Apparent proportion, or the sensible relation of quantities, is one of the most important means of obtaining unity between things which otherwise must have remained distinct in similarity, and as it may consist with every other kind of unity, and persist when every other means of it fails, it may be considered as lying at the root of most of our impressions of the beautiful. There is no sense of rightness, or wrongness connected with it, no sense of utility, propriety, or expediency. These ideas enter only where the proportion of quantities has reference to some function to be performed by them. It cannot be asserted that it is right or that it is wrong that A should be to B, as B to C; unless A, B, and C have some desirable operation dependent on that relation. But nevertheless it may be highly agreeable to the eye that A, B, and C, if visible things, should have visible connection of ratio, even though nothing be accomplished by such connection. On the other hand, constructive proportion, or the adaptation of quantities to functions, is agreeable not to the eye, but to the mind, which is cognizant of the function to be performed. Thus the pleasantness or rightness of the proportions of a column depends not on the mere relation of diameter and height, (which is not proportion at all, for proportion is between three terms at least,) but on three other involved terms, the strength of materials, the weight to be borne, and the scale of the building. The proportions of a wooden column are wrong in a stone one, and of a small building wrong in a large one,[20] and this owing solely to mechanical considerations, which have no more to do with ideas of beauty, than the relation between the arms of a lever, adapted to the raising of a given weight; and yet it is highly agreeable to perceive that such constructive proportion has been duly observed, as it is agreeable to see that anything is fit for its purpose or for ours, and also that it has been the result of intelligence in the workman of it, so that we sometimes feel a pleasure in apparent non-adaptation, if it be a sign of ingenuity; as in the unnatural and seemingly impossible lightness of Gothic spires and roofs. Now, the errors against which I would caution the reader in this matter are three. The first, is the overlooking or denial of the power of apparent proportion, of which power neither Burke nor any other writer whose works I have met with, take cognizance. The second, is the attribution of _beauty_ to the appearances of constructive proportion. The third, the denial with Burke of _any_ value or agreeableness in constructive proportion. Sec. 11. The value of apparent proportion in curvature. Now, the full proof of the influence of apparent proportion, I must reserve for illustration by diagram; one or two instances however may be given at present for the better understanding of its nature.