How Zenophon Proposed a Falsehood
Proportional subtraction keeps a ratio; equal subtraction does not
This page states the rule underlying the Zenophon argument. Subtracting from unequal quantities amounts that themselves share the wholes' ratio keeps the remainder in that same ratio, but subtracting equal amounts leaves the remainder unequal in a different ratio. Leonardo works the example with wholes 2 and 4 (a double), removing 1 and 2 to leave 1 and 2, still in the original proportion, and directs the reader to turn the page.
On this page
The rule of proportional versus equal subtraction
Leonardo distinguishes two cases: if the amounts removed from unequal quantities are in the same ratio as those quantities, the remainders keep the original ratio; if equal amounts are removed, the remainders stay unequal but in a new ratio. He demonstrates with 2 and 4, taking 1 from 2 and 2 from 4 to leave 1 and 2, preserving the double.