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The Magic Squares of Manuel Moschopoulos - Notes

Peter G. Brown

1. Rhabdas means staff or stick. Its meaning here is unknown. Sir Thomas Heath [7] seems to regard it as a man's name. Conceivably it could mean son of Rhabdas, a common use of \(\tau\omicron\overline\upsilon\) + genitive

2. An evenly-even number is a positive integer of the form \(2^n,\) for some positive integer \(n.\)

3. Literally 'the monad', Greek \(\mu\omicron\nu\acute{\alpha}\varsigma.\) Early Greek arithmetical theory regarded the number 1, the monad, as a special object, from which all numbers emanated, rather than as a number in its own right. I have translated the term as 'the number one' throughout. See [7], Vol. 1, pp. 69f.

4. An evenly-odd number is generally taken to refer to a number of the form \(2(2m+1).\) See [7], Vol. 1, p. 72. Moschopoulos may, however, be refering to those numbers traditionally called oddly-even, which have the form \(2^n (2m+1),\) for \(n, m\ge 1.\) 

5. Literally, an 'equal sided quadrangle'. Moschopoulos abbreviates this to 'quadrangle' in what follows. I translate this as 'square' or 'number square'. Square integers are still thought of geometrically, and the 'side' of a square refers, of course, to the square root. This leads to some confusion in the Greek which I have attempted to overcome in translation.
The term 'magic square' does not appear in Moschopoulos. The exact origin of the term is not known but may have been derived from the fact that medieval astrologers engraved such magic squares on charms which supposedly possessed magical properties. There is not the slightest suggestion that Moschopoulos attached any magical or spiritual power to these squares or their construction. See [4], p. 122.

6. Number represented by Greek letters are simply transposed into modern numerals, while those written out in Greek are translated by their equivalent English terms.

7. i.e. square root.

8. Moschopoulos here uses the term 'side', \(\pi\lambda\epsilon\upsilon\rho\acute{\alpha},\)  to refer to the sum of the numbers on any 'side' (i.e. row or column). He often confuses this use of the word 'side' with the 'side length' of the square. Thus the 'side' of a three-by-three magic square is 15, but the 'side of the square' is 3. To avoid this confusion I have used the term Side for the sum of the numbers along any side and 'side' for the side length.

9. The word translated as 'cell' is \(\tau\acute{\omicron}\pi\omicron\varsigma,\) meaning 'a place' or 'position'.

10. In fact, Moschopoulos only speaks of the construction of squares of odd length sides, and those whose sides are evenly-even, that is, a power of 2.

11. Literally, 'the amount corresponding to each monad of the sides'.

12. Literally, 'it occurs by necessity.' Moschopoulos is, of course, using the fact that \[\Sigma_{i=1}^n i = \frac{1}{2}n^2 + \frac{1}{2}n.\] Nowhere in his writings does he attempt to prove any of his claims. This result was well known in antiquity, since for each positive integer \(n\) this formula gives the triangular numbers. See [14], Vol II, p. 499.

13. Tannery adds in the words, \(\tau\omicron\overline{\upsilon}\) '\(\eta\mu\acute{\iota}\sigma\epsilon\omicron\varsigma\) \(\mu\acute{\epsilon}\rho\omicron\upsilon\varsigma.\)

14. That is, odd squares.

15. Literally, 'when the cells are filled it is always necessary to return to their beginning'.

16. Literally, 'when we vary from one side to another side'.

17. Literally 'signs', \(\sigma\eta\mu\epsilon\overline{\iota}\alpha.\)

18. Literally, 'we lead' or 'carry' dots.

19. Omitted in some manuscripts.

20. It is not clear why M. chose this archetype rather than the one constructed using the earlier method. It has been suggested that M. is simply copying some earlier material of which he has little understanding.

21. Literally 'signs'.

22. Compared with the verbose descriptions of the earlier methods, this section is remarkably terse, and would be difficult to follow without an example.

23. As shown, we move from left to right and add 8, 16, and 24, respectively, to the entries in the basic square, as we do so.

24. Adding, of course, 24, 32, 40, and 48, respectively, to the basic square as we reverse back.

25. This also holds for the diagonal sum of each sub-square.

Peter G. Brown, "The Magic Squares of Manuel Moschopoulos - Notes," Convergence (July 2012)