Arithmetic, Geometry, and Music

the heptagon, 15 is joined to the hexagon and the triangular ternary the number 3; and 34 is formed from the 29, which is the hexagon, and the triangular senary the number 6. And it is permitted to find this occurring without error in all cases. Do you see, therefore, how the first triangle creates all the sums, and is mixed into the generations of all things?

A reflection regarding the description of figured numbers. Chapter 19.

If, however, all of these are compared according to their breadth—that is, triangles to quadrilaterals squares, or quadrilaterals to pentagons, or pentagons to hexagons, or these again to heptagons—they will surpass one another without any doubt by the triangles. For if you compare a triangular ternary to a quaternary, or a quadrilateral quaternary to a pentagonal quinary, or a pentagonal quinary to a hexagonal senary, or a hexagonal senary to a heptagonal septenary: they surpass one another first by a triangle, that is, they increase by only a single unit. But if the senary is compared against the nonary 9, or this against 12, or this against 15, or the 5 pentagonal 5 against 18 for the sake of finding the differences, they will surpass one another by a second triangle, that is, by the ternary. If you compose the ten to 16, and 16 to 22, and 22 to 28, and 28 to 34, they will exceed one another by a third triangle, that is, by the senary. And this will be duly noted in all others following one another, as has been perceived, and all will be preceded by the triangles. Therefore, it has been demonstrated, I believe, perfectly that the principle and element of all forms is the triangle.

Regarding solid numbers. Chapter 20.

It is easier to move on to solid figures because, having previously understood what a number is in plane figures, and what the quantity itself naturally performs, there will be no hesitation regarding solid numbers. For just as we added another interval to the length of numbers—that is, a surface, as breadth was shown—so now, if one adds to the breadth what is called by some altitude, by others thickness, or depth, it will complete the solid body of the number.

That the pyramid is the principle of solid figures just as the triangle is of plane figures. Chapter 21.

It seems that just as the triangle is the first number in plane figures, so that which is called a pyramid is the principle of depth. For it is necessary in the numbers of figures to find the foundations. A pyramid is, at times, one that rises from a triangular base into altitude; at other times, from a tetragonal, or a pentagonal [base], and is raised according to the multitude of the following angles to a single vertex of the summit. For if a triangle is placed and arranged, and if individual straight lines are placed standing through the three angles, and these three are inclined so that they meet at one middle point for the vertices, a pyramid is made. When it has proceeded from a triangular base, it is enclosed by three triangles along the sides in this way. Let the triangle be a, b, c. If lines are raised through the three angles of this triangle and they incline toward one point, which is d, such that point d is not on the plane but suspended, those lines erected to that vertex will make a kind of summit d. And the base a, b, c will be one triangle, while through the sides there will be three triangles: namely, one triangle a, d, b, another b, d, c, and a third c, d, a.

A geometric diagram of a triangular pyramid (tetrahedron) is labeled with points a, b, and c at the base and d at the apex. Lines connect the apex to each corner of the triangular base.

Regarding those pyramids that proceed from squares or other multi-angled figures. Chapter 22.

Also, if it proceeds from a tetragonal base, and lines are directed to one vertex, it will be a pyramid of four triangles along the sides, with only one quadrilateral placed at the base, upon which the figure itself is founded. And if lines rise from a pentagon, the pyramid will be contained by five triangles again; if from a hexagon, by six triangles no less. And however many angles the figure upon which the pyramid rests may have, it is contained by that many triangles along its sides, as is evident in the descriptions placed below.

Four geometric diagrams of pyramids with different bases: a triangular base (tetrahedron), a square base, a pentagonal base, and a hexagonal base. Each shows lines converging from the base vertices to a single apex.

The generation of solid numbers. Chapter 23.

Pyramids of this kind are spoken of in this way. The first pyramid is from a triangle, the second pyramid from a quadrilateral, the third from a pentagon, the fourth from a hexagon, the fifth from a heptagon. The same holds true for other numbers. For when we spoke of linear numbers, which proceeded from one to infinity, as they are:

These, when arranged in order and joined to one another with distance, gave birth to surfaces. For if you joined one and two, the first triangle was born—that is, 3—and if you added to these a third, namely the ternary, a hexagonal triangle would occur again. And after these, quadrilaterals were born by skipping one, pentagons by skipping two, hexagons by three, heptagons by leaving four. But now, for the procreation of solid bodies, the naturally figured surfaces themselves will come to us. And to make pyramids from a triangle, triangles themselves must be composed by us. To prepare pyramids from a quadrilateral, quadrilaterals must be coupled, and to those that are from a pentagon, pentagons must be coupled; and those that are from a hexagon or heptagon will be born only by the coupling of hexagons or heptagons. Therefore, the first triangle by power is unity, and we shall also set this as the pyramid by virtue. The second triangle is the ternary, which if you join with the first—that is, with unity—a quaternary depth of the pyramid grows for me. But if you join the third, the senary, the denary 10 height of the pyramid will be provided. If you join the denary twice, a pyramid of 20 numbers will come. And thus, in all others, the same ratio of coupling exists.