**References**

Gerhardt, C.J. *Das Rechenbuch des Maximus Planudes*, Eislebes, 1865.

Ifrah, G. *From One to Zero; A Universal History of Numbers*, New York, Penguin Books, 1985.

Wäschke, H. *Das Rechenbuch des Maximus Planudes*, Halle, 1878.

Wendel, C. *Planudea* in *Byzantinische Zeitschrift*, 40, 1940, pp. 406-445.

Wilson, N.G., *An Anthology of Byzantine Prose*, 1971, pp. 126-129.

**Notes**

1. For ease of reading, I will use modern forms for the symbols.

2. I will use numerals when they are given in the Greek and names when Greek names for numbers are given.

3. Note that this is one word in Greek, as with the other multiples of a hundred. This will be important in understanding P.'s description of the four types later in the work.

4. lit. "monads"

5. Greek uses the word \(\mu\upsilon\rho\iota\acute{\alpha}\delta\epsilon\varsigma\) or "myriads" for tens of thousands.

6. It appears that the word \(\mu\upsilon\rho\iota\acute{\alpha}\delta\omega\nu\) has dropped out of the text here.

7. I have everywhere replaced the first person singular with the first person plural.

8. i.e. multiples of a thousand

9. This and the following passage arise from a quirk of the Greek naming of numbers, rather than anything implicit in the number system. P. is saying that there are four types (lit. signs): units, tens, thousands and myriads. After this, we repeat the types as **units** of myriads, **tens** of myriads, **hundreds** of myriads and **thousands** of myriads. Then, we again repeat the types as **units** of myriads of myriads, **tens** of myriads of myriads and so on. The difficulty arises because Greek uses one word for say three thousand but two for three hundred thousand (three tens-of-myriads).

10. lit. "seven myriadically"

11. The Greek is verbose and clumsy; lit. "it is placed at the extreme in the direction of the smallest numbers."

12. Lit. "To speak simply." Wilson reads '\(\epsilon\iota\pi\epsilon\tilde{\iota}\nu\) here for Gerhardt's '\(\epsilon\iota\pi\grave{\omega}\nu.\)

13. Gk. \(\sigma\upsilon\mu\beta\alpha\lambda\lambda\omicron\mu\acute{\epsilon}\nu\omega\nu.\) Representing numbers is not strictly speaking an "operation", but there is no simple English equivalent that captures the force of the Greek.

14. Lit. "the signs or schemata."

15. lit. "taking the side of each number as though it were a square."

16. lit. "monadics"

17. lit. "take hold of"

18. lit. "take on board," a nautical term

19. lit. "behold"

20. This is the old method of *casting out nines*.

21. This section in square brackets has been restored to the text by Gerhardt.

22. Gk. Here the word \(\sigma\eta\mu\epsilon\tilde{\iota}\omicron\nu\) or "sign" is used instead of the the usual \(\sigma\chi\tilde{\eta\mu\alpha}.\) I have translated both as "digit".

23. We are here subtracting 35843 from 54612 giving 18769. The carried units are written underneath and the larger number is written again above, which will be used as part of the checking process later.

24. Literally "as decadic"

25. The first 1 on the bottom line appears to have dropped out of the text.

26. Read \(`\omicron\,\,\kappa\alpha\tau\omega\tau\acute{\epsilon}\rho\omega\,\,\sigma\tau\acute{\iota}\chi\omicron\varsigma\,\,\tau\omicron\tilde{\upsilon}\,\,'\alpha\nu\omega\tau\acute{\epsilon}\rho\omega,\) ... instead of \(`\omicron\,\,\kappa\alpha\tau\omega\tau\acute{\epsilon}\rho\omega\,\,\sigma\tau\acute{\iota}\chi\omicron\varsigma,\,\,\tau\tilde{\omega}\iota\,\,'\alpha\nu\omega\tau\grave{\epsilon}\rho\omega\iota.\)

27. This almost incomprehensible sentence seems to be simply saying that when we have noting to subtract from a digit, we just copy the digit down.

28. Read \(`\epsilon\nu\omega\theta\tilde{\omega}\sigma\iota\nu\) instead of \('\epsilon\nu\omega\theta\tilde{\omega}\sigma\iota\nu.\)

29. lit. *monadic*

30. lit. *decadic*

31. lit. *nothing*

32. that is, in cross formation like the Greek letter chi \(\chi\)

33. i.e. as well of course as adding on any units that were carried

34. lit. nothing

35. This section in square brackets has been restored to the text by Gerhardt.

36. Greek: \(\mu\omicron\acute{\iota}\rho\omega\nu =\) divisions, \(\lambda\epsilon\pi\tau\tilde{\omega}\nu\,\,\pi\rho\acute{\omega}\tau\omega\nu =\) primary parts = minutes, \((\lambda\epsilon\pi\tau\tilde{\omega}\nu)\,\,\delta\epsilon\upsilon\tau\acute{\epsilon}\rho\omega\nu =\) - secondary parts = seconds.

37. The Greek word here is illegible and not included in the text. Possibly it is some word for *data.*

38. lit. *smallest*.

39. Recall that 12 signs is 360˚^{ }.

40. lit. *first constellation of the ram*, Greek \(\kappa\rho\iota\omicron\tilde{\upsilon}\)

41. i.e. nearest but less than

42. lit. *side*.

43. Symbolically, \(\sqrt{a^2 +\epsilon}\approx a+\frac{\epsilon}{2a}.\)

44. This seems to have dropped out of the Greek, restored by Gerhardt.

45. This is confusing. He means that you double the root, not the remainder.

46. i.e. 35.

47. Greek \(`\alpha\nu\alpha\acute{\iota}\rho\epsilon\sigma\iota\varsigma\)

48. The text has 15 which I do not understand.

49. Planudes has made the algorithm very complicated by not using place value. An easier way to express what he is doing is to say that we seek the largest integer *x* so that (20+*x*)*x* < 135. Here *x*=6 doesn't work, but *x*=5 does.

50. i.e. 135 - 5×25 = 10.