Jöran Friberg is both ambitious and confident in his assessment of the link between Babylonian and Egyptian mathematics. Unfortunately, his findings are neither well thought out or substantiated with solid evidence.
The intended audience for the book is unclear. One unnerving feature is the author’s transliterations of the mathematics found on the clay tablets, making it virtually impossible to check the author’s work for accuracy. Readers are rarely shown the original cuneiform or hieratic text. Thus, the book could not be aimed at historians of mathematicians, Egyptologists, or Babylonian scholars. But the book is also not intended for the lay reader, for it consists of a mind-numbing series of comparisons of Egyptian and Babylonian writings via problems found on papyri and clay tablets. Who is left?
The observation that problems found on Babylonian clay tablets from the ancient city of Mari are similar to those found in Egyptian texts propelled the author to compare many cuneiform texts and papyri in order to conclusively establish a clear link between Babylonian and Egyptian mathematics. He maintains that his search has been “unexpectedly successful” in turning up numerous candidates for such connections and the book is the result of his efforts. The author claims that, up to now, no serious attempt has been undertaken to establish mathematical connections between Babylonian cuneiform texts and Egyptian papyri. Those who have studied clay tablets have had “pessimistic attitudes.” Adding that it is not likely that their decryptions are correct and they have undoubtedly “poorly understood the material.” Further, some have “absolutely no clue what is going on.”
The seminal clay tablet of interest, Mari 7857, purportedly contains a five-term geometric progression with first term 99 and common ratio 9. The crucial “link” being to a geometric progression with 7 as the first term and 7 as the common ratio found in the Rhind papyrus, the latter being akin to that found Fibonacci’s Liber abaci and the St. Ives riddle. The author concludes, using an argumentum ad verecundiu, or an appeal to improper authority, that it, is “inconceivable” that the two were devised independently, thus declaring that he has found a “definite connection” between the mathematics of the two civilizations. That this link might be a mere coincidence seems never to have crossed his mind.
With respect to Egyptian papyri, a hidden use of the sexagesimal fractions in the Cairo papyrus that has eluded scholars for generations is revealed. In spite of modern arguments to the contrary, Plimpton 322 is described as a table of Pythagorean triples. Cuneiform text YBC 4652 contains an algebraic problem and its solution but no solution procedure. The author shows that the method of false position, employed by the Egyptians, can be used to obtain the answer.
In a geometric transversal problem found in the Rhind Papyrus, that scholars have tried to make sense of for years, and where some have even had the courage to admit that they do not understand what is going on, the author notes some errors in the text, identifies its geometric-algebraic nature, solves it, and identifies links to similar mathematical problems found on clay tablets. Time and time again, the author exhibits an uncanny knack for reconstructing problems that are missing significant parts and then discovering that his reconstructions display unexpected Babylonian-Egyptian connections. To go through all the so-called links highlighted in the book would be to invite mental paralysis.
From the author’s extensive research, he concludes that the level and extent of mathematical knowledge must have been comparable in Egypt and Mesopotamia in the earlier part of the second millennium BCE. He asserts further that there are certainly unexpectedly “close connections” between Egyptian and Babylonian mathematics. In fact, there are so many parallels of a similar nature that their mathematics must have influenced each other in decisive ways. He adds that the obvious similarities cannot be explained away as due to independent development. In conclusion, he asserts, using an argumentum ad ignorantium, or an appeal to lack of evidence, that the burden of proof ought to be on those who say that this was not the case.
A few intriguing items are noted but not fully explored in the book. For example, if the author’s transliterations are correct, it is remarkable that the numbers on Mari 7857 are expressed using three different number systems, the sexagesimal, decimal-sexagesmal, and centesimal place value systems. If the figures on clay tablet P.BM 10520 are translated correctly, it is remarkable that this early presence of triangular and pyramidal numbers is not mentioned. Further, the method of multiplying using a doubling and halving algorithm was employed by the Babylonians and the Egyptians. It is astonishing that no connection is claimed.
For those interested, an excellent, well-researched account of Babylonian mathematics can be found in Eleanor Robson’s article on Mesopotamian Mathematics in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by Victor Katz. For a well-informed account of Egyptian mathematics see Annette Imhausen’s article in Katz’s book. A scholarly account of mathematics on clay tablets can be found in Robson’s Mathematics in Ancient Irag: A Social History.
Similar but independent discoveries and advances have occurred in many intellectual fields, including mathematics. It might be hard to explain this phenomenon, but the evidence is there time and time again. While, I agree with the author that some clay tablets and papyri have comparable tables of contents and many of the worked problems are similar, could not these coincidences be explained by the simple nature of the problems involved? It is no surprise that there are similarities in the mathematics of ancient cultures. There may be significant “links” between Babylonian and Egyptian mathematics, but this book does not highlight any credible connections.
Jim Tattersall is Professor of Mathematics at Providence College, in Providence, RI.