On Paradoxical Machines

20 But indeed, let there be three numbers, A, B, and C, and let their subtraction be less than the dyadThe number two but greater than the sub-dyad. Let these be the smallest values of the stated ratio. And let the base-numbersoriginal: "pythmenes"; in ancient arithmetic, these are the single-digit "roots" of a number (1–9) stripped of their power of ten (e.g., the base of 500 is 5). of A, B, and C be H, Th, and K. Suppose that A, B, and C are even, and the base-numbers H, Th, and K correspond to A, B, and C.

A diagram depicts three vertical parallel lines of equal height. Above the lines are the letters K, Th, H (representing 10, 9, and 8). Below the lines are the letters G, B, A (3, 2, 1). Each vertical line has a small Greek letter (gamma, beta, alpha) in its center. Two horizontal connecting brackets bridge the gaps between the lines, labeled with the letters alpha and beta to indicate the ratios or intervals between the columns.

Since, therefore, the stated numbers A, B, and C are equal to the numbers I have mentioned for the sake of the ratio—H for A, Th for B, and K for C—and since C is 6, B is 6, and A is 6, and the base-numbers are H, Th, and K, the even number is equal to five hundred. And again, regarding the base-numbers H, Th, and K, their values for A, B, and C become one thousand unitsoriginal: "monades"; that is, the number A becomes five myriadsoriginal: "myriados"; a unit of 10,000. Five myriads equals 50,000. and 6. :~

A similar diagram follows with three vertical lines. The top labels remain K, Th, H, and the bottom labels are G, B, A. Greek letters alpha, beta, and gamma are placed inside the lines to denote segments. Horizontal markings indicate the specific proportions being calculated between the numerical lines.

21 But indeed, let there be more than three, namely A, B, C, and D. And let the smaller be greater than the dyad, and the larger be greater than the sub-dyad. Let the smaller and the larger of the sub-dyad be the total quantity.

This diagram features four vertical parallel lines representing the four variables. Above them are the letters L, K, Th, H (30, 10, 9, 8). Below them are D, G, B, A (4, 3, 2, 1). The diagram includes horizontal connectors and annotations such as "2nd bearer" and "thousands" near the base of the lines. Small Greek letters designate the proportional segments of the mathematical proof.

Of the numbers A, B, C, and D, let the first be measured according to the traditionoriginal: "paradosis"; likely referring to the established mathematical rules or "handing down" of the theorem. of O. And let the base-numbers of A and B be K, L, M, and N. Let the smaller A, B, C, D and Z, H, Th be a number equal to as many myriad-parts as there are units in the following. Let K, L, M, N be applied to the smaller Z, H, Th, and let it transmit the number of the named A—for the sake of the ratio, let I be for A, K for B, L for C, and M for D. And of the base-numbers K, L, M, N, let A, B, C, and D be the smaller number of the same, as 21. And the smaller A, B, C, D and Z, H, Th is again 141 myriads. And the smaller of the base-numbers K, L, M, N is 21. This, when applied to the smaller and having five myriads, the completed number of the smaller A, B, C, D and Z, H, Th is rendered, and in them it measures A, B, C, and D. :~