Studying ancient Egyptian metrology is as difficult as it is necessary for anyone working on Egyptian mathematical texts, as Otto Neugebauer pointed out over 70 year ago:
Die genaue Lektüre ägyptischer geometrischer Texte ist, ganz abgesehen von allen terminologischen Schwierigkeiten, keine bequeme oder unterhaltsame Angelegenheit; […]. Vor allem muss man durch eigene Erfahrung die Beschwerung fast aller dieser Beispiele mit rein metrologischen Dingen kennenlernen, […].
(Close reading of Egyptian geometric texts is — apart from difficulties arising from terminology — neither easy nor enjoyable; […]. Most notably one experiences that all these examples are complicated by purely metrological issues.)
(Otto Neugebauer, “Die Geometrie der ägyptischen mathematischen Texte”, in: O. Neugebauer et al., Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik , Abteilung B: Studien, Vol. 1: p. 414, Berlin: Springer 1931)
This holds especially for the capacity measures (“Hohlmaße”). Usually written in abbreviated forms that vary (also in respective sizes) over time, they are a significant obstacle when dealing with Egyptian geometry (which still holds a prominent place within Egyptian mathematics, e.g. in the calculations of volumes of cylindrical or rectangular granaries). There were only few publications a historian of mathematics or an Egyptologist could turn to — most notably a series of articles by Francis Llewellyn Griffith (“Notes on Egyptian Weights and Measures”, in: Proceedings of the Society of Biblical Archaeology 13 (1891): pp. 301–316 and 14 (1892) 403–450) — published more than 100 years ago —, an article by Walter Friedrich Reineke (“Der Zusammenhang der altägyptischen Hohl- und Längenmaße”, in: Mitteilungen des Instituts für Orientforschung 9 (1963): pp. 145–163 as well as the overview of New Kingdom grain measures given by Anthony Spalinger (“The Grain System of Dynasty 18”, in: Studien zur Altägyptischen Kultur 14 (1987): pp. 283-311).
As Pommerening indicates in her introduction (p. 6), no research so far had been done which incorporated simultaneously evidence from reliefs, textual evidence and archaeological evidence. She herself is predestined to accomplish this task by her training as a pharmacist, Egyptologist, and historian of science (from the school of Fritz Krafft, who supervised her dissertation and has also written a detailed review of the book; see “Neue Erkenntnisse über altägyptische Hohlmaße”, in: Berichte zur Wissenschaftsgeschichte 29 (2006): 345–356. This is what she set out to do for her doctoral dissertation, the result of which is this 500 page book. The publication has been awarded the “Dalberg-Preis für transdisziplinäre Nachwuchsforschung“ of the ‘Akademie gemeinnütziger Wissenschaften zu Erfurt in Verbindung mit den Thüringer Hochschulen’ indicating the impressive outcome of her project.
The book is structured in three parts, the analytic-theoretical section (pp. 3–277) is followed by a catalogue (pp. 281–418) and a section detailing abbreviations, several indexes, and the bibliography (pp. 421–504). Her introduction indicates that she made an effort to find relevant artefacts in various museums in Europe and the Egyptian museum in Cairo — an indispensable project, as she noted, since many measuring vessels had not been recognized as such in various collections. Collections in the USA, which Pommerening has not yet visited, are likely to hold further discoveries of this kind. But even though, the material she has collected and presents in her catalogue is vast and a good basis for her theoretical analysis.
In order to use her book, it is essential for every reader to familiarize himself with the terminology that Pommerening uses as defined in her introduction, i.e., the distinction between various technical terms that usually have been summarized under the heading “measure” (Maß) in previous literature: measuring-vessel (Meßgefäß), vessel (Alltagsgefäß), and measuring unit (Maßeinheit). She retains the most general term “measure” only when she is not able to identify one of the more specific terms (p. 8).
The main part of the analytic-theoretical section comprises seven chapters (2-8) followed by a summary of the main results (9) and a short annex about Egyptian length measures (10). The individual chapters present a thorough analysis of various aspects related to capacity measures in various contexts.
Chapter 2 (“Verkörperung der Maßeinheiten: das Meßgefåß”; “Embodiment of measuring units: the measuring vessel”) examines the available evidence for measuring containers in the form of their depiction in reliefs and paintings (usually in the context of tombs), their models within the models depicting scenes from everyday life (also from tombs), extant actual containers, and finally their textual attestations. Based on their physical shapes, Pommerening distinguishes three types of measuring vessels: cylindrical vessels, conic ones (only attested in the context of baking and brewing) and sacks (used not only to measure, but also for transport). The analysis of actual measuring vessels is complicated by the basic problem of distinguishing measuring vessels from other types of vessels. Pommerening examined extant objects in 15 museums, the result of which can be found in chapter 13.
Chapter 3 then takes a look at another type of vessel called “Vorratsgefåß” (storing container). Since these containers bear inscriptions pertaining to their volume, they are also valuable evidence in the study of capacity measures. The chapter is subdivided chronologically into four sections (3.1–3.4) from the Early Dynastic Period to the Late Period, and followed by a discussion of the accuracy of measuring units derived from the analysis of these containers (3.5). It includes the hin measure, and the lesser known units of Dwt, possibly derived from a measure for meat, a measuring unit for ointments, a varying unit for a “portion”, and a unit designated by a + sign.
Sometimes vessels that were used as storing or packaging containers later became measuring containers as is detailed for the case of the Egyptian beer vessel ds (Chapter 4).
Chapter 5 describes the development of the grain measures throughout pharaonic history (2nd Dynasty till Late Period) in detail. Further examples of prominent measures are analysed in chapters 6–8: Chapters 6 and 7 examine the hin (which was already introduced in chapter 3), a measuring unit which evolved from a (measuring) vessel, and its third part, which occurs as a separate unit in the Choiak text of the temple of Hathor at Dendera, a text describing the making of a statue of the god Sokar, which involves numerous metrological details. Chapter 8 then analyses the dja-measure (8.1: evidence on measuring cups; 8.2: relation between dja- and hin, 8.3: further evidence), its possible identification with the +-measure (8.4), an attempt of a mythological background of this measure and the measuring vessels (8.5), and finally a chronological discussion of this measure (8.6).
Throughout her work, Pommerening uses evidence from various contexts in her analysis of ancient Egyptian capacity measures. Many specific observations are made within the individual chapters; the summary of main results given in chapter 9 presents a synthesis of her work. This includes a typology for measuring vessels consisting of four types (I: vessels with graduation and explicit indications of measuring unit; II: vessels with graduation without indications of measuring unit and series of vessels with a systematic change in volume; III: stone measuring vessels in form of a bushel — standard measures IV: possible measuring vessels. A deviation worth mentioning is that of 5% or more. Standard measures from temples often deviate from “normal” measures. A concise 1 1/2 page summary is given of the results of chapter 5: grain measures hold a central place in the system of capacity measures. These comprise the hekat and its multiples, the oipe, the khar, the dyadic fractions of hekat and oipe and the ro-subdivisions of 1/64 hekat. Usage of individual units varied through time. While the Old Kingdom indicated measured grain in single hekat, the New Kingdom uses the quadruple hekat called oipe (jp.t) after the calibrated measuring bushel. During the Late Period, the process of measuring grain changed significantly with the arrival of the artabe.
The reader of this review should be aware that this summary is selective; Pommerening’s book comprises many more observations, indicated with great care to detail. Despite the fact that the most prominent use of this book is likely to be that of a reference work, and for this purpose the various indices (sources, names — personal and deities — subjects, Egyptian words) will prove most useful, it may also serve as an indicator of the complexity of a topic as dry as “metrology”. It is through the synthesis of various strands of evidence that this comes to life, and anyone who fancies research or teaching of Egyptian mathematics should take note of it.
In a work like this, it is only natural that minor mistakes or omissions creep in; e.g. the omission of pRhind, no. 74 (corn measure in title and in the workings of the problem), the reference to pMoscow, no. 10 in the index, when the actual passage refers to line 10 of problem no. 9 (problem 10 is the famous problem which allegedly calculates the surface of a hemisphere) or to pMoscow, no. 26, which does not exist (pMoscow contains only 25 problems), the historian of mathematics Jim Ritter, who is abbreviated correctly (e.g. p. 151, note 278), becomes the Egyptologist James Ritner in the bibliography.
Annette Imhausen is a historian of ancient mathematics. She is currently Juniorprofessor for History of Mathematics at Johannes Gutenberg-Universität Mainz (Germany).
Teil 1: Analytisch-theoretischer Teil
Einleitung - Verkörperung der Maßeinheiten: das Messgefäß - Vorratsgefäße mit Maßangaben - Vom ds-Gefäß zum ds-Maß und zurück zur Verpackungseinheit - Das Kornmaßsystem - Vom Hin-Gefäß zum Hin-Maß - Das Maß zu 1/3 Hin im Choiaktext - Das Dja-Maß und seine möglichen Vorläufer - Zusammenfassung der Hauptergebnisse - Anhang: Das Längenmaßsystem.
Bildliche Darstellungen von Messgefäßen - Modelle von Messgefäßen und Schreibtafeln - Reale Messgefäße und ihre Abgrenzung - Markierte Vorratsgefäße.
Teil 3: Verzeichnisse
Abkürzungen, Literatur, Tabellen, Bildnachweis, Indizes.