Quadrivium Miscellany with Rithmomachia

...[be]hind the larger white ones, there are eight pieces. These are of the same color [white] but are of a more complex type than those mentioned above, being of the superpartient kind superpartientia; a complex ratio where the larger number contains the smaller plus more than one fractional part of the smaller, such as 5:3. Among these are the sesquiquartans (5:4) as in the fourths, the sesquiquintans (6:5), and then the sesquiseptans (8:7). There are also those described as "super-seven-partient," the sesquiseptimans (8:7), the sesquinonans (10:9), and those that are "super-five-partient."

If these are thus arranged and moved from the opposing side, they may capture all species of the multiple class The "multiple" class includes pieces with values like 2, 4, 8, etc. by moving forward, backward, to the right, to the left, or diagonally. [The movement rules are as follows]: the multiples move into the second space original: in eam punctum secundum; this means they move one square, landing on the second point; the superparticulars move into the third space; and the superpartients move into the fourth space.

And as we read in the philosophers likely referring to Boethius, the primary authority on medieval arithmetic, if a piece is moved and encounters a number, it may capture it. For instance, if the number of intervening spaces, when multiplied by the piece's own value, produces a sum equal to the opponent's piece, they may take it This is the "Capture by Comparison" or "Multiplication" rule in Rithmomachia.

Or if, by various means, they surround a piece on its corners or its sides with other pieces whose values—when either multiplied or added together—yield that same sum, they may take it This describes the "Siege" rule, where a piece is captured if the surrounding pieces' values mathematically equal its own.

Whichever number, in its legitimate move, encounters another piece of the exact same value, it captures it. In the section where all the proportions are named, the number 91 is placed on one side; this is the perfect pyramid pyramis perfecta; a special game piece representing a sum of squares, in this case: 1² + 2² + 3² + 4² + 5² + 6² = 91. If the number 36—which is the pyramid's base—fights in the opposing camp and encounters the pyramid through its legitimate moves, it captures not only the pyramid itself...