4. Evenly uneven is that which admits of division into equal parts, but its parts soon remain indivisible, as VI, X, XVIII, XXX, and L, for presently, when you divide such a number, you run upon a number which you cannot halve. 5. Unevenly even number is that whose halves can be divided again, but do not go on to unity, as XXIV. For this number being divided in half makes XII, divided again VI, and again, III; and this part does not admit of further division, but before unity a limit is found which you cannot halve. 6. Unevenly uneven is that which is measured unevenly by an uneven number, as XXV, XLIX; which, being uneven numbers, are divided also by uneven factors, as, seven times seven, XLIX, and five times five, XXV. Of odd numbers some are prime, some compounded, some mean (_mediocris_). 7. Prime numbers are those which have no other factor except unity alone, as three has only a third, five only a fifth, seven only a seventh, for these have only one factor. Compound numbers are they which are not only measured by unity, but are produced by another number, as IX, XV, XXI, XXV. For we say three times three are nine, and seven times three are XXI, and three times five are XV, and five times five are XXV. 8. Mean (_mediocris_) numbers are those which in a certain fashion seem prime and uncompounded and in another fashion secondary and compounded. For example, when IX is compared with XXV, it is prime and uncompounded, because it has no common factor except unity alone, but if it is compared with XV it is secondary and compounded, since there is in it a common factor in addition to unity, that is, III. Because three times three make nine, and three times five make fifteen.[246] [246] The division into even, odd, and numbers sharing the characteristics of even and odd numbers goes back to Nicomachus. It is not a logical division, as the second class contains the third. See Gow, p. 90. 9. Likewise of even numbers some are excessive, others defective, others perfect.[247] Excessive are those whose factors being added together exceed its total, as for example, XII. For it has five factors: a twelfth, which is one; a sixth, which is two; a fourth, which is three; a third, which is four; a half, which is six. For one and two and three and four and six being added together make XVI, which is far in excess of twelve.... [247] _Superflui, diminuti, perfecti._ 10. Defective numbers are those which being reckoned by their factors make a less total, as for example, ten.... 11. The perfect number is that which is equalled by its factors, as VI.... The perfect numbers are, under ten, VI; under a hundred, XXVIII; under a thousand, CCCCXCVI. Chapter 6. On the second division of all number.