The Rule of Three Applied to a Bombard's Ball
Why proportion misleads, and a warning to trust experience over imagination
Leonardo works a proportion problem: if 1000 pounds of ball is thrown a fixed distance of 3 miles in 20 units of time, in how many units would the same power throw one pound? By the rule of three he finds 20/1000 of a time-unit, which he says makes about 1/513 of a time-unit. He then exposes the absurdity: taken literally, a single grain of powder's weight would fly a million miles in the time the thousand-pound ball went 3 miles, whereas experience shows it would go no farther than the smoke. He closes by warning speculators to trust not authors who reason by imagination alone, but only those who exercise their minds on the effects of nature's experiences.
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Rule of three: from 1000 pounds to one
Holding the 3-mile distance fixed, Leonardo sets the proportion: if 1000 pounds of ball is thrown in 20 time-units, in how many will the same power throw one pound? 'If 1000 gives me 20, what will one give me?' He finds 20/1000 of a time-unit, which he says makes about 1/513 of a time-unit.
The absurd result exposes the false rule
Taken to the limit, the weight of a single grain of powder would fly a million miles in the time the thousand-pound ball went 3 miles, yet experience shows it would carry no farther than the bombard's smoke at the first instant of firing. The mismatch shows the proportional rule cannot govern this motion.
Trust experience, not imagination
Leonardo warns speculators not to trust authors who, with imagination alone, would set themselves up as interpreters between nature and man, but only those who have exercised their wits on the effects of nature's own experiences.