...(incommensurables)

Incommensurables had only been recently discovered, reportedly by Hippasus of Metapontum

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...made

at the expense of algebra. All mathematicians were driven to geometry. The consequence persists. Even today, we still speak of and as x squared and x cubed.

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...given

Note the assumption that is superfluous in the language of the time as there were only positive numbers. So, the assumption is implicit

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...cube

This is a modern word; the original word as given by Archimedes is triple.

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...famous

Try to prove either of these without calculus!! Yet, the Egyptians seems to have been aware of the formulas. Levels of understanding progress.

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