In this book I have attempted to trace the development of numerical analysis during the period in which the foundations of the modern theory were being laid. To do this I have had to exercise a certain amount of selectivity in choosing and in rejecting both authors and papers. I have rather arbitrarily chosen, in the main, the most famous mathematicians of the period in question and have concentrated on their major works in numerical analysis at the expense, perhaps, of other lesser known but capable analysts. This selectivity results from the need to choose from a large body of literature, and from my feeling that almost by definition the great masters of mathematics were the ones responsible for the most significant accomplishments. In any event I must accept full responsibility for the choices. I would particularly like to acknowledge my thanks to Professor Otto Neugebauer for his help and inspiration in the preparation of this book. This consisted of many friendly discussions that I will always value. I should also like to express my deep appreciation to the International Business Machines Corporation of which I have the honor of being a Fellow and in particular to Dr. Ralph E. Gomory, its Vice-President for Research, for permitting me to undertake the writing of this book and for helping make it possible by his continuing encouragement and support.

1. The Sixteenth and Early Seventeenth Centuries.- 1.1. Introduction.- 1.2. Napier and Logarithms.- 1.3. Briggs and His Logarithms.- 1.4. Bürgi and His Antilogarithms.- 1.5. Interpolation.- 1.6. Vieta and Briggs.- 1.7. Kepler.- 2. The Age of Newton.- 2.1. Introduction.- 2.2. Logarithms and Finite Differences.- 2.3. Trigonometric Tables.- 2.4. The Newton-Raphson and Other Iterative Methods.- 2.5. Finite Differences and Interpolation.- 2.6. Maclaurin on the Euler-Maclaurin Formula.- 2.7. Stirling.- 2.8. Leibniz.- 3. Euler and Lagrange.- 3.1. Introduction.- 3.2. Summation of Series.- 3.3. Euler on the Euler-Maclaurin Formula.- 3.4. Applications of the Summation Formula.- 3.5. Euler on Interpolation.- 3.6. Lunar Theory.- 3.7. Lagrange on Difference Equations.- 3.8. Lagrange on Functional Equations.- 3.9. Lagrange on Fourier Series.- 3.10. Lagrange on Partial Difference Equations.- 3.11. Lagrange on Finite Differences and Interpolation.- 3.12. Lagrange on Hidden Periodicities.- 3.13. Lagrange on Trigonometric Interpolation.- 4. Laplace, Legendre, and Gauss.- 4.1. Introduction.- 4.2. Laplace on Interpolation.- 4.3. Laplace on Finite Differences.- 4.4. Laplace Summation Formula.- 4.5. Laplace on Functional Equations.- 4.6. Laplace on Finite Sums and Integrals.- 4.7. Laplace on Difference Equations.- 4.8. Laplace Transforms.- 4.9. Method of Least Squares.- 4.10. Gauss on Least Squares.- 4.11. Gauss on Numerical Integration.- 4.12. Gauss on Interpolation.- 4.13. Gauss on Rounding Errors.- 5. Other Nineteenth Century Figures.- 5.1. Introduction.- 5.2. Jacobi on Numerical Integration.- 5.3. Jacobi on the Euler-Maclaurin Formula.- 5.4. Jacobi on Linear Equations.- 5.5. Cauchy on Interpolation.- 5.6. Cauchy on the Newton-Raphson Method.- 5.7. Cauchy on Operational Methods.- 5.8.Other Nineteenth Century Results.- 5.9. Integration of Differential Equations.- 5.10. Successive Approximation Methods.- 5.11. Hermite.- 5.12. Sums.