What Have the Greeks Done for Modern Civilisation?

Note 33 — VI SCIENCE: GRAMMAR—LOGIC—MATHEMATICS—MEDICINE

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There is even an arithmetical proof mentioned by Aristotle that the ratio of side to diagonal of a square cannot be one of whole numbers.

If it were, the ratio will be that of two numbers in its lowest terms, and hence one must be even and the other odd, else both were still divisible by 2.

here b² = 2a² ∴ a is odd and b even.

[image unavailble: box split diagonally]

Now let b = 2c then 4c² = 2a² and 2c² = a² ∴ a is even and b odd, which is absurd.

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