In the twelfth and thirteenth centuries the ingenuity of the greatest French architects--the greatest school perhaps the world has ever seen--was taxed to the utmost to devise means for obviating all the difficulties attendant on the plan of an apse with an aisle or aisles continued round it.[234] The arrangement of the central vault is easy enough; but the great flying-buttresses which support this have to be carried in part on the columns which form the divisions of the aisles surrounding the apse. From the centre of the apse, therefore, a number of lines drawn through its angles represent the lines of the flying-buttresses, and mark the position for the outer orders of columns. These lines diverge so rapidly from each other that the compartments enclosed within them become extremely irregular in their outline; and this renders it very difficult to cover them with vaults which shall look thoroughly well, and in which the arched ribs shall not be crippled or irregular in their lines. The French architects had from the first realized the necessity for making the diagonal vaulting rib a semi-circle. They saw that the line thus obtained was a continuous line of the utmost value, leading the eye on in succession from one bay of vaulting to another without any interruption--gradually from one end of the vastest vault to the other. Whenever this form is given up the effect of vaulting is half destroyed; and it matters not whether we turn to the domical pointed vaults of the Angevine architects, or the vaults of some of our own cathedrals, with their pointed diagonal ribs, we shall at once see how inferior they are to the old French mode.[235] In these unequal vaulting bays in the apse it was impossible to make a straight diagonal rib a semi-circle, for then (I) the highest part of the vault would be higher than the intersection of the ribs, and the connexion of the intersection with the highest part of the transverse arch would be extremely bad, and all but unmanageable. To get over this difficulty, we find the architect of Bourges (A.D. 1230) planning his diagonal ribs on a curve (II); whilst at Chartres (A.D. 1220) the architect planned this rib on a broken line (III). The architect of the choir of Le Mans (just later in date than Chartres--circa A.D. 1230) improved enormously upon what his brethren had done by the introduction of a triangular compartment in the outer aisle, which enabled him to make the vaulting bays between them nearly square, and to obtain a light between each of the chapels of the apse, which vastly increased its beauty. The architect of Bourges had indeed introduced triangular-vaulting compartments in his outer aisle, but so clumsily, that he had increased rather than diminished the difficulty with which he was dealing; and the earlier architect of Notre Dame, Paris (A.D. 1170), had ingeniously planned almost all the vaults of his apse in triangular compartments, with great gain over the systems of those who