How a common measure works
If A contains n copies of C and B contains m copies, positive integers n and m express both against one measure. Lengths six and nine, for example, share a measure of three as well as smaller common measures.
The magnitudes must be of the same kind: a length unit cannot measure an area directly. Naming a conventional unit such as a centimetre is a different matter; changing display units does not change whether two lengths are commensurable.
References: [1]
Integer ratios and a square’s diagonal
A common measure makes the ratio of two magnitudes a ratio of positive integers. A square’s diagonal and side supply an incommensurable pair, with ratio the square root of two in modern notation.
If that ratio were p/q in lowest terms, p squared would equal twice q squared. Then p would be even, and substitution would force q even too, contradicting lowest terms. Computing many decimal places is a different procedure and does not supply that argument.
References: [1]
Lengths and their squares can differ
Book X also defines commensurability in square: the squares on two lines have a common area measure. A square’s side and diagonal are incommensurable as lengths but commensurable in square because their squared ratio is two.
Euclid’s subsequent use of a rational line does not coincide exactly with modern rational-number terminology and depends on a chosen reference line. Reading the text requires identifying whether it concerns lengths, square areas or modern integer ratios.