The mystical Eye of Horus / capacity system submultiples

Here is a story about number systems:

The wd3t is the eye of the falcon-god Horus, which was torn into fragments by the wicked god Seth.  Its hieroglyphic sign is made up of the fractional powers of 2 from 1/2 to 1/64, which sum to 63/64.  Later, the ibis-god Thoth miraculously ‘filled’ or ‘completed’ the eye, joining together the parts, whereby the eye regained its title to be called the wd3t, ‘the sound eye’.   Presumably the missing 1/64 was supplied magically by Thoth.



This is my retelling, using many of the same phrases, of Sir Alan Gardiner’s account of the ‘eye of Horus’ symbol used for notating measures of corn and land in his classic Egyptian Grammar (§ 266.1; 1927: 197).   It’s a nice story, and it is repeated again and again, not only in wacky Egypto-mystical websites but in a lot of serious scholarly work up to the present day.   I talk about it in Numerical Notation.   But is it true? Well, that depends what you mean by ‘true’, but mostly the answer is: not really.  As I mentioned in a post back in 2010, this is certainly not the origin of the symbols.  Jim Ritter (2002) has conclusively shown that these are ‘capacity system submultiples’, which originated in hieratic texts, not hieroglyphic ones, and appear to have had non-religious meanings originally.     Even while insisting on the mythico-religious origin of the Horus-eye fractions, Gardiner himself (1927: 198) is crystal clear that all the earliest ‘corn measures’ are hieratic.  The hieratic script is very different in appearance and character than the hieroglyphs, being the everyday cursive script of Egyptian scribes, rather than the monumental and more formal hieroglyphs.   Ritter shows conclusively that in their origin, and their written form, and their everyday use, the capacity system submultiples have nothing to do with the Eye of Horus.

Ritter distinguishes this “strong” thesis from a “weak” version, in which, many centuries after their invention, the hieratic capacity system submultiples were imported into the hieroglyphic script and that some scribe or scribes wrote about them as if they could be combined into the wedjat hieroglyph.  This weak version has more evidence for it, but as Ritter points out (2002: 311), this “does not automatically mean that ‘the Egyptians’ thought like that; for example, those Egyptians whose task it was to engrave hieroglyphic inscriptions on temple walls.  Theological or any other constructs of one community do not necessarily propagate to every other; the Egyptians were no more liable than any other people to speak with a single voice.”  This is a sociolinguistically-complex, reflective view that I think is essentially correct, and which I adopt in my work (although I would rewrite it today to be even clearer, as I hope I have above).   Ritter is not fully convinced by the weak thesis either, but acknowledges that it is tenable.

Ultimately, as Ritter concludes (correctly), our willingness to buy into the ‘Horus-eye fractions’ model tells us a lot about how we view the hieroglyphs, and Egyptian writing in general, as mythically-imbued and pictorial in nature, and ultimately reflects a mythologized view of Egyptians as a ‘mystical’ people, an ideology that goes back to the Renaissance and earlier in Western thought (Iversen 1961).  But I would go further, because it is about more than just Egypt.   We like stories that give numerological explanations for numerical phenomena, regardless of their veracity, and especially where the numerical system under consideration is from societies we conceptualize as having a more mystical or mysterious relationship with the world than we purportedly do.   Very often we are projecting our image of what is going on.  This isn’t to say that Gardiner’s description is wrong – he knew the texts better than almost anyone, and correctly identifies how the system worked and the texts in which it was found.  But it’s important that when (some) Egyptians transliterated the capacity system submultiples from hieratic to hieroglyphic writing and formed them into the wedjat, they were repurposing and transforming a pre-existing set of signs that had no mystical origin whatsoever.   It deserves our attention, both for what it tells us about Egyptian life  and also for its importance for the historiography of science, mathematics, and religion in non-Western societies.

(Thanks to Dan Milton, who as the winner of the contest last week asked the question that motivates this post.)

Gardiner, Alan H. 1927. Egyptian grammar: being an introduction to the study of hieroglyphs. Oxford: Clarendon Press.
Iversen, Erik. 1961. The myth of Egypt and its hieroglyphs in European tradition. Copenhagen: Gad.
Ritter, Jim. 2002. “Closing the Eye of Horus: The Rise and Fall of ‘Horus-eye fractions’.” In Under One Sky: Astronomy and Mathematics in the Ancient Near East, edited by John M. Steele and Annette Imhausen, 297-323. Münster: Ugarit-Verlag.

Human Expeditions: Inspired by Bruce Trigger

small title imageHuman Expeditions: Inspired by Bruce Trigger

Stephen Chrisomalis and Andre Costopoulos, editors

Human Expeditions: Inspired by Bruce Trigger, which was published recently by the University of Toronto Press, is a book that Andre Costopoulos and I envisioned shortly after the death of our friend and mentor Bruce Trigger, who was my dissertation supervisor.  In helping to sort through his papers, we became aware that he had developed, over several decades, a network of eclectic and important scholars, including many of his own students, whose work did not fit into conventional theoretical or disciplinary categories or whose serious ideas had not received adequate attention.  We also were reminded of how unconventional much of Trigger’s own work was, with articles such as ‘Brecht and ethnohistory’ and ‘Akhenaten and Durkheim’ among his eclectic works.  But the book was created not as a sterile memorial to Trigger, but rather, as a way to think about scholarship that is, “unfinished, unbegun, or even unthinkable, in the present intellectual climate”.

Human Expeditions is a decidedly ‘unfashionable’ book, and we are proud of that fact.  We identified people whose work is poorly characterized by ‘isms’, and asked them to share work that filled gaps in present thinking.  The contributors to the volume come from the philosophy of science, history, and Egyptology as well as anthropology and archaeology.  The result, we hope, will give greater depth to anthropological insights and greater conceptual breadth to the humanistic social sciences.  A number of contributors are very senior figures in their fields, but we believe it is just as critical to include contributions from early-stage scholars, including several who were students of Trigger in his final years.  At minimum, we want to provide immediate venues for important scholarship that lies outside disciplinary norms.  At its most utopian, Human Expeditions allows us to envision, “an alternate history of the social sciences in which conformity to convention is not an expectation,” and to think about different configurations of disciplines than those currently in fashion.   We think that Trigger would have approved heartily.

Screws, hammers, and Roman numerals: An allegorical complaint

Let’s imagine that you have a toolbox in your garage, full of all sorts of different useful things, and I’m your annoying neighbor.  One day I drop by while you’re working.  I rummage around, pick up a screwdriver, and say to you, “Gosh, that’s not a very good hammer, is it?”  Naturally, you protest that it isn’t a hammer at all.  Next, I hold the screwdriver by the head instead of the handle and say, “Well, of course, you could use it like this to bang in nails, but it would be very cumbersome.”  You look at me, wondering whether I didn’t hear you properly, and say, “No, really.  It’s not a hammer. I have a hammer, but it’s in the trunk of my car, and that’s not it.”  I turn to you and say, “Well, I’ve never seen your hammer, and it would really be a lot easier if you just used the handle of a screwdriver to bang in nails.  Except that it’s no good for that.”

Now let’s turn from this surreal Pythonesque world to another scenario.

You’re an epigrapher and you find some inscriptions with some Roman numerals.  You look at them and say, “Gosh, those things aren’t very good for math, are they?”  Of course, the writer is dead, so he/she doesn’t say anything.  Next, you fiddle around with the numerals and think to yourself, “Well, look at that!  You could use those for arithmetic if you wanted to, but it would be very cumbersome.”  Again, the writer is not around to protest, although as it turns out, someone else dug up an abacus a few kilometers away.   You think of that, though, and say, “Well, it would really be a lot easier if they had just used numerals to do arithmetic, except that their numerals are no good for that.”

So this is the world I live in, and this is the battle I fight.

The problem is a cognitive and ideological one. We are so attached to the idea that numerals are for arithmetic that it’s very hard to stop and ask whether number symbols were actually used for doing calculations in a given society.  There’s essentially no evidence that Romans or anyone else ever lined up or computed with Roman numerals on papyrus or slate or sand or anything else, while there’s abundant evidence that they used an abacus along with finger-computation.  This should give us pause, but our cognitive bias in favour of the numeral/math functional association overpowers it.    For almost all numerical notation systems used over the past 5000 years, there’s precious little evidence that numerals were manipulated arithmetically.  You might have a multiplication table, or you might write results, but you wouldn’t line up numbers, break long numerals into powers to work with them, or anything of the sort.   And since we don’t know that much about abaci and other arithmetic technologies, even though they were obviously used for arithmetic, we assume (wrongly) that they certainly could never be equally good as written numbers.  And thus we conclude (finally, wrongly, again) that Romans were hopeless at arithmetic.   We might even blame their (purported) lack of mathematical proficiency on their lack of a ‘good’, ‘efficient’ numeral system.

It’s a casual, all-too-easy ethnocentrism, and hard to detect.  It’s not the nativistic, “our ways are good, your ways are bad” ethnocentrism that we mostly know to avoid.     Because arithmetic as it is presently taught almost everywhere relies on the structure of the positional decimal numerals, lined up and manipulated as needed, it takes on a naturalness that is deceptively difficult to untangle.   Yes, the Roman numerals are quite difficult to use if you presume that the way to use them is to break them apart, line them up, and do arithmetic in something like the way we were taught.   This isn’t to say that the functions of technologies aren’t relevant, but if we decide in advance what their functions must be, we are likely to miss out on what they actually were, and our judgements will be compromised.

To hammer the point home: if we do that, we’re screwed.

Martin Bernal (1937-2013)

I heard the sad news today that Martin Bernal, the political scientist/historian/classicist and ‘humanist’ in every relevant sense of the word, passed away on Monday at the age of 76 (follow the link for his obituary in the Ithaca Journal).  At the time of his death, he was Professor Emeritus of Government and Near Eastern Studies at Cornell.  One can only imagine that he was, and will be, the only figure to hold simultaneous professorships in these two fields.  He was also a genial, considerate scholar, and it was my honour to have worked with him.

Bernal is, of course, best known for his three-volume Black Athena (Bernal 1987, 1991, 2006), a massive attempt to show the indebtedness of classical civilization to Egyptian and Phoenician influences and that Greek civilization was only secondarily Indo-European but principally an African and Near Eastern civilization which, due to racism among European early modern scholars, was not recognized as such.  To say that it was controversial is a gross understatement – few claims in the study of the ancient world have attracted as much scorn, including an entire edited volume dedicated to its refutation.  The scholarly consensus today is that Bernal’s linguistic, archaeological and historical evidence is too rough-and-ready and that he was too willing to take coincidence as evidence when considering similarities in the languages and symbolic lives of Greeks and Egyptians.  The Greek pantheon is not simply a set of African deities with a European veneer, any more than the Greek language is some sort of bizarre mixed language full of Semitic and Afro-Asiatic roots.

These are serious problems, and to his credit, Bernal did attempt to address them, not always successfully. I do not, however, agree with the assessment of some that he was credulous.  Rather, I view his work as a conscious attempt to provide a counterpoint to mainstream views, even when – or especially because – the ideas he was proposing were so challenging. He was unafraid to be wrong if the alternative was to be silent.  Unlike, say, Erich von Daniken, to whom he has been most unjustly compared, Bernal’s work was meticulously thorough in its citation, and rather than simply postulating massive conspiracies, asking ridiculous rhetorical questions, and dishonestly ignoring all contrary evidence, he was a very serious scholar, sometimes out of his depth, but never out of his mind.

I corresponded with Martin over the past five years (we never met in person) during the preparation of Human Expeditions: Inspired by Bruce Trigger.  His chapter in that volume, ‘The Impact of Blackness on the Formation of Classics’ (Bernal 2013) will surely prove to be one of his final published works.  I do not know if he had an opportunity to read the entire volume prior to his death, but I want to echo the remarks I made in the introduction: there is enormous value in Bernal’s demonstration of the role of implicit and explicit Eurocentric biases in shaping the course of classical scholarship from the 17th century to the present day.  His chapter expands on his earlier work by addressing the role of some key scholars, such as James Bruce, in shaping views of Greece and of Africa, and their interrelations.   He insists that postulating links between societies does not stigmatize borrowing: “I do not accept that hybridity leads to sterility; a culture is not a mule!” (Bernal 2013: 14).  Throughout the process of editing and revision and production, Martin was cheerful, thoughtful, and open, responsive to editorial comment and a true professional.

I wrote Bernal to invite him to participate in the volume, knowing full well that Trigger had been publicly quite critical of Black Athena and its revisionist assertions.  Indeed, that was part of the point.  Bernal exemplified an unflinching willingness to spend decades on an extreme version of a position that, over time and with much refinement and revision, has come to limited acceptance in some scholarly communities, at the cost of great scorn from others.  You might count this in the ‘loss’ column, if scholarship were measured in wins and losses.  But this was never the point.  The point is the process, and both Bernal and Trigger exemplified the principled willingness to present a point of view regardless of its conformity to some present orthodoxy.  In my view, too much anthropology and archaeology, and quite a lot of linguistics, lack this principled willingness to challenge, to doubt even when doubting has costs, to publish the unpublishable – a luxury his position afforded him, but alas, too few of us today enjoy such freedom.

In 2002, when I was just finishing my PhD and writing my first published article, ‘The Egyptian origin of the Greek alphabetic numerals’, I un-courageously felt the need to distance myself from Bernal in writing, “This theory is a further contribution towards delineating the economic and intellectual transfers between Egypt and Greece in antiquity, though of course at a later date and with a more secure contextual foundation than that found in Bernal’s (1987) analysis of the subject.” (Chrisomalis 2003: 58).  It’s a true statement, but at the time, what I failed to recognize is that I could not have had my hypothesis considered fairly without the sort of framing that Bernal’s body of scholarship afforded me.  I am grateful, a decade later, to have had a chance to know the man and return the favour.


Bernal, Martin. 1987. Black Athena : the Afroasiatic roots of classical civilization. Vol. 1: The Fabrication of Ancient Greece, 1785-1985. New Brunswick, N.J.: Rutgers University Press.

Bernal, Martin. 1991. Black Athena : the Afroasiatic roots of classical civilization. Vol. 2: The Archaeological and Documentary Evidence. New Brunswick, N.J.: Rutgers University Press.

Bernal, Martin. 2006. Black Athena : the Afroasiatic roots of classical civilization. Vol. 3: The Linguistic Evidence. New Brunswick, N.J.: Rutgers University Press.

Bernal, Martin. 2013. The impact of blackness on the formation of classics.  In S. Chrisomalis and A. Costopoulos (eds.), Human Expeditions: Inspired by Bruce Trigger, pp. 12-30.  Toronto: University of Toronto Press.

Chrisomalis, Stephen. 2003. The Egyptian origin of the Greek alphabetic numerals. Antiquity 77 (297): 485-496.

Three new anthro-blogs

Readers of Glossographia may be interested in three new anthropology blogs that have popped up over the past month:

Archaeogaming focuses on the intersection of archaeology and video games, and promises to be a lively discussion – it’s brand new this week!  Those of you who may not be up-to-date on differences between artifact types in the Elder Scrolls series may nonetheless be fascinated by the most recent post, ‘Video Game Archaeology in Meatspace‘, where Andrew Reinhard deals with the science and ethics of the recently announced excavation/looting to take place in Alamogordo, New Mexico at the site of the infamous Atari Dump Site where, purportedly, 14 truckloads of unsold E.T. cartridges were discarded and cemented over in 1983.

Bone Broke is authored by Jess Beck, a former student of mine who now studies bioarchaeology at the University of Michigan and whose blog will feature material on osteology and other related topics.  I particularly like her post ‘Taylorism and Teaching‘ where she develops a research protocol for evaluating the evolution of the hand and/or the qualifications of undergraduates for industrial line work (hold the comments on the job prospects of anthro majors, please).

The Human Family seems at first glance to be what would result if you brought Zombie Lewis Henry Morgan to life and sat him in front of a computer.  But far from just recapitulating classic theory in kinship and social organization, the author brings it into contemporary relevance with his discussion of ‘Pedigrees, genealogies, and same-sex parents‘, showing the continued practical applicability of kinship studies for modern biological and social relationships.

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