Proposition X.111 — The apotome is not the same as the binomial.

Proof

A binomial has a rational sum of squares plus a medial rectangle; an apotome has a rational difference of squares minus a medial rectangle; if they coincided, the two combinations would coincide, forcing the medial part to be rational — contradiction.

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Depends on (2)

X.36Proposition X.36If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be…
X.73Proposition X.73If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square…

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