Source — Euclid's Elements, encoded as an rrxiv paper

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X.9 Proposition X.9

The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length.

Proof

If

a : b = m : n

(integers) then

a^{2} : b^{2} = m^{2} : n^{2}

; converse holds by VIII.14 applied to integers and VI.22 applied to magnitudes.

lines 74–74 in main.tex