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GROIN, in architecture, the hollow formed by the inter section of two or more simple vaults, crossing each other at the same height.

In the geometrical point of view, the centre of a groin is formed by the entire meeting of the surfaces of two or more eylindrits; that is, such, that every straight line around the whole circumference, on the surface of the one cylindrit, will meet every straight line around the circumference of the one adjoining.

Hence the sections parallel to the axes of all the cylindrits which form the groin are in the plane of their spring ing ; otherwise the surfaces could not meet each other entirely.

Hence, also, the axes of all the cylindrits are also in the same plane, and cut each other in the same point.

In the above definition of a groin, it must, however, be observed, that its surface is no portion of that of the solid which would be contained by the surthces of the cylindrits and a plane passing through their axes, but only that part of the whole which is formed on the outside of the space which would be thus enclosed in the centre of the groin and form a polygonal dome. The surface of the groin is therefore equal to the whole of the cylindritic surfaces, deducting that of the dome.

The surface of any cylindrit is either that of a cylinder or cylindroid.

When the cylindrits which form the groin are all cylinders, the two vaults are of equal breadth.

In any simple vault of a groin, the planes which are tan gents to the surfaces at the springing, have equal inclinations with each respective wall. When all the openings of a groht are equal, the groin is termed an equilateral.

The branches of a groin are each of the two opposite parts of each simple vault.

The invention of groins must have been subsequent to that of simple vaulting, and probably originated from arched pas sages, when it was necessary to occupy the whole height. At what time they were first introduced in architecture, is uncertain ; the remains of antiquity show that they arc of very remote date, which, however, cannot be traced beyond the times of Roman power and grandeur. Use or necessity was, without doubt, the occasion of their invention, but in process of time they were used as ornaments, and became thshionable at the decline of the Roman empire ; they are to be found in the amphitheatre at Rome, formed at the inter sections of the radiating and elliptic passages. In the temple of Peace, and baths of Diolcetian, at the same place, instead of massive piers, they are supported upon columns,the most feeble of all supports, and which would be incapable of resisting the lateral pressure of the arches, were it not for the auxiliary support of the walls immediately behind them, at the sides and angles of the building, which act as buttresses.

Groins continued to be used after the dissolution of the Boman empire, in ecclesiastical structures ; and wherever grandeur or decoration was required, they were never omitted ; they became the most principal ornament of the time, and formed the most conspicuous features in the edifices in which they were employed : at first they were used in the same manner as by the Romans, but in after-times the groins were supported upon ribs, which sprung from cylindrical or polygonal pillars, with capitals of the same tbrin ; this pro.

(bleed a necessary change in the figure of the vaulting, as the bottoms of the ribs rose from the circumference of a circle, instead of the angles of a square, with its sides parallel to the walls ; and as the spaces between and over the ribs were vaulted in a twisted or winding surface, so as to coincide in every part with a straight line level between the ribs, the angles of the groined surface were thus very obtuse at the bottom, but diminished continually upwards. and ended in a right angle at the summit of the ceiling. Afterwards, when the pillars were f 'lamed upon a square plan, the sides of w hich were obliquely disposed with regard to the sides of the building, and decorated with vertical mouldings, or small attached columns, and the number of ribs increased, the first idea of fan-work would be presented at the springing of the ribs ; but in this the architects would soon perceive an incongruity of limn in the surface, as it approached the summit of the vaulting; the ribs would be formed all of equal radii, and disposed around, to support a concavity, which might be generated by revolving a curve round an axis w hich was in the centre of the pillars; and being accus tomed to groins meeting in lines crossing each other, it was natural to suppose they would at first permit the ribs to rum out and meet each other, which would then be of unequal lengths. It' the difference between the opening; was not very great, the intersection thus formed by the meeting of the opposite sides of the would not differ materially from straight lines, but would not be parallel to the horizon, as they would ruin upwards towards the centre of the groin ; but this would depend on the angle formed by two opposite ribs in the same plane. Thus, if the tangents formed at the vertex of the opposite curves contained an angle of 120 degrees, the apex line on the ceiling would forma curve in receding from the vertical aside of the said ribs, of a very decided convexity; but in going progressively forward, the curvature would change into a concavity, and then would begin again to descend. The idea of intersecting the ribs thus dispo:ed ill vcrtieal planes aromul a common axis, by circular horizontal ribs, was natural ; and thus again generate another idea of supporting the upper ends of the ribs, by a circular ring concentric with the axis of the pillar, and this being done from four pillars, would leave a space enclosed by four convex are; of circles : nothing fitrther was required to complete this system of vaulting than to till up the space, mid the whole would be key ed together. In this manner, by slow and imperceptible changes, a species of vaulting was invented, very difierent from that of the Greeks and Itoinans. Instead of closing the space, it' suppose another ring, forming a complete circunif•renee, to be built interiorly to touch the former aces, and the four t riangidar curved spaces closed and wedged together masonry, the will stand equally lirni as if the middle had been solid, and thus an api.U'llire for light will he formed the as in donn•vaulting. This species of vaulting has also another property, that it eau be carried up from a square plan with less hazard than the common !node of gniiiiing.

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