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How do you pronounce the name Xenocrates

  1. Jul 21, 2005 #1
    How do you pronounce the name "Xenocrates"

    Hi. Question. How do you pronounce the name "Xenocrates" ?

    2) I was reading about the first combinatorial problem which was formulated by Xenocrates. It asks how many syllables can be made from the letters of the alphabet. And the solution was 1,002,000,000,000. (Source from the Mactutor history of mathematics website). I was wondering what method was used to solve such a complicated permutation? Its not like they had calculators so I wonder is it all that complicated? Also there is no mention of what alphabet was used. I'd like to know more.
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  3. Jul 21, 2005 #2


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    My guess, zee-nok-rate-ees. How do you pronounce Xena (the Warrior Princess)? Or, xenophobe and xenophile? I think a "Z" is substituted for the leading "X".

    Did Xeno actually solve it, or just throw out the question? It's not like they had zeroes, either, so I am pretty sure that whatever he might have done, Xeno himself did not come up with the 1,002,000,000,000 number. My guess is, if he solved it, he would have expressed it as a sum product of distinct terms, each term being some power of "the number of letters in the ancient Greek alphabet (say, N) minus k," where k = 0, ..., N-1.
    Last edited: Jul 21, 2005
  4. Jul 21, 2005 #3


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    ?? Define "syllable".
  5. Jul 21, 2005 #4
    Uh, just because you need the concept of zero to write 1,002,000,000,000 in the decimal system, doesn't mean you need zero to be able to express it in some other system (see, for example, Roman numerals (and yes, I'm quite aware of the difference between the ancient Greeks and Romans, I'm just illustrating a point ;)).
    Last edited: Jul 21, 2005
  6. Jul 21, 2005 #5


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    So how do you write "1,002,000,000,000 Roman soldiers" in Roman? To make it easy on you I am not asking to write it in Greek numerals. Just give me the Roman number, okay?
  7. Jul 21, 2005 #6

    He was Greek. If you add up the possible combinations of letters and vowel/diphthong-sounds, you get the number of syllables. Given that there is one vowel/diphthong per syllable. A syllable is represented as a combination of vowels/diphthongs (v) and consonants (C) -- a diphthong is a vowel sound.

    So far, here are the syllable patterns:

    CvC or Cv or vC or v

    Where CvC is read, "consonant vowel consonant"

    But there are other cases. It can happen that two consonants run together. So we have to add in those kinds of syllables (and I don't know enough Attic Greek to say which consonants can never go together, so let's assume they all can). We then add:

    CCvC or CCv or CvCC or vCC

    Rearranging all my options I’ve given, I get this:

    Two syllable types having three consonants and one vowel/diphthong
    Three syllable types having two consonants and one vowel/diphthong
    Two syllable types with two consonants plus vowel/diphthong
    One syllables types of having only one vowel/diphthong.

    Assuming eighteen consonants (being generous and giving an extra for the digamma), seven long vowels, plus five short vowels, plus seven diphthongs, we get:

    C = 18 choices, v = 19 choices.

    From that, I don’t get Xenocrates’ answer. So I don’t know how he counted a syllable.

    Xenocrates might have been interested in poetic meter when he came up with his calculation. If so, I think I have the right estimate of what he was defining as a syllable. But since I am not getting his answer, I must be wrong!
  8. Jul 21, 2005 #7
    Surely Roman numerals are ill-suited for such a task, but I assure you that it can be done (take a string of 1,002,000,000,000 I:s and you have a valid representation, although not optimal). If all else failed, Xenocrates could've just written it out using words.

    And btw, (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html):

    So just a hundred years (at most) after Xenocrates' death, the ancient Greeks were capable of expressing numbers roughly 10^52 times larger than 1,002,000,000,000...
  9. Jul 21, 2005 #8
    The story of Xenocrates may be incorrect:

    "In Greece, one of the very few references to combinatorics is a statement by Plutarch about the number of compound statements from 10 simple propositions; Plutarch quotes Chrysippus, Hipparchus, and Xenocrates on the subject, so all apparently had some interest in the subject. (Plutarch's statement is also discussed in a recent article in the Monthly.)" -- http://math.truman.edu/~thammond/history/Euler.html

    That could explain why I could not get even close to the finding the total number of syllables. I think the original web site you found had the wrong problem.
  10. Jul 21, 2005 #9


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    Here it says "The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to 8x1016 in modern notation." I am not a math historian, but to me these numbers look soft. I understand that as a principle it is possible to express large numbers in any numerical system that is not especially designed to avoid them, just like we would be if somebody had not invented the number zero.
  11. Jul 21, 2005 #10
    Here is how Plutarch wrote what Xenocrates said:

    "myriad-and-twenty times a myriad-myriad",

    Greek Science of the Hellenistic Era: A Sourcebook,
    by Georgia L Irby-Massie, Paul T Keyser
    p. 35

    Myriad is defined in the Greek lexicons as 10,000. The same is true of Ugaritic (a more ancient language)! Myriad can also mean, "more than you can imagine!"

    What is interesting is that by my calculations (which I hinted at in an earlier post), Xenocrates is way off -- his answer is way too large.

    At best, we might say he was estimating and not calculating.
  12. Jul 21, 2005 #11


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    I have read that in olden cultures people used large numbers to represent infinity, and vice versa. So if somebody wrote "10,000 generations passed before this happened," they didn't literally mean that somebody counted 10,000 generations, nor did they care whether so many generations having passed is even logically possible within the context. They just meant "countless."

    Alternatively, someone might have said or implied "countless," but the scribe could have written "10,000" (perhaps because that particular scribe did not think it possible to count beyond 10,000, so expressed "countless" as a specific number).

    Do you think this may be one of those instances?
  13. Jul 21, 2005 #12
    I would want to preserve the fact that countless and infinity are not always synonymous.

    Where Plutarch is giving Xenocrates' answer, he is dealing with specific values. It is easy to turn the Greek "myriad" into a specific 10,000 by the addition of the article -- "the myriad". In the case we are looking at here, even if the article is missing in the Greek, the context is still one of specific numbers. So no, this would not be the case where the author is using the term to mean "many." I suspect he was being specific.

    What I would like to know is how the word "myriad" was ever able to take on the specific meaning of 10,000. I would wager that it has to do with geographical needs (size of land). I recall in the Ugaritic that one text used myriad (the word there is rbt) in terms of distance or number of fields. But think of it: What does "10,000" mean? Who has ever come across 10,000 of anything, as to know that such a thing is made up of 10,000? If we are talking about a square field, 100x100, then we can talk about 10,000!

    I happen to know that geometry -- the study of land measures -- came about, in part, from Greek city survey. I wonder if myriad is a survey term that takes on specific meaning so that mathematicians can use it in general?

    I am just thinking out loud.
    Last edited: Jul 21, 2005
  14. Jul 21, 2005 #13


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    Especially after Cantor's publishings. :smile:

    My uneducated guess, too, would be that that specific number would have arisen as 1002.
  15. Jul 21, 2005 #14
    Yes, and as I think of it, population counting would easily make myriad a useful word to have around! Even theater seating in the Roman world would measure in the myriads. At Pompeii, the theater held two myriad.

    This is beyond the original question, but it is interesting to think about the meaning and use of number at a time that did not have the mathematical tools we enjoy.
  16. Jul 22, 2005 #15
    The largest Roman numeral is m = 1,000,000.
    M = 1000, so

    mMM = 1,002,000 thus

    1,002,000,000,000 = 1 million "m" followed by 1 million (MM), that is:
    = mmmmm...mmmMMMMMMMMMMM...MM
    Last edited: Jul 22, 2005
  17. Jul 22, 2005 #16


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    "I'm lovin' it!" :smile:
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