Dear David, your analysis is correct, as far as it goes. Where I see a crack in your point of view is the publication of new ancient documents, or broader readings of well known Egyptian documents. Let me stress Hana Vymazalova's 2002 paper as an example. Daressy in 1906 was unable to fully read the Akhmim Wooden Tablet, leaving out several scribal methods of partitioning a hekat by divisors 3, 7, 10, 11 and 13, and then proving that each partitioning was correct, returning the orginal problem to a hekat unity, 1 = 64/64. The actual ancient text as read by anyone at any time must govern the scope and depth of scribal mathematics, and not the incomplete readings documents, as has taken place several points in our modern history. That is, the incomplete suggestion by Peet in 1923, where he mentioned that ro only meant 1/320th of a hekat, was as incomplete as Daressy's view of the AWT in 1906. The modern view of (64/64)/n = Q/64 + (5R/n)*ro has been confirmed, and in publication (a citation will appear in Dec. 2006), as the actual remainder method used by the AWT and RMP scribes. Clearly the confirmation of a particular ancient method in one text should be confirmed in at least one other text, as the two-part quotient and remainder method was been recently and fairly parsed from the AWT, RMP and other texts. Best Regards, Milo Gardner
Dear David, Struik's view of clumsy Egyptian fractions may have come from his misunderstanding of Hultsch-Bruins, or a related issue. The H-B n/p conversion issue, and Boyer's views of 2/pq views were fairly reviewed by Brown. If you have a problem with Brown, let's discuss it. Yet, Brown's and Boyer's version for the 2/pq conversion can be written in the modern form 2/pq = 2/A times A/pq where A = (p + 1), a neat and precise method. Note that Ahmes had improved upon this method in the n = 35 and n = 91 cases (as Brown may have over-analyzed). Ahmes may have used a version of n/pq = 1/pr + 1/qr, where r = (p + q), as I first read as a problem taken from the 300 AD Akhmim Papyrus (not the AWT)disucssed in Howard Eve's 1961 History of Math textbook, a topic not reviewed by Brown. Given that the added EMLR connection of 1/pq values, like 1/8 and 1/16 clearly were practice to the optimal RMP table, i.e. A = 5 and A = 25 test cases, and other special cae examples (discussed by many over the years). Egyptian fractian's history was varied, taken from several ancient texts. In addition, Egyptian fractions was a notation system that may be seen as clumsy to a few scholars, but I even doubt that conclusion is valid in the ancient context.
To conclude that the two basic EMLR and RMP algebraic identities were clumsy to ancients, as Neugebauer attempted to without citing direct evidence in Exact Sciences in Antiquity is at least odd. Neugebauer's limited view of the RMP 2/nth table and its optimized data, seen as signs of intellectual decline, is more than odd (to me). That view is disrespectful of Egyptian and Greek scribal math, and its theorerical arithmetic, on the highest level. The fact that the 2/nth table has been confirmed by Brown, Boyer, Hultsch-Bruins and many others as showing signs of intellectual advancement, within a remainder arithmetic structure, that lasted for 3,600 years, should be fairly noted, and not dismissed before being rigorously reviewed. Neugebauer and other classical scholars that have attempted to prematurely dismiss Egyptian fraction as meaningless, in a terse manner should be 'taken with a large grain of salt'! Other than the Hibeh Papyrus (HP), written by a Greek in 400 BC, no rational number was rounded off by Egyptians or Greeks when converting to Egyptian frection series. Taking all this evidence into account, Struik, like Neugebuar, was grossly incorrect in his 1967 terse dismissal of Egyptian fraction methodologies, at least in my view.
In contrast, Babylonians always rounded off in all of its published 'inverse tables', and continued doing so for 2,000 years. Whenever mutiples of 2, 3 or 5 were in the denominator of the rational number being converted rounding off took place. The same rounding off, an unaccepted practice in Egypt, was true for all 'inverted' Babylonian counting numbers published in cuneiform texts. Babylonians scribes always rounded off, as Egyptians after 2,000 BCE did not, except in one case. So which culture, Babylonians or Egyptian, was 'most' clumsy in their unit fraction methods?
I vote for the Babylonians, since they consistently cited inept approximations for 1/91 as 1/90 for all of Babylon's math history. I can only find one example of an Egyptian or Greek text, when properly read (as the AWT was corrected in 2002), that cited an inexact (rounded off)conversion of a rational number. And that Hibeh P. example was a time based context, well after Pythagoras, and the proven existance of irrational numbers. Let me stop with the point that the HP data may have originated in a water clock time reading, or calculated therefrom. Best Regards, Milo 11/24/06.
- I have no particular problem with Brown. I consider him mathematically knowledgeable and competent. But I think that is not enough for a reliable citation on the history of ancient mathematics. I think that sort of scholarship is built around concensus between many scholars, as developed in the record of academic publications, and a web page is just not part of that concensus no matter how competently researched and written it may be. —David Eppstein 22:02, 26
November 2006 (UTC)
Brown is an excellent mathematician, someone that is willing to ponder the ancient texts as written by scribes. Consensus in the academic community is changing fast these days. Read Hana Vymazalova, Tanja Pemmerening, myself, and others. The days of the Peet and the 1920's are long gone. Heinz Leuneburg and the Liber Abbaci says that same thing. Greeks and Egyptians both computed with rational numnbers, or vulgar fractions (if you prefer), with medievals stressing the later, with Egyptian fractions primarily being used to encode remainders into units for later use for weights, measures and other numeric purposes. Best Regards, Milo Gardner 12/2/06.