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