Here’s a book that does exactly what it says on the cover. Annette Imhausen, professor of the history of science at Frankfurt’s Goethe University, sets out what specialists such as herself know about the mathematical systems and techniques used in Ancient Egypt, traced through that civilisation’s long history. As it presents the most up-to-date academic understanding of the subject, from a Magonian perspective it has some relevance when it comes to the theories advanced by the ‘alternative Egyptology’ camp, which is what I’ll concentrate on in this review. (Well, that’s my excuse for ducking a critique of the mathematical parts.)
In the nineteenth century and for most of the twentieth the prevailing view among Egyptologists was that the Ancient Egyptians’ grasp of mathematics was hopelessly unsophisticated compared to that of other ancient civilisations, Greece especially – lacking, for example, an understanding of concepts such as pi, or an equivalent of algebra. It was therefore written off as a mathematically, and therefore scientifically, primitive culture.
The obvious mismatch between that image and the in-your-face achievements such as the Giza pyramids, which clearly required a mastery of all kinds of mathematical and geometric skills in their planning, project management and construction, is partly what fuelled, and continues to fuel, ‘fringe’ theories about the civilisation’s origins; perhaps those dunderhead Egyptians didn’t really build these things at all, but merely inherited them from an older, lost civilisation. It’s therefore instructive to see what the latest scholarly estimation is of their mathematical skills.
In her introduction, Imhausen reviews past studies – which began in 1877 with the publication of the Rhind mathematical papyrus, still the major source on the subject – and the progress that has been made since, not just in Egyptology but mathematical history generally. She concludes that, ‘Due to developments in the history of mathematics of the last 40 years, it has now become obvious that many “statements” about Egyptian mathematics that were made a long time ago and that have since been accepted as “truths” need to be reassessed.’
|THE RHIND MATHEMATICAL PAPYRUS|
That reassessment forms the backbone of the book. Imhausen presents a straightforward chronological account, devoting a section to each of the major periods of Ancient Egyptian history – Prehistoric/Early Dynastic, Old, Middle and New Kingdoms, and Greco-Roman. She begins each with an overview of the period’s history for context, before detailing what’s known about the maths employed during it, ending with a useful summary of the essential points.
Imhausen writes clearly and concisely, if with academic dryness. Although she assumes familiarity with some technical mathematical terms, it’s only when she sets the ancient and modern methods of working through a problem side by side, in order to illustrate the conceptual differences between them, that the going gets hard for a non-mathematician like myself. Fortunately, those parts can be skipped over without losing her argument.
One feature of the Ancient Egyptians’ mathematics that was singled out for derision, and to which Imhausen frequently returns, was their way of handling fractions, which appears childishly clunky to today’s eyes. She shows that this misconception arose from a failure, born of the smug belief that modern Western ways of doing things are always the best, to appreciate that there are alternative, equally valid, approaches to doing maths: ‘The Egyptian concept of fractions… was fundamentally different from our modern understanding. This difference is so elementary that it has often led to a distorted analysis of Egyptian fraction reckoning, viewed solely through the eyes of modern mathematicians, who marveled at the Egyptian inability to understand fractions like we do.’
The same goes for their lack of a ‘number’ for zero, another source of modern condescension. As for pi, the Rhind papyrus shows that the Ancient Egyptians could calculate the area of a circle perfectly well, they just did it another way, without using, or needing, that ratio.
In short, the Ancient Egyptians conceptualised numbers in a very different manner to our culture. What comes across from Imhausen’s analysis is the intensely practical nature of their maths - its use for accounting, surveying, building, calendar reckoning and so on. Their system, unlike ours, didn’t use number symbols as abstractions, divorced from the things they represent. Where we would use an equation to find a value, they followed a series of practical, logical steps to get the answer.
The old view of Ancient Egyptian primitiveness is wrong, then. So what do we now know about how far their mathematical understanding extended?
The answer is: precious little. The sources are extraordinarily scarce. A mere 25 mathematical texts – nearly all incomplete – have been found from the whole of Ancient Egypt’s three-millennia-plus history. About half are from the Middle Kingdom (2055-1650 BC) and the rest from the Ptolemaic and Roman Periods, a void of some 1500 years – and nothing at all from the Old Kingdom’s pyramid age. That’s it. And even that meagre collection isn’t entirely revealing to the modern researcher: as Imhausen remarks wryly, these texts ‘are very explicit and detailed (and still they sometimes puzzle us!)’.
Those few specifically mathematical texts, setting out principles and techniques, are, fortunately, supplemented by other sources – administrative documents, wall-paintings and the ‘autobiographies’ inscribed in the tombs of officials, particularly scribes (who had to be proficient in mathematics as well as literate) – which show the maths being applied, allowing some deductions about the underlying concepts and methods. But even so, there’s not much to go on. Small wonder Imhausen admits that, although her book is the fruit of ten years’ research, its findings are ‘temporary at best’ because so much about the subject remains to be discovered.
There’s nothing in the surviving texts that suggests the Egyptians had a concept of pure mathematics, or a Pythagorean-style metaphysics based on number, as many of the alternative theories assume – but with so little to go on it’s impossible to say for sure that they didn’t. I got the impression – although Imhausen doesn’t say it in so many words – that while the Ancient Egyptian’s mathematical and scientific mindset is now appreciated as being different it’s still not really understood.
The big gap in the record is in the Old Kingdom, which is particularly frustrating when it comes to the alternative theories as that’s where the real mystery lies. As Imhausen summarises, ‘Unfortunately, the surviving written evidence is extremely scarce. No mathematical texts are extant, and we do not have any documentation of a building project except for construction marks on the actual site.’ Ultimately, ‘How mathematical techniques to administer quantities of grain, to plan building projects, and others developed or what they were exactly at this time, we cannot say.’ The lack of evidence makes it impossible to track the evolution of Ancient Egyptian maths, either from its beginnings to the Old Kingdom or from that period on.
So, on the one hand, Imhausen shows that the old view of the Ancient Egyptians’ mathematical incompetence is out of date, removing one of the apparent anomalies that stimulated alternative theorists. On the other, however, the negligible amount of available information throws them a lifeline, as it’s impossible to say anything with certainty about the scope of the pyramid-builders’ mathematical and scientific knowledge, or how they came by it.
Imhausen doesn’t address the alternative theories – it’s not part of her remit - beyond lamenting ‘the publication of unfounded speculation about Egyptian mathematics (or even Egyptian science in general), which also had a recent boom with the Internet as a platform.’ However, while dismissive of such speculation, she is candid in her summing up of the current academic position:
‘The scarce source material has not been, and presumably never will be, able to answer every question asked in modern times. This has encouraged rather speculative theories founded on practically no evidence; in fact, it is exactly the lack of evidence that has enabled speculations of this kind. Classic examples are the methods that were used to align and build the pyramids as well as the way Egyptian mathematical knowledge was discovered. The honest and academically responsible answer to most of these questions (at least at the present moment) would simply be that we don’t know.’
Imhausen says, rather optimistically, that one of her book’s aims ‘is to encourage its readers to judge speculations about Egyptian mathematics with a critical and informed eye.’ However, with so little to inform the eye, any attempt – academic or fringe - to address such questions as the mathematical abilities of the pyramid builders is inevitably ‘founded on practically no evidence’.
In the end, Mathematics in Ancient Egypt is a book about how little we know about mathematics in Ancient Egypt. Good news for alternative Egyptologists, as at the very least the assumptions on which their theories are based can’t be proven wrong… -- Clive Prince