Arithmetic, Geometry, and Music

eFollowing the same order: and from these the forms of hexagons are born.

qYou will note these described according to the mode above, clearly for the preceding orders.

Regarding heptagons and their generations: and a common rule for finding the generation of all figures and the description of the figures. Chapter 16.

sThe figure of seven angles is formed when you join to the superior one in the same order of progression by skipping one more number than in the figure of 6 angles. For if you add by interposing four, which exceed each other by five, the figure of the heptagon will be born immediately. Let these numbers be their roots and, as has been said above, foundations.

nThe form of angles produces according to the same order, such that the first numbers of them also differ according to an equal progression. For in the triangular numbers, which is the first figure of surface, the numbers proceed by only one, which clearly complete their nature and description. In the tetragon, however, which is the second, the numbers joined exceed each other by two, and in the pentagon by three, and in the hexagon by 4, and in the heptagon by 5, and there is no limit to this matter. But the descriptions of the forms placed below will teach us.

A diagram depicts a series of four nested triangles illustrating triangular numbers 1, 3, 6, and 10. Each triangle contains dots representing the units, with labels above indicating the summation (e.g., 1.2.3.4. for 10).

A diagram depicts a series of three nested squares illustrating square numbers 1, 4, 9, and 16. The squares are divided into smaller unit squares with dots, labeled with summations like 1.3.5.7. for 16.

A diagram depicts a series of nested pentagons illustrating pentagonal numbers 1, 5, 12, and 22. The diagrams show how each larger pentagon is built upon the previous one by adding layers of dots.

A diagram depicts a series of nested hexagons illustrating hexagonal numbers 1, 6, 15, and 28. The diagrams use dots and connecting lines to show the geometric progression of the figures.

Description of figured numbers in order. Chapter 17.

sSimilarly, it will be permitted to form others that are contained by more angles and to ascribe the quantities. But because eyes retain what is placed before them more easily, the numerosity of the aforementioned forms is placed in the description below.

Which figured numbers are made from which figured numbers; and that the triangular number is, as it were, the beginning of all the rest. Chapter 18.

bThese things therefore being thus, we shall investigate what follows in this matter. For all tetragons, which are arranged below the triangles in natural order, are produced from the triangles above them: and by their collection the figure of the square is proposed. For the quaternary 4 square is made from one and three, that is, from the two triangles above it. Nine is made from three and 6, but both are triangles. Likewise 16 from 6 and 10, and 25 from 10 and 15. And the same is found constant and immutable in the following order of squares. But the sums of pentagons are completed from one tetragon above itself and another triangle established second. For the pentagon 5 is aggregated from the tetragon placed above it and from the one which is placed in the order of triangles. But the pentagon 12 is born from the novenary 9 square above it and the ternary 3 second triangle. Twenty-two, however, from 16 and 9, clearly the square and the triangle, and 35 from 25 and 10, and in order in the same way, no hesitation of contrariety will hinder one looking at it. But if you observe the hexagons with balanced examination, they are produced from the same triangles and the pentagons placed above them. For the hexagon 6 is born from the pentagon 5 and the one which is arranged in the order of triangles. Nor is there any other origin for the hexagon 15 than from the pentagon 12 and the triangle 3. But if you learn from which superiors the hexagon 28 is again born, you will find none except the pentagon 22 and the triangle 6. And this in the others. Nor will the creation of heptagons refute this order of generation. For they are produced from the hexagons above them and the triangles placed beyond them. For the heptagon 7 is born from the hexagon 6 and the one triangle placed before it. 18, however...