(Excerpt from Introduction to the Calculus In (b) and (c...)
Excerpt from Introduction to the Calculus In (b) and (c), h denotes the altitude and r, the radius of the base; V is here a function of the two independent variables, r and h. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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(Excerpt from Plane and Solid Analytic Geometry Algebra, ...)
Excerpt from Plane and Solid Analytic Geometry Algebra, on the other hand, was unknown to the Greeks. Its beginnings are found among the Hindus, to whom the so into Western Europe late, and not till the close of the middle ages was it carried to the point which is marked by any school book of today that treats this subject. When scholars had once possessed themselves of these two subjects Geometry and Algebra the next step was quickly taken. The renowned philosopher and mathematician, René Descartes, in his Geometrie of 1637, showed how the methods of algebra could be applied to the study of geometry. He thus became the founder of Analytic Geometry.* The originals and the locus problems of Elementary Geometry depend for their solution almost wholly on ingenu ity. There are no general methods whereby one can be sure of solving a new problem of this class. Analytic Geometry. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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(Excerpt from Lehrbuch der Funktionentheorie, Vol. 2: Erst...)
Excerpt from Lehrbuch der Funktionentheorie, Vol. 2: Erste Lieferung Die allgemeine Theorie der Funktionen mehrerer komplexen Großen hat seit der Wende des Jahrhunderts bedeutende Fortschritte gemacht. Während bis dahin, außer den unmittelbaren Übertragungen bekannter Sätze im Falle einer unabhängigen Veränderlichen, kaum mehr als defweierstraßsche Vorbereitungssatz und die Sätze von Cousin zu verzeichnen waren, erfuhr die Theorie bald darauf durch die Untersuchungen von Harto gs und E. E. Levi eine wesentliche eweiter1mg. Auch die Veröffentlichung des zweiten Weierstraß schen Satzes bezüglich impliziter Funktionen, nebst den Untersuchun gen amerikanischer Mathematiker betreffend die Umkehrung einer Transformation bei verschwindender Jacobischer Determinante, fiel in diese Zeit. Es tut jetzt not, diesen ganzen Stoff zu schichten und zu einem einheitlichen Ganzen zu verarbeiten, zumal deshalb, weil die Einfachheit manches Ergebnisses erst dann klar hervortritt, wenn dasselbe im Rahmen einer systematischen Darstellung ver wandter Erscheinungen steht. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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William Fogg Osgod was born on March 10, 1864, in Boston, Massachussets, the son of William and Mary Rogers (Gannett) Osgood, natives, respectively, of Kensington, New Hampshire, and Cohasset, Massachussets. His father, a Harvard graduate and a Unitarian, was the fourth in a direct line of general practitioners of medicine in New England and was known for his concern for the poor of the community. Except for twin sons who died in infancy, William was an only child.
William Osgood prepared for college at the Boston Latin School, entered Harvard in 1882, and was graduated with the A. B. degree in 1886, summa cum laude. As an undergraduate, Osgood had devoted his first two years largely to the classics, in which he received second-year honors. Influenced, however, by the mathematical physicist Benjamin Osgood Peirce - one of his favorite teachers - and by Frank Nelson Cole, who began lecturing at Harvard in Osgood's senior year, his interests turned to mathematics. He remained at Harvard for one year of graduate work in that field, receiving the degree of A. M. in 1887, before going on for further study at the University of Göttingen in Germany with Felix Klein, under whom Cole had recently studied. Two schools of thought were rivals in the stimulating mathematical atmosphere of Europe at that time. One, as represented by Bernhard Riemann, employed intuition and arguments borrowed from the physical sciences; the other, as represented by Karl Weierstrass, stressed strict, rigorous proof. Osgood throughout his career chose the best from the two schools, using intuition in its proper place to suggest results and their proofs, but relying ultimately on rigorous logical demonstrations. The influence of Klein on the "arithmetizing of mathematics" remained with Osgood throughout his later life. After two years at Göttingen he went to Erlangen, where he received his Ph. D. degree in 1890. His dissertation was a study of Abelian integrals of the first, second, and third kinds based on previous work by Klein and Max Noether. The subject was part of the theory of functions, to which Osgood was to devote so much of his career.
During the early years much of Osgood's scientific writing was done in German. Upon his return from Germany in 1890, Osgood joined the department of mathematics at Harvard, where he remained for the rest of his life, becoming assistant professor in 1893 and professor in 1903. Like other young Americans of his generation who had studied in Germany, he was ambitious to raise the scientific level of mathematics in the United States. There was no spirit of research at Harvard then except Osgood's own, but a year later Maxime Bocher was appointed to the department, and the two were close friends both personally and scientifically until Bocher's death in 1918. Osgood's scientific articles were of impressively high quality from the start. In 1897 he published a paper on uniform convergence of sequences of real continuous functions; this strongly influenced the later development of the subject. The next year he brought out a paper on the solutions of the differential equation y' = f(x, y) that is now known as a classic.
In 1900 Osgood established, by methods derived from Henri Poincare and others, the Riemann mapping theorem, namely that an arbitrary, simply connected region with at least two boundary points can be mapped uniformly and conformally onto the interior of a circle. This is a theorem of great importance, long conjectured to be true, but until then without a satisfactory proof. Osgood always did his research on problems that were both intrinsically important and classical in origin - "problems with a pedigree, " as he used to say. Felix Klein, as one of the editors, invited Osgood to contribute to the Encyklopädie der Mathematischen Wissenschaften, and Osgood's article "Allgemeine Theorie der analytischen Funktionen a) einer und b) mehreren komplexen Groessen" appeared in 1901. This was a deep, scholarly historical report on mathematical analysis; the writing of it gave Osgood an unparalleled familiarity with the literature of the field.
Osgood loved to teach at all levels. His exposition, though not always thoroughly transparent, was accurate, rigorous, and stimulating, invariably with emphasis on classical problems and results. Ranking with his teaching were his numerous books, notably his great Lehrbuch der Funktionentheorie (vol. I, 1906-07, and five later editions; vol. II, 1924-32). Its purpose was to present systematically and thoroughly the fundamental methods and results of analysis, with applications to the theory of functions of a real and of a complex variable. More systematic and more rigorous than French traités d'analyse, a monument to the care, orderliness, rigor, and didactic skill of its author, the book became a standard work wherever higher mathematics was studied. Osgood's text on the differential and integral calculus (1907) showed deep originality, especially in weaving the material into a single whole by anticipating at any stage more advanced material.
Osgood's influence throughout the mathematical world was very great through the soundness and depth of his Funktionentheorie, through the results of his research, through his stimulating yet painstaking teaching of both undergraduate and graduate students (though he did not direct many Ph. D. theses), and through his scholarly textbooks. He was elected to the National Academy of Sciences in 1904 and was president of the American Mathematical Society in 1904-05. He owned a summer cottage at Silver Lake, New Hampshire, and ordinarily spent his summers there. Although to outsiders he seemed reserved and somewhat formal, his friends found him warm and tender. Osgood retired from Harvard in 1933 and for the next two years taught in China at the National University of Peking. He died in 1943, at the age of seventy-nine, at his home in Belmont, Massachussets, of acute pyelonephritis. Cremation followed at Forest Hills Cemetery, Boston.
William Osgood's books: Introduction to Infinite Series (1897); Plane and solid analytic geometry (1921); Lehrbuch der Funktionentheorie (1907); First Course in Differential and Integral Calculus (1907); Elementary calculus (1921); Mechanics (1937). Osgood worked a lot with complex analysis, in particular conformal mapping and uniformization of analytic functions, and calculus of variations. William served as editor of the Annals of Mathematics (1899-1905) and Transactions (1909–1910).
(Excerpt from Introduction to the Calculus In (b) and (c...)
( This work has been selected by scholars as being cultur...)
(Excerpt from Plane and Solid Analytic Geometry Algebra, ...)
(Excerpt from Lehrbuch der Funktionentheorie, Vol. 2: Erst...)
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William Osgood was a member of the National Academy of Sciences and president of the American Mathematical Society (1904–1905).
Tall, spare, alert, and keen-eyed, Osgood during his middle years wore a square black beard.
On July 17, 1890, before his return from Germany, William Osgood married Therese Anna Amalie Elise Ruprecht of Göttingen. They had three children: William Ruprecht, Frieda Bertha, and Rudolf Ruprecht. Osgood's first marriage ended in divorce; on August 19, 1932, he married Mrs. Céleste Phelps Morse.
