Arithmetic, Geometry, and Music

are born: they were arranged in the first row. Again, those that were born by the five multiplying them are set in the second place. After these, those which the seven created by multiplying the others, we wrote down in the third place, and we performed the same in the rest of the description.

¶ In the following formula, the similarity of the evenly even and the evenly odd to the unevenly even is shown.

A complex mathematical diagram consisting of four overlapping ellipses or circles inscribed within a circular frame, which is itself placed inside a rectangular border. The figure is used to illustrate Boethian arithmetic, specifically relationships between evenly even pariter paris and other number types.

The central area is a 4x4 grid containing the following numbers:
Row 1: .12. | 24. | 48. | 96.
Row 2: .20. | 40. | 80. | 160.
Row 3: .28. | 56. | 112. | 224.
Row 4: .36. | 72. | 144. | 288.

Between the central grid and the outer rings, the labels "half" medietas are placed. The horizontal axis is labeled "Length" Longitudo (split as Longi/tudo) and the vertical axis "Width" Latitudo (split as Lati/tudo).

In the outer segments of the circles, large product numbers are displayed:
Top: 576, 2304.
Sides: 1600, 3200, 6400.
The bottom half of the diagram mirrors the top but the numbers and labels (Longi / tudo, medietas) are printed upside-down. These inverted numbers include: 3136, 12544, 20736, 9216, 10368, and 1536.

¶ An exposition of the description pertaining to the natures of the unevenly even in width and the evenly even in length. Chapter 12.

The reasoning for the description digested above is this. If you look at the width, where there is one half of two terms, and you join those same terms, you will find them to be double their own half, as 36 and 20 make 56, whose half is 28, which is established as the middle term between them. And again, if you join 28 and 12, they make 40, of which 20 is the half, found in the middle of those terms. But where they have two halves, both extremities joined become equal to both halves, as when you have joined 12 and 36, they become 48; if you apply the halves to these, that is, 20 and 28, it will be the same. And in the other part of the width, the numbers that are made are noted in the same order. And this iteration does not differ from either width, and you will note the same in the same order among the other numbers; and this happens according to the form of the evenly odd number, in which this property is held as was already written above. Again,

if you look to the length, where two terms have one half, which is done from the multiplied extremities, this happens if the middle term takes the augmentations of its plurality. For twelve times 48 makes 576. But if the middle term of them, that is 24, is multiplied by the same again, it will create 576. And again, if you multiply 24 by 96, they make 2304. If the middle term of these, that is 48, is led into itself, the same 2304 will be created. But where two terms enclose two halves, which is done by multiplying the extremities, this same result is yielded if they are led into either sum of the halves. For twelve times 96 multiplied creates 1152; but the two halves of them, that is 24 and 48, if multiplied into themselves, will restore the same 1152. And this is by the imitation and relationship of the evenly even number, from which this participation is extracted and recognized as a property in the generated number. And on the other side of the length, the same reasoning and description is noted. Wherefore it is manifest that this number was created from the former two, since it retains their properties.