perfect every form, and to eliminate every footstep of barbaric element; to rid their work of all rudeness of execution; and in every way within their reach to raise the architecture they were developing into a really high art.

These earnest and restless strivings, however, had the effect of rendering apparent to them a defect, both structural and artistic, in the conditions prescribed by a round-arched style. They had freed themselves from the trammels of the arbitrary rules of proportion, and might render their structures lofty or the reverse at pleasure: their columns might be as short and sturdy as the most archaic Doric, or might outdo the most elegant Corinthian in the lightness of their proportions; yet the arch they were condemned to carry was limited in height to one-half of its own diameter; or, if allowed to exceed this, by means of stilting, this was evidently but a clumsy expedient, and only suited to particular positions.

The whole tendency, too, of the onward movement of the art was towards increased height; and, while walls and pillars might avail themselves to the full of this upward striving, it was hard that the arch—the very essence of the style—should be condemned to unalterable stuntiness. Proportion evidently claimed that the arch should have its fair share in the increasing height of the buildings, yet the inexorable semicircle said, “Nay, my proportions are fixed. You may lengthen your straight lines as you please; but by no law of science can my height exceed one-half of my width.”

A geometrician might reply that the semicircle might be stretched upwards into a semi-ellipse with its major axis upright. I do not think that our Mediæval builders ever tried this dismal experiment, nor do I know that it was ever attempted, except by the barbarous Parthians, in a building you will find figured in Mr. Fergusson’s Handbook; and so hideous was the result that one may well suppose it to have been handed down as a warning to subsequent generations!

Nor was this craving after a loftier arch the result of taste alone. Constructive motives pointed in the same direction; for it was found that round arches, when carrying great loads, as those sustaining towers, etc., were apt to overcome the resistance of their piers; and many failures were the result. The same was found to result from vaulting over wide spaces. True it is that the Romans, in the great halls of their baths, had vaulted over spans of double the width of the naves of our Norman cathedrals; but this had been effected at the expense of the utility of their aisles, which were cut up into short lengths by the ponderous abutments needed to sustain the tremendous pressure of the central vault. Besides which, the Mediæval builders aimed at raising the springing of these vaulted naves to a height out of the reach of the abutment of the aisles. An arch of less lateral pressure was therefore desired.

Another motive might have led to a similar aim. We have seen what difficulties and contrivances resulted from the exigencies of vaulting over irregular spaces where it was desirable that the crowns and springers of the surrounding arches should range on the same levels, though their spans might differ to any extent. It was clear, then, that an arch of more elastic proportions was the grand desideratum.

The claims, then, of proportion, of construction, and of geometrical convenience, all took the same direction, and demanded an arch of variable proportions.

This three-fold demand was met by the introduction of the Pointed arch.

To apply this to our main subject of vaulting, we at once see that, in addition to constructive advantages, the arch could now he proportioned in height to its supporting piers, and the unequal sides of the vaulted spaces could now be arched in such a manner as to satisfy the exigencies of the vaulting without the necessity of resorting to awkward contrivances; so that an accession was obtained at once of strength, beauty, and facility of application.

I have called the use of diagonal ribs the Magna Charta of the art of vaulting; but it must share this honour with the Pointed arch. Let us now proceed to trace the progress of the art under this double charter of liberty.

The first introduction of the Pointed arch into vaulting seems to have been made without a full consciousness of its advantages, and rather with a view to strength and general beauty than to the convenience of covering irregular spaces, for in many early specimens—as originally in the Cathedral of Sens,[46] and in the work of William of Sens at Canterbury,[47]—the round arch continued to be used in the narrow bay against the walls, while the pointed arch was used for the wider spans. In nearly all English specimens, however, full advantage was at once taken of the newly-attained freedom: thus, at St. Joseph’s Chapel, at Glastonbury,—a work otherwise purely round-arched,—the groining assumes throughout the pointed form, the narrow bays being especially acute.[48] The same is the case at St. Cross, another very early transitional work,[49] and in the nave and transepts of St. David’s Cathedral[50] (erected about 1180), though the groining was never carried out, we have the preparations for it with pointed wall-ribs in the sides, while the round arch is mainly used beneath. I shall, therefore, disregard this occasional inconsistency.

Before going further, I will, to prevent mistake, give the names of the parts of a groined compartment (Fig. 349). The main ribs from wall to wall are called by us transverse ribs; by the French, arcs doubleaux. Those which pass from angle to angle, intersecting in the middle, we call diagonal ribs; the French, arcs ogives. Those which adhere to the wall, we call wall ribs; the French, formerets. If there is a rib or moulding along the apex, we call it a ridge rib; the French, a lierne. The latter, however, does not exist in early examples. Other features appear as we proceed, but I limit my first list to the simpler forms of vaulting. The French names are found in the treatise of Philibert de l’Orme, a work of the sixteenth century; whether they have been traditionally kept up I do not know, but they are now universally adopted by French writers on the subject.

I will just go over our leading cases, already treated of, showing the changes effected in them by the use of the pointed arch.

In the square groined space with level ridges there was no alteration excepting in the form of the arch, and in the more finished mouldings made use of. The diagonal ribs often took the form of a round arch, but this depended wholly on the proportions of the surrounding pointed arches.

As the diagonals were not formed by elliptical curves, it followed that the vaulting surfaces were not portions of cylinders, and that an error had to be thrown into them. In fact, they were filled in from rib to rib without any view to purely geometrical forms.

When the pointed arch is applied to an oblong compartment, or to the sides of a polygonal apse, its advantage becomes more manifest; for the power of making the narrow arches against the walls as high as we please wholly removes the difficulty which we encountered while limited to the round arch, and that without the necessity of stilting, though the convenience which the last-named method offered for the introduction of windows still led to its frequent use.

The irregular compartments of an apsidal aisle ceased now to present difficulties, as all their arches could be made of equal height.

It is curious that, while we have in London two specimens of such aisles in the round-arched style (those in the Tower of London and St. Bartholomew’s), so have we also two in the pointed arched style, and those very different indeed in their treatment. The aisle round the apse of Westminster Abbey has compartments enormously wider on one side than on the other, and this is met simply by the varied proportions of the arches (Fig. 350); while that surrounding the round portion of the Temple Church[51] has double as many compartments as there are pillars in the arcade, and, consequently, behind every arch of the great arcade is a groined compartment which is nearly square, while behind every pillar is one of a triangular plan, vaulted in a peculiar manner from its corners without any ribs between the transverse ribs.

The vaulting of a polygon with a central pillar assumed now a form of exquisite beauty. Its two special types in its simpler form are the Chapter-houses at Salisbury and Westminster,—truly a par nobile fratrum,—and claiming special attention as showing the extraordinary beauty attained by the use of ribbed vaulting united with the pointed arch.

I have already mentioned that in this form of vaulting

there is a choice between two methods of effecting it: either by supposing the main vault to span from wall to pillar or from angle to pillar (Figs. 351, 352).

The former is, on a primâ facie view, the more natural, but it has the disadvantages of breaking the chief side of the vaulting compartment which rises from the corners into a resalient angle, and also of rendering the main ribs from these angles across to the pillar, in one half of their length diagonal ribs, and in the other transverse; and of making one half represent a projecting and the other a receding angle, while the angle ribs of the outer half meet the transverse ribs of the inner half of the vault.

These objections are entirely removed by supposing the main vaults to run directly from the angle to the pillar. In either case the ridge which surrounds that half of the vault which springs from the pillar takes the form of an inner octagon.

In the first case, the sides of this are parallel to the walls, while in the second they take an intermediate direction; the angles of the inner octagon being opposite the centres of the sides of the outer one, and vice versâ.

The vaulting compartments which rise from the angles of the great octagon are precisely similar to the opposite ones which rise from the pillar, and the ribs which rise from the angles to the pillar are throughout transverse ribs, while the angle ribs from each side duly meet one another.

I exhibit a view of the interior of the Chapter-house, Westminster (Fig. 353), to show the beauty of this form of vaulting. Few forms, in fact, in any style of architecture present such beauties as an octagon vaulted in this manner; and I am happy to think that our London specimen, which has been lost for the last century or more, will now very shortly be restored to its original form and condition.

I have already mentioned that in all these forms of vaulting,—that is to say, those with level ridges,—owing to a geometrical error resulting from the use of circular curves for all the ribs, the filling in of the vaulted spaces must be artificially shaped to fit those curves.

The use, however, of a form of vaulting analogous to that before described as having raised ridges would obviate this inaccuracy.

Suppose, for example, an oblong compartment with pointed arches of similar proportions on all its sides and on its diagonals, and the vaults of each cell generated by the motion of the curve of the surrounding arches towards the point of intersection, guided by the diagonals, we obtain at once a vault with pointed arches and raised ridges, the precise correlative of that before described with round arches and raised ridges, and one in which the filling of the vaulted spaces assumes a systematic and accurately geometrical form (Fig. 354).