The ancient Egyptian system of measures provides another example of number signs conceived as a coherent system. In the so- called “Horus-eye fractions”1, the designer of a numerical sequence linked its members also into a unified whole derived from a myth, just as in the series of numerals for the powers of ten.
Gay Robins and Charles Shute describe this series of measures in their book on “The Rhind Mathematical Papyrus”, an ancient Egyptian mathematics text for apprentice scribes :
“The common unit of volume, used for measuring amounts of grain or flour, was the hekat, approximately equal to 4.8 litres or just over a [British] gallon. (...)
For smaller amounts, the hekat could be progressively halved to give /2, /4, /8, /16, /32, and /64 fractions. (...) [These ] are known as the Horus-eye fractions, because they were written with distinctive signs that resemble the parts of the eye of the falcon- headed god Horus, known as the wedjat-eye.
In Egyptian mythology the eye of Horus was wounded, wrenched out or eaten by the fearsome god Seth. Later it was restored and made whole, according to spell 17 in the Book of the Dead, by the ibis- headed god Thoth, the originator of mathematics, who ‘did this with his fingers’.
The question has been raised whether this phrase referred to ‘finger- counting’, and whether it could relate in any way to the Horus- eye fractions.
These fractions in modern terminology form a convergent geometrical progression of six terms with the first term equal to the common ratio. Problem no. 79 of the Rhind Papyrus makes it clear that (...) the Egyptians would have been able to sum the series (...) to 63/64.
They could also have shown that the sum was short of 1 by 1/64. (...) If the Horus - Seth - Thoth story really had a mathematical connotation, it could be that the damaged Horus- eye was magically made whole by the restoration of the missing 1/64.”2
The magic required to complete this Eye made from numbers introduces us to the theme of this book -- the ancient function of the intangible yet knowable numbers as bridges to the equally intangible but more mysterious realm of gods and souls.
The commonly used term “wedjat” for the Horus eye meant “the sound Eye” and so refers to it after its magic restoration.
This so completed Eye had great symbolic importance. It provided protection in amulets; it guided boats that carried it on their bow; pairs of it painted on coffins allowed the deceased to see in the afterlife; its damage and healing reflected the waning and waxing of the moon; and it stood also for the religious act of offering as well as for an offering itself3.
The Pyramid Texts from the late Old Kingdom, for instance, have Thoth the number god bring this Eye as an offering table, followed by long litanies in which almost every Utterance exhorts the dead king to “take the Eye of Horus” as an offering which is then identified with various grave goods4.
Significantly, the Horus- Eye hieroglyph and its hekat- components always omit the pupil which would have been needed to complete the eye.
The corresponding graphic element, a circle like that for the quarter- hekat iris but smaller, was used to designate whole hekats from one to nine5, so the missing part stood for the entire unit which only it could complete,
This is a typical example for the pars- pro- toto method of distilling things to their essential part in which the hieroglyph designers excelled greatly. The pupil was clearly the sum and substance of the Eye because only the pupil lets light through for seeing, and looking into someone’s pupils gives the impression of revealing their innermost depth.
The pupil or iris of the eye sign in the hieroglyph- based proto- alphabet of the Sinai Script also led to the circular “o” of the Phoenician and Hebrew alphabets from which the letters you read here are derived6.
The pupil of the Horus Eye had special magical powers for reviving the deceased, to judge from a Pyramid Text Utterance where the priest tells the dead king to “take the pupil which is in the Eye of Horus, for your mouth is split open by means of it.”7
The ceremony of “opening the mouth” formed the core of the funerary rites because it restored breath and life to the mummy, and the pupil was probably needed for that magic because it seems to have been connected with the beyond.
The symbologist Manfred Lurker draws attention to a peculiarity which suggests this proposed association:
“It is strange that Egyptians described the pupil of the eye as ‘white’ although this part of the eye is not at all that color. (...) White was the color of purity and sanctity.”8
The actual color of the pupil is black. Although black could mean regeneration and fertility, as in “the Black Land” which was the ancient Egyptian name of Egypt, black was above all the color of night and death and the netherworld, and a circle filled in with black represented the black hole of non-being in which the world will dissolve at the end of time9.
The gist of the elaborate Egyptian funerary rites was to deny death by constantly reassuring the deceased that s/he was not dead but alive, and by replacing all allusions to death with euphemisms such as “the good day” for the day of death10 or "the beautiful house" for the smelly embalming shack11.
The unnatural denial of the real color for the pupil is therefore a giveaway clue that the pupil of the Horus Eye was a window to that feared infinite darkness which the replacement of its color with the opposite may have helped to hold at bay.
The prominent role of the mathematically composed Eye of Horus in these rites makes sense in the logic of magical thinking because the missing part produced by the god had restored the badly damaged divine Eye and therefore would also be able to restore the equally damaged dead.
Moreover, the method for obtaining this powerful pupil linked it again to that boundless beyond: the progression of ever smaller successive halvings adds up to this missing amount only when it continues to awe- inspiring infinity -- that is, presumably, to where the gods dwelt and where the mummy owner’s soul was going.
Admittedly, nothing proves that the ancient number investigators knew the series must continue to infinity to add up to the missing part.
On the other hand, this missing part amounts to about 75 cc or 2.64 oz -- too much to not need further subdividing in practice when scribes had to apportion grain to hungry workers. Indeed. the users of the hekat system continued the splitting with the "ro-measure" or "mouthful" of about half a fluid ounce which turns up frequently in the medical papyri and was also consistently the measure of all the tablespoons studied from the first to the eighteenth Dynasties12.
Five ro amounted to the missing 1/64 of the hekat13, and one ro could be halved again and again, like the original hekat, in a second series of six consecutive terms that allowed the expression of morsels even smaller than individual grains.
In so continuing their halvings, the ancient scribes must also have realized what is now obvious to us, even without doing the math: that adding up the successive halvings of any quantity will come ever closer to the original unit so halved, but will never reach it exactly because the last remainder fraction so obtained can always be halved again, and again, without end, just as any integer can be doubled and redoubled as often as desired.
There is no a priori reason why people a mere five thousand years ago, with the same brain capacities as ours and with a well attested interest in numbers, would not have seen that the progressions of further halvings could be continued to infinity.
That the Horus Eye series includes only six components and needs a magically obtained seventh part to make this divine unit complete was therefore not due to any lack of computing skills among the pharaonic scribes. This system may be based on the same reasons why the common cubit was six palms long and needed a seventh palm to make it a royal and thus perfect or complete cubit, the measuring unit used for all sacred construction.
The magical power of the seventh part to complete a higher and holier unit than the common one composed of six parts may also be the reason why the later Hebrews, who relied on Egyptian mathematical methods and measuring concepts, had a week of six work days plus a seventh day belonging to God that made it complete and perfect.
This same magical power of the seventh part suggests also that the scribes would have interpreted its mind- boggling infinite divisibility in their religious framework, and that they would therefore have viewed that elusive pupil as connected with infinity and the gods.
The magical powers of the Horus- Eye survive today because it is still a popular design for amuletic jewelry, as illustrated by these modern examples made from gold in Egypt and offered in a shop that also sells crystals and accessories for divination.
In any case, whether the ancient Egyptians had recognized this mathematical link to infinity or not, it is clear that the signs for the successive Horus-Eye fractions were designed together as a system, and that this numerical system was related to a major mythical event.
You will now see that the hieroglyphic numeral signs for the powers of ten were also designed together as a system with six steps beyond unity plus a seventh that represented the divine realm, and that this system was also based on a major mythical event.
Return to Introduction
For a collection of excellent books on many aspects of mathematics, including "Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics by David Reimer", go to Princeton University Press at http://press.princeton.edu/math/
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