Commensurable Sides and Square Numbers
When square surfaces are proportional as square numbers, their sides are commensurable
This text-only folio states, in three linked propositions, the theorem on the commensurability of the sides of square surfaces (Euclid, Book X). If two square surfaces are in proportion as one square number to another, their sides are commensurable in length; conversely, if their sides are commensurable in length, the proportion between the squares is as a square number to a square number. Finally, if two square surfaces are not in proportion as square number to square number, their sides are not commensurable in length.
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Square-number proportion implies commensurable sides
If two square surfaces are in proportion to one another as a square number is to a square number, then their sides will be communicant, that is, commensurable in length.
Converse: commensurable sides give square-number proportion
And if there are two square surfaces whose sides are commensurable in length, it follows that the proportion between them will be as a square number to a square number.
No square-number proportion implies incommensurable sides
And if two square surfaces are not in proportion to one another as a square number to a square number, then their sides will not be commensurable in length.