Flying buttresses were not a necessity. The Merveille had none; the Angevin school rather affected to do without them; Albi had none; Assisi stands up independent; but they did give support wherever the architect wanted it and nowhere else; they were probably cheap; and they were graceful. Whatever expression they gave to a church, at least it was not that of a fortress. Amiens and Albi are different religions. The expression concerns us; the construction concerns the Beaux Arts. The problem of permanent equilibrium which distresses the builder of arches is a technical matter which does not worry, but only amuses, us who sit in the audience and look with delight at the theatrical stage-decoration of the Gothic vault; the astonishing feat of building up a skeleton of stone ribs and vertebrae, on which every pound of weight is adjusted, divided, and carried down from level to level till it touches ground at a distance as a bird would alight. If any stone in any part, from apex to foundation, weathers or gives way, the whole must yield, and the charge for repairs is probably great, but, on the best building the Ecole des Beaux Arts can build, the charge for repairs is not to be wholly ignored, and at least the Cathedral of Chartres, in spite of terribly hard usage, is as solid to-day as when it was built, and as plumb, without crack or crevice. Even the towering fragment at Beauvais, poorly built from the first, which has broken down oftener than most Gothic structures, and seems ready to crumble again whenever the wind blows over its windy plains, has managed to survive, after a fashion, six or seven hundred years, which is all that our generation had a right to ask. The vault of Beauvais is nearly one hundred and sixty feet high (48 metres), and was cheaply built. The vault of Saint Peter's at Rome is nearly one hundred and fifty feet (45 metres). That of Amiens is one hundred and forty-four feet (44 metres). Rheims, Bourges, and Chartres are nearly the same height; at the entrance, one hundred and twenty-two feet. Paris is one hundred and ten feet. The Abbe Bulteau is responsible for these measurements; but at Chartres, as in several very old churches, the nave slopes down to the entrance, because--as is said--pilgrims came in such swarms that they were obliged to sleep in the church, and the nave had to be sluiced with water to clean it. The true height of Chartres, at the croisee of nave and transept, is as near as possible one hundred and twenty feet (36.55 metres).