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Rh covered so as to spring out of the angles. In this case it is clear that the dome as seen from within gradually expands from the four lowest points and spreads on the walls as it grows upward, forming concave triangles having curved lines against the four walls. These pieces of the domical surface running down into the angles are called pendentives. When the circular basis required for the dome is formed by these pendentives it is possible to set a complete semispherical done on them, and there will be a break in the curvature where such a dome springs from the pendentives; or it is possible to carry on the curvature of the pendentives, forming in this case a flatter dome with the surface continuous to the angles. The first would be a dome on pendentives, and the other we might call a pendentive dome. Again a third variety is obtained by building a circular ring of wall, a "drum," above the pendentives, and on that the dome at a higher level. This was a later fashion. It is rather difficult to see the geometry of all this without a model; but if an apple be cut into halves, and then one half is laid on its cut surface and four vertical cuts are made in pairs opposite to one another so as to reduce the circular base to a square, we shall obtain a model of a dome with continuous pendentives.

The methods of building ordinary vaults with inclined courses as described above were practised in Egypt in the early dynasties, and also in Mesopotamia. Evidence is accumulating which suggests that domes, even domes with pendentives, were used in these countries long before the Christian era. A dome with pendentives has been found over an Egyptian tomb which seems to have been built about 1500 years B.C. When Alexander built his new Greek capital in Egypt it must have been a city of brick buildings covered with vaults, save for a few chief structures which were built in the usual manner of Greek temples. A Latin author, writing about the year B.C. 50, says that the houses of Alexandria were put together without timber, being constructed with vaults covered over with concrete or stone slabs. The scarcity of timber in Egypt, the cause behind the development of vaulted structures, is again brought before us in a letter written by St Gregory to Eulogius, the Patriarch of Alexandria, in regard to timber which was sent to him all the way from Italy. It was doubtless from the new Hellenistic capital, and possibly from Western Asia as well, that the art of building vaulted structures spread to Pompeii and Rome. Later, it was almost certainly from Alexandria that Constantinople obtained the more developed traditions of brick building by which it was possible to erect the great church of St Sophia. It seems to be equally true that decorative ideas and processes were largely derived from Alexandria. In addition to the facts mentioned in the first volume, reference may be made to a painted catacomb chamber at Palmyra illustrated by Strzygowski, who assigned it to the third century. Amongst the subjects are Victories carrying medallions like those on consular ivories of the fifth century. There are