1. Number is divided into even and odd. Even number is divided into
the following: evenly even, evenly uneven, and unevenly even, and
unevenly uneven.[244] Odd number is divided into the following: prime
and uncompounded, compounded, and a third class which comes between
(_mediocris_) which in a certain way is prime and uncompounded, but
in another way secondary and compounded.
[244] _Pariter par, et pariter impar, et impariter par et
impariter impar._ Since these all profess to be divisions of even
number, the word odd is not used in the translation.
2. An even number is that which can be divided into two equal parts,
as II, IV, VIII.[245] An odd number is that which cannot be divided
into equal parts, there being one in the middle which is either too
little or too much, as III, V, VII, IX, and so on.
[245] To remind the reader of Isidore’s notation Roman numerals
are kept wherever he used them.
3. Evenly even number is that which is divided equally into even
number, until it comes to indivisible unity, as for example, LXIV has
a half XXXII, this again XVI; XVI, VIII; VIII, IV; IV, II; II, I,
which is single and indivisible.
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.