Date: Jan 1, 2007 11:55 AM Author: Robert (Bob) Eldon Taylor Subject: [HM] Al-Khayyam's Commentary on Euclid's Elements.

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