Acknowledgements
Contents
1 Cultures of Computation and Quantification in the Ancient World: An Introduction
1.1 What is at Stake?
1.1.1 Number and Operation in Recent Syntheses on the History of Mathematics in Various Regions of the World
1.1.2 Further Issues at Stake in Numbers and Computations
1.2 Tacit Assumptions in the Standard Histories of Number and Computation
1.2.1 How the History of Numerical Signs Has Been Tied to the History of Language and Writing
1.2.2 Has Computation Played no Major Part in the History of Numerical Signs?
1.2.3 Can We Decontextualize Numerical Signs Without Loss?
1.2.4 Do All Types of Numerals Express Quantities?
1.2.5 Are Measurement Values and Units not Part of the History of Mathematics?
1.2.6 Were the Executions of Operations Mere Tools?
1.3 Cultures of Computation and Quantification: Some First Results
1.3.1 Shaping Quantities and Relating Them to Numbers
1.3.2 Interpreting Numbers and Quantities in Texts: Different Notations, Locations, Uses, and Meanings
1.3.3 Working with Operations and Algorithms
1.3.4 Different Cultures of Computation and Quantification
1.4 Conclusion: Bridging the Gap Between the Two Branches of the Historiography of Number
References
Part I Shaping Quantities and Relating them to Numbers
2 Carrying Bricks and Bundling Reed in Theory and Practice
2.1 Carrying Bricks
2.1.1 Counting Bricks
2.1.2 Brick Types According to OB Mathematical Texts
2.1.3 Brick Types in Mathematical Texts and Their Labels in the Literature
2.1.4 Carrying of Bricks
2.1.5 Carriage (Nazbalum)
2.1.6 Actual Dimensions of Ur III and OB Bricks
2.1.7 Carrying Bricks in Administrative Records of the Ur III Period
2.1.8 Reconstructing Bricks Mentioned in Records of Brick Carrying
2.1.9 Brick Carrying from the Point of View of the Garshana Administrator
2.2 The Bundle Measure GuLAGAB
2.2.1 Ur III Bundles
2.2.2 GuLAGAB and Load
2.2.3 Labor Norms Defined by GuLAGAB
2.2.4 Number of Bundles in a GuLAGAB
2.2.5 ‘Extracted from GuLAGAB’
2.2.6 The Royal Bundle
2.2.7 The Reed Bundle of the OB Mathematical Text BM 85196 I 8-12
2.2.8 The Reed Bundle of BPOA 6 759
2.3 In Conclusion: Mathematical Texts Viewed from the Evidence of Administrative Records
References
3 Measuring Grain in Early Bronze Age Mesopotamia: Form, Use, and Control of the Bariga Container in the Twenty-First Century BCE
3.1 The System of Capacity Measures and the Bariga Standard
3.2 Clay Vessels as Standard Containers
3.3 Operating Standards of Capacity Measures in Ur III
3.4 Measurement by the Bariga
3.5 The Measuring Container Bariga as an Object: A Bitumen and Leather-Coated Reed Vessel with Bronze Reinforcements
3.6 Other Measuring Containers Made of Wood and Bronze
3.7 The Control of Measuring Containers
3.8 The Royal Edict of Ur-Namma Concerning Capacity Measures
References
4 Volume, Brickage and Capacity in Old Babylonian Mathematical Texts from Southern Mesopotamia
4.1 Introduction
4.1.1 Quantifying Space
4.1.2 The Sexagesimal Place-Value Notation (SPVN)
4.2 Catalogue Texts and Related Procedure Texts
4.3 Calculating a Volume: Procedure Text YBC 4663 #1–2 (Text 2)
4.3.1 General Comment
4.3.2 Problem #1
4.3.3 Problem #2
4.4 Brick Metrology: Catalogue YBC 4607 (Text 3)
4.4.1 Structure of the Text
4.4.2 Group I
4.4.3 Group II
4.4.4 Relation Between Volume and Brickage: The Nalbanum
4.5 Capacities and Standard Vessels: YBC 4669 #1–9 (Text 4)
4.6 Conclusion
Appendix 1: Main Texts
Text 1. Catalogue Text YBC 4657 #1–2 and Colophon
Transliteration
Translation
Text 2. Procedure Text YBC 4663 #1–2
Transliteration
Translation
Text 3. Catalogue YBC 4607
Translation
Text 4. YBC 4669 #1–9
Transliteration and Translation
Text 5. BM 85194 #30
Transliteration of #30
Translation
Appendix 2: Tables
Table 1-UET 7–114
Transliteration obv. #1–16
Translation obv. #1–16
Appendix 3: Composite Text of Metrological Tables
References
Websites
Bibliography
Part II Interpreting Numbers and Quantities in Texts
5 Place-Value Notations in the Ur III Period: Marginal Numbers in Administrative Texts
5.1 Introduction
5.1.1 In Search of the Origins of SPVN
5.1.2 System S
5.1.3 Paleography
5.1.4 Ur III Mathematical Texts
5.1.5 Positional Notations in Margins: Sources
5.2 Discussion of Individual Texts
5.2.1 About the Evidence
5.2.2 Umma
5.2.3 Puzriš-Dagan
5.2.4 Girsu
5.2.5 Unknown Provenience
5.3 Numbers and Operations
5.3.1 Common Features: Material Aspects of the Margins
5.3.2 Differences: Shaping Numbers for Operations
5.3.3 Patterns of Calculations
Appendix 1: Notations and Chronology
Transliterations and Translations
Location and Orientation
Chronology
Appendix 2: System S
A Broad Definition of System S
Distinguishing System S from SPVN
Appendix 3: Catalogue of Ur III Administrative Texts with Positional Notations
Appendix 4: Texts
Mathematical Text
Administrative Texts
References
6 The Nazbalum in Old Babylonian Mesopotamia: An Absolute Number or an Administrative Tool?
6.1 Introduction
6.2 The Coefficients in Problem Texts and Tables
6.2.1 Distance and Brick Load as a Definition of the Nazbalum
6.2.2 Parts of Work
6.3 Interpretation
6.3.1 What Defined the Nazbalum?
6.3.2 An Example
Appendix 1. Brick Typology and Parameters
1.1 Brickage
1.2 Brick Types as Presented in YBC 4607
1.3 Important Brick Coefficients
1.4 The Nalbanum
1.5 Construction of a Wall
1.6 The Brick System Revisited
Appendix 2. Texts
2.1 Practice Tablet MS 2221
2.2 AO 8862
2.3 YNER 4, 101, NBC 6339, P286528, Sumuel of Larsa Year 16, Month 7, Day 2
References
Part III Working with Operations and Algorithms
7 Computing Tools and Representations of Arithmetic
7.1 Introduction
7.2 The Representation of Numbers
7.2.1 The Chinese Abacus
7.2.2 The Japanese Abacus
7.2.3 The Russian Abacus
7.2.4 The French Abacus
7.2.5 Representation of Numbers as Structured Data
7.3 The Representation of Operations
7.3.1 Addition
7.3.2 Multiplication
7.3.3 Currying
7.4 The Representation of Arithmetical Properties
7.4.1 Reading Arithmetical Formulae as Programs
7.4.2 Left-Associativity and Syntactically Invalid Programs
7.4.3 Operation as Results, Operations as Techniques
7.5 Conclusion
7.6 Appendix Chapter 7: Sources
References
8 Working on and with Division in Early China, Third Century BCE—Seventh Century CE
8.1 Introduction
8.2 The Documents
8.3 A Change in the Terminology as an Indication of a Key Transformation with Respect to Division
8.4 Division in The Gnomon of the Zhou [Dynasty] as a Key to the Introduction of Several Types of Phases in the Execution of the Operation
8.4.1 Reconstructing an Older Execution of Division
8.4.2 Reconstructing the More Recent Execution of Division to Which the Commentator Refers
8.4.3 A Hypothetical Reconstruction of the Older Execution of Division as Attested to in the Gnomon of the Zhou [Dynasty]
8.5 Focus on Type 2 and Type 3 Phases, and Their Transformation: Determining the Core of the Result
8.5.1 Type 2 and Type 3 Phases in the Manuscripts
8.5.2 Type 2 and Type 3 Phases Vanish from the Nine Chapters
8.5.3 Related Theoretical Transformations Evidenced in the Nine Chapters
8.6 Focus on the History of Type 1 Phase of a Division: Determining the Operands
8.6.1 Type 1 Phase According to the Manuscripts
8.7 Conclusions
Appendix
Bibliography
9 Multiplying Integers: On the Diverse Practices of Medieval Sanskrit Authors
9.1 Introduction
9.1.1 A Variety of Methods for the Execution of One Operation
9.1.2 Sources
9.1.3 Different Terms for Multiplication
9.2 Brahmagupta and Pṛthūdaka's Commentary
9.2.1 A Multiplier Subdivided into `Portions' (khaṇḍa)
9.2.2 Multiplier Subdivided into `Parts' (bheda)
9.2.3 Reconciling Manuscripts and the Commentary?
9.2.4 Reflecting on the Historiography—Part 1
9.2.5 Conclusion on Pṛthūdaka and Brahmagupta's Methods of Executing Multiplications
9.3 The Gaṇitasārasaṃgraha and Its Anonymous Commentaries
9.3.1 A Procedural Rule (GSS 2.1)
9.3.2 Understanding the Vocabulary
9.3.3 The `Notes from Karanja'
9.3.4 Reflecting on the Historiography—Part 2
9.4 Śrīdhara and His Anonymous, Undated Commentary (PGT)
9.4.1 A `Door-Junction' (kavāṭa-sandhi)
9.4.2 A `Units' Partition' (rūpa-vibhāga)
9.4.3 A `
Place Partition' (sthāna-vibhāga)
9.4.4 Reflecting on the Historiography—Part 3
9.5 Conclusion
9.5.1 Brahmagupta, Mahāvīra and Śrīdhara
9.5.2 Describing a Multiplication?
9.5.3 Back to the `Resources' of Decimal Place-Value Notation
9.5.4 Several Multiplications: Easy, Quick or General?
9.5.5 Historiography and the Practices of Operations
References
Part IV Different Cultures of Computation and Quantification
10 Another Culture of Computation from Seventh-Century China
10.1 Introduction
10.1.1 Historical Background for These Two Types of Scholarship in Seventh-Century China
10.1.2 Aims and Sources
10.2 Analysis
10.2.1 Computational Methods (Suanfa 筭法)
10.2.2 The Shaping of, and Operation with, Numbers and Quantities
10.2.3 Use of Geometrical Representations of Quantities
10.2.4 Use of Tables
10.2.5 Writing Modes
10.3 Conclusion
10.3.1 Differences Between the Two Cultures of Computation
10.3.2 Scholars and Scholarship in the Early Tang Period
10.4 Appendix of this Chapter: Documents
References
11 The Characteristics of Mathematical Methods in the Wu Cao Suanjing and Its Social Background
11.1 Introduction
11.2 The Characteristics of the Methods in the Wu Cao
11.2.1 Computation Operations in the Wu Cao
11.2.2 Preference for Smaller Decimal Measurement Units Rather Than Fractions
11.2.3 Excluding Fractions from the Whole Book
11.2.4 Preference of Approximate or Even Mistaken Methods
11.2.5 Preference for Methods Expressed Using Specific Data
11.2.6 Significant Increase in the Number of Shapes and Names of Fields in the Wu Cao
11.3 Accounting for the Wu Cao’s Characteristics in Terms of the Social Background
11.3.1 Zhen Luan and His Purpose When Compiling the Wu Cao
11.3.2 The Social Background of Chaps. 2–5 of the Wu Cao
11.3.3 The Wu Cao’s Characteristics Fit the Needs of Local Government in the Northern and Southern Dynasties
11.3.4 The Characteristics of the Wu Cao and the Equal-Field System
11.4 A Brief Conclusion
Appendix of Chapter 11: Units
References
12 Weighing Units and Weights in the Context of Trade Between Upper Mesopotamia and Anatolia (Nineteenth and Eighteenth Centuries BCE)
12.1 A Progressive Metrological List of Weights
12.1.1 An Old Assyrian Metrological List
12.1.2 Comparison with Contemporaneous Old Babylonian Lists and Tables of Weights
12.1.3 How Quantities Were Shaped: Fractions Linked to Weight Units
12.2 Weighing and Weight Standards
12.2.1 Weight Standards
12.2.2 The Verification of Weight Standards, Discrepancies and Corrections
12.3 Two Cultures of Quantification
12.3.1 The Mesopotamian and the Anatolian Mana
12.3.2 Weighing Tin in Mesopotamia, Copper in Anatolia
12.3.3 The Anatolian Gin and Other Regional Standards
12.3.4 A Collection of Weights from Kültepe
12.4 Conclusion
Primary Sources: Old Assyrian Texts References
References
13 Quantification and Computation in the Mathematical Texts of Old Babylonian Diyala
13.1 Introduction
13.1.1 A Brief Review of Measurement Values and Abstract Numbers
13.1.2 Where and When? The Diyala Region in the Old Babylonian Period
13.1.3 A Methodological Standpoint: Using General Characteristics of Mesopotamian Mathematical Practices in a Study of Regional Mathematical Practices
13.2 The Corpus of Tablets
13.3 Analysis of the Sources
13.3.1 Measurement Units and Tablets That Contain Them
13.3.2 Correspondence Between Measurement Values and Abstract Numbers
13.3.3 Repertoire of Associations
13.4 Conclusions
13.4.1 Scribal Education in the Diyala
13.4.2 A Different Style from the South
13.5 Final Remarks
13.6 Appendix 13.1: Description of the Corpus of the Mathematical Tablets from the Diyala
13.7 Appendix 13.2: Transliterations and Translations
References
Annex A Conventions
A.1 Conventions for the Representation of Numbers and Measurement Values Used in Cuneiform Texts—SAW Project
A.2 Conventions for Numbers, Measurement Values and Measurement Units in Chinese Texts—SAW Project
Bibliography
A.3 The Representation of Quantities, Numbers and Measuring Units in Transliterations and Translations of South Asian Languages—SAW Project Reflections
Bibliography
Annex B Maps
Index 
