Refuting Zenophon: Equal Subtraction Alters a Ratio
Taking equal amounts from 2 and 4 leaves a triple, not a double
Leonardo continues an argument against a figure he names Zenophon, proving that subtracting equal amounts from two unequal quantities does not preserve their ratio. Removing one from 2 and one from 4 leaves 1 and 3 — a triple, greater than the original double. He concludes that Zenophon, who took equal parts from unequal wholes expecting the same proportion, deceived himself.
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Equal subtraction turns a double into a triple
Starting from the ratio 2:1 in the numbers 2 and 4, Leonardo shows that removing an equal amount (one) from each term leaves 1 and 3, a triple rather than a double. The remainder no longer holds the first proportion, exposing the error of assuming equal subtraction keeps a ratio fixed.