"Closing the Eye of Horus: The Rise and Fall of Horus-Eye Fractions" in J. Steele & A. Imhausen...

Alter Orient und Altes Testament Veröffentlichungen zur Kultur und Geschichte des Alten Orients

und des Alten Testaments Herausgeber: Manfried Dietrich • Oswald Loretz

John M. Steele − Annette Imhausen (Hrsg.)

Under One Sky Astronomy and Mathematics

in the Ancient Near East

Ugarit-Verlag

Münster

Closing the Eye of Horus:The Rise and Fall of ‘Horus-eye Fractions’

Jim Ritter, Paris

For nearly a century now students of Egyptology as well as of the History of Sciencehave paused in wonder when learning of the unique attitudes of Egyptians “religiousbeyond measure, more than any other people” (Herodotus, Histories, ch. 37); eventhat most mundane of subjects, weights and measures, was apparently imbued bythem with deep mystical purport. For part of their system of capacity measures wasapparently a product of theological speculation, based on one of their most potentmythological symbols, the Eye of Horus, dismembered by his evil brother Seth andthen restored to soundness (wDAt) by Thoth. Each part of the broken Eye in thisperspective was assigned a dimidiated fractional value—in descending order: 1/2,1/4, 1/8, 1/16, 1/32, 1/64, representing parts of the basic capacity unit, the HoAt. Thedefinitive form of this view was given by Alan Gardiner in the first edition of hisEgyptian Grammar:1

This is the account, and the image, that has continued to be used, often withoutattribution, in almost every mention of the Horus-eye-as-capacity-measure storyuntil very recent times.

I shall argue that just as the Holy Roman Empire was “neither holy nor Romannor an empire,”2 so the “Horus-eye fractions” were, in their origin at least, neitherfractions nor associated with the Eye of Horus. I have discussed elsewhere3 the firstquestion and I shall thus avoid speaking about fractions in this context—I shall use

1 GARDINER (1927), p. 197.2 “Ce corps qui s’appelait et qui s’appelle encore le saint empire romain n’était en aucunemanière ni saint, ni romain, ni empire.” Voltaire, Essai sur l’histoire générale et sur lesmœurs et l’esprit des nations (1756), ch. 70.3 RITTER (1992).

298 J. Ritter

rather the term ‘capacity system submultiples’ (CSM) and, in the absence of anyevidence for their phonetic values (pace the conjecture of SETHE (1916), p. 74), Ishall refer to them by the use of quotation marks around their purported fractionalvalues.

Since the question of sign forms—hieratic and hieroglyphic—are at the base ofthe following discussion, it is important to note the differences in evidential weightthat will be accorded to the signs in question. Of the six hieratic signs, two:(“1/16”) and (“1/32”) have shapes which are diagnostic, that is, are sufficientlycomplex and differentiated from other hieratic signs to enable one to map the rangeof synchronic variations and of diachronic development (the latter being ofparticular importance for us), while three: (“1/4”), (“1/8”) and (“1/64”) aretoo simple and/or too close to others to be of equal value for these purposes. Thethird: (“1/2”), falls somewhere between the two classes on the diagnostic scale.4

Similarly for the purported hieroglyphic equivalents, but here the diagnostic signsare (“1/2”), (“1/16”), (“1/32”) and (“1/64”); while (“1/4”) and (“1/8”)still remain less helpful.

If I am right that there is no solid evidence for the Horus-eye interpretation of theCSM signs the question arises just why the story has had such an imaginative holdon succeeding generations. This in turn opens up two questions, one specific to theEgyptological community and the other of broader application. The first questionconcerns the nature of Egyptian writing and how the Egyptological literaturetraditionally treats it. This will be a constant concern of ours during our examinationof the relationships between the Eye of Horus and the dimidiated capacity units andwe shall return to it explicitly at the end of our enquiry.

The second question concerns how conjectures, originally put forward in atentative fashion, lose their provisional character over time, even in the absence ofany change in the quantity or quality of the evidence in their favour. As I hope toshow in what follows, the case of the Eye of Horus and capacity units provides anearly paradigmatic example. But to do so requires us to go back to the origins andearly history of the hypothesis—or rather the two hypotheses—which lie at the basisof the story.

I. Origins and Reception

In 1911, Georg Möller at the Berlin Museum was nearing completion of hismonumental Hieratische Paläographie. The first two volumes had already beenpublished in 1909, covering the period from the Old Kingdom to the end of the NewKingdom. The structure of the work, with chronologically ordered examples ofhieratic script organized by hieroglyphic sign, necessitated a choice of hieroglyphic“original” for each hieratic sign.

The CSM hieratic signs posed no problem, they were known from a number oftexts: accounting, mathematical and medical.5 Their place in the capacity system had

4 All the examples are drawn from New Kingdom sources, see Table I.5 Sources in these categories used by Möller in the first two volumes of his HieratischePaläographie are: Administration: various Illahun fragments, pBoulaq 18, pLouvre E 3226,pRollin, pHarris; Mathematics: pRhind; Medicine: pEbers, pBerlin 3038.

Closing the Eye of Horus: The Rise and Fall of ‘Horus-eye Fractions’ 299

been authoritatively summed up twenty years earlier by Griffith.6 However theunique full set of the hieroglyphic CSM signs known at the time—from the festivalcalendar on the walls of Ramesses III’s temple in Medinet Habu—were availableonly in the rather mediocre hand copies due to Johannes Dümichen, andtypographical forms of these had had to serve in the first two volumes.7

But current work on photographs of votive cubits by Heinrich Schäfer and,especially, Ludwig Borchardt8 revealed the unexpected presence of hieratic CSMsigns integrated into the hieroglyphic inscriptions on the back of the cubits; and “onthe same cubits”, the hieroglyphic signs , , and . Since the last—decomposable into + — was recognizable as the underpart of the eye of Horus,the other signs could be identified with the same object. Thus the enigmatic hieraticcapacity-system signs and the until now equally enigmatic hieroglyphic signs fromMedinet Habu seemed to be intimately linked to the culturally significant wDAt-eyeof Horus. The third volume of Möller’s Hieratische Paläographie could now bepublished with the new hieroglyphic equivalents in the “Bruchteile vom ”section.9

To sum up rapidly here, Möller presents essentially two arguments for theidentification of the CSM signs with parts of the completed right eye of Horus: 1°the proximity of some hieratic and some hieroglyphic forms of the signs on twounpublished votive cubits and 2° the visual similarities of the remaining signs,which “can be traced back to parts of the ”.10 He finishes the article by adding afurther two points: 1° that the unpublished Old Kingdom Berlin papyrus 1050011

shows the antiquity of the system and 2° that “doubtless”, its origin is to be soughtin the mythological tradition of the wDAt-eye. Möller then reconstructs the outline ofthe myth from a number of disparate sources. In a final note, Möller underlines thetentative nature of this last: “We are conscious that, in these last parts of ourdiscussion, we have several times trodden upon the domain of hypothesis. In sodoing we have only intended to urge that further allusions in the mythological textsin question be explored.”12 However the emphasis on the age of the relationshipbetween CSM and the Eye of Horus is, for Möller, certain and a further proof of hisbelief that “we must count the birth of the greater part of the hieroglyphic script assymbolic games (Spielereien).” I shall call this original argument, that the CSM–Horus-eye equivalence is early and reflective of the development of Egyptianwriting, the strong form of the Eye of Horus fractions thesis.

6 GRIFFITH (1892), p. 426.7 MÖLLER (1909 a) and (1909 b).

The Medinet Habu calendar forms were introduced in BRUGSCH (1883) and the printingfonts based on these were used in GRIFFITH (1892). The photographic edition of MedinetHabu was only to appear forty years later (EPIGRAPHIC SURVEY (1934)).8 Borchardt’s preliminary results were published in the same issue of ZÄS as Möller’sarticle (BORCHARDT (1911)).9 MÖLLER (1912), sect. DD c.10 MÖLLER (1911), p. 100.11 This Sixth Dynasty accounting text, the so-called ‘Sharuna’ papyrus, remains unpublishedstill. Its hieratic forms for the CSM were not published until 1936, in the posthumousErgänzungsheft to the Hieratische Paläographie due to Grapow and Przybylla. (“MÖLLER”(1936), p. 1 s.v. Elephantine nos. 708, 710).12 MÖLLER (1911), p. 101, n2.

300 J. Ritter

For some dozen years after its publication, Möller’s thesis stirred little interest;for this period I have been able to locate only two minor references. The first is thatof Kurt Sethe in his influential Von Zahlen und Zahlworten.13 This is limited to apassing mention (p. 74) of the Horus-eye interpretation in the context of the questionof the Egyptian names for the area and capacity submultiples. The only details givenhere, both coming from Möller, are the “strange Spielerei” at the origin of this use14

and a résumé in note of the myth of the Eye. The second occurs in the last line of adiscussion by Battiscombe Gunn of the connection in Pyramid Spell 359 betweenthe Eye of Horus and a finger-counting ritual; “one is involuntarily reminded of themathematical properties of the WDAt.”15

It was only in 1923 that substantive comment was forthcoming. The publicationin that year by T. Eric Peet of the second, and best, of the Rhind mathematicalpapyrus editions included a commentary on the Möller thesis.16 Möller’s sketch isreproduced along with the various fractional values assigned to the eye parts. Butthis is followed by a denial of the antiquity of such an identification on two grounds:1° the hieratic forms “show but slight resemblance to the parts of the eye”17 and2° the hieroglyphic forms are much later, they do not predate Dynasty 18 or, forsome forms, Dynasty 20. The frequent use of the hieratic forms in the medical textsleads Peet to speculate that the Horus-eye identification was “an ingenious discoveryof some [medical] scribe” since the sacred eye was “so closely connected inEgyptian eyes with healing power.” This concludes Peet’s treatment and is the firstappearance in print of what I shall call the weak thesis: the hieratic CSM signs areearly but bear no relation to the Eye of Horus; the hieroglyphic versions do bearsuch a relation, but are of New Kingdom date at the earliest and the result of laterscribal speculation.18

However, the year 1927 saw the publication of the first volume of yet anotheredition of the Rhind papyrus, the work of Arnold Chace, Henry Manning andRaymond Archibald.19 The last two were mathematicians, the principal author abusinessman and chancellor of Brown University; the work was published by anassociation of American mathematics teachers and was aimed explicitly at“mathematicians and the general public” (p. III), rather than Egyptologists. Thediscussion of capacity measures includes a simple reference to “‘Horus-eye’fractions” (p. 31) with reference given to Möller’s article. None of Peet’sreservations about the origins of the CSM are even mentioned.

The existence of two near-contemporaneous editions of the Rhind Papyrus wasto have an important impact on the future of studies of Egyptian mathematics. Theexistence in later years of two distinct communities in this field, Egyptologists and

13 SETHE (1916).14 This is reinforced by a later (p. 80) mention of the Sixth Dynasty presence of thecomplete set of six CSM; an error, since no OK source known at the time contains thesmallest two CSM. The references given in note by Sethe are in fact discussions of Ptolemaicevidence for small-end area submultiples.15 GUNN (1922), p. 72.16 PEET (1923a).17 All citations are from PEET (1923 a), p. 25–26.18 Peet repeated the weak thesis the same year in his discussion of the MK writing boardswith metrological exercises from Akhmim, PEET (1923 b), p. 92.19 CHACE et al. (1927).


historians of mathematics, each with its own ‘Rhind papyrus’ has resulted in anunfortunate split, to which we shall return at the end of this section.20

It was in this same year that the definitive form was to be given of the strongthesis. It opens the section of Alan Gardiner’s Egyptian Grammar dedicated toweights and measures (the initial subsection is reproduced at the beginning of thisarticle).21 For the first time, and without commentary, the Eye is drawn in reversedorientation—contra Möller, it is now a left eye.22 None of Möller’s arguments arereproduced, only his mythological reconstruction is taken up, constituting essentiallythe whole of the subsection. These two aspects, the left-orientation of the Eye and adiscussion limited to the associated myth, are to be found in virtually every latermention of the Horus-eye thesis, whether they explicitly acknowledge Gardiner ornot.23

At roughly the same time, the Eye of Horus also makes its entry into AdolfErman’s work, in the first volume of the Berlin Wörterbuch24 and, two years later, inthe fourth edition of his Ägyptische Grammatik,25 represented in both by a copy ofMöller’s sketch of the Eye but, like Gardiner’s, reversed in orientation (and,exceptionally, without any mention of the myth).26 The Wörterbuch gives a Neo-Egyptian date for the metrological use of the Eye of Horus and the Belegstelle27

cites, beside the Möller article, only the late composition “The Teaching ofAmenope”. This implicit endorsement of the weak thesis was to receive furthersupport a few years later.

The publication in 1930 of the second great mathematical papyrus, that ofMoscow,28 was the occasion for the editor, Vasilij Struve, to give a brief reminder ofthe Horus-eye nature of the CSM (citing Peet and Gardiner) and a suggestion that anidentification of the Eye of Ra with the jpt-grain-measure (Amenope 18, 23) was atthe origin of the identification of the HoAt-grain-measure with the Eye of Horus.29

For this last, Struve, following Peet, points to a medical papyrus (pHearst 14, 2–3).If Struve’s mention of a source for the Eye signs of the CSM in a medical

papyrus of the New Kingdom can be understood as implicit support for the weakthesis, it was Otto Neugebauer’s article of the same year that provided the onlydetailed arguments for this position.30 In the first substantive discussion of thestructure of the Egyptian capacity measures since Griffith’s classic 1892 article,Neugebauer argues (p. 46), on the basis of calculational practices in the

20 See the discussion in IMHAUSEN, this volume and references therein.21 GARDINER (1927), § 266.22 In addition to the change in orientation, Gardiner modifies the form of and ,replacing Möller’s outline drawings by line drawings for these two elements.23 A partial exception is Richard Gillings’s influential 1972 work on Egyptian mathematics,where he returns, without argument, to a right Eye (GILLINGS (1972), p. 210–211); henonetheless takes up other parts of Gardiner’s modifications of Möller’s original drawing and,like Gardiner, omits all but the mythological justification for the identification.24 ERMAN and GRAPOW (1926 a), p. 402.25 ERMAN (1928), p. 105.26 To complete the three classic grammars, Gustave Lefebvre’s Grammaire of 1940 followsGardiner on both points, image of left Eye and mythical reference (LEFEBVRE (1940), p. 111).27 ERMAN and GRAPOW (1926 b), p. 67.28 STRUVE (1930).29 STRUVE (1930), p. 51.30 NEUGEBAUER (1930).

302 J. Ritter

mathematical papyri, that there were originally two independent systems of capacitymeasure: one based on the HoAt (and hnw) and the other on the rA. Some of theHorus-eye signs are to be attached to the second and not the first, i.e., they wereoriginally multiples of rA and not fractions of a HoAt. Thus the hieratic equivalent of

is to be understood not as 1/32 HoAt but as 10 rA, and (probably) the hieraticequivalent of as 20 rA. When the two systems were put together, the largemetrological ‘space’ between the rA and the HoAt was filled with additional hieraticsigns, presumably drawn from a Horus-eye interpretation, in a systematized dyadicstructure into which the two earlier rA -multiples were integrated.

In a last section on the paleography of the capacity signs, Neugebauer reinforceshis argument by pointing out that the hieratic signs for “1/16” and “1/32” bear norelation to their supposed Horus-eye equivalents but do appear to resemblereconstructed hieratic forms of hieroglyphic (20 [rA]) and (10 rA) respectively.Adding to this the fact that all “1/2” hieroglyphic forms can only be correlated totheir hieroglyphic Horus-eye counterpart by means of a reflection (right eye versusleft eye), he concludes that the introduction of the Eye of Horus in this context, likea number of the fractions themselves, is of late date.

The only reaction to Neugebauer’s assault on Möller’s strong thesis I know ofwas a brief note in the following issue of the same journal by another historian ofancient mathematics, Kurt Vogel.31 And even here the main thrust of his defense ofthe traditional understanding is based on a denial of the independence of separateHoAt and rA systems; noting that the occasional writing of hieratic “1/16” and “1/32”with superfluous dots above them show their essentially fractional nature. Neither henor any later writer save one would ever respond to the question of the validity ofthe resemblance of the Horus-eye parts and the CSM signs.32

Thus by the beginning of the nineteen-thirties, the Horus-eye interpretation hadentered the two canonical grammars and, especially in the form given to it byGardiner, was to become the standard interpretation for Egyptologists and historiansof science in the succeeding decades. This was true very early; indeed neither Struvenor Neugebauer nor Vogel cites Möller’s article but only Gardiner. Moreover, sincethe presentation in the grammars was necessarily abbreviated with, at most, a shortrecapitulation of the proposed mythical background but without Möller’s arguments,these last disappeared completely from the literature.33 The authors of the onlysubstantive article on capacity metrology before the nineteen-sixties contentthemselves with copying both Gardiner’s picture and part of his text (p. 90–91).34

31 VOGEL (1930).32 The only later reference to Neugebauer’s publication is the final phrase of the article onthe Eye of Horus in the Lexikon der Ägyptologie (MÜLLER-WOLLERMANN (1985), col. 824)where the author defends the weak thesis, citing specifically the paleographical section.33 Later references to Möller are almost invariably “priority references”; citations ofhistorical interest for purposes of acknowledging priority, but without use (or often realknowledge) of the original. The illustration, if there is one, and the accompanying text arealmost invariably Gardiner’s.34 LUCAS and ROWE (1940). The reason for the excursus by archeologist Alan Rowe onHorus-eye CSM signs in this article is not clear since, despite his statement on p. 91, suchsigns do not appear on any of the inscribed vessels under discussion.


If we try to understand the dynamics of the vicissitudes of the Horus-eye thesisover this twenty-year period, we are led, I feel, to two specificities of the domain.On the one hand, the absence of new data or even the publication of the old, led to afreezing of the state of the debate once initial positions had been staked out. Theenshrinement of an abbreviated version of Möller’s strong thesis in the form of adrawing and a little mythological story in the necessarily confined context of agrammar settled the question for the great majority of mainstream Egyptologists,particularly as, for most of them, their interest did not generally run in this period toaccounting texts or the like.

This ebb of interest in matters metrological was accentuated by the rise ofspecialization within Egyptology, with matters of ‘science’ (including metrology) inEgypt coming increasingly under the exclusive purview of historians of science orof medicine, often without first-hand knowledge of the Egyptological literature.Coming from a different domain, with questions and tools developed elsewhere, theinterests of these last, until very recently, simply did not bear on metrologicalquestions, viewed only as impedimenta to the study of the ‘serious’ conceptualmathematical or medical theories extractable from the texts.

The mathematician’s Rhind papyrus was that of Chace et al., Peet’s edition—andhis reservations—remaining for the most part unknown; thus practically alltreatments of Egyptian mathematics, if they mention weights and measures at all,uncritically accept the strong Horus-eye thesis.

In medicine the situation is even simpler. Almost none of the major publicationson Egyptian medicine even mention metrological questions and those few who dooften make no mention of the Eye of Horus. Even the massive Grundriss derMedizin der alten Ägypter reserves its whole discussion of capacity units to the finalsupplementary volume35 where it ignores completely the nature of the CSM signs, ablackout extended to any mention of capacity units at all in the 250 pageintroduction to the most recent edition of medical text translations.36

The division of the field and the delegation of interest in the CSM signs tohistorians of science for nearly half a century had another effect. When the oldsources used by Möller in his arguments were finally published and new evidenceconcerning the signs started to appear in mainstream Egyptological publications, ithad almost no impact. Those who read such things did not have metrologicalquestions in mind and historians of science, even with the rebirth of interest in thenineteen-eighties of questions concerning metrology and the development ofwriting, had no accesss to this new information. It is thus to a survey of this newmaterial that we now turn.

II. New Evidence

It was the publication of new texts—and the reinterpretation of previously-knowndocuments—in the 1960s and 1970s which permits a new look at the question.These new approaches open up a glimpse of the situation in the third millennium aswell as access to some of the then-unpublished evidence cited by Möller. We shallsee that the results of both tend to the same end, a weakening of the whole idea of

35VON DEINES et al. (1973), p. 1–16.

36 BARDINET (1995).

304 J. Ritter

“Horus-eye fractions”. In this section we shall resume the new evidence, bothhieroglyphic and hieratic, stemming from third-millennium sources.

The discovery and publication37 of the Fifth Dynasty tomb of Nj-cnh~-H ;nm andH ;nm-h9tp at Saqqara, a particularly rich source of metrological information,38 gavethe first secure glimpse of a hieroglyphic Horus-eye fraction in the Old Kingdom. Inparticular, the well-preserved market scene on the north wall of the “gate room”shows a client, carrying a number of objects, approaching a fruit vendor:39

The fruit seller states: “Give (me) this grain-measure so that (I) may measure outthis in it.”. Though the detailed analysis of the customer’s reply is unclear,40 thedouble presence of a clearly marked “1/2” CSM sign following the grain-measuresign in both parts of the dialogue is certain.

Possible evidence for hieroglyphic CSM had been in fact available even earlier,in the publications of Archaic period tombs at Saqqara and Helwan during the inter-war period. These revealed Second Dynasty offering stelae on which were inscribedbread lists of a new kind, one in which capacity system signs, including and ,apparently appeared as part of the name of the bread.41 As was first suggested bySelim Hassan, among these the “1/2” CSM occasionally figures.42 The earliest clearexample of this sign is on the stela of a Second Dynasty royal princess named Sj-h9fnr:43

37 MOUSSA and ALTENMÜLLER (1977).38 This includes not only capacity measures but also information on length and weightsystems, see RITTER (forthcoming).39 MOUSSA and ALTENMÜLLER (1977), fig. 10 and pl. 24.40 Moussa and Altenmüller suggest (p. 84), rather unconvincingly, that the shopper replies:“Oh foreigner(?), should I give (you) the director’s half grain measure?”41 The identification of breadstuffs by content in grain quantity recalls the psw-baking ratiosin later mathematical texts and offering scenes. For the former, see now IMHAUSEN

(forthcoming); the latter will be discussed in RITTER (forthcoming).42 HASSAN (1944), p. 95, n2. Followed in this by Winfried BARTA (1963), p. 16 and 27, andJochem KAHL (1994), p. 443 s.v. D 11. Note that this interpretation is combated by PeterKAPLONY (1963), p. 248, who sees here a form of the sign , to be read wDb. However, ashis argument is based precisely on the dissimilarity between the sign and its Horus-eyecounterpart, it begs the question raised here.43 QUIBELL (1923), pl. 27.


The appearance of “1/2” in alternance with the clear capacity-measure sign and thebread name dp(t), well-known from later sources, renders this interpretationplausible at least. The difficulty is that this sort of bread name disappears early inthe Third Dynasty, so that later phonetic or other clues are unavailable. However ifthis interpretation is correct, there are four sets of the hieroglyphic form of the “1/2”sign at a very early stage.44 (See Table I).

Thus the third-millennium hieroglyphic evidence—like every hieroglyphicinstance until Dynasty 19—involves only the largest CSM. The mutability of form,even in those cases where the same text contains more than one copy of the sign(inscriptions of Sj-h9fnr and, especially, of Nj-cnh~-H ;nm and H ;nm-h9tp) shows thekind of variability that normally is typical of attempts to render in hieroglyphic formsigns coming from a hieratic metrological context.45 But none of the variants showthe broad opening and pointed tail, typical of the small white of the Eye of Horus, ,which, in the New-Kingdom examples of this CSM sign, had so largely impressedMöller and others.

Turning to hieratic CSM signs in the third millennium, practically all theevidence is again comparatively recently available. We recall that the only third-millennium evidence known to Möller was an unpublished sixth-dynasty accountingpapyrus (pBerlin 10500 A & B), examples of whose CSM signs were first publishedin 1936, in the Ergänzungsheft to the Hieratische Paläographie.46 Examining theexamples drawn from these papyri47 and shown in Table I, the diagnostic “1/16”sign, , is the only one to show a possibly significant difference with its latercounterparts, such as the Dynasty 12 form, .48 That this difference is indeedsignificant will become apparent as we now turn to even earlier forms.

44 These are the Second Dynasty stele of Sj-h 9fnr (QUIBELL (1923), pl. 27) and Jmtj (SMITH

and SIMPSON (1981), pl. 32) from Saqqara and Dw3t from Helwan (SAAD (1957), pl. 6 andKÖHLER and BIRRELL (forthcoming)); and the Third Dynasty stela of Jry-stt / Jc-Hjpw (?) fromSaqqara (KAPLONY (1966), pl. 4). Hassan’s proposal for the presence of this sign on the“Bankfield” stela (GARDINER (1917), pl. 55) is not convincing.45 Compare the hieroglyphic forms of the area submultiples, e.g., the Old Kingdomexamples in SCHÄFER (1902), p. 34–36 and JACQUET-GORDON (1962), p. 3 n2 and passim. Ingeneral, see BAER (1956) and RITTER (forthcoming).46 Examples of these signs (for all but the smallest two CSM) have been republished inGOEDICKE (1988), p. 70–71.47 I should like to thank the Staatliche Museen zu Berlin for providing me with photographsof these papyri.48 I have been unable to locate an appearance of the sign “1/32” in this papyrus, apparently

306 J. Ritter

It was not until 1968 that an edition of Old Kingdom accounting texts waspublished,49 with numerous fragments from the archives of the mortuary temple ofthe fifth-dynasty pharaoh Nfr-jr-k3-Rc K3-k3⋅j in Abusir. Only the largest three ofthe CSM signs—the least diagnostic—are represented in these fragments and theiressential identity with those of the Sharuna papyri is clear.50

The oldest known accounting papyri are those found at the site of Gebelein byGiulio Farina in 1935 and dated to the Fourth Dynasty.51 Though still unpublished,the paleography of their capacity system signs has been the subject of an importantarticle by Paule Posener-Kriéger,52 including a table of sign forms from which theCSM entries for the Fourth Dynasty in Table I have been taken. Here the full rangeof CSM signs is available, a situation which has no parallel until the MiddleKingdom.

The information contained in these third-millennium hieratic sources—andespecially in the Gebelein papyri—seems to me decisive against the strong thesis ofMöller. Not only do these papyri confirm clearly the line of evolution of the “1/16”sign, already suggested in the Sharuna papyrus, but they show that the “1/32” and“1/64” too are quite different than what would be expected from the Middle andNew Kingdom instances. Even the non-diagnostic large CSM signs reserve somesurprises; the “1/8” shows no sign of curvature, the “1/4” is more reminiscent of asolid dot than the open versions occurring later. All this points to the fact that noneof the early CSM hieratic signs—with the conceivable exception of “1/2”—resemble their putative hieroglyphic Horus-eye counterparts.

Another possibly useful avenue of comparison might have been that betweenhieratic and hieroglyphic CSM signs and those representing parts of the (Horus) eyein non-metrological third-millennium contexts. Here however there is a difficulty;neither on the hieroglyphic side, where only the “1/2” sign (D 11) comes intoquestion, nor on the hieratic, where all the eyeparts (D 11–D 16) are relevant, do weknow of any non-metrological use of these signs before the Middle Kingdom at theearliest.53 And this in itself is a difficulty for the strong thesis. If the use of parts ofthe Horus-eye were indeed, as Möller had it, “Spielereien”, then their total absencefrom the early written record in any other use than as CSM indicators robs the gameof much of its point.

The trend of the evidence of third-millennium sign forms that have accumulatedsince the end of the debate in the nineteen-thirties is clear. The further back we gothe further the hieratic signs diverge from their supposed Horus-eye counterparts.The inconsistency between the hieratic signs for “1/16” and “1/32” and their

referred to in POSENER-KRIÉGER (1994), p. 270. Nor is the sign indicated for this papyruseither in “MÖLLER” (1936) or in GOEDICKE (1988).49 POSENER-KRIÉGER and DE CÉNIVAL (1968), POSENER-KRIÉGER (1976).50 POSENER-KRIÉGER (1994 b), p. 269, states that the Rc-nfr⋅f archives contain examples notonly of these three but also of the next two, “1/16” and “1/32”. Pending publication, nothingfurther can be said.51 No significant report was ever published by the excavator. See for now POSENER-KRIÉGER

(1975) and DONADONI ROVERI (1993). I should like to take this opportunity to acknowledgemy deep gratitude to the late Mme. Posener-Kriéger for her generosity in sharing her copiesof the Gebelein papyri with me.52 POSENER-KRIÉGER (1994 b).53 For the hieroglyphic case, cf KAHL (1994), p. 443; GARDINER (1957), p. 451–452; for thehieratic, cf GOEDICKE (1988), p. 6.


purported hieroglyphic originals, already signaled by Neugebauer (see last section),is even greater for the original third-millennium forms and the others reveal similardifficulties. The hieroglyphic “1/2” signs vary widely among themselves and alldiffer from their New Kingdom descendants. Were one to offer a preliminarysynthesis of the third-millennium evidence, it would be to underscore the non-representational nature of the hieratic signs and the fluidity and consequent probablesecondary nature of the sole venture into creating a hieroglyphic equivalent of thisessentially abstract metrological structure.

III. A New Look at Old Evidence

If, as I think, the evidence presented in the last section disposes of Möller’s strongthesis, it is still possible that the weak thesis, proposed by Peet and Neugebauer, istrue; that, though the CSM signs and (parts of) the Eye of Horus were ofindependent third-millennium origins, in a later period an identification was madebetween them. In view of what has been said in the first section, all positivearguments for such an identification—at any period of Egyptian history—go back toMöller’s original article.

We recall that the principal textual evidence adduced by Möller for theidentification of the Horus-eye parts and the CSM was a section of the inscriptionsfound on votive cubits, dating from the New Kingdom and later. These stone cubit-long objects with a characteristic polygonal cross section are inscribed with a largenumber of texts relating geographical, metrological and cultic information: linearmeasurements of Egypt, shadow-clock and clepsydre tables, relations of gods withsubdivisions of the cubit, etc.54 Since their mention by Möller, a number of newcubits have been found, but their publication has had to await Adelheid Schlott-Schwab’s thesis (1973, 1981) and subsequent publications.55 The section to whichMöller referred is “Text j” in Schlott-Schwab’s terminology56 and consists of some14 cases (representing “finger” divisions of the cubit), numbered 7–20 by her. Thedifficulty of the section is such however that she renounces any attempt at aconnected analysis;57 indeed that the section is a unity seems doubtful — we shallhere only consider the first nine cases, 7–15, those on which the claim for the weakthesis rests.

Photographs or copies of the 11 more-or-less extant versions of this subsectioncan be found in Table II.58 As can be seen there, each case save 11 and the lastcontains one, two or three hieroglyphs, generally drawn from parts of the body(Gardiner’s Signlist Section D), at the top followed by one or more numerical signsin the abstract system—discrete number signs and fractional signs.59

54 BORCHARDT (1920), SCHLOTT-SCHWAB (1981).55 The thesis was first published in 1969. The 1981 reedition contains new photographicplates and a supplement to the bibliography (in the Vorwort) but is otherwise unchanged fromthe original publication. Subsequent publications that provide additional parts of Text j andare used here are GABRA (1969) and ZIVIE (1972).56 SCHLOTT-SCHWAB (1981), p. 51–52.57 “Ich kann den Text nicht erklären.” SCHLOTT-SCHWAB (1981), p. 52.58 The last three cubit fragments (IX–XI) contain neither the hieratic signs of case 15 nor thehieroglyphic signs of 7–14 and will be considered here only when the question of numericalvalues arises. For the full section see SCHLOTT-SCHWAB (1981), p. 51.59 For this system see RITTER (2001), p. 121.

308 J. Ritter

The hieroglyphs of cases 7–10 are apparently in part identical with those usedfor parts of the (left) Eye of Horus (though III has a right eye). The non-numericalhieroglyphs present are the lower half of the Eye of Horus (case 7), “1/8” (8), “1/4”(9), “1/2” and “1/4” (10). The following case, 11, contains only numerical signs(with the possible exception of the much damaged VI, which may contain “1/16”).Case 12, written right to left, has an apparently feminine word written with the hairsign, . Case 13 resumes the left to right orientation with the (ordinary) eyejr(t), ‘eye ?; making ?’ The next-to-last case of our subsection, 14, contains rdj ,‘highlands ? joy ?’ Case 15 breaks with this structure: jpt jrt m/mAa (var. inVIII) Damw mAa, ‘jpt (-grain measure) made of N units (?) of true electrum’ with acapacity quantity N written using hieratic signs.

The first thing that strikes one, in view of Möller’s insistence that the cubits givea clear connection between the hieratic and hieroglyphic CSM forms, is that theapparent hieratic signs for capacity measures are found in case 15, well removedfrom those of the hieroglyphic candidates, cases 7–10. Moreover the particularhieratic combination in case 15, “1/8” + “1/64”, or its variant “1/64” (?), is notrepresented in any of the other compartments.

Another problem is presented by the sign in case 7; the sign appearing there is aseparate and well-known hieroglyph (D 17), representing the bottom half of the Eyeof Horus and attested since the Middle Kingdom as a logogram, tjt, or determinativein words concerned with ‘image’ and, in mathematical papyri, with ‘fraction’.Though Möller claims, no doubt on graphical grounds, that here the sign representsa ligature of “1/32 + 1/64”, in metrological contexts the two signs, when appearingtogether, are never written in ligature—cf. Medinet Habu 374, 386, 447, 985.60

Finally we note that, based on their non-numerical signs, cases 12–14 seem toseparate themselves from those preceding: the reversed orientation of 12 and theiruse of common signs, unconnected with the wDAt eye. Though ignored by Möller,they too are associated with numbers and seem to play the same functional role asthe undoubted Horus-eye parts. The impression given by the whole subsection of thevotive cubit appears to be an association of abstract numbers and hieroglyphsrepresenting parts of the head, but the relation of such forms with case 15, the onlypossible reference to CSM, is far from obvious.61

Turning now to the question of the values of the numbers appearing in cases 7–14, we face difficulties at least equal to those presented by the non-numericalhieroglyphs. Numerical variants among the different cubits, ambiguities ofgroupings of signs and of notational technique (additive or multiplicative) preventany immediate understanding.

Let us first deal with cases 12–14. They appear to form a unity; to the reasonsadduced above on the basis of their non-numerical sign repertory can be added thefact that they are the only cases to contain high-end numerical signs (and onlythese); even sums of the other cases leave us far from such large values. Unlikecases 7–11, the numerical values for these cases are the same in nearly all the votive

60 EPIGRAPHIC SURVEY (1934). Similarly, the ligature in II 10 for “1/2” + “1/4” is otherwiseunknown.61 If the strange use of hieratic signs in an otherwise hieroglyphic context is to beunderstood as a definition of the relation between hieratic CSM and Horus-eye signs, whythen the choice of non-matching values for the two sides?


cubits on which they are completely (I, VII) or partially (VIII) extant—only XIpresents a variant reading. Case 12 with 1 000 000 and 14 with a clear multiplicativewriting of 6 × 100 000, that is 600 000 (except for XI which apparently has amisreading of as as well as only 5 strokes beneath )62. Case 13 howeverpresents problems; All texts have 3 , each followed by a stroke, as well as agrouping of 4 strokes. The temptation is strong to read this also as a multiplicativenumber, 4 × 100 000, for then 12 would be the sum of 13 and 14. Such a readingthough, with a plural being multiplied, is contrary to standard practices.63

But if 12–14 pose some difficulties, 7–11 are nearly impenetrable. First of all,the cubits show numerous variations; 7 alone offers various groupings of the 6 unitstrokes which yield 300 ( + stroke) + 3 (part of 1/13 ? = I, II); 300 ( ) + 3 + 3 (partof 1/13 ? = IV); 300 ( ) + 6 (part of 1/16 ? = IX, X or part of 1/10 + 1/6 = V)—orjust conceivably in a multiplicative writing 3 (or 6) times 100 (written as a plural,see above). Secondly it is not clear how many numbers are written in each case; is 8in II to be read 1040 — 22 or 1062 (I has 1000 + (2+n) x 10 — 10 + 2(+n) × 1 or astwo numbers, 1040 and 22 (II) or as 1000 + (2+n) × 10 + 12(+)?

Given so much ambiguity it is difficult to advance far in the comprehension ofthe numerical scheme. However certain suggestive points can be discerned. Takingseriously the division into two sets of numbers for each compartment, we can read7–10 in the following way:64

7: 1300 (V, I, II, IX*, X*) — 1/10 1/6 (V) or 1/16 (X, IV*, IX*, I*, II*)8: 1/8 1040 (II, I) — 22 (II, IV*)9: 1/4 2100 (II, I, X*) — 42 (II, I*, X*, XI*)10: 1/2 + 1/4 6300 (II) — 1/8 (II) or 1/11 (XI, I*)

We concentrate here on the CSM signs heading each compartment and the first,largest number in each. Now the hieroglyph in 8 is 1/2 the CSM value 1/4 in 9; andindeed the number 1040 is quite close to one-half the value 2100. Similarly, theCSM hieroglyphic signs in 10, 1/2 + 1/4 = 3/4, i.e., 3 times the value of the CSMsign in 9, precede the number 6300 = 3 x 2100. Finally such an understanding of thetext implies that for the CSM sign in 7 to be consistent with 8, it must bear the sameproportion to 1/8 that 1300 does to 1040, i.e., 1300/1040 = 5/4. Thus

= 5/4 × 1/8 = 5/32 = 4/32 + 1/32 = 1/8 + 1/32.

In the votive cubits then, the numerical hapax would represent the fraction1/8 + 1/32—and not 1/32 + 1/64 as proposed by Möller. Though not what one mightexpect, these are nonetheless dimidiated fractions. Finally, note that the numbergiven in 11, 30 450 (I, VII*), is not the sum of the proceeding cases.

However any such attempt at an understanding of this difficult section requires acareful choice of cubit and reading, selecting other combinations of readings from

62 I have not been able to collate this cubit for which, like Schlott-Schwab, I rely onLepsius’s old copy (LEPSIUS (1865), fig. III b).63 One might conjecture a reading along the lines of Hfnw 4, ‘hundred-thousands, 4’.64 The Roman numerals in parentheses following each number give the cubits which justifythe reading; an asterisk following the numeral means “damaged but consistent with thereading.” Note that the 2100 in 9 and 6300 in 10 are based on multiplicative readings,justified by the otherwise incomprehensible position of the unit strokes.

310 J. Ritter

other cubits would have produced quite different results. These very inconsistenciesamong the different copies are indicative of the lack of clear comprehension bythose who produced these objects of the sense of the information contained in them;a situation rather typical of esoteric material which has little meaning outside arestricted circle undergoing recopying over long periods of time but not whollyconsonant with a supposed widespread and general revision of an understanding ofthe nature of important elements of writing.

Thus, even were we able to show that the tantalizing but completely inconclusiveevidence of the votive cubits did represent a reinterpretation of the CSM signs asHorus-eye components sometime in the New Kingdom, the question would still beopen as to the use of such a new interpretation in actual metrological contexts. Toexamine this question we must turn to the evidence from New Kingdom CSM use—which means essentially hieratic administrative and medical texts and inhieroglyphic temple festival inscriptions. As Table I shows, the objections torelations between the hieratic signs and their putative hieroglyphic equivalentsremain in force; no remodeling of the hieratic signs for the diagnostic “1/16” and“1/32” to more closely match their hieroglyphic equivalents takes place. The onlysignificant difference comes in the change in “1/8” between the Middle and NewKingdoms,65 when the the previously straight diagonal line develops a curve or kinkwhich might be associated with an attempt to mimic the corresponding non-linearityin the eyebrow hieroglyph. However, even in this case, no attempt is made tomodify the vertical orientation to bring it in line with the strictly horizontalalignment of the Eye of Horus eyebrow and the other non-metrological hieraticrepresentations of the sign.66

Furthermore, changes in the hieroglyphic signs present further evidence againstsuch an innovation. The replacement of the older CSM “1/2” sign by the standardfractional (gs) is seen for the first time in capacity unit inscriptions in preciselythis period—in Amenophis IV’s calendar at Karnak (Dynasty 18), Ramesses II’s atAbydos (Dynasty 19) and Ramesses IV’s wall stela at Karnak (Dynasty 20). Further,the apparent reversal of orientation of the classic CSM “1/2” sign in Ramesses II’scalendar inscription at Karnak may have been influenced by just such an alternativeuse of . But neither the use nor the influence of this last is compatible with anunderstanding by the Egyptian engraver of the CSM units as essential elements ofthe “sound Eye” of Horus.

As far as the limited number of published sources allows us to judge, the use ofthe special CSM signs appears to diminish in the Late and Ptolemaic periods. Thehieroglyphic forms seem not to have been used. The administrative papyri show noessential change in the sign forms,67 while the Ptolemaic medical (Rubensohn)fragments illustrate the virtually total replacement of the classic CSM signs by their

65 Or, more exactly, Dynasty 17, for the Rhind papyrus examples already show the NewKingdom form.66 The unique diagonal instance, appearing in MÖLLER (1909 a), no. 89 (Sinuhe B 58), is aconjectural identification of a problematic determinative in the word anw, which everywhereelse is determined by or (D7 or D8). He has not been followed in this by thedictionaries, nor shall we here do so.67 The Dynasty 22 accounting papyrus now Berlin 23252+23253 (formerly Berlin 3081) isused by MÖLLER (1912), nos. 708–709, 713, as the sole example of Late period CSM signs.


ordinary fraction equivalents except in combinations of fractions.68 When thehieratic CSM signs are used, however, they differ in no way from their NewKingdom forms.

The evidence for the weak thesis, it seems to me, stands as follows. If it, unlikethe strong thesis, cannot be ruled out completely, the positive evidence for it seemsto be meagre in the extreme. The difficulties and ambiguities of the votive-cubitsubsection provides only suggestive evidence for a possible attempt to understandthe traditional hieratic CSM signs as parts of the Eye of Horus. The New Kingdomsign forms, as evidenced in the festival inscriptions and the metrological andmedical papyri, point in rather a different direction.

The proof of word play at the origin of concepts even in so unlikely a domain asthat of capacity measures, that Möller had seen in the close and early relationbetween the myth of the Eye of Horus and capacity submultiples can certainly nolonger be considered tenable; the early hieratic form of the CSM signs, thederivative forms and fluctuations in the rare third-millennium hieroglyphicequivalents for “1/2”, all rule the strong thesis out of court, as had already beensuspected by a number of critics in the nineteen-twenties, whose arguments,ironically along with Möller’s original positive ones, were for the most part laterforgotten.

The weak thesis, developed by these critics that, as a result of later second-millennium speculation, a connection was created between the Eye of Horus myth,with its healing or measuring properties, and the dimidiated series of the capacitysystem. But even here we have seen there are serious problems. The beginning ofthe replacement of the special CSM signs by their general fractional equivalents isto be located just at this period when, in the weak thesis, the idea that the classicalCSM signs form an important conceptual and religious unity begins its career. Thevotive cubits that are supposed to enshrine this idea have proven to be much moreambiguous and unclear than foreseen, and this for the Egyptians themselves as theinnumerable variants or misunderstandings of sign disposition and content indicate.And nowhere more so than precisely in the section which contains, at some distance,a possible though strange hieratic writing of a particular combination of CSM unitsand a series of hieroglyphs involving undoubted parts of the Eye of Horus, thoughnot always those combinations which occur in metrological contexts, along withhieroglyphs coming from quite different horizons; the whole accompanied byabstract, non-volumetric, numbers with no obvious ties to the capacity system.

But even if the cubits did represent an attempt to reinterpret the CSM signs interms of Horus-eye parts, a speculation in the milieu which produced votive cubitsdoes not automatically mean that “the Egyptians” thought like that; for example,those Egyptians whose task it was to engrave hieroglyphic inscriptions on templewalls. Theological or any other constructs of one community do not necessarilypropagate to every other; the Egyptians were no more liable than any other people tospeak with a single voice.

The “Horus-eye fractions” are a paradigmatic case of the nature of relationsbetween writing and other areas of thought in Egypt. The linkage, felt to be natural,

68 The large fragment has been published by WESTENDORF (1974) while the CSM signs inthe small fragment are listed in MÖLLER (1912), nos. 708, 711–713. For earlier examples ofthis development, with a somewhat different interpretation, see VON DEINES et al. (1973),p. 4–6.

312 J. Ritter

especially in Egypt, between sign creation and other facts of linguistic and religiousexperience, true for a number of signs, was natural in an Egyptological researchcommunity itself irrigated by literary and theological studies. In particular, theprimacy of word play and mythical speculation in creating script and in the formthat script might take renders the signs and concepts involved in numerical andmetrological development secondary at best. This, in turn, is related to what, inanalogy with the situation in molecular biology, I shall call the “central dogma” ofour ideas about Egyptan writing: all hieratic signs go back ultimately to pictographicsigns.69 The idea of organizing hieratic signlists by hieroglyph is one expression ofthis and, as we have seen, it was precisely in just such a context that the search forthe “hieroglyphic originals” of the capacity metrological signs led to the Horus-eyehypothesis. But it is just the metrological signs which here cast doubt on thisapproach. Recent work on the origins of writing in early Mesopotamia has shownthe importance of treating numerical and non-numerical signs quite differentlybecause they reflect quite different elements both in the origin and in thedevelopment of writing.70 The case of the “Horus-eye fractions” has revealed theshortcomings of approaches which ignore this difference and underlines the interestin globally reexamining the Egyptian case.

References

BAER, Klaus. 1956. “A Note on Egyptian Units of Area in the Old Kingdom”.Journal of Near Eastern Studies 15: 113–117.

BARDINET, Thierry. 1995. Les papyrus médicaux de l’Égypte pharonique. Paris:Fayard.

BARTA, Winfried. 1963. Die altägyptische Opferliste von der Frühzeit bis zurgriechisch-römischen Epoche (Münchner Ägyptologische Studien 3). Berlin: B.Hessling.

BORCHARDT, Ludwig. 1911. “Altägyptische Sonnenuhren”. Zeitschrift fürÄgyptische Sprache und Altertumskunde 48: 9–17.

1920. Die altägyptische Zeitmessung. Berlin: de Gruyter.BRUGSCH, Heinrich. 1883. Thesaurus Inscriptionem Aegypticarum II:

Kalendarische Inschriften altägyptischer Denkmäler. Leipzig: J. C. Hinrichs.CHACE, Arnold B.; MANNING, Henry P.; and ARCHIBALD, Raymond C. 1927. The

Rhind Mathematical Papyrus. Volume I: Free Translation and Commentary.Oberlin [OH, USA]: Mathematical Asssociation of America.

DAVIES, Norman deGaries. 1922. The Tomb of Puyemrê at Thebes. I: The Hall ofMemories. New York: Metropolitan Museum of Art.

VON DEINES, Hildegard, Hermann GRAPOW & Wolfhart WESTENDORF. 1973.Ergänzungen: Drogenquanten, Sachgruppen, Nachträge, Bibliographie,Generalregister (Grundriss der Medizin der alten Ägypter 9). Berlin: Akademie-Verlag.

69 In its original biological context the reference is to the proposed rule that all proteins areexpressions of genes encoded in nuclear DNA (genetic information flows in only onedirection); see e.g., James Watson et al., Molecular Biology of the Gene4, 1987. In this fieldtoo, there are serious doubts concerning the universal validity of such a position.70 NISSEN et al. (1994) and RITTER (1999).


DONADONI ROVERI, Anna Maria. 1993. “Gebelein”. In Anna Maria DONADONI

ROVERI et al., Il Museo egizio di Torino2: 243–250. Turin: Instituto geographicoDe Agostini.

EPIGRAPHIC SURVEY. 1934. Medinet Habu III. The Calender, the “Slaughterhouse,”and Minor Records of Ramesses III (Oriental Institute Publications 23). Chicago:Oriental Institute.

ERMAN, Adolf. 1928. Ägyptische Grammatik4. Berlin: Reuther & Reichard. and GRAPOW, Hermann. 1926 a. Wörterbuch der Ägyptischen Sprache I. Leipzig:

J. C. Hinrichs. and GRAPOW, Hermann. 1926 b. Wörterbuch der Ägyptischen Sprache. Die

Belegstellen I. Leipzig: J. C. Hinrichs.GABRA, Sami. 1969. “Coudée votive de Touna el Gebel Hermopolis Ouest, la

Khemenow pa Meket des Égyptiens”. Mitteilungen des DeutschenArchäologischen Instituts. Abteilung Kairo 24: 129–135.

GARDINER, Alan H. 1917. “An Archaic Funerary Stele”. Journal of EgyptianArchæology 4: 256–260.

1927. Egyptian Grammar1. Oxford: Clarendon Press.GILLINGS, Richard J. 1972. Mathematics in the Time of the Pharaohs. Cambridge

[US]: MIT Press.GOEDICKE, Hans. 1988. Old Hieratic Paleography. Baltimore: Halgo.GRIFFITH, F. Llewellyn. 1892. “Notes on Egyptian Weights and Measures”.

Proceedings of the Society for Biblical Archæology 14: 403–450 (Supplement:PSBA 15 (1893) 301–316).

GUNN, Battiscombe. 1922. “‘Finger-Numbering in the Pyramid Texts”. Zeitschriftfür Ägyptische Sprache und Altertumskunde 57: 71–72.

HASSAN, Selim. 1944. Excavations at Giza V: 1933–1934. Cairo: GovernmentPress.

IMHAUSEN, Annette. (Forthcoming). Ägyptische Algorithmen (ÄgyptologischeAbhandlungen).

JACQUET-GORDON, Helen K. 1962. Les Noms des domaines funéraires sousl’Ancien Empire égyptien. Cairo: Institut français d’Archéologie orientale.

JAMES, Thomas G. H. 1962. The H 9ek 9anakhte Papers and Other Early MiddleKingdom Documents (Egyptian Expedition 19). New York: The MetropolitanMuseum of Art.

KAHL, Jochem. 1994. Das System der ägyptischen Hieroglyphenschrift in der 0.–3.Dynastie (Göttinger Orientforschungen IV. Reihe Ägypten 29). Wiesbaden:Harrassowitz.

KAPLONY, Peter. 1963. Die Inschriften der ägyptischen Frühzeit (ÄgyptologischeAbhandlungen 8). Wiesbaden: Harrassowitz.

1966. Kleine Beiträge zu den Inschriften der ägyptischen Frühzeit(Ägyptologische Abhandlungen 15). Wiesbaden: Harrassowitz.

KITCHEN, Kenneth. 1983. “The West-Theban Calendar of Festivals: Medinet HabuVersion – Ramesseum Fragments”. In Ramesside Inscriptions. Historical andBiographical: vol. V, p. 115–184. Oxford: Oxford University Press.

KÖHLER, E. Christiana and BIRRELL, Michael. (Forthcoming). Helwan II. The EarlyDynastic Stelae.

LEFEBVRE, Gustave. 1940. Grammaire de l’Égyptien classique1. Cairo: Institutfrançais d’Archéologie orientale.

314 J. Ritter

LEPSIUS, Karl Richard. 1865. Über die altägyptische Elle und ihre Eintheilung.Berlin: Königliche Akademie der Wissenschaften.

LUCAS, Alfred and ROWE, Alan. 1940. “Ancient Egyptian Measures of Capacity”.Annales du Service des Antiquités de l’Égypte 40: 69–92.

MEGALLY, Mounir. 1971 a. Considérations sur les variations et la transformationdes formes hiératiques du papyrus E. 3226 du Louvre (Bibliothèque d’Étude 49).Cairo: Institut Français d’Archéologie Orientale.

1971 b. Le Papyrus hiératique comptable E. 3226 du Louvre (Bibliothèqued’Étude 53). Cairo: Institut Français d’Archéologie Orientale.

MÖLLER, Georg. 1909 a. Hieratische Paläographie. Die ägyptische Buchschrift…I:Bis zum Beginn der achtzehnten Dynastie1. Leipzig: J. C. Hinrichs.

1909 b. Hieratische Paläographie. Die ägyptische Buchschrift…II: Von der ZeitThutmosis’ III bis zum Ende der einundzwanzigsten Dynastie1. Leipzig: J. C.Hinrichs.

1911. “Die Zeichen für die Bruchteile des Hohlmaßes und das Uzatauge”.Zeitschrift für Ägyptische Sprache und Altertumskunde 48: 99–101.

1912. Hieratische Paläographie. Die ägyptische Buchschrift…III: Von derzweiundzwanzigsten Dynastie bis zum dritten Jahrhundert nach Chr.1. Leipzig:J. C. Hinrichs.

“MÖLLER, Georg” [H. GRAPOW and H. PRZYBYLLA]. 1936. HieratischePaläographie. Die ägyptische Buchschrift…Ergänzungsheft zu Band I und II.Leipzig: J. C. Hinrichs.

MOUSSA, Ahmed M. and ALTENMÜLLER, Hartwig. 1977. Das Grab desNianchchnum und Chnumhotep (Archäologische Veröffentlichungen desDeutschen Archäologischen Instituts. Abteilung Kairo 21). Mainz: von Zabern.

MÜLLER-WOLLERMANN, Renate. 1985. “Udjatauge”. In: Wolfgang HELCK andWolfhart WESTENDORF (eds.), Lexikon der Ägyptologie: vol. VI: 824–826.Wiesbaden: Otto Harrassowitz.

NELSON, Harold H. and HÖLSCHER, Evo. 1934. Work in Western Thebes 1931–1933(OIC 18). Chicago: University of Chicago Press.

NEUGEBAUER, Otto. 1930. “Über den Scheffel und seine Teile”. Zeitschrift fürÄgyptische Sprache und Altertumskunde 65: 42–48.

NISSEN, Hans; DAMEROW, Peter, and ENGLUND, Robert K. 1994. ArchaicBookkeeping. Writing and Techniques of Economic Administration in theAncient Near East. Chicago: University of Chicago Press.

PEET, T. Eric. 1923a. The Rhind Mathematical Papyrus, British Museum 10057 and10058. London: Hodder & Stoughton (for Liverpool University Press).

1923b. “Arithmetic in the Middle Kingdom”. Journal of Egyptian Archaeology9: 91–95.

PETRIE, W. M. Flinders. 1926. Ancient Weights and Measures. London: UniversityCollege.

POSENER-KRIÉGER, Paule. 1975. “Les Papyrus de Gébélein. Remarquespréliminaires”. Revue d’Égyptologie 27: 211–221.

1976. Les Archives du temple funéraire de Néferirkarê-Kakaï (Bibliothèqued’Étude 65; 2 vols.). Cairo: Institut Français d’Archéologie Orientale.

1994. “Les Mesures de grain dans les papyrus de Gébélein”. In Christopher EYRE

et al. (eds.), The Unbroken Reed. Studies in the Culture and Heritage of Ancient


Egypt in Honour of A. F. Shore (EES Occasional Publication 11): 269–272.London: Egypt Exploration Society.

and DE CENIVAL, Jean Louis. 1968. The Abu Sir Papyri (Hieratic Papyri in theBritish Museum. Fifth Series). London: British Museum.

QUIBELL, James. 1923. Archaic Mastabas. (Service des Antiquités de l’Égypte.Excavations at Saqqara 6). Cairo: Institut Français d’Archéologie Orientale.

RITTER, Jim. 1992. “Metrology and the Prehistory of Fractions”. In: Paul BENOIT,Karine CHEMLA and Jim RITTER (eds.). Histoire des fractions, fractionsd’histoire: 19–32. Basel: Birkhäuser.

1999. “Metrology, Writing and Mathematics in Mesopotamia”. In: JaroslavFOLTA (ed.), Calculi 1929–1999 (Fs Novy): 215–241. Prague: Prague Studies inthe History of Science and Technology.

2001. “Egyptian Mathematics”. In: Helaine SELIN (ed.), Mathematics acrossCultures: The History of Non-Western Mathematics: 115–136. Dordrecht:Kluwer.

(Forthcoming). Egyptian Metrology in the Third Millennium.ROBINS, Gay and SHUTE, Charles. 1987. The Rhind Mathematical Papyrus: An

Ancient Egyptian Text. London: British Museum.SAAD, Ramadan and MANNICHE, Lise. 1971. “A Unique Offering List of

Amenophis IV Recently Found at Karnak”. Journal of Egyptian Archaeology 57:70–72.

SAAD, Zaky Y. 1957. Ceiling Stelae in Second Dynasty Tombs from the Excavationsat Helwan (Annales du Service des antiquités de l’Égypte Supplement, Cahier21). Cairo: Institut Français d’Archéologie Orientale.

SCHÄFER, Heinrich. 1902. Ein Bruchstück altägyptischer Annalen. Berlin:Königliche Akademie der Wissenschaften.

SCHLOTT-SCHWAB, Adelheid. 1973. “Altägyptische Texte über die AusmasseÄgyptens” Mitteilungen des Deutschen Archäologischesn Instituts. AbteilungKairo 28: 109–113.

1981. Die Ausmasse Ägyptens nach altägyptischen Texten2 (Ägypten und AltesTestament 3). Wiesbaden: Harrassowitz.

SCOTT, Nora E. 1942−1943. “Egyptian Cubit Rods”, Bulletin of the MetropolitanMuseum of Art (NS) 1: 70−75.

SETHE, Kurt. 1906–1909. Urkunden der 18. Dynastie (Heft 1–16). Leipzig: J. C.Hinrichs.

1916. Von Zahlen und Zahlworten bei den alten Ägypten und was für andereVölker und Sprachen daraus zu lernen ist (Schriften der WissenschaftlichenGesellschaft Straßburg 25). Strasbourg: Trübner.

SMITH, W. Stephenson and SIMPSON, William K. 1981. The Art and Architecture ofAncient Egypt2. New Haven: Yale University Press.

STRUVE, Vasilij Vasilevitch. 1930. Mathematischer Papyrus des StaatlichenMuseums der schönen Kunst in Moskau (Quellen und Studien A1). Berlin:Springer.

VOGEL, Kurt. 1930. “Zur Frage der Scheffelteile”. Zeitschrift für ÄgyptischeSprache und Altertumskunde 66: 33–35.

WESTENDORF, Wolfhart. 1974. “Papyrus Berlin 10456. Ein Fragment deswiederendeckten medizinischen Papyrus Rubensohn”. In Festschrift zum

316 J. Ritter

150 jährigen Bestehen des Berliner Ägyptischen Museums: 247–254. Berlin:Akademie-Verlag.

ZIVIE, Alain-Pierre. 1972. “Un Fragment inédit de coudée votive”. Bulletin del’Institut Français d’Archéologie Orientale 71: 181–188.


Tab

leI.

CSM

Sign

Lis

t.

318 J. Ritter

Tab

leI.

CSM

Sign

Lis

t(co

ntin

ued)

.


Legend to Table I

For each CSM typical sign forms, both hieratic and hieroglyphic, are presented inchornological order by dynasty. The hieratic forms are presented on the left andhieroglyphs on the right of each column. The hieroglyphic signs have had theiroriginal orientation changed if necessary to conform to a right to left readingdirection, facilitating comparison with the hieratic and permitting a determination ofchanges in orientation over time.

The Arabic numbers are keyed to the source list below. An asterisk (*) indicatesthat the sign form is attested but is unpublished.

Where a photograph was not available or was of insufficient contrast I have useda published hand copy.

Sources:

Dynasty 21. Sjh 9fnr (Saqqara): QUIBELL (1923), pl. XXVI.2. Jmty (Saqqara): SMITH and SIMPSON (1981), pl. 32.3. Dw3t (Helwan): photograph by courtesy of Christiana Köhler.

Dynasty 31. Jrj-Stt: KAPLONY (1966), Tf. 4 (833).

Dynasty 41. Gebelein papyri: POSENER-KRIÉGER (1994), fig. 1.

Dynasty 51. Abusir papyri: POSENER-KRIÉGER and DE CENIVAL (1968), Pal. pl. XVI.2. Tomb of Nj-cnh }-H;nm & H;nm-h 9tp (Saqqara): MOUSSA and ALTENMÜLLER (1977),

Abb. 10.

Dynasty 61. pBerlin 10500 A & B: GOEDICKE (1988), p. 70–71.

Dynasty 111. Account papyrus: JAMES (1962), pl. 16, IX, 19.

Dynasty 121. Illahun papyri: MÖLLER (1909 a), nos. 708–713.

Dynasty 171. pRhind 47: 7 (2, 8); 8 (4); 10 (16, 32, 64): ROBINS and SHUTE (1987), pl. 15.

Dynasty 181. pLouvre E 3226: MEGALLY (1971) b, pl. LXXI, B r° V: 5.2. Tuthmosis III campaign (Karnak): SETHE (1906–1909), 756, 8.3. Tuthmosis III calendar (Karnak): SETHE (1906–1909), 762, 6.4. Tomb of Pwj-m-Rc (Tuthmosis III, Thebes): DAVIES (1922), pl. I, 36.5. Amenophis IV Calendar (Karnak): SAAD and MANNICHE (1971), pl. XXI.

320 J. Ritter

Dynasty 191. pBerlin 3038 (medical): MÖLLER (1909 b), nos. 708–713.2. Ramesses II Festival calendar (Abydos): photographs by courtesy of Kenneth Kitchen,

Z 21–Z 27.3. Ramesses II Festival calendar (Karnak): NELSON and HÖLSCHER (1934), Fig. 11.

(right 5) R24+26 (2, 4, 16, 32); KITCHEN 1983, R3.2 (8, 64).

Dynasty 201. pRollin (Ramesses III): MÖLLER (1909 b), nos. 708–713.2. Ramesses III Festival calendar (Medinet Habu): KITCHEN (1983), MH 970 (2 4 8 16

32) MH 301 (64).3. Ramsses IV wall-stela (Karnak): photograph courtesy of Centre Golenischeff, A XI


Table II. Votive Cubits.

322 J. Ritter

Table II. Votive Cubits (continued).


Legend to Table II

All photographs are shown in negative to increase contrast. No VIII has beenreversed in this printing to coincide with the direction of writing of the other votivecubits. The three cubits (IX–XI) for which I have no photographs have been printedin hand copy taken from their original publication.

Sources

(a royal name mentioned elsewhere on the votive cubit is included in parentheses followingthe museum number):

I BMA 71.38 + Cairo 31/12//22/1 (Nectanebo II)BMA 71.38: photograph courtesy of the Brooklyn Museum of Art (Charles EdwinWilbour Fund).Cairo 31/12//22/1: photograph courtesy of the Centre Golénischeff (Archive Lacau),OI 24.

II Cairo 31/12//22/2 (Osorkon N)photograph courtesy of the Centre Golénischeff (Archive Lacau), OI 24.

III (Petrie 2)photograph courtesy of the Petrie Museum of Egyptian Archaeology.

IV Cairo 31/12//22/5photograph courtesy of the Centre Golénischeff (Archive Lacau), OI 23.

V (Tuna el-Gebel) (Amenhotep III)GABRA (1969), pl. XVIII A.

VI Cairo 31/12//22/4photograph courtesy of the Centre Golénischeff (Archive Lacau), OI 22.

VII Cairo 31/12//22/3photograph courtesy of the Centre Golénischeff (Archive Lacau), OI 24

VIII Turin Nr. Suppl. 2681(Sheshonq I)photograph courtesy of the Centre Golénischeff (Archive Lacau)

IX MMA 25.7.41SCOTT (1942−1943), p. 72

X (Petrie 1)PETRIE (1926), pl. XXIV.1

XI BM 36 656LEPSIUS (1865), tf. III b

"Closing the Eye of Horus: The Rise and Fall of Horus-Eye Fractions" in J. Steele & A. Imhausen...

Documents

Transcript of "Closing the Eye of Horus: The Rise and Fall of Horus-Eye Fractions" in J. Steele & A. Imhausen...