On the Art of Building

...for one is whole, another is diminished, another is composite. A whole [arc] is that which occupies half of a circle, that is, one whose chord is the diameter of the whole circle. A diminished [arc] is one whose chord is smaller than the diameter, and indeed this same diminished arc is a part of a semicircle. A composite [arc] consists of two diminished ones, and for this reason, it creates an angle at the top where the arcs intersect each other in mutual conjunction, which does not happen in a whole or a diminished [arc]. With these things known, we shall proceed with the matter thus. One area is angular, another is circular. Of the angular ones, one is entirely enclosed by straight lines, another is enclosed by a mixture of straight and curved lines. Indeed, I do not remember having found in the buildings of the ancients an angular area that is made with many curved lines without any straight ones interspersed. But in these, one must observe those things which are vehemently criticized in the entire parts of a building if they are lacking, and are not without grace and convenience if they are present; that is, that there should be a certain variety in the angles, in the lines, and in all the parts, so that it is not too frequent, nor entirely rare, but placed for use and grace in such a way that the whole corresponds to the whole and the equal to the equal. They use right angles quite conveniently. No one has used acute angles, and they are neglected in very small areas unless unwillingly, or when compelled by the reason and manner of the places or the dignity of the areas. They considered obtuse angles to be quite decent, but they would preserve this, that they should be nowhere unequal in number. They affirm that the most capacious area of all, and one in which there is less expense for fencing either with an embankment or a wall, is that which is made round. They think the next to this is the one that has the most projecting angles. But the angles must be entirely comparable, mutually corresponding, and equal in every area. And indeed, they approve of those primarily that they understand to be suitable for raising walls correctly to the height of the work, such as one that has six or eight angles. We have also seen an area with ten angles that is most suitable and graceful. It is also possible for an area of twelve and likewise sixteen angles to be placed correctly. We have indeed seen an area of twenty-four angles, but such are rarer. The lines must be such that those opposite to one another are equal, and the lines should never be excessively long, and they will not be driven to be produced as the shortest in the entire work; but there will be a just and fitting proportion between them relative to the nature of the things...