Book contents
- Frontmatter
- PREFACE
- LIST OF THE PRINCIPAL WORKS CONSULTED
- Contents
- INTRODUCTION
- CHAPTER I ARCHIMEDES
- CHAPTER II MANUSCRIPTS AND PRINCIPAL EDITIONS—ORDER OF COMPOSITION—DIALECT—LOST WORKS
- CHAPTER III RELATION OF ARCHIMEDES TO HIS PREDECESSORS
- CHAPTER IV ARITHMETIC IN ARCHIMEDES
- CHAPTER V ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ
- CHAPTER VI CUBIC EQUATIONS
- CHAPTER VII ANTICIPATIONS BY ARCHIMEDES OF THE INTEGRAL CALCULUS
- CHAPTER VIII THE TERMINOLOGY OF ARCHIMEDES
- THE WORKS OF ARCHIMEDES
CHAPTER IV - ARITHMETIC IN ARCHIMEDES
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- PREFACE
- LIST OF THE PRINCIPAL WORKS CONSULTED
- Contents
- INTRODUCTION
- CHAPTER I ARCHIMEDES
- CHAPTER II MANUSCRIPTS AND PRINCIPAL EDITIONS—ORDER OF COMPOSITION—DIALECT—LOST WORKS
- CHAPTER III RELATION OF ARCHIMEDES TO HIS PREDECESSORS
- CHAPTER IV ARITHMETIC IN ARCHIMEDES
- CHAPTER V ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ
- CHAPTER VI CUBIC EQUATIONS
- CHAPTER VII ANTICIPATIONS BY ARCHIMEDES OF THE INTEGRAL CALCULUS
- CHAPTER VIII THE TERMINOLOGY OF ARCHIMEDES
- THE WORKS OF ARCHIMEDES
Summary
Two of the treatises, the Measurement of a circle and the Sand-reckoner, are mostly arithmetical in content. Of the Sandreckoner nothing need be said here, because the system for expressing numbers of any magnitude which it unfolds and applies cannot be better described than in the book itself; in the Measurement of a circle, however, which involves a great deal of manipulation of numbers of considerable size though expressible by means of the ordinary Greek notation for numerals, Archimedes merely gives the results of the various arithmetical operations, multiplication, extraction of the square root, etc., without setting out any of the operations themselves. Various interesting questions are accordingly involved, and, for the convenience of the reader, I shall first give a short account of the Greek system of numerals and of the methods by which other Greek mathematicians usually performed the various operations included under the general term λογιστική (the art of calculating), in order to lead up to an explanation (1) of the way in which Archimedes worked out approximations to the square roots of large numbers, (2) of his method of arriving at the two approximate values of which he simply sets down without any hint as to how they were obtained.
Greek numeral system.
It is well known that the Greeks expressed all numbers from 1 to 999 by means of the letters of the alphabet reinforced by the addition of three other signs, according to the following scheme, in which however the accent on each letter might be replaced by a short horizontal stroke above it, as ā.
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- Information
- The Works of ArchimedesEdited in Modern Notation with Introductory Chapters, pp. lxviii - xcixPublisher: Cambridge University PressPrint publication year: 2009First published in: 1897