I now continue my comments on Book II of Al Khayyam's Commentary on Euclid's Elements. In book two, Khayyam attempts to revise the definition of proportion for magnitudes given in E. book V. Recall that Euclid gives two different definitions of ratio and proportion. In book V, he develops a theory of proportions for continuous magnitudes based on definitions attributed to Eudoxus (a contemporary of Plato). In book VII he develops a theory of ratio and proportion for numbers which is considered to be essentially Pythagorean. Ratio for the Pythagoreans was a sort of sub-classification of inequality. Thus, of two numbers: they are equal, which case has no sub-divisions, or they are not equal, which case has three sub-divisions: the first is a multiple of the second, thus: it is twice the second it is thrice the second etc. or, the first is a part of the second, i.e. it measures or divides it, thus: it is half, it is a third, it is a quarter, etc. or, the first is parts of the second, thus: it is two thirds, it is three quarters, it is two, or three, or four fifths, it is five sixths, etc. Now you can see that every ratio of numbers must fall in one and only one of these categories, and if two numbers fall in one of these classes, and two different numbers also fall in the same class, they are the _same_ ratio.
For us moderns, the third category would encompass the first two, but that is not the Pythagorean way of thinking which is preserved in Euclid. Also note that, strictly speaking, equality is not a ratio.
Every serious student of Euclid has probably wondered why he didn't replace this Pythagorean definition of proportion for numbers given in book VII by the definition of proportional magnitudes given in book V (attributed to Eudoxus). But what Khayyam wants to do is the opposite, and adapt the Pythagorean classification scheme for ratios of magnitudes. This definition can easily be applied to commensurate magnitudes, but for incommensurate magnitudes, another way must be found. Khayyam uses an anthyphairetic definition, which he seems to regard as intuitively obvious.
Why does he want to do this. It is hard to say but it seems to be what I mentioned in my previous message on this subject, that he wants to observe the ratio directly, as a fact. For he says "the ratio between three and nine is the _fact_ of being one-third". He also raises the following objection:
Do you not see that if a questioner inquires, saying: Four magnitudes are proportional according to Euclidean proportionality, and the first is the half of the second, will the third then be the half of the fourth, or not?"
Since one can easily prove this from Eudoxus' definition, it seems that Khayyam's objection is precisely that one _must_ prove it, that it is not obvious, that is not clearly and directly observable, from the definition.
The principle involved in the anthyphairetic technique is this: if two unequal magnitudes have a common measure, there is a magnitude which measures both and that magnitude measures also the difference. Using the remainder and the smaller magnitude we can repeat the procedure as many times as necessary until we come to the common measure. But, as shown in Euclid X.2, if the given magnitudes are not commensurate, this procedure will not come to an end because there is no common measure to act as a unit to end the division of magnitudes.
His definition of "true" proportion is then as follows:
given four magnitudes, the first and second being commensurate, if the first is the same part or parts of the second as the third is of the fourth, they are in the same ratio. (He calls this "numeric" proportion.) But if the first and second are not commensurate, then if they have the same anthyphairetic expansion as the third and fourth, that is if the alternate subtractions are in the same order and the same number of repetitions, then they have the same ratio. (He refers to this as "geometric" proportion.)
It would seem that the anthyphairetic part of the definition could cover also numeric ratios, but this is not the Pythagorean mode of thought which Khayyam follows.
He now proposes to alter Euclid's book V by adding certain postulates and propositions in order that he may establish the equivalence of "true" proportion and "common" proportion. Among these is a proof of the existence of a fourth proportional, which he does by a sort of "binary search". But one cannot find what is not to be found so it seems to me he has assumed what he is required to prove, that is, given three magnitudes there exists a fourth such that the first is to the second as the third is to that fourth.
He then introduces book V proposition 1 which is required and can be proved here, but neglects to mention book V proposition 2 which is also needed, but only assumes it. He then establishes the equivalence of "true" and "common" proportion, which, because of his definition must be in "cases". That is, in the case of "numeric" proportions, he shows that true proportionality implies common proportionality and conversely. Then in the case of "geometric" proportion, common proportionality implies true proportionality and conversely.
The anthyphairetic expansion of a ratio corresponds to a so-called simple continuous fraction. These are notoriously intractable, however useful they may be in certain applications. One cannot define addition, subtraction, much less multiplication or division except by collapsing the continued fraction into a common fraction or an infinite sequence of approximations. It will be very difficult then to prove anything using an anthyphairetic definition, so we will depend on the equivalence of "true" and "common" proportionality to use the latter for any purpose we may have for proportions. This is why it is important for Khayyam to establish this equivalence.
The question then is why Khayyam wants to introduce this definition? I have attempted to answer this above, that he looks for direct intuitive understanding. But does he find it? Is his anthyphairetic technique clearer and more intuitive than Euclid's? I suspect not. I am not unsympathetic with his motives (if these are his motives). Mathematical proofs, even when one has followed them through completely, often seem unsatisfactory. One is forced to admit each step and at the end the truth of the whole, but with no understanding _why_ it is so. But how can we make progress while waiting for a why which may never come?
So I think the Old Tent Maker is tilting at windmills, and let's leave it at that.
Regards, Bob Robert Eldon Taylor philologos at mindspring dot com