Arithmetic, Geometry, and Music

natural number in this manner.

From these, therefore, if you take the first number, that is, the unit, I have the first triangle, which exists in strength and potentiality, though not yet in act and operation. If I add to this the second number, which is described in the natural arrangement of numbers, that is, the binary, the first triangle is born to me in operation and act, that is, the ternary. But if you add to this the third number in the natural order, the second triangle is created for me in operation and act. For if I add the third, that is, the ternary, to the one and the two, the senary 6 is extended, which is clearly the second triangle. If you place the following quaternary 4 under this, the denary 10 is unfolded, which is the third triangle in act. By arranging these by sides according to the model of the description above, you will note all triangular numbers without any errors of doubt. And however many units the last number has, which you add to the ones above, that is how many units the triangle itself that is formed will have on its side. For we made the ternary, which is the first triangle in act, by adding the binary to the unit, and this has two on its side. And we produced the senary by adding the ternary amount twice, whose side contains only three; and so in all the others, however many units the number you add to the previous ones has, its sides will be contained by that many units.

Regarding square numbers. Chapter 10.

A square number is one that also extends into breadth, but not in three angles like the previous form, but in four. It is also extended with an equal dimension of sides. These are of this kind.

A diagram depicts four square arrangements composed of small black squares (units) representing the first four square numbers: 1, 4, 9, and 16. The squares are arranged in a grid format to demonstrate geometric area.

Regarding their sides. Chapter 11.

But in these, too, the augmentations of the sides increase according to the natural number. For the first square in strength and potentiality, that is, the unit, has one on its side. The second, which is the first in act, that is, 4, is contained by two placed along the sides. The third, that is, nine, which is the second in operation, is aggregated by three placed on the side. And all proceed according to the same sequence.

Regarding the generation of squares and again regarding their sides. Chapter 12.

Such numbers are born from the arrangement of the natural number, not in the way the triangles above are formed, where numbers are gathered by being ordered against each other; rather, one is always left out, and if the one that follows is joined with the previous one or ones, it will produce squares ordered from themselves. For let the natural number be arranged in this way.

From these, therefore, if you look at the one, the first square in nature and potentiality is mine. But if, leaving one, I join the third to the first, the second square is made for me. For if I leave out the binary and add the ternary, the quaternary square arises for me. But if again, leaving out the middle quaternary, I similarly add the quinary 5, the third square, that is, the novenary 9, is produced for me. For one and 3 and 5 gather into nine. But if you join the septenary 7 after passing over the senary 6 twice, the total grows into 16, that is, the quantity of the fourth square. And so that the appearance of this form of generation may be brief: if all odd numbers are added to themselves, with the natural number placed out, the order of squares is maintained. There is also in these this subtlety of nature and immovable order: each square will retain as many units on its side as there were numbers added to the joining of its own. For in the first

square, which is made from one, there is one on the side. In the second, that is, the quaternary, which is produced from one and three, which are two terms, the side is covered by the binary. And in the novenary, since it is produced by three numbers, the side is contained by the ternary. And the same can be seen in the others.

Regarding pentagons and their sides. Chapter 13.

A pentagonal number is one that extends itself into breadth and is contained by five angles described according to units, with all its sides clearly arranged with equal dimension. These are they.

From where also their sides increase in this way. For the first unit is to me the pentagon in potentiality, that is, the unit; that is, one holds the space of the side. But the second, the quinary, which is the first pentagon in act and operation, has two fixed along the sides. The third, that is, 12, is increased to three on the side. The fourth, 22, is extended by the quantity of 4 numbers on the side. And the same tends toward the others according to the progression of the unit in the natural number, clearly following the increments of the figures above.

A diagram depicts two pentagonal representations for the numbers 22 and 35. The diagrams use dots and connecting lines to show how pentagonal numbers are built up geometrically from a central point, with labels '22' and '35' above each.

Regarding the generation of pentagons. Chapter 14.

But these numbers, which extend five angles when erected in breadth, are born from the same quantity of the natural number gathered into itself, such that it adds to the one to which it is to be joined the number that exceeds the previous one or ones by three, always skipping two numbers. For if you join four to the unit after skipping two and three, the four itself exceeds the unit by three, and the pentagonal quinary will be produced. After 4, if you add 7 after skipping the quinary and the senary, you will produce the duodenary 12 pentagon. For the unit and 4 and 7 will complete the number 12. This will also happen in the others. For if you add 10, or 13, or 16, or 19, or 22, or 25 to all the previous ones, pentagons will be made in the same way as above, according to the description above.

Regarding hexagons and their generations. Chapter 15.

Hexagons, which are contained by six angles, and heptagons, which are contained by 7 sides, have their sides increase in this way. For in the generation of triangular numbers in nature, we joined those numbers that followed one another in natural order, passing from their end by a unit; the generation of square, that is, tetragon numbers, was made from numbers that were joined with one skipped, by which it exceeded by a binary; but pentagons were born from two interposed, leaving those that exceeded each other by a ternary. According to such augmentations, you will complete hexagons, or octagons, or figures of 9 sides, or 10, or any others with the same tempering. For just as in the pentagon we joined those that exceeded each other by a ternary after skipping two, so now in the hexagon we will join those that exceed each other by a quaternary after skipping three. And they will be their roots and foundations, from which, when joined, all hexagons are born.