Arithmetic, Geometry, and Music

the whole more than once, and with it either two, or three, or however many more parts referring to the fractional remainders in a ratio according to the figure of the superpartient a ratio exceeding the whole by more than one part number. In this also, because of the aforementioned reason, there will not be two halves, nor nine fourths, nor two sixths, but [rather] thirds, or two fifths, or sevenths, following the previous similar consequence. It is not difficult, however, according to the examples of the positions given earlier, to find these numbers beyond our examples. And they will be called, according to their own parts, double superpartient, or double supertripartient, or double superquadripartient. And again, triple superbipartient, and triple supertripartient, and triple superquadripartient, and similarly. For example, 8 compared to 3 makes a double superbipartient, and 16 to 6, and all five starting from 8 increase by the number eight, compared to those starting from 3 which increase by the quantity of the triad. Nor will it be difficult for those who are diligent to find its other parts according to the stated mode. Here also we must remember that the lesser and attendant [numbers] are named not without the preposition sub under, so that it may be sub-double superbipartient, [or] sub-double supertripartient.

Demonstration of how every inequality has proceeded from equality. Chapter 32.

t

It remains for us to hand down a very profound discipline which pertains with the greatest reason to the whole force of nature and the integrity of things. There is great fruit in this science: if one is not ignorant that goodness is defined and falls under knowledge, and is the first nature referring to the inherent goodness of original unity always lovable and perceptible to the mind, and perpetual in the beauty of its own substance. But to know the infinite is a disgrace, resting on no proper principles, but by nature fleeing from the definition of the good, it is composed only by some sign of an optimal figure from the beginning, and is retained from that fluctuation of error. For a pure intelligence, strengthened by the mind of the righteous, binds excessive greed and the immoderate unbridling of anger, as if by certain forms; and inequality, tempered by goodness, constitutes these things. This, however, will be clear if we understand that all species of inequality have grown from the beginnings of equality, so that equity itself, holding the force of a mother and root, pours forth all species of inequality in order. For let there be for us three equal terms: that is, three units, or three twos, or three threes, or three fours, or however many one wishes to place. For what happens in one set of three terms, the same occurs in the others. Therefore, in the order of our precept, you may see first that multiples are born, and among these, doubles first, then triples, then quadruples, and those following in the same order. Again, if multiples are converted, superparticular exceeding by one part ratios arise from them: and indeed from doubles, sesquialter one and a half; from triples, sesquitertius one and a third; from quadruples, sesquiquartus one and a fourth; and others in this mode. But it is necessary that superpartients are born from converted superparticulars. So that from a sesquialter, a superbipartient is born; the sesquitertius signifies the supertripartient; and from a sesquiquartus, the superquadripartient. But if the prior superparticulars are placed straight and not converted, multiple superparticulars arise. If the superpartients are straight, they will make multiple superpartients. These three precepts are: first, you make the first number equal to the first; second, to the first and the second; third, to the first, two seconds, and the third. Therefore, when you have done this in equal terms, they will be double those that are born. If you do the same from those doubles, triples are created, and from these, quadruples. And so it will explain all forms of multiple numbers to infinity. Let there be, therefore, three equal terms.

Thus, first, the first equal is born, that is, one.
p

The second is equal to the first and the second, that is, 2. The third, however, should be equal to the first, two seconds, and the third: that is, one and two ones and one, which are 4, as in the description.

So that the following order may be woven by a double proposition. Do the same again concerning the doubles, so that the first is equal to the first, that is, one. The second is the first and the second, that is, one and two, which are 3. The third is the first, that is, one, two seconds, that is, 4, and the third, that is, four. Which together are 9. And this form comes about.

Again, if you do the same from the triples, the quadruple will immediately be created. For if the first is equal to the first, that is, 1; let the second be equal to the first and second, that is, 4; let the third be equal to the first, two seconds, and the third, that is, 16.

c

In the others, indeed, we will use these three precepts for this form. But if we arrange those multiples that were born from equals, and we turn them according to these precepts, so that they are converted in order, the sesquialter will be created from the doubles; the sesquitertius from the triple; the sesquiquartus from the quadruple. For let there be three double terms that were created from equals, and let the last be placed as the first, of this kind.

And it is established in this order: the first is equal, that is, 4; the second is equal to the first and the second, that is, 6; the third is equal to the first, two seconds, and the third, that is, 9.

Behold for you that sesquialter quantity arises from the term of doubleness. Let us now see in the same mode what is born from the triple. For let the triple terms be arranged, converted of course in order, if they are double, this is also the order arranged.

Therefore, let the first be equal to the first, that is, 9; the second to the first and the second, that is, 12; the third equal to the first, two seconds, and the third, that is, 16.

Again, the second species of superparticular number, that is, sesquitertia, is created. But if anyone wishes to do the same from the quadruple, the sesquiquartus will be born immediately, as the following description will show.

And if anyone does the same concerning all parts multiplied to infinity, they will conveniently find the order of superparticularity. But if one converts the converted superparticulars according to these precepts, one may immediately see superpartients increase. And from the sesquialter, indeed, the superbipartient; from the sesquitertius, the supertripartient is created; and the others will be born according to the common species of denomination without any interpolation of order. Let them be arranged, therefore, thus.

Therefore, in the first [row] of the upper description, let the first equal number be ascribed, that is, 9; the second, the first and the second, that is, 15; the third, the first, two seconds, and the third, that is, 25.

If, therefore, I turn the sesquitertius in the same way, the order of supertripartient is found. For the first proposition of the sesquitertius is made.