Carl Ludwig Siegel gave a course of lectures on the Geometry of Numbers at New York University during the academic year 1945-46, when there were hardly any books on the subject other than Minkowski's original one. This volume stems from Siegel's requirements of accuracy in detail, both in the text and in the illustrations, but involving no changes in the structure and style of the lectures as originally delivered. This book is an enticing introduction to Minkowski's great work. It also reveals the workings of a remarkable mind, such as Siegel's with its precision and power and aesthetic charm. It is of interest to the aspiring as well as the established mathematician, with its unique blend of arithmetic, algebra, geometry, and analysis, and its easy readability.

From the Preface by K. Chandrasekharan: "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL. In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time...this publication...will no doubt stimulate generations of scholars to come." Volume III collects Siegel's papers from 1945 to 1964.

From the Preface by K. Chandrasekharan: "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL, who is rising seventy. In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time...this publication...will no doubt stimulate generations of scholars to come." Volume II includes Siegel's papers written between 1937 and 1944.

From the Preface by K. Chandrasekharan: "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL... In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time... this publication...will no doubt stimulate generations of scholars to come." Volume IV collects Siegels papers from 1968 to 1975.

Carl Ludwig Siegel gave a course of lectures on the Geometry of Numbers at New York University during the academic year 1945-46, when there were hardly any books on the subject other than Minkowski's original one. This volume stems from Siegel's requirements of accuracy in detail, both in the text and in the illustrations, but involving no changes in the structure and style of the lectures as originally delivered. This book is an enticing introduction to Minkowski's great work. It also reveals the workings of a remarkable mind, such as Siegel's with its precision and power and aesthetic charm. It is of interest to the aspiring as well as the established mathematician, with its unique blend of arithmetic, algebra, geometry, and analysis, and its easy readability.

The description for this book, Transcendental Numbers. (AM-16), will be forthcoming.

From the Preface (K. Chandrasekharan, 1966): "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL, who is rising seventy.In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time."Volume I includes Siegel's papers written between 1921 and 1937.

Uber die im folgenden behandelten Fragen der Himmelsmechanik habe ich in Frankfurt am Main und Baltimore sowie wiederholt in Gottingen und Princeton gelesen, am ausftihrlichsten in einem vier- sttindigen Gottinger Kolleg des Wintersemesters 1951/52. Herr Dr. J. MOSER, jetzt in ew York, hat damals eine sorgfaltige Xachschrift angefertigt, welche dieser Veroffentlichung zugrunde liegt. Ich bin kein Astronom yon Fach und habe deshalb auch keinen Versuch gemacht, die tiblichen Methoden zur praktischen Bahnbestim- mung erneut darzustellen, tiber die es bekanntlich gute Lehrbticher gibt. Es wird sich yielmehr yorwiegend darum handeln, einige Ideen und Resultate zu entwickeln, welche im Laufe der letzten 70 Jahre tiber das Verhalten der Losungen von Differentialgleichungen im groBen ent- standen sind, wobei allerdings die Anwendungen auf HAMILToNsche Systeme und insbesondere die Bewegungsgleichungen des Dreikorper- problems einen wichtigen Platz einnehmen. Auch hier habe ich keine Vollstandigkeit angestrebt, sondern die Auswahl so getroffen, wie sie durch personliches Interesse und die Hoffnung auf Anregung der Horer im Rahmen einer Vorlesung geboten wurde. Nach einleitenden Betrachtungen zur Transformationstheorie der Differentialgleichungen ist das Ziel des ersten Kapitels eine Darstellung der wichtigen Ergebnisse von K. F. SUNDMAN zum Dreikorperproblem. Obwohl die SUNDMANschen Satze bald 50 Jahre alt sind, so sind sie nur in klein em Kreise bekannt geworden und haben auf die spatere Entwicklung kaum gewirkt. Nachst POINCARES Leistungen zur Theorie der Differentialgleichungen gehoren SUNDMANs Arbeiten trotz ihres speziellen Charakters vielleicht zu den bedeutendsten neueren Ergeb- nissen auf dies em Gebiet.