Arithmetic, Geometry, and Music

Let the first be placed according to the previous mode, equal to the first, that is, 16; the second to the first and the second, that is, 28; the third to the first, two seconds, and the third, that is, 49. Therefore, every sum arranged will effect supertripartients.

Again, if you convert the sesquiquartus in the same way, the superquadripartient quantity will immediately be procreated, as is that form which you see placed below.

a

It remains to show how multiple superpartients are born from superparticulars and superpartients. I shall make only two descriptions of these. For if we place the sesquialter straight, not converted, a double superparticular grows; let it be in this way.

Let it be placed according to the superior mode: the first equal to the first, that is, 4; the second to the first and the second, that is, 10; the third equal to the first, two seconds, and the third, that is, 25.

a

This law, indeed, has produced a double sesquialter sum. But if we place the sesquitertius not converted, a double sesquitertius is found, as the subject description teaches.

a

But if we turn our mind to the superpartients, and arrange them in order according to the superior precepts, we direct the ordered multiple superpartients that have been born. For let this formula of superpartient be arranged.

a

s

Therefore, let the first be written equal to the first, that is, 9. The second to the first and the second, that is, 24. The third to the first, two seconds, and the third, that is, 64.

n

The same, so that from a superpartient, a double superpartient has arisen. But indeed, if we place the supertripartient, a double supertripartient is found, as is clear in the subject description.

g

Thus, therefore, multiple superparticulars or multiple superpartients arise from superparticulars or superpartients. Wherefore it is evident that equality is the principle of all inequalities. For all unequal things are born from the same [equality]. And I believed that we must discourse about these things thus far, lest we pursue things that are infinite, or, by detaining the minds of those entering [this study] with most obscure matters, we be delayed from useful things.

The chapters of the second book begin.
How every inequality is reduced to equality. Chapter 1.
On finding in each number how many numbers of the same proportion may precede it, and their description and the exposition of the description. Chapter 2.
That a multiple interval is made from which superparticulars [are formed] by placing a mean between the intervals, and the rule for finding it. Chapter 3.
On quantity constant in itself, which is considered in geometric figures, the total ratio of all magnitudes. Chapter 4.
On linear number. Chapter 5.
On plane rectilinear figures, and that the triangle is the principle of them. Chapter 6.

Arrangement of triangular numbers. Chapter 7.
On the sides of triangular numbers. Chapter 8.
On the generation of triangular numbers. Chapter 9.
On square numbers. Chapter 10.
On their sides. Chapter 11.
On the generation of square numbers and again on their sides. Chapter 12.
On pentagons and their sides. Chapter 13.
On the generation of pentagons. Chapter 14.
On hexagons and their generations. Chapter 15.
On heptagons and their generations, and the common rule for finding the generation of all figures and the description of the figures. Chapter 16.
Description of figurate numbers in order. Chapter 17.
Which figurate numbers are made from which figurate numbers, and that the triangular number is the principle of all the rest. Chapter 18.
Speculation pertaining to the description of figurate numbers. Chapter 19.
On solid numbers. Chapter 20.
On the pyramid, that it is the principle of solid figures just as the triangle is of plane ones. Chapter 21.
On those pyramids which proceed from squares or other multi-angled figures. Chapter 22.
Generation of solid numbers. Chapter 23.
On truncated pyramids. Chapter 24.
On cubic, or oblong, or side-related, or wedge-shaped, or spherical, or parallelepiped numbers. Chapter 25.
On numbers longer by one part referring to numbers where one side is larger than the other by a unit and their generations. Chapter 26.
On antelonger numbers, and on the name of the number longer by one part. Chapter 27.
That squares are made from odd numbers, and numbers longer by one part are made from even numbers. Chapter 28.
On the generation of their sides and their definition. Chapter 29.
On circular or spherical numbers. Chapter 30.
On that nature of things which is said to be of the same nature, and that which is said to be of another nature, and which numbers are born to which points. Chapter 31.
That all things consist of the same nature and another nature, and that this is seen first in numbers. Chapter 32.
That all conditions of proportions consist of the nature of the same and another number, which are the square and the one longer by one part. Chapter 33.
That every ratio of forms consists of squares and numbers longer by one part. Chapter 34.
How squares are made from numbers longer by one part, or numbers longer by one part from squares. Chapter 35.
That, principally, the unit is indeed of the same nature as substance; in the second place, even numbers; in the third, squares. And that quality is principally of another substance; in the second place, even numbers; in the third, numbers longer by one part. Chapter 36.
With squares and numbers longer by one part placed alternately, what is their agreement in differences and in proportions. Chapter 37.
Proof that squares are of the same nature. Chapter 38.
That cubes participate in the same substance because they are born from odd numbers. Chapter 39.
On proportions. Chapter 40.
What proportionality was among the ancients, and which ones the later [authors] added. Chapter 41.
That one must first speak of that which is called arithmetic proportionality. Chapter 42.
On arithmetic mean and its properties. Chapter 43.
On geometric mean and its properties. Chapter 44.
Which mean is compared to which states of public affairs. Chapter 45.
That one surface [mean] is joined in proportionalities by one mean, but solid numbers are placed in the middle by two means. Chapter 46.
On harmonic mean and its properties. Chapter 47.
Why the mean which is digested is called harmonic. Chapter 48.
On geometric harmony. Chapter 49.
How, with two terms constituted externally, arithmetic...