Book:Euclid/The Elements/Book X/Linguistic Note

Linguistic Note on Book $\text X$ of Euclid's The Elements

The term rational (ῥητός) is often used in Book $\text X$ of Euclid's The Elements.

However, its meaning differs from the modern-day usage of rationality.

Therefore, the following nomenclature is used instead in $\mathsf{Pr} \infty \mathsf{fWiki}$:

• If $x \in \Q$, $x$ is called rational (ῥητός)
• If $x^2 \in \Q$, $x$ is called rationally expressible (ῥητός)
• If $x \notin \Q$ but $x^2 \in \Q$, $x$ is called linearly irrational (ῥητός)
• If $x \notin \Q$ and $x^2 \notin \Q$, $x$ is called squarely irrational (ἄλογος)

where $x$ is specified as being strictly positive.