Discourses on Fortifications

On various figures and forms of enclosures, and their perfection and imperfection.
Chapter Three.

BUT because the sharpness of many things is a cause for bad results, one will mainly have to avoid not only triangular enclosures but quadrangular ones as well, even if they had equal angles. Because by constituting the bastions at the angles of such figures so that their faces are cleared by the flanks, it is necessary that the exterior angles of such bastions come to be very acute. Indeed, on whatever figure, the exterior angles will always be more imperfect or less good than the interior angles of the enclosure. And therefore, a circular figure, and one of five sides and equal angles, having just distances from flank to flank, and every other thing proportioned, as will be said in its place, is still forced to come out with acute bastions.

A bastion, even if (according to the whim of some, or according to necessity) it may have many faces, nevertheless, as far as my purpose now concerns, I intend it to be of two. These, at their conjunction, can cause the exterior angles to be acute, right, or obtuse. The acute ones are not good; better than these are the right ones; and most perfect are the obtuse ones, for being the strongest of all. It being known, without my saying it, that the acute is that which is less than the right, and the obtuse that which is greater than this, as is seen by the three manners of angles in the following plans.

A diagram shows three bastion angles: acute, right, and obtuse.

Just as therefore we know that the ancients avoided the angles, even though they erred in many things, so we, not being able to avoid them so as not to leave any space of the enclosure undefended, will have to guard ourselves as much as we can. Even if the fortress had more than four angles, the plan of it should not be so poorly divided that the bastion or other building of similar intention turns out acute, for the reasons that will be told in their place. This sharpness or lesser perfection can happen for two other reasons, even if the interior angles were obtuse: the first is the drawing of the direction of their faces from some very near place; the second is the length of the flanks. And to be better understood, I have made the following bastions on equal angles, where, considering what is said, one can see their difference.