means by certis intervallorum rationibus pauciores; "fewer by definite reasons of intervals."
Book IV 243 added, but come to the exact sum of the number itself, they are much fewer, and only found at ever-increasing intervals.6 Of these the first is six. The number one, you see, has no parts; I mean, we use the word "one" as a number in counting in such a way that it has no half or any other part, but is really and truly and simply one. Two has one as its part, its half part, and no other. Three has two parts; one part, about which it can be said how many times, and that is the number one, which is a third of it; and another bigger part about which it cannot be said how many times, and that is the number two. So neither of this number three can the parts in the sense we now mean, about which you can say how many times, be added up. Going on to four, it has two such parts, because one is a quarter and two is a half of it; but add them up, that is one and two together, and they make three, not four. So its parts do not complete or perfect it, because added together they come to less. Five only has one such part, that is the number one, which is a fifth of it, because neither of two nor three, its lesser and larger parts, can it be said how many times. Six, on the otherhand has three such parts; a sixth, a third, and a half; a sixth of it being one, a third two, and a half three. Now add these parts together, that is one and two and three, and together they complete and perfect six.
4. Coming now to the number seven, the only such part it has is a seventh, which is one; eight has three, an eighth, a quarter and a half, that is one, two and four. But these likewise added up make seven, stopping below the number they are parts of; so they do not complete the number eight. The number nine has two such parts, a ninth, which is one, and a third, which is three; together these only come to four, a long way short of nine. The number ten has three such parts; one as its tenth, two its fifth, five its half; add them up, they reach eight, not ten. Eleven has only an eleventh, as seven only has a seventh, five only a fifth, three only a third, and two only a half; in all cases the number one.
Twelve, however, when its parts of this sort are added up, is not completed but exceeded; they make in fact a greater number than twelve, reaching as far as sixteen. It has, you see, five such parts; a twelfth, a sixth, a quarter, a third, a half.
I mean, its twelfth part is one, its sixth two, its quarter three, its third four, its half six; but one and two and three and four and six added up to a total make sixteen.
5. And not to waste time, in the infinite series of numbers a great many are found which only have one such part, like the numbers three and five, and the rest of that sort; or else they have several such parts which added up together either fall short of their total, like eight and nine and a great many others, or
6. This last phrase is not a translation, but only a rather desperate attempt to say roughly what he means by certis intervallorum rationibus pauciores; "fewer by definite reasons of intervals." The Latin certus means "certain" in English only in the sense of definite, sure, never in the vague sense of "there are certain things you must not do in polite society," or "it was with a certain sense of relief that we saw the last of him." The Latin here would be quidam.