Measuring a tower by its shadow; strength of bundled ropes

Measuring a height from the sun's shadow

Fix a cane one braccio long and wait until the sun makes its shadow two braccia long, then measure the tower's shadow. If that shadow is one hundred braccia, the tower is fifty high. Leonardo calls it a good rule.

The tower's height from the ratio b c to a c

A second shadow diagram states the proportion directly: as many times as b c goes into a c, so many times does the height of the tower enter into a n. It generalizes the shadow rule into a ratio of similar triangles.

How much will ten twisted ropes bear?

A sketch hangs a large weight from a many-twisted rope beside a smaller weight on a simple rope. If one rope bears one hundred pounds, how many pounds will ten such ropes bear when twisted firmly together?

How much will ten bound pieces of wood bear?

Two weights rest on supports, one on a single stick and one on a bundle of sticks wrapped together. If a piece of wood bears one hundred pounds, how much will ten such pieces bear when firmly bound together?

A saw cuts the narrow section sooner

A lumber saw is drawn over a piece of wood wide at a b and narrow at c d. As many times as the wide part a b contains the narrow part c d, so many times sooner will c d be sawed through than a b.

Diagram showing the sun casting the shadow of a tower, producing a triangle proportional to the triangular shadow of a cane. The height and base of the cane are labeled a and b. The horizontal limit, and third vertex, of the cane's shadow is labeled c. Labels:

Another diagram showing the sun casting the shadow of a tower, whose base is labeled n. The horizontal limit of the tower's shadow is labeled a; within the shadow zone, another cane, whose base is fixed at point c; the top of the cane, not within the shadow, is labeled b. Labels:

Near the right margin, a sketch of two weights suspended from a crossbar by ropes. The larger weight, on the left, is suspended by a rope with many twists; the other, smaller weight is suspended by a simple rope. [III]

Two weights on supports: the smaller weight, on a single stick; the larger weight, on a bundle of sticks wrapped together. [IV]

A lumber saw positioned over a triangular section of wood. The horizontal limits of the upper, broader portion of the wood are labeled a and b; the limits of a lower, narrower width of the piece of wood are labeled c and d. Labels:

Measuring a height by the sun's shadow

The tower's height from the ratio of the shadows

On twisted ropes

On bound pieces of wood

On sawing wood of unequal width