Why are perfect numbers called perfect numbers?

Question

A perfect number is a number than can be expressed as a sum of its factors. For example,

28 = 1 + 2 + 4 + 7 + 14

Why is this property important? What is so perfect about perfect numbers?

Answer 1

From the Earliest Known Uses of Some of the Words of Mathematics pages ...

PERFECT NUMBER. According to Smith (vol. 2, page 21), the Pythagoreans used this term in another sense, because apparently 10 was considered by them to be a perfect number.

Proposition 36 of Book IX of Euclid’s Elements is: "If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect."

The Greek poet and grammarian Euphorion (born c. 275 BC?) used the phrase ". . . equal to his [or their] limbs, with the result that they are called perfect." This is an apparent reference to perfect numbers, according to J. L. Lightfoot, "An early reference to perfect numbers? Some notes on Euphorion, SH 417," Classical quarterly 48 (1998), 187-194.

The term was used by Nicomachus around A. D. 100 in Introductio Arithmetica (Burton, page 475). One translation is:

Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. Nichomachus identified 6, 28, 496, and 8128 as perfect numbers. St. Augustine of Hippo (354-430) wrote De senarii numeri perfectione ("Of the perfection of the number six") in De Civitate Dei. He wrote, in translation: "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist."

Perfect number appears in English in 1570 in Sir Henry Billingsley’s translation of Euclid.

In 1674, Samuel Jeake wrote in Arithmetic (1696) "Perfect Numbers are almost as rare as perfect Men" (OED2).