What does measure mean in Euclid's Elements?

Definition 14: Numbers relatively composite are those which are measured by some number as a common measure. Definition 15: A number is said to multiply a number when the latter is added as many times as there are units in the former. read more

It has to be a whole multiple, though. If [math]B[/math] is two and a half times [math]A[/math], then [math]A[/math] doesn’t measure [math]B[/math], and [math]B[/math] is not a multiple of [math]A[/math]. Indeed, Euclid did not treat numbers axiomatically as he did geometry. read more

The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 read more

Euclid, however, supplies none. Missing postulates occurs as early as proposition VII.2. In its proof, Euclid constructs a decreasing sequence of whole positive numbers, and, apparently, uses a principle to conclude that the sequence must stop, that is, there cannot be an infinite decreasing sequence of numbers. read more