Arithmetic, Geometry, and Music

A small decorative woodcut initial 'I' marks the start of the first column.

In this conjunction, therefore, it is necessary that whoever is the last of the joined numbers is, as it were, a base. For it is found to be broader than all others, and it is necessary that the numbers joined before it be smaller, until the calculated subtraction arrives at unity, which holds the place of a vertex point in a certain way. For in the 10-pyramid, 3 and 1 are added above 6, as the senary exceeds the ternary quantity; these three themselves transcend one in plurality, and this one shows the extreme end of the progression. A similar ratio can also be seen in others if you wish to examine their procreation more diligently. Those that are from a quadrilateral, those pyramids, are born from the composition of quadrilaterals upon themselves. For when all quadrilaterals are described, that is:

If I take the first unity from this arrangement, the unity itself will be a pyramid by power, not yet by work and act. But if I place a quadrilateral upon this—that is, 4—a pyramid of five numbers will be born, which is contained by only two numbers placed along the sides. But if I add the following 9 to these, a pyramid form of 14 numbers will be made for me, which is concluded by three units along the sides. And if I place the following quadrilateral, 16, upon this, a pyramid form of 30 is produced for me. In all these pyramids, there will be as many units along the sides as there were quantities of numbers aggregated in them. For the unit, which is the first pyramid, carries only one—that is, itself—on the side. But five, which consists of one and 4, is designated by two along the sides, and 14, which is made from three composed numbers, is constituted by the ternary number placed on the side. The description below shows this generation of pyramids.

In the same manner, all forms proceeding from other multi-angled figures are produced into the spaces of a higher sum. For every form of many angles proceeds to infinity from the figure of its own kind placed upon unity, beginning from one to constitute the figures of pyramids. And from this, it must appear that triangular forms are the principle of the other figures, because every pyramid, from whatever base it is projected—whether from a square, or a pentagon, or a hexagon, or a heptagon, or from any similar [figure]—is contained only by triangles up to the vertex.

Regarding truncated pyramids. Chapter 24.

A small decorative woodcut initial 'S' marks the start of the chapter.

One must know which pyramids are truncated curte shortened/truncated, or twice-truncated, or thrice-truncated, or four times, and so on according to the addition of numbers. For a perfect pyramid is one that, proceeding from any base, arrives at the first unit by power of the pyramids. But if that altitude has not arrived at unity starting from any base, it is called truncated. A pyramid of this kind is rightly signified by such a naming if it has not arrived at the extremity and the point. This, however, is such that if one adds 9 to the 16-quadrilateral, and 4 to this, and suspends itself from the further addition of unity, it is indeed a figure of a pyramid, but because it has not grown to the final summit, it will be called truncated, and it will have a summit not as a point, which is unity, but a surface, which

is any number stretched according to the angles of that base, and the last aggregated number. For if the base were tetragonal, the square diminution always ascends; if the base is pentagonal, similarly; and if hexagonal, that last surface will also be hexagonal. Therefore, in a truncated pyramid, there will be as many angles of the surface as there are in the base. If, however, that pyramid has not only not arrived at unity and the extremity, but also not even to the first work and act of a multi-angled [figure] of the kind of which its base is, it will be called twice-truncated. As if, starting from the 16-quadrilateral, it ends at 9 and does not grow to 4, and however many quadrilaterals are missing, we say it is truncated that many times. Where unity is missing, the first square, it is a truncated [pyramid], which the Greeks call kaoluron truncated. But if it lacks two quadrilaterals—that is, the unity and the one that follows—it is called twice-truncated, which the Greeks call dikaoluron twice-truncated. But if it is missing three quadrilaterals, it is called thrice-truncated, which the Greeks name trikaolyron thrice-truncated; and however many quadrilaterals are lacking, we propose that the pyramid is truncated that many times. This can be observed not only in the pyramid of a quadrilateral, but in all those proceeding from a multi-angled base.

Regarding cubes, or planks, or bricks, or wedges, or spheres, and parallelepipeds. Chapter 25.

A small decorative woodcut initial 'L' marks the start of the chapter.

We have spoken of solids that obtain the form of a pyramid, increasing equally and proceeding from their own root as if from a multi-angled figure. There is again another orderable composition of solid bodies: those which are called cubes, or planks asseres beams/planks, or bricks laterculi small bricks/tiles, or wedges cunei wedges, or spheres, or parallelepipeds, which are [formed] as often as surfaces are against each other and, carried to infinity, will never concur. For with quadrilaterals arranged in order:

Since these have sorted out only length and breadth and lack altitude, if they receive only one multiplication through the sides, they will produce an equal depth. For the 4-quadrilateral has two on the side, and is born from two times two. For two times two make four. Therefore, if you multiply these two from its side equally, the form of a cube will be born. For if you multiply two by two twice, the quantity of eight grows, and this is the first cube. But the 9-quadrilateral, since it has 3 on the side, and is made from three multiplied into itself, if you add one multiplication of the side, another cube grows again with an equal formation of sides. For if you multiply three times three a third time, the figure of the 27-cube is produced. And 16, which is from 4, if it is increased four times, the 64-cube will be thickened by an equal dimension of sides. And the following quadrilaterals will be advanced according to the same mode of multiplication made. It is necessary that a cube has as many units on the side as the first quadrilateral from which it was produced had. For since the 4-quadrilateral has only two numbers on the side, the 8-cube also has two. And since the 9-quadrilateral was figured by three units per side, the side of the 27-cube is strengthened by the ternary alone. And since the 16-quadrilateral had a side of four units, the 64-cube will carry as many units on the side. Therefore, also by power and virtue, the cube that is unity will be one on the side. For every quadrilateral is one surface of four angles and as many sides. Every cube, however, which has grown from the surface of quadrilaterals into the depth of a body, multiplied through the side of the quadrilateral, will have 6 surfaces, the flatness of each of which is equal to that prior quadrilateral. The sides are 12, each of which is equal to those that were of the prior quadrilateral; and as we demonstrated above, it is of as many units. The angles are 8, each of which is contained under three of this kind, such as the prior quadrilaterals were from which the cube itself was produced. Therefore, from