Contents. Introduction Algorithms for Arithmetic Operations Magic Squares Methods of False Position... 83

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1 Contents Introduction Algorithms for Arithmetic Operations Sumerian Division A Babylonian Algorithm for Calculating Inverses Egyptian Algorithms for Arithmetic Tableau Multiplication Optimising Calculations Simple Division by Difference on a Counting Board Division on the Chinese Abacus Numbers Written as Decimals Binary Arithmetic Computer Arithmetic Bibliography Magic Squares Squares with Borders The Marking Cells Method Proceeding by 2 and by Arnauld s Borders Method Bibliography Methods of False Position Mesopotamia: a Geometric False Position Egypt: Problem 26 of the Rhind Papyrus China: Chapter VII of the Jiuzhang Suanshu India: Bha skara and the Rule of Simple False Position Qusṭa Ibn Lu qa : A Geometric Justification Ibn al-banna : The Method of the Scales Fibonacci: the Elchatayn rule Pellos: The Rule of Three and The Method of Simple False Position Clavius: Solving a System of Equations Bibliography

2 VI Contents 4 Euclid s Algorithm Euclid s Algorithm Comparing Ratios Bézout s Identity Continued Fractions The Number of Roots of an Equation Bibliography From Measuring the Circle to Calculating π Geometric Approaches The Circumference of the Circle The Area of the Circle in the Jiuzhang Suanshu The Method of Isoperimeters Analytic Approaches Arithmetic Quadrature Using Series Epilogue Bibliography Newton s Methods The Tangent Method Straight Line Approximations Recurrence Formulas Initial Conditions Measure of Convergence Complex Roots Newton s Polygon The Ruler and Small Parallelograms Bibliography Solving Equations by Successive Approximations Extraction of Square Roots The Method of Heron of Alexandria The Method of Theon of Alexandria Mediaeval Binomial Algorithms Numerical Solutions of Equations Al-Ṭu s1 s Tables Viète s Method Kepler s Equation Bernoulli s Method of Recurrent Series Approximation by Continued Fractions

3 Contents VII Horner like Transformations of Polynomial Equations The Ruffini-Budan Schema Bibliography Algorithms in Arithmetic Factors and Multiples The Sieve of Eratosthenes Criteria For Divisibility Quadratic Residues Tests for Primality The Converse of Fermat s Theorem The Lucas Test Pépin s Test Factorisation Algorithms Factorisation by the Difference of Two Squares Factorisation by Quadratic Residues Factorisation by Continued Fractions The Pell-Fermat Equation The Arithmetica of Diophantus The Lagrange Result Bibliography Solving Systems of Linear Equations Cramer s Rule The Method of Least Squares The Gauss Pivot Method A Gauss Iterative Method Jacobi s Method Seidel s Method Nekrasov and the Rate of Convergence Cholesky s Method Epilogue Bibliography Tables and Interpolation Ptolemy s Chord Tables Briggs and Decimal Logarithms The Gregory-Newton Formula Newton s Interpolation Polynomial The Lagrange Interpolation Polynomial

4 VIII Contents 10.6 An Error Upper Bound Neville s Algorithm Bibliography Approximate Quadratures Gregory s Formula Newton s Three-Eighths Rule The Newton-Cotes Formulas Stirling s Correction Formulas Simpson s Rule The Gauss Quadrature Formulas Chebyshev s Choice Epilogue Bibliography Approximate Solutions of Differential Equations Euler s Method The Existence of a Solution Runge s Methods Heun s Methods Kutta s Methods John Adams and the Use of Finite Differences Epilogue Bibliography Approximation of Functions Uniform Approximation Taylor s Formula The Lagrange Remainder Chebyshev s Polynomial of Best Approximation Spline-Fitting Mean Quadratic Approximation Fourier Series The Fast Fourier Transform Bibliography Acceleration of Convergence Stirling s Method for Series The Euler-Maclaurin Summation Formula The Euler Constant

5 Contents IX 14.4 Aitken s Method Richardson s Extrapolation Method Romberg s Integration Method Bibliography Towards the Concept of Algorithm Recursive Functions and Computable Functions The 1931 Definition General Gödel Recursive Functions Alonzo Church and Effective Calculability Recursive Functions in the Kleene Sense Machines The Turing Machine Post s Machine Conclusion Bibliography Biographies General Index Index of Names

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