On the strength and quality of arches, and where they are strong or weak, and similarly for columns.

z c n a – t – e p o – m f r b S

[II]

When triangle a z n falls, it drives [the lower] two-thirds of each semi-arch backward, that is, the parts from a to S and from z to m; and the reason is that a falls plumb on top of b and, similarly, z on top of f.

The part of the arch that is most level will make the least resistance to the weight placed on top of it.

Any semi-arch, when overcome by an excessive weight, will break at two-thirds of its height, which corresponds to the point perpendicularly over the middle of its base, as appears at a b.  And this happens because the weight tends to fall and to pass through point r.1  And if it tended to fall, against its nature, through point S,2 then arch n S would break at its exact middle.  And if arch3 n S were of one single piece of wood, a weight placed at n would tend to fall4 toward m, which would cause arch c m5 to break at its middle.6  Otherwise it would break at the point one-third7 from the top, at point a, because the arch is more level from a to n8 than it is from a to o, or than it is from o to S;9 and as much greater as p t10 is than t n, so much stronger is a o than a n;11 and, similarly, S o is as much stronger than o a, as r p is greater than p t.

An arch that been doubled in width and squared12 will bear four times as much weight as the simple one bore, and accordingly more as the diameter of its cross section13 enters a smaller number of times into its length.  That is, if the width of the simple arch enters ten14 times into its length, the width of the doubled arch will enter five times into its length.  Therefore, as the width of the doubled arch enters half as many times into its length as does15 that of the simple arch into its own, it is reasonable for it to bear twice as much weight16 than it ought to have borne,17 had it had the proportions of the simple arch.  So, since this doubled arch is four times the quantity18 of the simple arch, it would appear that it ought to bear four times as much weight, and the aforementioned rule demonstrates that it will support exactly eight times as much.

a b – c d – e f

[V]

The pillar burdened more unequally with weight will tend more quickly to collapse.19

Column c d20 is more permanent since it is equally burdened by its sum.  And the other two, outer columns need as much weight from their center on out as they have from their center on in, that is, from the center of the column to the middle of the arch.

x y – f – n m – a b c - g

[VI]

Arches that stand through the support of chains are not permanent.

An arch is longer lasting when it has a proper opposition to its thrust.  The arch, of itself, tends to fall; and if the arch is thirty braccia and the interval between the walls that support it is twenty, we know that the thirty will not pass through the twenty unless the twenty, also, becomes thirty.  So, when the arch is overcome by an excessive weight, it straightens and the poorly resisting walls open to give entry between their space, to the ruin of the arch.  But if you do not want to strengthen the arch with an iron cord, you will need to provide it with such shoulders as will make resistance to its thrust, which you will do as follows: Load spandrels21 m and n with stones22 in such a way that the lines made by their junctures are directed toward the center of the circle made by the arch.  And the reason this will make the arch permanent is as follows.  We know clearly that when the arch is loaded over quarter a b with an excessive weight, wall f g is thrust [outwards], because the arch tends to straighten itself;23 and if the other quarter, b c, were loaded, this would draw wall f g inwards, were it not for the line of stones, x y, that resists this.