Ancient Egyptian Science: A Source Book, Vol. 3: Ancient Egyptian Mathematics by MarshallClagettReview by: Anthony SpalingerJournal of the American Oriental Society, Vol. 121, No. 1 (Jan. - Mar., 2001), pp. 133-134Published by: American Oriental SocietyStable URL: http://www.jstor.org/stable/606755 .

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Reviews of Books Reviews of Books

La ville de Shuruppak ha l'poque de ces tablettes comptait en- viron 30,000 personnes, etait situee sur un canal et donc proba- blement un important centre de navigation et de commerce et enfin avait cree un puissant lien de federation avec d'autres

metropoles du sud pour constituer la sexapole. La ville connut

apres une imposante mobilisation generale une fin violente dans un enorme incendie dont elle ne se releva plus jamais.

L'administration de Shuruppak est divisee en deux poles: l'e-

gal, le palais qui comporte les courtisans et les grands respon- sables. Chaque secteur dans le palais est sous les ordres d'un ugula (contremaitre). Cet ensemble comprend environ 200 personnes.

L'autre p61e s'appelait le e-uru, la maison de la ville, qui re-

groupait l'ensemble des metiers. L'auteur est oblige a juste titre d'inclure un e-geme-ensi, la maison de la femme de l'ensi

qui en fait assure la coordination entre les deux mentionnees

plus haut. C'etait un peu le bureau central de l'administration de la ville. Meme si ce e-g6me-ensi assurait le lien entre les deux

principaux centres il est sOr qu'a chaque niveau subalterne des

ponts devaient exister entre les personnes servant dans l'une ou l'autre administration. Ainsi il est des gens responsables de la

navigation dans les deux administrations et le nimgir-kisal doit assurer la relation entre les deux. I1 en va de meme pour les engar. Cette interpenetration des administrations est particulierement interessante.

Ces recherches strictement economiques de comptabilisation de gurush menent toujours plus loin qu'on ne le pense. Car dans le cas present elles ont permis de mieux comprendre les rela- tions avec d'autres villes de Babylonie.

I1 apparait qu'un grand nombre de personnes ont ete trans- ferees des secteurs productifs de la ville a l'armee, au cours des dernieres ann6es de fonctionnement de la ville. Ce serait la pre- miere attestation indirecte d'une mobilisation generale de tous les hommes en 6tat de porter les armes pour la guerre. I1 s'en suit

qu'il fallait pourvoir aux besoins de ces gens, notamment aux be- soins alimentaires. Cela est atteste par les tablettes. On ne voit

pas contre quel ennemi on s'est arm6 mais cela a du etre une mo- bilisation sans pr6ecdent si l'interpretation est juste.

II est aussi possible qu'une simple mais importante modifica- tion du cours d'eau ait enleve a la ville le contrOle du commerce maritime qui a passe a d'autres villes comme Ur, ville en train de monter. Mais de toute faqon quelles que furent les causes de la dis-

parition de la ville, la mise en place d'une organisation adminis- trative aussi avancee et complexe que celle de Shuruppak, surtout dans ses relations avec les autres villes, a dO prendre un certain nombre d'annees pour se mettre en place et nous fait ainsi re- monter dans l'histoire jusqu'au debut de l'age dynastique.

Dans la limite de la documentation disponible, le systeme py- ramidal avait a sa tete le ensi-gar-gal. Le personnel a son service etait reparti en unit6s administratives, les imru. I1 semblerait que les biens et la terre arable etaient en sa possession. Une stratifica- tion de la societe existait pour autant que les courroies de trans- mission devaient permettre a partir de la tete l'execution des differents ordres. Les gurush et les lu2-ri-ri-ga etaient au bas de

La ville de Shuruppak ha l'poque de ces tablettes comptait en- viron 30,000 personnes, etait situee sur un canal et donc proba- blement un important centre de navigation et de commerce et enfin avait cree un puissant lien de federation avec d'autres

metropoles du sud pour constituer la sexapole. La ville connut

apres une imposante mobilisation generale une fin violente dans un enorme incendie dont elle ne se releva plus jamais.

L'administration de Shuruppak est divisee en deux poles: l'e-

gal, le palais qui comporte les courtisans et les grands respon- sables. Chaque secteur dans le palais est sous les ordres d'un ugula (contremaitre). Cet ensemble comprend environ 200 personnes.

L'autre p61e s'appelait le e-uru, la maison de la ville, qui re-

groupait l'ensemble des metiers. L'auteur est oblige a juste titre d'inclure un e-geme-ensi, la maison de la femme de l'ensi

qui en fait assure la coordination entre les deux mentionnees

plus haut. C'etait un peu le bureau central de l'administration de la ville. Meme si ce e-g6me-ensi assurait le lien entre les deux

principaux centres il est sOr qu'a chaque niveau subalterne des

ponts devaient exister entre les personnes servant dans l'une ou l'autre administration. Ainsi il est des gens responsables de la

navigation dans les deux administrations et le nimgir-kisal doit assurer la relation entre les deux. I1 en va de meme pour les engar. Cette interpenetration des administrations est particulierement interessante.

Ces recherches strictement economiques de comptabilisation de gurush menent toujours plus loin qu'on ne le pense. Car dans le cas present elles ont permis de mieux comprendre les rela- tions avec d'autres villes de Babylonie.

I1 apparait qu'un grand nombre de personnes ont ete trans- ferees des secteurs productifs de la ville a l'armee, au cours des dernieres ann6es de fonctionnement de la ville. Ce serait la pre- miere attestation indirecte d'une mobilisation generale de tous les hommes en 6tat de porter les armes pour la guerre. I1 s'en suit

qu'il fallait pourvoir aux besoins de ces gens, notamment aux be- soins alimentaires. Cela est atteste par les tablettes. On ne voit

pas contre quel ennemi on s'est arm6 mais cela a du etre une mo- bilisation sans pr6ecdent si l'interpretation est juste.

II est aussi possible qu'une simple mais importante modifica- tion du cours d'eau ait enleve a la ville le contrOle du commerce maritime qui a passe a d'autres villes comme Ur, ville en train de monter. Mais de toute faqon quelles que furent les causes de la dis-

parition de la ville, la mise en place d'une organisation adminis- trative aussi avancee et complexe que celle de Shuruppak, surtout dans ses relations avec les autres villes, a dO prendre un certain nombre d'annees pour se mettre en place et nous fait ainsi re- monter dans l'histoire jusqu'au debut de l'age dynastique.

Dans la limite de la documentation disponible, le systeme py- ramidal avait a sa tete le ensi-gar-gal. Le personnel a son service etait reparti en unit6s administratives, les imru. I1 semblerait que les biens et la terre arable etaient en sa possession. Une stratifica- tion de la societe existait pour autant que les courroies de trans- mission devaient permettre a partir de la tete l'execution des differents ordres. Les gurush et les lu2-ri-ri-ga etaient au bas de

l'Fchelle. L'auteur montre que le role des structures economiques des temples etaient r6duites et surtout int6ressees par l'elevage, peut-etre pour les besoins du temple et du palais.

Ce livre est a recommander; sa traduction en anglais est ex- cellente; moins de soin a ete apporte a la bibliographie des livres

franqais et allemands oh abondent les fautes.

MARCEL SIGRIST ECOLE BIBLIQUE

Ancient Egyptian Science: A Source Book, vol. 3: Ancient

Egyptian Mathematics. By MARSHALL CLAGETT. Philadel-

phia: AMERICAN PHILOSOPHICAL SOCIETY, 1999. Pp. x +

462, illustrations. $30.

This work, the third in a series of detailed but basic studies about ancient Egyptian mathematics, is far less extensive than

Clagett's previous contributions to the scientific perception of

pharaonic Egypt. Clagett is at his best when he is explaining the

practical side of things, but here he too often merely compiles existing ideas. This orientation, which pervades the introduction and the entire first chapter, actually lessens the utility of this "source book."

Clagett begins by writing at length about the Rhind Mathemat- ical Papyrus, our major extant compendium, thereby spending an inordinate amount of time rehashing previous non-Egyptological and Egyptological contributions to the series of 2/n calculations. This approach has blinded him to the last two decades of perti- nent Egyptological research on the Rhind Papyrus and other mathematical tractates from the Nile Valley.

Besides not engaging the latest scholarship, Clagett also avoids discussion of the day-to-day calculations of the Egyptians. Account papyri and ostraca are overlooked, even though we pos- sess quite a large number of such mundane yet mathematically revealing texts. Surprising results have been forthcoming from such lowly sources, results that include helpful and revealing points of arithmetical applications. A discussion of these texts

belong in the type of book for which Clagett aims, but unfortu-

nately the author has not bothered to turn to the mass of Egypto- logical primary material, such as the ostraca from the workmen's

village of Deir el Medineh (Dynasties XIX-XX) or the numer- ous mathematical calculations (but not treatises) from Illahun (late Dynasty XII).

This reviewer was disappointed to find Clagett ignorant of the complex calculations involved with dilution of liquids ("Dates in Ancient Egypt," Studien zur altagyptischen Kultur 15 [1988]: 255-76), and those used for the varying measures of

grain ("The Grain System of Dynasty 18," Studien zur altagyp- tischen Kultur 14 [1984]: 283-311, and "Baking During the

Reign of Seti I," Bulletin de l'lnstitut francais d'Archeologie

l'Fchelle. L'auteur montre que le role des structures economiques des temples etaient r6duites et surtout int6ressees par l'elevage, peut-etre pour les besoins du temple et du palais.

Ce livre est a recommander; sa traduction en anglais est ex- cellente; moins de soin a ete apporte a la bibliographie des livres

franqais et allemands oh abondent les fautes.

MARCEL SIGRIST ECOLE BIBLIQUE

Ancient Egyptian Science: A Source Book, vol. 3: Ancient

Egyptian Mathematics. By MARSHALL CLAGETT. Philadel-

phia: AMERICAN PHILOSOPHICAL SOCIETY, 1999. Pp. x +

462, illustrations. $30.

This work, the third in a series of detailed but basic studies about ancient Egyptian mathematics, is far less extensive than

Clagett's previous contributions to the scientific perception of

pharaonic Egypt. Clagett is at his best when he is explaining the

practical side of things, but here he too often merely compiles existing ideas. This orientation, which pervades the introduction and the entire first chapter, actually lessens the utility of this "source book."

Clagett begins by writing at length about the Rhind Mathemat- ical Papyrus, our major extant compendium, thereby spending an inordinate amount of time rehashing previous non-Egyptological and Egyptological contributions to the series of 2/n calculations. This approach has blinded him to the last two decades of perti- nent Egyptological research on the Rhind Papyrus and other mathematical tractates from the Nile Valley.

Besides not engaging the latest scholarship, Clagett also avoids discussion of the day-to-day calculations of the Egyptians. Account papyri and ostraca are overlooked, even though we pos- sess quite a large number of such mundane yet mathematically revealing texts. Surprising results have been forthcoming from such lowly sources, results that include helpful and revealing points of arithmetical applications. A discussion of these texts

belong in the type of book for which Clagett aims, but unfortu-

nately the author has not bothered to turn to the mass of Egypto- logical primary material, such as the ostraca from the workmen's

village of Deir el Medineh (Dynasties XIX-XX) or the numer- ous mathematical calculations (but not treatises) from Illahun (late Dynasty XII).

This reviewer was disappointed to find Clagett ignorant of the complex calculations involved with dilution of liquids ("Dates in Ancient Egypt," Studien zur altagyptischen Kultur 15 [1988]: 255-76), and those used for the varying measures of

grain ("The Grain System of Dynasty 18," Studien zur altagyp- tischen Kultur 14 [1984]: 283-311, and "Baking During the

Reign of Seti I," Bulletin de l'lnstitut francais d'Archeologie

133 133

This content downloaded from 62.122.76.60 on Wed, 18 Jun 2014 02:38:45 AMAll use subject to JSTOR Terms and Conditions

Journal of the American Oriental Society 121.1 (2001) Journal of the American Oriental Society 121.1 (2001)

Orientale 86 [1968]: 307-52), to take two key examples. Even the recent study on the background layout and reuse of the fa- mous Rhind Mathematical Papyrus in "The Rhind Mathematical

Papyrus as a Historical Source," Studien zur altigyptischen Kul- tur 17 (1990): 295-338 is not cited. Instead, we are treated noch einmal to the rather second-level remarks of R. J. Gillings and the hyper-mathematical comments of B. Van der Waerden. It is as if all Egyptologists since T E. Peet were dismissed, or at least

placed to the side. Such an obvious bias is not acceptable. If Clagett wants to

discuss the inserts of Rhind, then he must do so from a modern

historiographical point of view. This he has not accomplished. Similarly, if he discusses the brewing measure psw, then I would have expected a dialogue with modern researchers rather than a

cursory and simplistic overview. Indeed, the entire development of the hekat system of grain measurement is, surprisingly, not addressed. Clagett should have dealt with the various hekats in Rhind which were the basis of the different grain measures

(double and triple, not to mention the quadruple measure). Yet this is not attempted.

Why, for example, refer to the epoch-making and too-

neglected study of Otto Neugebauer ("Zur iigyptischen Bruch-

rechnung," Zeitschrift fur dgyptische Sprache 64 [1929]: 44-48) without adding recent studies of Egyptian fractions? Ex-

cluding my own work, I can refer to Paule Posener-Kri6ger, "Les mesures de grain dans les papyus de G6b6lein," in The Unbroken

Reed, ed. C. Eyre et al. (London: Egypt Exploration Society, 1994), 269-72. Her analysis of the hieratic notations of the dim- idiated fractions commencing with 1/2 must be read in conjunction with Neugebauer's conclusions regarding the superimposition of such a system upon an original decimal one. Specifically, Neuge- bauer showed that the dimidiated fractions 1/16 hekat and 1/32 hekat were, as the hieratic proved, originally connected to a dec- imal system. Its origins have now been traced back to the heyday of the Egyptian Old Kingdom by Posener-Kri6ger.

There were other combinations of measures that combined

simpler ones by a multiplicative factor of two, three, and finally four (the latter was the latest). For example, the Rhind Papyrus most certainly demonstrates the existence of a "triple hekat." Evidence from an earlier time (the Hekanakhte papyri) indicates that a triple sack (three khar) existed as a unit, comprising thirty hekats. This triple sack was separate from thirty hekats, which would be normally assembled through three sacks, each of ten khar. The use of a multiplicative factor of three can be found in the Old Kingdom as well, in the Sharuna papyri-conveniently, see Hans Goedicke, Old Hieratic Palaeography (Baltimore: Halgo, 1988), 70 (bottom) and 71 (top). Goedicke notes the two

separate signs for three sacks; I confirmed this on the original papyrus now in Berlin.

The evolution of the grain measures reveals the development of the Rhind Papyrus itself. Originally a master tract dated to a Middle Kingdom (Dynasty XII) exemplar, internal data prove that the quadruple hekat grain measures entered somewhat later

Orientale 86 [1968]: 307-52), to take two key examples. Even the recent study on the background layout and reuse of the fa- mous Rhind Mathematical Papyrus in "The Rhind Mathematical

Papyrus as a Historical Source," Studien zur altigyptischen Kul- tur 17 (1990): 295-338 is not cited. Instead, we are treated noch einmal to the rather second-level remarks of R. J. Gillings and the hyper-mathematical comments of B. Van der Waerden. It is as if all Egyptologists since T E. Peet were dismissed, or at least

placed to the side. Such an obvious bias is not acceptable. If Clagett wants to

discuss the inserts of Rhind, then he must do so from a modern

historiographical point of view. This he has not accomplished. Similarly, if he discusses the brewing measure psw, then I would have expected a dialogue with modern researchers rather than a

cursory and simplistic overview. Indeed, the entire development of the hekat system of grain measurement is, surprisingly, not addressed. Clagett should have dealt with the various hekats in Rhind which were the basis of the different grain measures

(double and triple, not to mention the quadruple measure). Yet this is not attempted.

Why, for example, refer to the epoch-making and too-

neglected study of Otto Neugebauer ("Zur iigyptischen Bruch-

rechnung," Zeitschrift fur dgyptische Sprache 64 [1929]: 44-48) without adding recent studies of Egyptian fractions? Ex-

cluding my own work, I can refer to Paule Posener-Kri6ger, "Les mesures de grain dans les papyus de G6b6lein," in The Unbroken

Reed, ed. C. Eyre et al. (London: Egypt Exploration Society, 1994), 269-72. Her analysis of the hieratic notations of the dim- idiated fractions commencing with 1/2 must be read in conjunction with Neugebauer's conclusions regarding the superimposition of such a system upon an original decimal one. Specifically, Neuge- bauer showed that the dimidiated fractions 1/16 hekat and 1/32 hekat were, as the hieratic proved, originally connected to a dec- imal system. Its origins have now been traced back to the heyday of the Egyptian Old Kingdom by Posener-Kri6ger.

There were other combinations of measures that combined

simpler ones by a multiplicative factor of two, three, and finally four (the latter was the latest). For example, the Rhind Papyrus most certainly demonstrates the existence of a "triple hekat." Evidence from an earlier time (the Hekanakhte papyri) indicates that a triple sack (three khar) existed as a unit, comprising thirty hekats. This triple sack was separate from thirty hekats, which would be normally assembled through three sacks, each of ten khar. The use of a multiplicative factor of three can be found in the Old Kingdom as well, in the Sharuna papyri-conveniently, see Hans Goedicke, Old Hieratic Palaeography (Baltimore: Halgo, 1988), 70 (bottom) and 71 (top). Goedicke notes the two

separate signs for three sacks; I confirmed this on the original papyrus now in Berlin.

The evolution of the grain measures reveals the development of the Rhind Papyrus itself. Originally a master tract dated to a Middle Kingdom (Dynasty XII) exemplar, internal data prove that the quadruple hekat grain measures entered somewhat later

than the original sack measure. The complex history of Rhind is not mentioned by Clagett. According to the analysis of Steven

Quirke, referred to by me in "Dates in Ancient Egypt," Studien

zur altdgyptischen Kultur 15 (1985): 255ff., it is not even certain that the Vorlage is to be dated to Amenemhet III.

It is baffling that the author neglected to delve into the math- ematics of the New Kingdom and later periods; e.g., the Demo- tic papyri, on which the major work has been accomplished by Richard A. Parker. Therefore, Clagett's work is missing the ac-

counting principles and the handy mathematical formulas for vol-

ume, weights, and fractions, most of which come from XVIIIth

Dynasty papyri and the later Ramesside ostraca of Deir el Medineh. For earlier texts, Posener-Kri6ger's useful commentary on the mathematical system employed for garments and clothing ("Les mesures des 6toffes a l'Ancien Empire," Revue d'Egypto- logie 29 [1977]: 86-96) should have been consulted for a better

understanding of the Egyptian world-view of measurement. Thus Clagett's book, while expanding the vista of ancient

Egyptian mathematics somewhat (e.g., see some comments with regard to the Reisner papyri), nevertheless remains an old- fashioned "source book," and one that is out of date as well. The self-imposed parameters of the book have limited its audi- ence to a general level. Those who are conversant with new

Egyptological research will be dismayed to find that it has been overlooked. There remains a certain dissatisfaction with the ab- sence of any historiographical or socio-historical analysis. One wishes that Clagett had offered a more complete schema of an- cient Egyptian mathematical thought. It is up to Egyptologists to do so.

ANTHONY SPALINGER

UNIVERSITY OF AUCKLAND

Proche-Orient ancien; Temps vecu, temps pense': Actes de la Table-Ronde du 15 novembre 1997 organisee par I'URA 1062 <Etudes Semitiques.>> Edited by FRANCOISE BRIQUEL- CHATONNET and HELENE LOZACHMEUR. Antiquit6s Semi-

tiques, vol. 3. Paris: JEAN MAISONNEUVE, 1998. Pp. 238, maps, illustrations. FF 260.

A large dictionary will indicate several meanings for English 'time,' French 'temps,' German 'Zeit,' etc., and dozens of idio- matic usages. In none of the ancient Semitic languages that I know, however, is a word attested designating the abstract con-

cept of 'time' as "indefinite and continuous duration regarded as that in which events succeed one another" (part of the first

gloss in the Random House Dictionary of the English Lan-

guage, 2nd unabr. ed., 1987). There are, of course, words for units of time as defined by the movements of the moon and of the sun ('day,' 'month,' 'year'; 'week,' which does not correspond

than the original sack measure. The complex history of Rhind is not mentioned by Clagett. According to the analysis of Steven

Quirke, referred to by me in "Dates in Ancient Egypt," Studien

zur altdgyptischen Kultur 15 (1985): 255ff., it is not even certain that the Vorlage is to be dated to Amenemhet III.

It is baffling that the author neglected to delve into the math- ematics of the New Kingdom and later periods; e.g., the Demo- tic papyri, on which the major work has been accomplished by Richard A. Parker. Therefore, Clagett's work is missing the ac-

counting principles and the handy mathematical formulas for vol-

ume, weights, and fractions, most of which come from XVIIIth

Dynasty papyri and the later Ramesside ostraca of Deir el Medineh. For earlier texts, Posener-Kri6ger's useful commentary on the mathematical system employed for garments and clothing ("Les mesures des 6toffes a l'Ancien Empire," Revue d'Egypto- logie 29 [1977]: 86-96) should have been consulted for a better

understanding of the Egyptian world-view of measurement. Thus Clagett's book, while expanding the vista of ancient

Egyptian mathematics somewhat (e.g., see some comments with regard to the Reisner papyri), nevertheless remains an old- fashioned "source book," and one that is out of date as well. The self-imposed parameters of the book have limited its audi- ence to a general level. Those who are conversant with new

Egyptological research will be dismayed to find that it has been overlooked. There remains a certain dissatisfaction with the ab- sence of any historiographical or socio-historical analysis. One wishes that Clagett had offered a more complete schema of an- cient Egyptian mathematical thought. It is up to Egyptologists to do so.

ANTHONY SPALINGER

UNIVERSITY OF AUCKLAND

Proche-Orient ancien; Temps vecu, temps pense': Actes de la Table-Ronde du 15 novembre 1997 organisee par I'URA 1062 <Etudes Semitiques.>> Edited by FRANCOISE BRIQUEL- CHATONNET and HELENE LOZACHMEUR. Antiquit6s Semi-

tiques, vol. 3. Paris: JEAN MAISONNEUVE, 1998. Pp. 238, maps, illustrations. FF 260.

A large dictionary will indicate several meanings for English 'time,' French 'temps,' German 'Zeit,' etc., and dozens of idio- matic usages. In none of the ancient Semitic languages that I know, however, is a word attested designating the abstract con-

cept of 'time' as "indefinite and continuous duration regarded as that in which events succeed one another" (part of the first

gloss in the Random House Dictionary of the English Lan-

guage, 2nd unabr. ed., 1987). There are, of course, words for units of time as defined by the movements of the moon and of the sun ('day,' 'month,' 'year'; 'week,' which does not correspond

134 134

This content downloaded from 62.122.76.60 on Wed, 18 Jun 2014 02:38:45 AMAll use subject to JSTOR Terms and Conditions