Mathematics in Literature

In addition to the accounts, evidence for the significance of mathematics for scribal culture also comes from some literary texts, most notably, the Late Egyptian Miscellanies. The Miscellanies are a group of various texts, including model letters, excerpts of literary compositions, praises of the scribal profession, eulogies, and hymns to a teacher and others. During the Ramesside Period, scribes copied them on a regular basis. Several of these texts refer explicitly to the profession of the scribe, its demands and rewards. The so-called Satire of the Trades, extant in several versions, compares the duties and privileges of a scribe with those of other professions (e.g. that of a farmer, a soldier, and others). This chapter illustrates direct or indirect references to mathematics within these sources. Topics covered include mathematical education, mathematics in the scribe's daily work, and mathematics in Papyrus Anastasi I.

Mathematical Texts (I): The Mathematical Training of Scribes

This chapter discusses mathematical texts that originated from the Middle Kingdom. While this may well be caused by the vagaries of preservation, it might be that it reflects the actual situation, that is, that mathematical texts of the kind that we have from the Middle Kingdom did not exist in earlier periods. With the reestablishment of central power by the king in the Middle Kingdom also came about a complete new organization of the administrative apparatus that was designed to be much less independent than it had been at the end of the Old Kingdom. And this may well have entailed the organization of teaching mathematics to the future scribes in a centrally organized style, with prescribed problems and their solutions. The chapter considers extant hieratic mathematical texts, mathematical procedure texts, and types of mathematical problems.

Uses of Numbers and Their Contexts in Predynastic and Early Dynastic Times

This chapter reviews evidence on the use of the ancient Egyptian number system. There have been only three types of signs used in writing a number: a vertical stroke, a horizontal stroke, and a rope. Of these, the vertical stroke and the rope have successors in the later evidence of Egyptian number notations in the writing of units and hundreds, respectively. Early evidence for the use of number notation originates mostly from funerary or temple contexts. The usage of numbers in a funerary context can be described as administrative and may serve as a mirror for the usage of numbers in daily life at this time. Apart from the administration of goods, in which numbers and quantities have an inherent place, numbers also obtain a secondary, ritual meaning, which is shown by objects like the Narmer macehead and Narmer palette, but also by the offering lists. The writing of numbers is used to indicate the presence of something and to specify its quantities.

Summary

This chapter summarizes key discussions in chapters 9 to 15. The New Kingdom offers a variety of sources that provide insights into the uses and roles of mathematics at that time. As in previous periods, mathematics and the scribes who practiced it expertly continue to play a vital role in the administration. In order to illustrate the type and style of available sources, extracts of Papyrus Harris I and the Wilbour Papyrus have been used, both of which are as interesting as they are complex in terms of their usage of mathematical techniques and concepts. However, apart from these two outstanding papyri, it is the large number of smaller accounts written on papyri and ostraca that clearly documents the ongoing significance of administrative mathematics during the New Kingdom.

Further Aspects of Mathematics from New Kingdom Sources

This chapter proposes the following scenario: mathematics not only played a huge role within the education and work life of a scribe, but it also held an important role within ancient Egyptian culture in general. This role developed from the beginnings of the Egyptian state and the first uses of the number system (e.g., when the display of large numbers was used to express a certain power) and was by no means confined to the realm of the living. The discussions cover mathematics and wisdom literature, mathematics in the Duties of the Vizier, mathematics and death, and mathematics in architecture and art.

Notation of Fractions

This chapter discusses the development of the ancient Egyptian concept of fractions. The beginnings of fractions in ancient Egypt consisted of a small group of specific fractions written by special signs. These fractions are first attested within the context of metrological systems, but they retain their notation in later times as abstract fractions. The list of earliest fractions comprises 1/2, 1/3, and 1/4, and it may be inferred that fractions came to be understood as the inverses of integers. As a consequence, the Egyptian notation of fractions did not consist of numerator and denominator but rather of the respective integer of which the fraction was the inverse and a symbol to designate it as an inverse, that is, a fraction. Following the concept of fractions as inverses of integers, the next step would have been to express parts that consist of more than one of these inverses. This was done by (additive) juxtaposition of different inverses.

Summary

This chapter summarizes key discussions in chapters 1 to 3. Writing was developed in Egypt at the end of the fourth millennium BCE. Therefore, it is not surprising to find representations of numbers among the earliest evidence of written material from Egypt. The evidence from the tomb U-j reflects the administrative function of script and numbers, which by then had been taken over into the funerary context (in form of the “administration,” i.e., the recording of grave goods). The number system was fully developed even before the First Dynasty. It used a set of seven distinct hieroglyphic signs to represent powers of 10 in a decimal, nonpositional number system. The evidence presented indicates the close connection between literacy and numeracy, which will resurface again at later times in the history of Egyptian mathematics.

Introduction

This introductory chapter begins by providing the reader with a background to the historiography of Egyptian mathematics and the problems and possible approaches in the historiography of ancient mathematics (technical aspects vs. contextual, sociological, and cultural aspects), as well as indicates the specific difficulties inherent in Egyptian sources. It then sets out the book's purpose, which is to sketch the development of Egyptian mathematics from the invention of number notation, which occurred at approximately the same time as the invention of writing, until the Greco-Roman Periods using a variety of available sources, thereby also describing the context and cultural setting of Egyptian mathematics throughout pharaonic history. In order to achieve this, archaeological sources, administrative texts, autobiographies, and various literary texts will be used.

New Kingdom Mathematical Texts: Ostraca Senmut 153 and Turin 57170

This chapter considers mathematical texts from the New Kingdom. Almost no mathematical texts from the New Kingdom are extant. So far, only two fragmentary ostraca have been published: Senmut 153 and Turin 57170. The Senmut Ostracon comprises an incomplete (possible) title, followed by six lines of computation. The individual steps of the calculation are achieved with the aid of auxiliary numbers. The initial doubling may have been aided by the use of the 2 ÷ n table and the auxiliary number then introduced at a later stage. The ostracon, today kept in Turin, contains only a few (and incomplete) lines of text that seem to indicate solutions.

Metrological Systems

The development of metrology constituted the foundation of the quantitative control of agricultural resources, which then enabled the cultural achievements of ancient Egypt. Note that while some units remained the same and were used throughout Egyptian history, others became obsolete or changed. This chapter focuses on metrological units, which are attested in the Old Kingdom. Some metrological units were linked to a specific kind of object to be measured; thus it is to be expected that there were different units for assessing volumes of grain, liquids, or building materials. The discussions cover length units, area units, capacity units, and weights.