Ten Books on Architecture

of straight and right angles), as Bernardino Baldi, Abbot of Guastalla, correctly proves in his commentaries on Aristotle’s Mechanics; where I observe in passing that when something demonstrated mathematically is weak, it is even weaker mechanically; because errors occur more easily in the handling of solid materials than in linear delineations.

Theorem V. Just as semicircular chambers or hemispherical vaults are of all forms the most rounded and consequently the most firm, according to the preceding theorem; so too are they the most graceful, which, while precisely preserving the same height, are nevertheless extended one-fourteenth part longer than the entire diameter: which addition of extension will contribute much to their grace, and diminish only a little from their firmness. I find this observation in Leon Battista Alberti: but the practical method of preserving the same height, and yet extending the arms or terminals of the arch, is in Albrecht Dürer’s Geometry, who taught the Italians many and excellent lines, which are of great use in this Art.

Upon these five theorems all the discipline of vaulting is founded. As for the chambers, which our artisans call of the third and fourth point; and the Etruscan writers di terzo and di quarto acuto, because they always meet at an acute angle and spring from the division of the diameter into three, four, or more parts at will; I say that such, both because of the natural weakness of the acute angle itself, and because of their lack of beauty, should be rejected by judicious eyes, and left to their first inventors, the Goths or Lombards, along with the other relics of that barbaric age.

Since, from my initial division of the parts of every structure into five chapters, I have completed the first two and have been led by chance to this doctrine concerning chambers, there follow now in order Openings; under which notion I comprehend doors, windows, steps, chimneys, and other conduits: in short, all outlets or inlets; to which two cautions pertain.

First: that they be as few in number and dimension as can consist with other necessary considerations: for in one word, all openings weaken.

Second: that they do not approach too closely to the corners of the walls: for it would be an essential solecism to weaken that part from which the others ought to be strengthened: a precept well noted by the Italians, but poorly observed by them, especially in Venice, where I have observed various galleries or balconies (as Vitruvius seems to call those which are waxed or balustraded projections for the satisfaction of the curiosity of sight) very dangerously projected to the very edge of the mural corner. Furthermore, although I am hastening to the repartition of the whole work, (since it is in truth the highest perfection of this Art to distribute a well-chosen subject usefully and gracefully) yet I wish first to collect briefly under certain chapters the principal notes that pertain to these particular openings.

Concerning doors and windows. I join these inlets for people and light, because I find their dimensions related to one rule by Leon Alberti (a learned investigator), who, from the school of the Pythagoreans (in which the fundamental maxim was that the images of all things lie hidden under numbers), determines that the most decent proportion between widths and heights is if you reduce the symmetry of symphony and the harmony of sounds to the species of the harmony of sight, in this way: The two principal consonances, which most capture the ears by the consensus of nature, are the fifth and the octave: of which the first arises radically from the proportion between two and three: the other from the double interval between one and two, or between two and four, etc. Now, however, if we transfer these proportions from audible objects to visible ones, and apply them as aptly as possible; (the nature of the place always being considered) namely, in certain windows and doors, the symmetry of two to three, in their width and length; in others, double, as said before: without doubt, from either, there will result a graceful and harmonic delight to the eyes. Which contemplation, although