On the Institution of Arithmetic

not commonly or confusedly. For the prime number measures that which is placed after two in accordance with its own quantity. For the number three measures the nine in the third place. But if, after the nine, I leave two out, that which occurs to me after them must be measured by the first by means of the quantity of the second odd number, that is, by the five. For if after 9 you leave two out, that is, 11 and 13, the number three measures 15 by the quantity of the second number, that is, by the five, because three measures 15 five times. Again, if starting from 15 I intermit two, the first number that is placed behind is its measure by the plurality of the third odd number. For if after 15 I intermit 17 and 19, there occurs 21, which the number three measures in accordance with the seven. For three is the seventh part of the number 21. And doing this to infinity, I find the first number: if I intermit two, it measures all that follow after it in accordance with the quantity of the ordered odd numbers. But if the number five, which is established in the second place, wishes for one to find what its first and subsequent measure is, having passed over four odd numbers, the fifth occurs to him which it can measure. For let four odd numbers be intermitted, that is, 7, 9, 11, and 13; after these is the fifteenth, which the five measures, according to the quantity of the first, that is, the three. For five measures 15 three times. And subsequently, if it intermits four, it measures the one that is placed after them, the second, that is, the five, by its own quantity. For after 15, having intermitted 17, 19, 21, and 23, after them

25: I find those which the five, or number, measures by its own plurality. For five times five multiplied equals 25. If, however, after this one any four intermit, the same consistency of order being kept, that which follows them will be measured by the five in accordance with the sum of the third number, that is, the seven. And this is an infinite process. But if the third number, which it can measure, is sought, six are left in the middle, and that which the order shows as the seventh is to be measured by the quantity of the first number, that is, the three. And after that, six others being interposed, that which the series of numbers gives after them will be measured by the five, that is, the second of the third. But if someone again leaves six in the middle, the one that follows is to be measured by the seven, from that same seven, that is, by the quantity of the third. And this order proceeds correctly even to the end. Therefore, they will receive the vicissitude of measuring, just as they are naturally established in the order of odd numbers. But they will measure if they pass through even numbers, beginning from the two, intermitted by a fixed interval between themselves, as the first two, the second four, the third six, the fourth eight, the fifth ten. Or if they duplicate their places and intermit terms according to the duplication, as the three, which is the first number, and one (for every first is one) duplicates its place, and makes two times one. Since these are two, it passes over the two middle ones. Again, the second, that is, the five, if it duplicates its place, will explain four; it also intermits four. Likewise, if the seven, which is the third, duplicates its place, it will create six. For twice