Of the power of the arch in architecture.

The permanence of the arch built by the architect consists in its chord (tie) or in its shoulders (abutments).

Of the placing of the chord in the aforesaid arch.

The placing of the chord has its first necessity at the beginning of the arch and at the end of the straightness of the pillar on which it rests. This is proved by the 2nd (proposition) of the supports, which says: that part of the supports resists least which is most remote from the fastening of its whole. Therefore, the summit of the pillar being the last resting-place of its fastening, the like occurring at the opposite ends of the arch, which are the farthest distance from the middle, its true fastening, we have concluded that such a chord a b of necessity requires the placing of its opposite ends among the 4 aforesaid opposite ends.

The adversary says that such an arch should be more than a semicircle, and then it will not need a chord, because such ends will not push outward but inward, as is shown in the excess a c b d. Here it is answered that such an invention is bad for 5 reasons; and the first is as to strength, because it is proved that the circular parallel, being composed of two semicircles, only breaks where those semicircles join together, as the figure n m shows. Besides this, it follows that there is greater space between the ends of the semicircle than between the faces of the walls. The third is that the weight placed against the strength of the arch diminishes in weight by as much as the springing of the arch is wider than the said space interposed between the pillars; and the pillars are weakened by as much as their part c a b d bends backward in receiving the arch upon itself. The 5th is that all the expense and the weight of the arch that exceeds the half-circle is useless and harmful. And it is here to be noted that the weight placed upon the arch will break the arch more easily at a b, finding the curvature of the excess that is added to the half-circle, than if the pillar were straight up to the contact of the semicircle.

The arch which is loaded upon its middle will break at its right and left quarter. It is proved by the 7th of this, which says: the opposite extremities of the supports are equally burdened by the weight that is hung from them. Therefore the weight given at f is felt at b, c, that is, half at each end; and by the 3rd, which says: that part of the support of equal strength breaks soonest which is farthest from its fastening. Whence it follows that ... d being equidistant from the f c fastening ...

If the centering (framework) of the arch does not descend together with the descent of the arch, the mortar in drying shrinks into itself and detaches from one of the bricks between which it is interposed to bind them, and thus leaves them unbound; wherefore the vault remains disjoined and the rains soon ruin it.

The contacts c n are called the centres of the rotatable body, and in this case they do what the pole of the balance does to its balance.

Every weight burdens its support toward the centre of the world. This is proved by the two figures placed here in the margin, and it is also proved by the 3rd of this, which says: the centre of every suspended gravity falls beneath the centre of its support.

First make the treatise of the causes generating the breaking of walls, and then the treatise of the remedies, separately.

Parallel cracks are universally generated in those buildings which are built in mountainous places composed of stones bedded with oblique layering. And because into such obliquity water and other moisture often penetrates, carrying a certain greasy and slippery earth, and because such strata are not continued to the bottom of the valleys, such stones move by their obliquity, and never end their motion until they descend to the bottom of the valley, carrying with them, like a boat, that part of the building which is separated by them from its remainder. The remedy for this is to found close-set pillars under the wall that moves, with arches from one to the other and well braced, and let such pillars be founded and fixed in the strata which are not broken. To find the stable part of the aforesaid strata it is necessary to make a well under the foot of the wall to a great depth among those strata, and of such a well to polish with a smooth surface the width of a palm, from the top to the bottom on that side whence the mountain descends; and after some time this polished part made in the wall of the well will show a manifest sign of which part of the mountain moves.