Pyramidal Proof That Motion Grows in Arithmetic Proportion
A tall isosceles triangle ruled with parallels shows that time, motion and velocity increase pyramidally in the descent of heavy bodies.
Under the heading "On motion," Leonardo sets out to prove that the growth of time, motion and velocity in a falling body is pyramidal in shape, because each begins at nothing and increases by degrees in arithmetical proportion. A tall isosceles triangle drawn at the right of the page, ruled with parallels to its base, carries the demonstration: apex a, then the descending levels b/f, c/g, d/h and the base e/n, divided into equal units. Cutting the pyramid with any line parallel to the base makes the ratio of that cut's height to the whole height equal to the ratio of the cut's width to the base width, so that ab is one-quarter of ae just as the cut fb is one-quarter of the base ne.
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Why the increase of motion is called 'pyramidal'
The proof asserts that the proportion of time and motion, together with velocity, in the descent of heavy bodies is pyramidal in shape. The reason is that these powers are all pyramidal, since they begin at nothing and proceed increasing by degrees, in arithmetical proportion.
Cutting the pyramid gives proportional widths (labels a, b/f, c/g, d/h, e/n)
If you cut the pyramid at any degree of its height with a line parallel to its base, the proportion between that cut's distance from the base and the whole height equals the proportion between the width of the cut and the width of the base. Leonardo observes that ab is one-quarter of ae, and likewise that the cut fb is one-quarter of the base ne.