How Doubling a Column's Proportion Affects Its Load

A sketch of two vertical columns supporting two rectangular weights. Labels:

100 200

Just below the bases of the columns, a plan view of the same: the base of the column supporting 100 is shown as a simple circle; the other base is shown as another circle the same size as the first, encircled by a slightly larger, concentric circle. Labels:

bases of the columns

[I]

Any column1 that is twice the proportion of a smaller one will bear twice as much weight as the smaller one, if each one is, of itself, united and solid.

The reason for this is manifestly obvious, because although the one column is twice the quantity2 of the other, smaller one, and it would appear that the union of the doubling provides a greater force than it would if disunited, nonetheless — since it increases further in its height, and it is the weight, more than the height, that becomes excessive, and the length that weakens it — when all these opposing contributions are added together, the result is that all that is provided by this uniting will not bear any more weight than twice as much as does the smaller, exactly, which is confirmed3 by experience.

Nearly identical to figure 1, a sketch of two vertical columns supporting two rectangular weights. Labels:

100 240

Just below the bases of the columns, a plan view of the same: the base of the column supporting 100 is shown as a simple circle; the other base is shown as another circle the same size as the first, encircled by a slightly larger, concentric circle. Labels:

bases

[II]

Among any supports of equal material and width, the shortest one will sustain more than the longest one, in proportion as it is shorter than the longest one.4

Although it cannot be numerically shown in a determinate way5 how much the increase is of a body that is twice the quantity of another, it is still possible to come rather close to the truth. I find that its increase is two-fifths of the entire sum of the smaller one, for the height as well as for the diameter. And if one of the two columns is covered over by the other, so that the diameter of this combination increases by two-fifths of the first width, then, lacking two-fifths of the due proportion,6 this support will be of two-fifths more strength.7 And you will state it in this manner:8 If the first column bears one hundred pounds and the second, which is of twice the proportion, bears two hundred pounds, then, since the second does not increase the two-fifths in height that it did in width, it has two-fifths times more strength than it would if it had its due height in accordance with proportions of the smaller column.9

So, since the [additional] 2/5 make it more powerful than if it were unitary10, which would bear 200, take 2/5 of 200, which are 4011, and add that to 200, and say that 240 is the resistance of the doubled and united columns.

**A power that is equidistant from two other, different powers will resist the weight found numerically between the other two ends sustained by the supports.

Analogous to the two preceding drawings, a sketch of two rectangular weights, one labeled 100, supported by a simple square column, the other, labeled 200, supported by two adjacent columns like the first one. Labels:

100 200

Just below the bases of the columns, a plan view of the same: the base of the column supporting 100 is shown as a simple square; the other base is shown as two adjacent squares the same size as the first. Labels:

bases

Three vertical, rectangular supports, each the same height and each divided into various rectangular sections. The first, labeled 8, is a square column divided into eight cubes. The second, labeled 4, is two vertically juxtaposed columns like the first one; the side view of the pair is divided into eight cubes; the front face of the pair shows the vertical division between the two columns, each of which is then divided into four sections, each section equivalent to two of the cubes from the side view. The third support, labeled 16 and appearing from the side like a fine grid, is constructed by starting with one column like the first, then dividing a second, equal column into cubes small enough that the first column can be uniformly covered over by all of the cubes from the second column. Labels:

8 4 16

Just below the bases of the support, a plan view of the same: the base of the first support is shown as a simple square; the second base is shown as two adjacent squares the same size as the first; the third base is shown as one square the same size as the first, surrounded by a slightly larger, concentric square. Labels:

bases

[III]

If the first support is joined to the second support, without12 being surrounded by it, these will support exactly the same weight as before.

The reason is this. If one square column is united with another one, even though13 the overall height14 diminishes15 in the larger, making it half of its proportion[al height], it remains as though apart for its former weakness; and if the first bore one hundred, the two in this form will not bear more than two hundred.

A sketch of two vertical columns supporting two rectangular weights. Labels:

Just below the bases of the columns, a plan view of the same: the base of the column supporting 100 is shown as a simple circle; the other base is shown as another circle the same size as the first, encircled by a slightly larger, concentric circle. Labels:

Nearly identical to figure 1, a sketch of two vertical columns supporting two rectangular weights. Labels:

Just below the bases of the columns, a plan view of the same: the base of the column supporting 100 is shown as a simple circle; the other base is shown as another circle the same size as the first, encircled by a slightly larger, concentric circle. Labels:

Analogous to the two preceding drawings, a sketch of two rectangular weights, one labeled 100, supported by a simple square column, the other, labeled 200, supported by two adjacent columns like the first one. Labels:

Just below the bases of the columns, a plan view of the same: the base of the column supporting 100 is shown as a simple square; the other base is shown as two adjacent squares the same size as the first. Labels:

Two columns bearing 100 and 200

Bases of the columns; the rule that double proportion bears double weight

Two columns bearing 100 and 240

Bases; the shorter support gains two-fifths more strength (240)

One column bearing 100 and a doubled pair bearing 200

Bases: a single square and two adjacent squares

Three supports labeled 8, 4, and 16

Bases; joining supports side by side adds no strength

How Doubling a Column's Proportion Affects Its Load

On this page

A column of double proportion bears twice the weight (100 to 200)

A shorter proportion adds two-fifths more strength (to 240)

Joining two columns side by side adds no extra strength