On the Institution of Arithmetic

incomposite.

On the prime and incomposite. Chap. 14.

A woodcut depicts a decorative, ornate initial 'E' featuring scrolling vines and floral patterns.

The prime and incomposite is that which has no other parts except that which is denominated by the total quantity of the number, so that its part is nothing but the unit, as are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. In these, therefore, no other part will ever be found in each, except that which is denominated by them, and that part is only the unit, as has already been said above. For in three, there is one part only, that is, the third, which is denominated from three, and that third part is the unit. In the same way, the fifth part of the quinary is the unit, and the same will be found in each subsequent one. It is called prime and incomposite because no other number measures it, except the unit alone, which is the mother of all. For 2 does not number the ternary, because if you compare only two against three, they are fewer. But if you make the binary twice, it grows into 3, which is greater than three. A number measures a number as often as, whether once or twice or three times or as often as you like, the number is compared to a number and reaches the term of the compared number without the sum being diminished or increased. For example, if you compare two to 6, the binary number measures the senary three times. Therefore, no number measures the prime and incomposite numbers except the unit alone, because they are not composed from any other numbers but are created only from units increased and multiplied in themselves. For three times one made 3; seven times five made... original: "Ter eni un9 . 3 . 7 qnqes unus qnqʒ . 7 septies un9 . 7 . fecerunt" - The text provides a list of prime generation, though the Latin OCR contains some arithmetic discrepancies regarding the numbers 5 and 7.. And others whom we described above are born in the same way. These, however, multiplied into themselves make other numbers, just like the prime ones, and since they have always kept their primary substance, you will find them to be like certain elements of all numbers procreated by them. Because they are prime and incomposite, and formed by a simple generation, all numbers whatever that are generated from them are resolved into them, but they themselves are neither produced from others nor are they reduced into others.

On the secondary and composite. Chap. 15.

A woodcut depicts a decorative, ornate initial 'S' featuring scrolling vines and foliage motifs.

The secondary and composite, however, is also odd, because it is formed by the same property of the odd, but it retains no principal substance in itself and is composed from other numbers. And it has parts both denominated by itself and by an alien term. But you will find only the unit as the part denominated by itself in these. By an alien term, however, one or however many others there are, depending on how many numbers that composite number is procreated from. As are these: 9, 15, 21, 25, 27, 33, 39. Therefore, each of these has its own proper parts denominated by itself, namely the units; as 9 has the ninth, that is 1; 15 has the fifteenth, the same unit again. And in the others which we described above, the same is agreed. They also have parts from an alien term; as 9 has the third, that is the ternary; and 15 has the third, that is 5, and the fifth, that is 3; 21 has the third, that is 7, and the seventh, that is 3; and in all others the same sequence follows. This number is called secondary because it is not measured by the unit alone, but also by another number from which it is joined. Nor does it have in itself any principal understanding.