On Paradoxical Machines

21 Since it has been demonstrated in the 21st theorem that a value less than a hundred [ἑκατοντάδος] but measured by a ten [δεκάδος] produced a hundred; it has also been demonstrated through the number derived from them the order of the unitsoriginal: "monades"; the basic building blocks of the number system. of the 32 base-numbersoriginal: "pythmens." In ancient Greek arithmetic, the 'pythmen' is the single-digit root of a larger number (e.g., 3 is the pythmen of 30 and 300)., where those of the 20 and those of the hundred are equal to the numbers 23, 4, 5, just as the hundred of number A B G D E are equal starting from a thousand to the 30, and from 500 numbers through the numbers of the 1st [alpha] together [being] 40, the 2nd [beta] 820, and the 3rd [gamma] 32 [and] 5, and the 4th [delta] 37 [and] 7, and the 5th [epsilon] 37 [and] 7 and those from the base-numbers?. Similarly, the number woven together after the 80, the [value] 1432 produced by a thousand, is also equal to the hundred numbers A B G D E: —

This vertical diagram consists of five parallel vertical lines, likely representing a calculation board or an abacus. Each line is capped with short horizontal bars. At the top of the lines, from right to left, are the Greek letters Alpha, Beta, Gamma, Delta, and Epsilon, acting as column headers. Along the lines are various letters representing numerical values or positions, such as Rho (100), Tau (300), Phi (500), and Omicron (70).

That which is in the 27th theorem regarding the proposition and demonstration therefore? holds? such a reputation? similarly; for instance, numbers less than a thousand but measured by a hundred, and? other? numbers. The second? is that which is from a hundred, each measured by a hundred, but measured by a ten and? others? again, the remainder? of the numbers, however many are less than a hundred but measured by a ten and? others? again for each A B G D E Z H Th? number. The order of each [is] after the A B G D E base-numbers of the 27th?. The double [value] from A B 6 [is] the number A E itself; it is measured by a tetradA group of four. of days?original: "hemeron." While this usually means 'days', in a mathematical context it may refer to a specific unit of measure or may be a scribal error for 'parts' (meron). just as? before by the tetrad according to the 24th [number] under? the arrangement? having? for the A B a hundred, and those such as G D E a ten they are?, and? the double therefore of them? with? the multitude? is? measured by a tetrad according to the 24th and they number? the hundred onto? the hundred 20, 5, 5, 7, 8, 9 just as? that from the 30, 10, 40, 20, 9? 60, 20? numbers is? the 500 to the? hundred A B G D E Z H Th? number. They say there are so many myriads [μυριάδων] similarly to the 20 and? the unit? which they are?.

A second diagram appears below the main text featuring four vertical lines. The Greek letters Theta (9), Kappa (20), Iota (10), and Mu (40) are positioned at the top and middle points of the lines, indicating specific values assigned to each column in this calculation.

8 all? accepted? A B G D E?

A third, simpler diagram at the bottom right consists of four short vertical lines labeled with the Greek letters Alpha, Beta, Gamma, and Delta (1, 2, 3, 4), likely representing the four-part "tetrad" structure mentioned in the text.

Theorem, Hundred, Ten, Thousand, Base-number, Number, Myriad, Order