Arithmetic, Geometry, and Music

& breadth. And that which retains these is a surface. This surface, however, exceeds the dimension of a solid body by one of the three intervals, and it in turn surpasses a line by one interval. That which retains the nature of length is devoid of breadth. That line, because it has been allotted one interval, is surpassed by the surface by one interval, and by solidity by two spaces. Therefore, a point is surpassed by a line by another interval, namely that which remains in length. Therefore, if a point is exceeded by a line by one interval, it is surpassed by a surface by two, and it is left behind by solidity by three dimensions of intervals. It is clear that the point itself, without any magnitude of body or dimension of interval, since it is devoid of all intervals—length, breadth, and depth—is the beginning of all intervals and is by nature indivisible, which the Greeks call atomon indivisible/atom, that is, so diminished and very small that no part of it can be found. Therefore, the point is the beginning of the first interval, but not yet an interval, and the head of the line, but not yet a line. Just as the line is also the beginning of the surface, but is not the surface itself, and is the head of the second interval, but does not itself retain the interval. The same also applies to the concept of the surface, which is the natural starting point of the solid body and the triple interval, yet it does not extend with the triple dimension of the interval, nor does it solidify with any thickness.

Regarding the linear number. Chapter 5.

Decorative initial S with floral motifs.
Just as also in number, the unit, though it is not itself a linear number, is the beginning of the number extended in length. And the linear number, although it is devoid of all breadth, is the starting point of the number extended into another space of breadth. The surface of numbers, although it is not itself a solid body, when added to height, is the head of a solid body. This will be clearer from these examples. The linear number is a heap of quantity beginning from two, with the unit always added, explained in one and the same directed way, as is what we have placed below.

Regarding plane rectilinear figures, and that the triangle is their principle.

Decorative initial P with floral motifs.
A plane surface is found in numbers whenever, having begun from three and added the breadth of the description, the angles are dilated by the multitude of natural numbers following. Thus, the first is the triangular number, the second is the square, the third is that which is contained under five angles, which the Greeks name the pentagon, the fourth is the hexagon, that is, which is enclosed by six angles, the fifth is the heptagon, the sixth is the octagon, that is, which are dilated by the limits of 7 or 8 angles. And the others in the same way increase the angles one by one through the natural number in the plane description of the figures. These, however, begin from the number three because the number three is the only principle of breadth and surface. In geometry, the same is also found more clearly. For two straight lines do not contain a space, and every triangular figure, or square, or pentagon, or hexagon, or whatever is contained by more angles, if lines are drawn from the center through each angle, they divide it into as many triangles as the figure itself happens to have angles. For the square is divided into 4 by such drawn lines, the pentagon into 5, the hexagon into 6, the heptagon into 7, and the others into the measure of their own angles through triangles, as in the subjoined description.

A square divided by two diagonal lines into four triangles meeting at a central point.

Decorative initial A.
But truly, the triangular figure, when one has divided it in such a way, is not resolved into other figures except into itself, for it is dissipated into three triangles.

An equilateral triangle divided into three smaller triangles by lines drawn from each vertex to the center point.

Therefore, this figure is the prince of breadth, so that all other surfaces are resolved into this one, but it itself, because it is not subject to any principles, nor did it take its beginning from another breadth, is resolved into itself. The same thing also happens in numbers, as the following order of the work will show.

Arrangement of triangular numbers. Chapter 7.

Decorative initial S with floral motifs.
If therefore the first triangular number, which dissipates into only three units, according to the position of the surface—namely in a triangular description—and after this, whoever separates the equality of sides into three spaces of sides.

Three triangular arrangements of dots (units). The first is a single dot labeled 1. The second is three dots (base 2) labeled 3. The third is six dots (base 3) labeled 6. Each dot is represented by a small vertical stroke 'I'.

Two triangular arrangements of dots. The first has ten dots (base 4) labeled 10. The second has fifteen dots (base 5) labeled 15.

Two triangular arrangements of dots. The first has twenty-one dots (base 6) labeled 21. The second has twenty-eight dots (base 7) labeled 28.

Regarding the sides of triangular numbers. Chapter 8.

Decorative initial A.
In this manner is the infinite progression: all equilateral triangles will be created in order, and first of all they will place that which is born from the unit, so that by its own power this is a triangle, not in operation and act. For if it is the mother of all numbers, whatever is found in these numbers which are born from it, it is necessary that it contain them by some natural power. And the side of this triangle is the unit. The ternary, however, which is the first triangle in operation and act, with the unit growing, will have the binary number as its side. For by the power of the first triangle, that is, the unit, the unit is the side. But by the act and operation of the first triangle, that is, the ternary, the duality, which the Greeks call the dyad, is the side. For the second triangle, however, which is second in operation and act, that is, the senary, with the natural number growing in the sides, the ternary is found. The third, that is, the denary, contains the quaternary as its side, and the fourth, that is, 15, holds the quinarius as its side, and the fifth the senary. And the same to infinity.

Regarding the generation of triangular numbers. Chapter 9.

Decorative initial N.
Triangles are born by the ordered natural quantity of numbers if the multitude of succeeding numbers is always gathered to the previous ones. For let it be arranged...