Why did Euclid call 6 a perfect number?

Question

The old Greek did not consider $1$ a number. Nevertheless Euclid called $6 = 1+2+3$ a perfect number. How could he use $1$ which was not a number?

Answer 1

See :

• Thomas Heath, A History of Greek Mathematics : Vol.I (1921), page 69 :

Aristotle observes that the One is reasonably regarded as not being itself a number, because a measure is not the thing measured, but the measure or the One is the beginning (or principle) of number.

See also Euclid's Elements, Book VII, Defs.1&2 :

1. A unit is that by virtue of which each of the things that exist is called one.

2. A number is a multitude composed of units.

See also :

22. A perfect number is that which is equal to the sum its own parts.

and Def.3 :

3. A number is a part of a number, the less of the greater, when it measures the greater.

See Heath, page 74 :

There is no trace [...] anywhere before Euclid, of the perfect number in the well-known sense of Euclid's definition (VII, Def.22), a number, namely, which is "equal to (the sum of) its own parts" (i.e. all its factors including $1$).

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Probably, we see here an inconsistent terminology, due to the clash between the old Pythagorean numerology, where the numbers are "individuals" with names and virtues, and a more modern approach, reflected into Euclidean arithmetical books.