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[HM] Al-Khayyam's Commentary on Euclid's Elements.
Posted:
Jan 1, 2007 11:55 AM
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Dear Julio and All,
I have been studying Al-Khayyam's "Commentary on Euclid's Elements"
using the English translation (from Arabic) made by Roshdi Rashed and
Bijan Vahabzadeh and would like to post some comments now in hopes of
stirring up some interesting conversation. First a word about the
translation: Khayyam was a poet. Not every mathematician is a good
writer, but one expects a poet who is also a mathematician and statesman
to be at least a competent, if not elegant, writer on serious things.
Unfortunately there is no hint of that in this translation. The
translators, honest scholars they may be, are quite obviously not native
speakers of English. The punctuation in the translation is very
strange, which, together with various grammatical flaws and rather odd
selection of words, indicates the translators may not have the necessary
command of English to render the original appropriately. Whether this
causes a problem in understanding is another matter, although one
wonders whether the selection of certain technical terms is correct. In
any case, when dependent on translations it is always a problem whether
the translators adequately understand certain issues to allow the
correct choice of terms. This of course applies to Khayyam himself who
was dependent on translations from Greek into Arabic.
Please do not take this to be any lack of gratitude on my part to the
translator for his efforts. As I think we all know, translation is very
difficult and especially into a language not your own. Unfortunately,
due to my complete ignorance of Arabic, I am dependent upon this
translation and some help from Jeff Oakes and his colleague Haitham
Alkhateeb. For my purposes a more literal translation and a latinized
text of the Arabic would have been more useful.
How do we deal with Khayyam's references to Euclid? There are several
propositions quoted by Khayyam and attributed by him to Euclid which I
cannot identify in Heath's Euclid and, although Jeff Oakes points out
that we cannot assume Heath's Euclid to be the "right" one, which are
clearly interpolations, whether in the original Greek, the Arabic
translation or a subsequent copy. We cannot refer to the Heath
translation to understand what Khayyam is saying, because Khayyam was
using the Arabic translation. So when Khayyam speaks of a proportion as
the _equality_ of ratios, when Euclid only speaks of ratios being the
_same_, this may only be due to the way the Euclid propositions are
stated in the Arabic translation.
The commentary is in three rather independent parts. In book 2, which
I will discuss later, Khayyam criticizes the Eudoxan theory of
proportion used by Euclid, which he calls "Common Proportion" and
defines his own, which he calls "True Proportion". In book 3, Khayyam
deals with the compounding of ratios.
Book One concerns itself with the parallel postulate, an important and
interesting topic, about which I know nothing. There is however one
passage (p. 223) which is required in book two. This will perhaps serve
to illustrate some of the problems I have alluded to above. Khayyam writes:
"And as when he proves in the fifth Book, _that the ratio of the same
magnitude to two equal magnitudes is the same_. But as ratio falls
within magnitude _qua_ magnitude, why should this need a demonstration?
Since the two equal magnitudes are equal _qua_ measure, there is no
difference whatsoever between them; therefore they are from this
viewpoint truly the same: there is no alterity whatsoever between them,
except the alterity of number and no more."
The proposition quoted is from Euclid's _Elements_ the second part of
V.7, but Heiberg/Heath does not have the word "two", which has the
effect of limiting the generality of the proposition.
"Alterity" is a very obscure word meaning "otherness". According to
Jeff the Arabic word can mean "difference" or "distinction", which seem
to fit the meaning better.
I found the word "number" in the last line very puzzling. According to
Jeff the Arabic here is "`adad", which can only be translated "number".
We could only speculate what is meant.
The translator is fond of the word "_qua_" which he always italicized,
as though a foreign word. I do not know whether it is always used to
translate the same Arabic word and I am not sure what meaning is to be
attached to it other than an English preposition meaning "in the
capacity of". So - if a ratio is a relation between magnitudes, what
would it mean to say "ratio falls within magnitude _in the capacity of_
magnitude"? How are equal magnitudes equal _in the capacity of_
measure? I.e. in what other way would they be equal?
What is Khayyam saying? Perhaps something like this: Recall that, like
numbers, magnitudes can be equal while remaining different. If A is a
magnitude and BC and DE are equal but different magnitudes, then the
ratio of A to BC is the same as the ratio of A to DE, because (i.e. _by
cause_) ratio is concerned with size only and not with location or shape
or other attributes of the underlying geometric objects and equal
magnitudes can be said to be the _same_ magnitude if one considers only
the size (measure). That is, the size of the underlying geometric
object is the _same_. Thus two equal lines have the _same_ length.
Therefore the ratio of the same magnitude to equal magnitudes is the
ratio of the same magnitude to the same magnitude. This may be the
right interpretation, but it seems to me to press Euclid very hard.
Recall that Euclid consistently speaks of "same" ratios, but of "equal"
magnitudes and numbers. But Khayyam, at least in this translation, is
not consistent at all, but sometimes speaks of two ratios being _equal_
and even _similar_.
If the above is the correct understanding, then the answer to Khayyam's
question is that Euclid & the Greeks did not abstract (or did not so
completely abstract) magnitudes from the geometric figures. Further it
should be said that it requires some proof to say that magnitudes _can-_
be so abstracted, i.e. that there is such a property which resides in
the geometric structures which allows the abstraction.
Despite these difficulties it is apparent from the above that Khayyam is
a profound thinker who drives for the essence of things. The other two
books strengthen that impression.
From a sunny but soggy Comer, Georgia, wishing all my friends here a
very bright and prosperous New Year,
Bob
Robert Eldon Taylor
philologos at mindspring dot com
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