Early-Dynastic Tables from Southern Mesopotamia, or the Multiple Facets of the Quantification of Surfaces

Abstract

How were surfaces evaluated before the invention of the sexagesimal place value notation in Mesopotamia? This chapter examines a group of five tablets containing tables for surfaces of squares and rectangles dated to the Early Dynastic period (ca. 2600-2350 BCE) and unearthed in southern Mesopotamia. In order to capture the methods used by ancient scribes to quantify surfaces, special attention is paid to the layout and organization of the tables, as well as to the way in which measurement values are written down. It is argued that these methods vary according to the dimensions of the squares or rectangles concerned: the quantification of small surfaces does not use the same mathematical tools as the quantification of large parcels of land. The chapter shows a reciprocal influence between the metrological systems adopted by the ancient scribes and the methods of calculation of surface they implemented. Some methods may reflect ancient land-surveying practices, and others may testify the emergence of new mathematical concepts applied to all kinds of surfaces, large or small. Ultimately, several different conceptualizations of the notion of surface emerge from the examination of these tables.

Separating Space and Time for Dimensional Analysis and Euclidean Relational Modeling

“…where k can equal 0 [ 0  ] and k = 1 represents the first of N ordered geometric elements in the metric space of cycle fractions. Rt unit natural-number counts are consistent with Babylonian sexagesimal place-value notation, a 60-base system using finite sequences of sexagesimal digits 00, 01, …, 58, and 59, where both the first and last digits are non-null [52] [53], and the Mesoamerican vigesimal base-20 counting system (00, 01, …, 18, 19) [54]. In comparison to the Rt system, 00 corresponds to both 0 and 60 in a cycle, 1 0/60 = 60/60, or 1 0/20 = 20/20.…”

Relational and Euclidean Temporal Space

“…Whereas at the tenth line element of the second cycle, we now have 1 [(10)/ (10)] which is also equal in mathematics (given the described notation) and Rt description to 2 [(0)/ (10)]. The approach is consistent with Babylonian sexagesimal place-value notation, 0, 01, •••, 58, and 59, where both the first and last digits are non-null, [31] [32] as we interpret these findings to be consistent with the rishtar approach as discussed. Also consistent with the Mesoamerican vigesimal base-20 counting system (0, 01, •••, 18, 19).…”

Table understanding: Problem overview

“…Tables have been used to store and share information since ancient times. Proust (2020) described tables from southern Mesopotamia dated about 2600–2500 BCE (Figure 1a). Tallet and Marouard (2014) reported accounts on a papyrus organized in tables dated about 2500 BCE (Figure 1b).…”

“…Two ancient tables: Sumerian clay tablet with a table of surfaces known as VAT 12593 (Proust, 2020)—(a); account on a papyrus related to the construction of the Great Pyramid of Khufu at Giza (Tallet and Marouard, 2014)—(b).…”