Babylonian influences in some of the best known works of Greek mathematicians (1)

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The simplest way of explaining the many parallels that can be found between (certain parts of) Greek mathematics and Old or Late Babylonian mathematics is to assume that in ancient Greece elementary education in mathematics for young students (not necessarily intending to become mathematicians) was conducted in terms of metric algebra in the Babylonian style. Here “metric algebra” is a convenient name for the very special kind of mathematics, with an elaborate combination of geometry, metrology, and linear or quadratic equations, which is first documented in proto-Sumerian texts from the end of the fourth millennium BC, and which prevailed in Mesopotamia without much change to the Seleucid period close to the end of the first millennium BC. During the 2500 years of its existence already before the dawn of Greek mathematics, this kind of mathematics ought to have had ample opportunity to spread to more or less distant neighbors of Mesopotamia itself. That this hypothesis is correct in the case of Egypt was demonstrated in Unexpected links (Friberg 2005). To show that it may be correct also in the case of ancient Greece is the object of the discussion in Amazing traces (Friberg 2007).

It is important to understand that one of the obstacles in the way for a better understanding of possible relations between Greek and Babylonian mathematics is the circumstance that Greek mathematics is documented mainly through copies of copies of important manuscripts with advanced mathematics, while Old Babylonian mathematics is documented mainly through clay tablets with relatively low level mathematics, written by mediocre scribe school students, and Late Babylonian/Seleucid mathematics is documented only through a small number of texts, for the simple reason that in the second half of the first millennium BC clay tablets had been replaced by more easily perishable materials as the preferred medium for writing. For these reasons, it is difficult to know what Greek mathematics at a lower level was like, and equally difficult to find out how advanced Old and Late Babylonian mathematics at a higher level may have been. It is also important to understand that since the heated but inconclusive debate about Greek “geometric algebra” in the late 1970’s, much has happened in the study of Babylonian mathematics. Many new mathematical cuneiform mathematical texts have been published since then, several of them with unexpected and astonishing revelations about the scope of Babylonian and pre-Babylonian mathematics, and many of the earlier published mathematical cuneiform texts have been explained in new, and much more satisfactory ways. Therefore, it is now obvious that the aforementioned debate was conducted against a background of regrettably insufficient knowledge about the true nature of Babylonian mathematics.

One of the most important results of the discussion of possible relations between Greek and Babylonian mathematics in Amazing traces is the realization that, far from being Greek reformulations in geometric terms of Babylonian (non-geometric) algebra, the propositions in Euclid’s Elements 2 are abstract, non-metric reformulations of a well defined set of basic equations or systems of equations in Babylonian metric algebra, that is of quadratic and linear equations or systems of equations for the lengths and areas of geometric figures. Strictly speaking, Elements 2 is not about “geometry” at all, in the literal sense of the word, which is ‘land-measuring’. Characteristically, as a consequence of the different Greek and Babylonian approaches to geometry, diagrams illustrating non-metric propositions in the Elements are what may be called “lettered diagrams”, while diagrams illustrating Babylonian metric algebra problems are “metric algebra diagrams” with explicit indications of relevant lengths and areas. However, as a whole, Elements 2 is a well organized “theme text” of the same kind as similarly well organized Babylonian mathematical theme texts.

Another particularly important new observation in Amazing traces is that it is possible to find a completely new approach to the study of the notoriously difficult tenth book of the Elements. Thus, it may be shown that the theory of inexpressible straight lines in El. 10 is based on a number of fundamental lemmas and propositions such as the lemmas 10.28/29, 10.32/33, 10.41/42, and the propositions 10.17-18, 10.30, 10.33, 10.54, 10.57, 10.60, all of which can best be explained by use of Babylonian metric algebra. As a matter of fact, a particularly great role is played in El. 10 by “quadratic-rectangular systems of equations of type B5”, by which is meant problems where both the sum of the squares of two unknowns and the product of the unknowns are given. Such problems appear as well in Babylonian mathematics.

Source (list of abbreviations) (source links will open in a new browser window)
Euclid, Elements

Bibliography

Jöran Friberg

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