Mathematics in the Time of the Pharaohs

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More About This Textbook

Overview

In writing the first book-length study of ancient Egyptian mathematics, RichardGillings presents evidence that Egyptian achievements in this area are much more substantial than has been previously thought. He does so in a way that will interest not only historians of Egypt and of mathematics, but also people who simply like to manipulate numbers in novel ways. He examines all the extant sources, with particular attention to the most extensive of these—the Rhind MathematicalPapyrus, a collection of training exercises for scribes. This papyrus, besides dealing with the practical, commercial computations for which the Egyptians developed their mathematics, also includes a series of abstract numerical problems stated in a more general fashion.The mathematical operations used were extremely limited in number but were adaptable to a great many applications.

The Egyptian number system was decimal, with digits sequentially arranged (much like our own, but reading right to left), allowing them to add and subtract with ease. They could multiply any number by two, and to accomplish more extended multiplications made use of a binary process, successively multiplying results by two and adding those partial products that led to the correct result.

Division was done in a similar way. They could fully manipulate fractions, even though all of them (with one exception) were expressed in the unwieldy form of sumes of unit fractions—those having"1" as their numerator. (The exception was 2/3. The scribes recognized this as a very special quantity and took 2/3 of integral or fractional numbers whenever the change presented itself in the course of computation.) In expressing a rational quantity as a series of unit fractions, the scribes were generally able to choose a simple and direct solution from among the many—sometimes thousands—that are possible. Doing this without modern computers would seem quite as remarkable as building pyramids without modern machinery.The range of mathematical problems that were solved using these limited operational means is far wider than many historians of mathematics acknowledge.

Gillings gives examples showing that the Egyptians were able, for example, to solve problems in direct and inverse proportion; to evaluate certain square roots; to introduce the concept of a"harmonic mean" between two numbers; to solve linear equations of the first degree, and two simultaneous equations, one of the second degree; to find the sum of terms of arithmetic and geometric progressions; to calculate the area of a circle and of cylindrical (possibly even spherical) surfaces; to calculate the volumes of truncated pyramids and cylindrical granaries; and to make use of rudimentary trigonometric functions in describing the slopes of pyramids. TheEgyptian accomplishment that historians have tended to repeat uncritically, one after another, is one that Gillings can find no evidence to support: that the Egyptians knew the Pythagorean theorem,at least in the special case of the 3-4-5 right triangle.

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Product Details

  • ISBN-13: 9780262571791
  • Publisher: MIT Press
  • Publication date: 5/30/1972
  • Pages: 286
  • Product dimensions: 21.60 (w) x 27.70 (h) x 1.00 (d)

Table of Contents

PREFACE
Introduction
Hieroglyphic and Hieratic Writing and Numbers
The Four Arithmetic Operations
  ADDITION AND SUBTRACTION
  MULTIPLICATION
  DIVISION
  FRACTIONS
The Two-Thirds Table for Fractions
  PROBLEMS 61 AND 61B OF THE RHIND MATHEMATICAL PAPYRUS
  TWO-THIRDS OF AN EVEN FRACTION
  AN EXTENSION OF RMP 61B AS THE SCRIBE MAY HAVE DONE IT
  EXAMPLES FROM THE RHIND MATHEMATICAL PAPYRUS OF THE TWO-THIRDS TABLE
The G Rule in Egyptian Arithmetic
  FURTHER EXTENSIONS OF THE G RULE
The Recto of the Rhind Mathematical Papyrus
  THE DIVISION OF 2 BY THE ODD NUMBERS 3 TO 101
  CONCERNING PRIMES
  FURTHER COMPARISONS OF THE SCRIBE'S AND THE COMPUTERS DECOMPOSITIONS
The Recto Continued
  EVEN NUMBERS IN THE RECTO: 2 ÷ 13
  MULTIPLES OF DIVISORS IN THE RECTO
  TWO DIVIDED BY THIRTY-FIVE: THE SCRIBE DISCLOSES HIS METHOD
Problems in Completion and the Red Auxiliaries
  USE OF THE RED AUXILIARIES OR REFERENCE NUMBERS
  AN INTERESTING OSTRACON
The Egyptian Mathematical Leather Roll
  THE FIRST GROUP
  THE SECOND GROUP
  THE THIRD GROUP
  THE FOURTH GROUP
  THE NUMBER SEVEN
  LINE 10 OF THE FOURTH GROUP
  THE FIFTH GROUP
Unit-Fraction Tables
  UNIT-FRACTION TABLES OF THE RHIND MATHEMATICAL PAPYRUS
  PROBLEMS 7 TO 20 OF THE RHIND MATHEMATICAL PAPYRUS
Problems of Equitable Distribution and Accurate Measurement
  DIVISION OF THE NUMBERS 1 TO 9 BY 10
  CUTTING UP OF LOAVES
  SALARY DISTRIBUTION FOR THE PERSONNEL OF THE TEMPLE OF ILLAHUN
Pesu Problems
  EXCHANGE OF LOAVES OF DIFFERENT PESUS
Area and Volumes
  THE AREA OF A RECTANGLE
  THE AREA OF A TRIANGLE
  THE AREA OF A CIRCLE
  THE VOLUME OF A CYLINDRICAL GRANARY
  THE DETAILS OF KAHUN IV
Equations of the First and Second Degree
  THE FIRST GROUP
  SIMILAR PROBLEMS FROM OTHER PAPYRI
  THE SECOND AND THIRD GROUPS
  EQUATIONS OF THE SECOND DEGREE
  KAHUN LV
  "SUGGESTED RESTORATION OF MISSING LINES OF KAHUN LV 4, AND MODERNIZATION OF OTHERS"
Geometric and Arithmetic Progressions
  GEOMETRIC PROGRESSIONS: PROBLEM 79 OF THE RHIND MATHEMATICAL PAPYRUS
  ARITHMETIC PROGRESSIONS: PROBLEM 40 OF THE RHIND MATHEMATICAL PAPYRUS
  KAHUN IV
"Think of a Number" Problems"
  PROBLEM 28 OF THE RHIND MATHEMATICAL PAPYRUS
  PROBLEM 29 OF THE RHIND MATHEMATICAL PAPYRUS
Pyramids and Truncated Pyramids
  THE SEKED OF A PYRAMID
  THE VOLUME OF A TRUNCATED PYRAMID
The Area of a Semicylinder and the Area of a Hemisphere
Fractions of a Hekat
Egyptian Weights and Measures
Squares and Square Roots
The Reisner Papyri: The Superficial Cubit and Scales of Notation
APPENDIX 1
  The Nature of Proof
APPENDIX 2
  The Egyptian Calendar
APPENDIX 3
  Great Pyramid Mysticism
APPENDIX 4
  "Regarding Morris Kline's Views in Mathematics, A Cultural Approach"
APPENDIX 5
  The Pythagorean Theorem in Ancient Egypt
APPENDIX 6
  The Contents of the Rhind Mathematical Papyrus
APPENDIX 7
  The Contents of the Moscow Mathematical Papyrus
APPENDIX 8
  A Papyritic Memo Pad
APPENDIX 9
  "Horus-Eye Fractions in Terms of Hinu: Problems 80, 81 of the Rhind Mathematical Papyrus"
APPENDIX 10
  The Egyptian Equivalent of the Least Common Denominator
APPENDIX 11
  A Table of Two-Term Equalities for Egyptian Unit Fractions
APPENDIX 12
  "Tables of Hieratic Integers and Fractions, Showing Variations"
APPENDIX 13
  Chronology
APPENDIX 14
  A Map of Egypt
APPENDIX 15
  The Egyptian Mathematical Leather Roll
BIBLIOGRAPHY
INDEX
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