I have, in a previous lecture, described the series of tentative experiments, all of them dictated by constructive and practical requirements, by which it was attempted to avoid these difficulties. I will not weary you by recapitulating them. The two obvious desiderata were an arch of less pressure and an abutment of greater resistance; and these were the two objects aimed at in most of the succeeding developments. The first demand was met by the pointed arch; the second by the systematised use of the buttress, whether of the solid or arched description. It was perfectly well known that the outward thrust of an arch diminished as its height increased; that the resisting power of an abutment depended mainly on its extension in the direction of the pressure; and that where sufficient extension of abutment could not be obtained without inconvenience or dissight, the deficiency might be compensated by loading it from above: and by arguing on these three facts the constructive characteristics which distinguish Gothic from Romanesque, or the pointed-arched from the round-arched Gothic, were logically worked out. The strictly mathematical mode of increasing the height of an arch would, I suppose, be by using a semi-ellipse, its major axis being vertical. The form is, however, most unpleasing to the eye and troublesome in execution, from its constant variation of curvature, so that by far the most natural and practical means of effecting the object is the adoption of an arch of two centres, or what is commonly called the “pointed arch.” We accordingly find, as I have shown by ample evidence in a previous lecture, that this form was in the first instance used just in those situations in which a reduction of outward pressure or an increased power of bearing weight were of the greatest importance. I have shown that this form was not adopted at first as a matter of taste, of fashion, or of fancy; nor even, as has been suggested by a highly talented writer, as a means of meeting the difficulties arising from the varied heights of the arches of vaulting, but simply from _structural_ and _mechanical_ necessity. It matters not whether the form was new or old, whether it occurred to them without external suggestion, or whether they saw it in the East, in their own intersecting arcades, or in the first proposition of Euclid. It was not the seeing of it in any such manner which caused its introduction, but the simple fact that they had arrived in the course of their constructive development at a practical problem of vital importance, which _absolutely demanded_ the pointed arch for its solution.