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The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry.

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. I would particularly like to acknowledge my thanks to Professor Otto Neugebauer for his help and inspiration in the preparation of this book.

" In the meantime, however, Noel Swerdlow, also starting from Greek astronomy, not only extended his work into a deep analysis of De revolu tionibus, but also systematically investigated its sources and predecessors (Peurbach, Regiomontanus, etc.

Munk was the first modern scholar to draw attention to the significance of Levi ben Gerson's Astronomy, surely the most original work on astronomy written in Hebrew in the Middle Ages. Indeed, this is the first edition of the Hebrew text of any part of Levi's Astronomy but for the table of contents (Renan, 1893, pp.

Kepler's Physical Astronomy is an account of Kepler's reformulation of astronomy as a physical science, and of his successful use of (incorrect) physics as a guide in his astronomical discoveries.

In this work he announced the key criticality theorem 28 years before it was rediscovered in incomplete form by Galton and Watson (after whom the process was subsequently and erroneously named).

The calculus of variations is a subject whose beginning can be precisely dated. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem.

In this study we are concerned with Vibration Theory and the Problem of Dynamics during the half century that followed the publication of Newton's Principia. In fact, it was through problems posed by Vibration Theory that a general theory of Dynamics was motivated and discovered.

I first learned the theory of distributions from Professor Ebbe Thue Poulsen in an undergraduate course at Aarhus University.

This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. Though Zermelo's research has provided the focus for this book, much of it is devoted to the problems from which his work originated and to the later developments which, directly or indirectly, he inspired.

One of the pervasive phenomena in the history of science is the development of independent disciplines from the solution or attempted solutions of problems in other areas of science.

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.

Our interest in 1. J. Bienayme was kindled by the discovery of his paper of 1845 on simple branching processes as a model for extinction of family names. In this work he announced the key criticality theorem 28 years before it was rediscovered in incomplete form by Galton and Watson (after whom the process was subsequently and erroneously named). Bienayme was not an obscure figure in his time and he achieved a position of some eminence both as a civil servant and as an Academician. However, his is no longer widely known. There has been some recognition of his name work on least squares, and a gradually fading attribution in connection with the (Bienayme-) Chebyshev inequality, but little more. In fact, he made substantial contributions to most of the significant problems of probability and statistics which were of contemporary interest, and interacted with the major figures of the period. We have, over a period of years, collected his traceable scientific work and many interesting features have come to light. The present monograph has resulted from an attempt to describe his work in its historical context. Earlier progress reports have appeared in Heyde and Seneta (1972, to be reprinted in Studies in the History of Probability and Statistics, Volume 2, Griffin, London; 1975; 1976).

Ptolemy's Almagest shares with Euclid's Elements the glory of being the scientific text longest in use. The cautious emancipation of the late middle ages and the revolutionary creation of the new science in the 16th century are not conceivable without reference to the Almagest.