The great invention of the English architects in vaulting is the form usually known as fan-tracery. It is so beautiful in itself, and so exclusively English, that it may, perhaps, be worth while to retrace the steps by which it was arrived at. This may lead to a little repetition, but the stone vault is so essentially the governing modulus of the style that its principles cannot be made too clear. [Illustration: 813. Choir of Gloucester Cathedral. (Cath. Hb.)] The original form of the intersecting vault is that of two halves of a hollow-sided square pyramid placed opposite one another in an inverted position.[104] One half of such a vault is shown at A and A A (Woodcut No. 814, fig. 1). The English seem early to have tired of the endless repetition of these forms, and, after trying every mode of concealing their sameness by covering them with tracery, they hit on the happy expedient of cutting off their angles, as shown at B and B B. This left a flat square space in the centre, which would have been awkward in the central vault, though in a side-aisle it was easily got over, and its flatness concealed by ornament. Arrived at this stage it was easy to see that by again dividing each face into two, as at C, fig. 1, the principal original lines were restored, and the central space could be subdivided by constructive lines to any extent required. By this process the square pyramid had become a polygonal cone of 24 sides, which was practically so near a circle that it was impossible to resist the suggestion of making it one, which was accordingly done, as shown at D and D D, fig. 1. [Illustration: 814. Diagrams of Vaulting.] So far all was easy, but the fact of the flat central space resting on the four cones was still felt to be a defect, as indeed is apparent in such a vault as that of the cloisters at Gloucester (Woodcut No. 815), where a segment is used nearly equal to an equilateral spherical triangle. In this case they did not dare to employ a constructive decoration, but covered the space with circles so as to confuse and deceive the eye. At Windsor (Woodcut No. 816) the defect was obviated by using a low four-centred arch invented for the purpose, so that the outer tangent of the concoid was nearly flat, and the principal transverse rib was carried to the centre without being broken—as the others might have been had that mode of decoration been deemed expedient. This may be considered the perfection of this kind of vaulting, and is perhaps the most beautiful method ever invented. At Westminster (as shown in Woodcut No. 817) the difficulty was got over by reversing the curve by the introduction of pendants. This was a clever expedient, and produced a startling effect, but is so evidently a _tour de force_ that the result is never quite satisfactory; though on a small scale perfectly admissible.