Arithmetic, Geometry, and Music

the ternary; the fourth exceeds the fourth by the number of the quaternary: as 3 does the quaternary; and through the same sequence all pass one another by the plurality of the lesser. If, however, the third angle is observed, and starting from 9 it extends its length and breadth by thirty altrinsecus alternating/offset numbers, and this is compared with the first breadth and length, it arrives at the triple in its species of multiplicity, such that this comparison is made by the letter r. And these numbers will exceed one another according to the connection naturally made of parity. For the first exceeds the first by two: as 1 does 3; the second exceeds the second by a quaternary: as 2 does 6; the third exceeds the third by six: as 3 does 9. And the others increase according to the same mode of progression. Indeed, natural integrity itself has presented this to us, with us contriving nothing extraneous, as appears in the very module of the description.

If someone wishes to compare the limit of the fourth angle, which is marked by the quantity of the sixteen number, and which determines length and breadth in forty, to the things above by the form of the letter r, having collated the proportion, he will note the multitude of the quadruple. And for these, a progression is orderable over itself: so that the first exceeds the first by three: as 4 does unity. The second surpasses the second by six: as 8 does two. The third exceeds the third by nine: as 20 does three. And the following sums always leap over by an added quantity of three. And if one observes the lower angles, he will arrive at the tenth by the most ordered arrangement through all the species of multiplicity. But if one should seek the species of the superparticular a ratio of n+1 to n in this description, he will find it in such a manner. For if he notes the second angle, whose beginning is the quaternary, and its surface is the binary, and one fits the following in order, the sesqualter a ratio of 3:2 proportion will be declared. For the third is the sesqualter of the second: as 3 is to 2; or 6 to 4; or 9 to 6; or 12 to 8. Likewise, in the others that are in the same series of numbers, if such a conjugation is mixed, no dissimilarity of variety will creep in. Yet the exceeding of the sums in this is the same as it was in the doubles. For the first exceeds the first—that is, the ternary exceeds the binary—by one; the second exceeds the second by two; if the third, the third by three; and so forth. If, however, the fourth order is compared to the third, as 4 to 3, and you follow the others in the same order, the sesquitertial a ratio of 4:3 comparison is collected: as 4 to 3, or 8 to 6, and 12 to 9. Is it not seen how in all these the sesquitertial comparison is preserved? Furthermore, if you compare the following verses to each other for those that are under them, doing the same, you will recognize all the species of the superparticular without any impediment. But this is divine in this arrangement: that all the angular numbers are tetragonal square. A number is called tetragonal, to speak most briefly—which will be explained more broadly later—which two equal numbers multiply, as is also in this description. For one multiplied by once is one, and is by power tetragonal. Likewise, twice two are 4. Thrice 3 are 9. These are the multiplications into themselves that complete the first order. But around them—that is, around the angular ones—are the longilateri oblong numbers (n × n+1) numbers. I call those longilateri which numbers that exceed one another by one multiply. For around 4 are 2 and 6; but two are born from one and two when you have multiplied one by two; but unity is preceded by the binary by a unit. But six are from two and three; for twice three yield the senary. But 9 and 12 enclose the nonary. 9 and 12 are born from 3 and 4. For thrice 4 are 12. But the senary is from two and three. For twice 3 make 6. These are all generated by sides greater by one. For when 6 is born from the binary and the ternary, three exceeds the binary number by one, and all others are of the same mode: so that the first and second orders, with terms multiplied to one another, proceed such that that which is born from two longilateri placed altrisecus alternately, and twice the middle tetragonal, is a tetragonal; and again, that which is born from two tetragonals altrinsecus alternately, and one middle longilaterus made twice, is itself also a tetragonal. And that the angles of the whole description...

to the tetragonal angles placed, the first unit is of one angle, but 9 is opposite to the other. But the two altrinsecus alternating angles have the second units, and two angles of the tetragonals make equal what is contained under them to that which is made by one of those that is the altrinsecus angle. For there are many other things in this description that can be weighed as useful and admirable, which we allow to remain unknown for now for the sake of the brevity we are introducing. But now let us turn our proposed intent to the following.

On the third species of inequality which is called superpartiens a ratio of n+x to n (where x > 1); of its species and their generations. Chapter 28.

Therefore, after the first two relationships, the multiples and the superparticulars, and those that are under them, the submultiples and sub-superparticulars, the third species of inequality is found in order, which is called by us superpartiens. This is that which occurs when a number, compared to another, has the whole of it within itself, and over it some parts: either two, or 3, or 4, or however many the comparison brings. Which relationship begins from two third-parts. For if it has two halves, he who restricts that whole within himself is composed of a double plus a superpartiens. It will have either two thirds, or two fifths, or two sevenths, or two ninths. And as they progress, if it has only two parts of the lesser number over it, the greater sum transcends the lesser by these same parts in odd numbers. For if it has it whole and two of its fourths, it is necessarily found to be superparticular. For two fourths is a half, and a sesqualter comparison is made. If, however, two sixths, it is again superparticular. For two sixths is a third part. And if this is placed in comparison, it will effect the form of the sesquitertial relationship. After these are born the companions that are called sub-superpartientes, and these are they which are held by another number and two, or 3, or 4, or however many other parts of them. If, therefore, a number having another number within itself has two parts of it, it is named superbipartiens having 2/n parts over; if three, supertripartiens having 3/n parts over; if 4, superquadripartiens having 4/n parts over; and as they progress toward infinity, one may invent names. But their natural order is as often as all even and odd numbers are naturally set from three, and under these others are fitted which are all the odd numbers starting from five. These, therefore, being thus arranged, if the first is compared to the first, the second to the second, the third to the third, and the others to the others, the superbipartiens relationship is generated. Let the arrangement be in this mode:

If, therefore, the comparison of the quaternary number to the ternary is considered, it will be that superpartiens which is called superbipartiens. For the quaternary has three wholes in itself and two parts of them, that is, 2. If, however, the speculation is referred to the second order, the supertripartiens proportion will be known. And in the following, through all the arranged numbers, you will see all the species of this number that are suitable, ordered to infinity. But truly, in what manner individuals are generated, if someone cares to know to infinity, this is the mode: for the relationship of the superbipartiens, if it is doubled in both terms, is always generated by the proportion of a superbipartiens. For if one doubles 5, he will make 10; if three, he will make 6; which, 10 compared against the senary, make the superbipartiens relationship. And if you have doubled these same again, the same order of proportion increases. And if you do the same to infinity, it will not change the state of the prior relationship. But if you contend to find the supertripartientes, you will triple the first supertripartientes—that is, 7 is 4—and such things will be born. If, however, you produce those that are born from these by the multiplication of the ternary, they will effect the same again. But if you wish for those that are superquadripartientes, how they proceed to infinity...