On the support.

A pillar that is enlarged in width will gain as much more than1 its due strength as it remains short of its2 proportional height.

Example.

If a pillar ought to be nine widths high — meaning that if it is one braccio wide, the rule3 sets it at nine braccia high — and if you connect one hundred of them together, widthwise, this combination will be ten braccia wide and nine high.  And if the first pillar bore ten thousand pounds, then, because this second one is about one width in height, and since eight parts of its length are lacking, it will bear eight times more; that is, each connected pillar will bear eight times more than it would when not connected.  That is, if it bore ten thousand pounds before, now it will sustain ninety thousand.

Verbi gratia.  If you take a rush4 whose height is equal to one hundred of its widths, putting it in a perpendicular line, and give it one ounce of weight, it will sustain that.  Then bind one hundred rushes together with continuous binding, so that the rushes compose a body whose height is equal to five5 of its widths.  As many times as five enters into one hundred, so many times more weight will this body sustain than if its height were of one hundred widths.  And since five enters6 twenty times into one hundred, each7 rush will sustain twenty times more weight than before because of the aforesaid binding; that is, if before it bore one ounce, then afterwards it will bear twenty.

If a lance whose length is one hundred times its8 width bears twenty pounds, what will a lance of the same staff bear whose length is five of its widths?  As many times as five enters into one hundred, so many times less weight will the staff of one hundred widths bear than the staff of five widths.

a b – cd – e f

[II]

If rush e f bears one ounce and its height is equal to one hundred of its diametrical widths, then rush a b, which is also composed of one hundred diameters and one hundred rushes, will bear one hundred ounces, because9 it has10 the same proportions as rush e f.

If you take one hundred similar rushes, like e f, and connect them with a tight and continuous binding — since f e is one hundred widths, and c d is five11 widths, and five enters12 twenty times into one hundred — then each of the bound rushes will bear twenty times more than when not bound.  Therefore, if e f 13 bears one ounce, c d will bear two thousand ounces.