On Paradoxical Machines

A mathematical diagram at the top right represents numerical scales or place values. It consists of three horizontal rows of Greek alphabetic numerals connected by vertical lines, illustrating the relationship between units (1–8), tens (10–80), and a second scale of tens (20–80). The vertical lines suggest a method for multiplication or alignment of these values across different orders of magnitude.

22 Since the multitude of the A-B sides is measured by the tetradThe number four; the author is likely referring to a four-sided figure or a calculation grouped by fours., as was once shown by us to be the most excellent arrangement, the number to be placed from the same starting point will be a number of myriadsA myriad is 10,000. having the same name as those which are whole. And if the number seeks the complete [value], the resulting [sum] is universaloriginal: "oletikos"; likely referring to a totalizing sum in the calculation.. The total is whole throughout, and the number produced for us is a stout sum of four wholes. And if it results in thousands upon the complete [value], such a thing will be homonymous with the myriads. The linear [measurement] of the universal element is through the whole: —

From the 9 [$θ$], being measured by a thousand of hundreds, there remains less than a hundred. Of the 583 [$επγ$], from a mega-myriadoriginal: "hekatommyrias"; 100,000,000 or simply a very large power of ten. and two parts of the hundreds of numbers, it will be written below; for while the 9 [$θ$] is set before the 8 [$η$], and the wholes are 5, 2, 3, 4, 10, 6 [$εβγδιζ$] universal [units] 1, 2, 3, 4 [$αβγδ$], the number is revealed a hundredfold from the 7 [$ζ$]. And this is because of the number of the 1 [$α$] placed below: 502, 4, 300, 800, 20, 30, 8, 300, 7 [$φβδτωκλ\bar{η}τ\overline{ζ}$].

The [product] of 1 [$α$] by 3 [$γ$], and 3 [$γ$] by 4 [$δ$], and 4 [$δ$] by 1 [$α$]—for the [sum] produced by 1, 2, 3, 4 [$αβγδ$] is 40 [$μ$], while that produced by 5, 6, 7, 8 [$εζηθ$] is 1,280 [$κρπ$]The OCR reading "κρπ" is non-standard; in standard Greek notation, 1,280 would be ,ασπ. The author may be using a shorthand for 20 + 100 + 80 or a specific divisional notation.. This, when increased a hundredfold, becomes 1,140 [$αρμ$] linear measurement from the element: —

A second diagram on the middle right shows four vertical lines. The top ends are labeled with the variables A, B, G, D (1, 2, 3, 4) and the bottom ends are labeled E, Z, H, Th (5, 6, 7, 8), likely representing the distribution of the values discussed in the text.

23 And since the configuration of the 1 [$α$] placed below has a ratio also of the 2 [$β$] to the 3 [$γ$], and of the 3 [$γ$], 42 [$μβ$], and of the 4 [$δ$] to the 1 [$α$], and of the 5 [$ε$] to the 4 [$δ$], the number contains itself as simple myriads: 134,1100 [$ρλδαρ$].

The double of the multitude of the A-B parts [is taken] so that it reaches parts of the tetrad. For the base-numbersoriginal: "pythmenes"; the single-digit root of a number (e.g., the base of 70 is 7). of the sides 7 and 8 [$ζη$]... the number from the ratio is 131 [$ρλα$], and from the hundred, the number is 1, 2, 3, 4, 5 [$αβγδε$]. If the double of the multitude is not measured by the tetrad, it is clear that when measured according to the 5 [$ε$], it takes two of these. The number becomes hundreds of thousands; because of this, a hundred times a myriad of hundreds—as many as there are thousands—remains for the 20 [$κ$]. And from the hundred, the number 1, 2, 3, 4 [$αβγδ$] becomes homonymous with twenty thousand-myriads [$κ$]. The linear [calculation] is likewise reasoned: —

A third, smaller diagram at the bottom right consists of three vertical line segments, possibly representing the remaining units or the final ratio described in the text.