Water emptying from two vessels; the rise of rivers
A proof that two vessels of doubled height empty in ever-diverging proportion
The upper lines observe that the height of river water never rises the same from one time to the next and never stands fixed. The main demonstration concerns two cylindrical vessels of equal width but one twice as tall as the other, each pierced with an equal orifice (spiraculo) at its base; opened at the same instant, the ratio of their sinking water levels grows without bound within a finite time. Leonardo divides the taller vessel into 12 equal grades and the shorter into 6, subdividing each grade into 12 minutes, and reasons that the levels can never agree because the slower fall of the lighter head of water lags the faster one. A pair of vessels with issuing jets is drawn at right, with marginal figures (144, 72, 132 minutes).
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The height of river water is never constant
Leonardo states that the height of river water will never rise one time as it did another, but will fall short or overshoot. It will never stand fixed in its being.
Two vessels emptying through equal orifices at the base
Two vessels a b and c d of uniform and equal width but doubled height are each pierced with an equal orifice (spiraculo) at the lowest point, b and d. Opened at the same instant, the proportions of their emptying levels change continuously as the water runs out.
Emptying proportions grow without bound in a finite time
Starting from a doubled (dupla) ratio of outflow, Leonardo argues the proportion increases to infinite magnitude across the finite time in which the smaller vessel empties. When the greater vessel has lost one grade of height, the smaller has lost less than half a grade, because its motion slows as the weight of water above its orifice diminishes.
Marginal reckoning in minutes
The margin carries the figures 144 minutes and 72 minutes, then the note that removing 12 minutes from the greater water leaves 132 minutes. These numbers track the graded subdivisions used in the proof.
