Arithmetic, Geometry, and Music (1492) - Page 15

they increase. But if after this one, any person skips 4: with the same consistency of order preserved: the second that follows them will measure the third, that is, the sum of the number seven, by the number five. And this is an infinite progression. If, however, a third number is sought that it might be able to measure: six are left in the middle: and the one the order shows as the seventh: this is to be measured by the quantity of the first number, that is, the three. And after that one, with six others interposed: the one that the series of numbers will give after them: is to be measured by the five, that is, by the seven, from that same seven: that is, by the quantity of the third. And this established order proceeds thus to the end. They will therefore look up to the alternation of measuring: just as they are naturally established in the order of the odd. But these will measure through even numbers starting from two, positioned among themselves in such a way that they skip over the odd with a fixed interval: so that the first skips two: the second 4: the third 6: the fourth 8: the fifth 10. Or if they duplicate their places and skip terms according to the duplication, as the three which is the first number and one. (For twice the first is one) he may multiply his place twice: and make twice one. Who, since they are two: the first skips over two in the middle. Again the second, that is the five: if he duplicates his place he will explain 4: this one also skips 4. Likewise if the seven which is the third duplicates his place: 8 will increase. That one also skips 8. And this indeed is to be observed in the others. But the series itself will give the mode of measurement according to the order of those placed. For the first measures the first whom he counts: he counts the second first, that is, the second by himself. And the second measures the first whom he counts: by the second he counts. And the third by the third and the fourth likewise by the fourth. When the second has undertaken the measurement, he measures the first whom he counts by the second first. But the second whom he counts, by himself. That is, by the second and the third by the third. And in the others, the measure will consist in the same similarity. Therefore, if you look at others: or those that have measured others: or those that are themselves measured by others: you will find that there cannot be a common measure for all at the same time. Nor that all count any other at the same time. However, some of these can be measured by another, so that they are counted by only one. Others indeed so that they are counted even by many. Others, however, so that besides unity there is no measure for them. Those therefore that receive no measure besides unity: we judge these to be first primi prime and incomposite incompositi indivisible. Those indeed which are allotted the name of a foreign part in addition to a measure besides unity: we shall pronounce them second secundi and composite compositi. That third genus of those that are second and composite in themselves, but first and incomposite when compared to one another, this method of wits will find. For if you multiply any of those numbers according to the quantity of their own multiples in themselves: those that are created when compared to one another are joined by no communion of measure. For if you multiply three and 5: three times three make 9: and five times 5 will return 25. Therefore there is no kinship of common measure for these. Again, if you compare the 5 and 7 that they create: these also will be incommensurable. For five times five as it is said is 25, seven times 7 make 49. Of which there is no common measure, unless perhaps unity, the creator and mother of all these.

Concerning the discovery of those numbers which are second and composite to themselves, but first and incomposite when related to others.

Chapter 18.

But by what reason we might be able to find such numbers: if anyone proposes them to us and asks to recognize whether they are commensurable by any measure: or certainly if only unity measures both: the art of finding them is such. For with two unequal numbers given: it is necessary to remove the smaller from the larger. And the one that remains: if it is larger: to remove the smaller from it again: but if it is smaller: when it is removed from the remaining larger one. And this

must be done, naturally, until unity impedes the final act of retraction: or some odd number necessarily, if two odd numbers are proposed. But if you see the number that remains is equal. Therefore, if this subtraction in turn happens upon one: the numbers are necessarily first against each other and contain no other measure except unity alone. But if the end of the reduction has occurred to some number as was said above: that number will be the one that measures both sums. And we shall say that the same one which remained is the common measure for both. For let us have two proposed numbers: which we are commanded to recognize: whether any common measure measures them. And let these be 9 and 29. Therefore we shall do this by the method of reciprocal diminution. We shall take the smaller from the larger: that is: from 29 the nine will be taken, 20 will remain. From these 20, therefore, we shall again remove the smaller: that is 9, and 11 will remain. From these I again subtract 9: 2 are left. Which if I subtract from the nine: 7 are left. But if two are taken again from the seven: 5 remain. And from these the other two: three again exceed. Which, being diminished by the other two, only unity survives and emerges. Again if I take one from two: the term of subtraction will stick at one: which it is established is the only measure, and no other, of those two numbers, that is 9 and 29. We shall therefore call these first against each other. But let other numbers be proposed to us with the same condition: that is 21 and 9: so that we may investigate what kind these are when they have been compared to each other. Again I remove the quantity of the smaller number from the larger: that is: 9 from 21, 12 will remain. From these I again remove 9: 3 remain. Which if they are withdrawn from the nine: 6 will remain. From which, if one removes the three: 3 will remain. From which three cannot be removed. And this is equal to itself. For the 3 which were being removed reached the number three. From which, since they are equal: they cannot be removed or diminished. We shall therefore pronounce these commensurable, and the three that is left is their common measure.

Another division of even numbers according to perfect, imperfect, and beyond perfect.

Chapter 19.

The brevity allowed by the introduction has explained as much as it could concerning odd numbers. Again, the second division of even numbers is made thus. For some of them are superfluous superflui abundant, others are diminished diminuti deficient, according to both conditions of inequality. For in these inequality is considered in either the larger or the smaller. For those, in a certain immoderate fullness, precede the measure of their own body by the number of their parts. Those, however, as if oppressed by poverty and needy, a certain lack of their nature makes the sum of their parts less than they themselves are. And indeed those whose parts have extended themselves beyond what is enough: are named superfluous: as are 12 or 24. For where they are compared to their parts, they are larger than the sum of the parts sorted out by the whole body. For the half of twelve is 6, the third part is 4, the fourth part is 3, the sixth part is 2, the twelfth part is 1. With these this accumulation overflows to 16, and they exceed the multitude of their whole body. Again, the half of the number 24 is 12, the third is 8, the fourth 6, the sixth 4, the eighth three, the twelfth 2, the twenty-fourth one, which all together repay thirty-six. In which matter it is manifest that the sum of the parts is larger: and overflows beyond its own body. And this indeed, because the composite parts exceed the sum of the whole number: is called superfluous. But that is diminished whose composite parts are exceeded in the same way by the multitude of the whole term: as 8 or 14. For eight has a half part: that is 4, and it has a fourth: that is two. It has an eighth: that is one, which all brought into one collect to 7: containing a sum smaller than the whole body. Again 14 has a half: that is seven. It has a seventh: that is 2, it has a fourteenth: that is 1, which if they are collected into one: the sum of the number ten will be created, that is, the whole term.