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The VNR Concise Encyclopedia of Mathematics PDF

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THEVNR CONCISE ENCYCLOPEDIA OF MATHEMATICS ~ THEVNR CONCISE ENCYCLOPEDIA OF MATHEMATICS ~~~~~~ WGellert· S.Gottwald M. Hellwich . H. Kastner· H. KOstner Editors KA.Hirsch· H.Reichardt Scientific Advisors rnDBI VAN NOSTRAND REINHOLD ~ ____ New York © VEB Bibliographisches Institut Leipzig, 1975 Softrover repri1t oft he hardoover 1st edition 1975 Mathematics at a Glance First American Edition 1977 Second American Edition 1989 Library of Congress Catalog Card Number 88-26992 1SBN-13: 978-94-011-69844 e-lSBN-13: 978-94-011-6982-0 001: 10.1007!978-94-011-6982-O All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means - graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems - without written permission of the publisher. Made in the German Democratic Republic. Published by Van Nostrand Reinhold 115 Fifth Avenue New York, New York 10003 Van Nostrand Reinhold International Company Limited 11 New Fetter Lane London EC4P 4EE, England Van Nostrand Reinhold 480 La Trobe Street Melbourne, Victoria 3000, Australia Macmillan of Canada Division of Canada Publishing Corporation 164 Commander Boulevard Agincourt, Ontario MIS 3C7, Canada 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloglng-In-Publlcation Data Main entry under title: The VNR concise encyclopedia of mathematics. First published under title: Mathematics at a glance. Includes index. I. Mathematics-Handbooks, manuals, etc. I. Gottwald, S. II. Van Nostrand Reinhold Company. QA40.v18 1989 510-dc19 88-26992 Contents Introduction ..................•.....•...•..•.••.....•.......••.........•......... 11 I. Elementary mathematics 1. Fundamental operations on rational numbers. . . . • . . . . . . . . . . . . . . . . . . . . . . . . . • . . . • 17 2. Higher arithmetical operations............................... .•...... ........• 47 3. Development of the number system............................................ 69 4. Algebraic equations ............•...•..•...•..•.........•...................• 80 5. Functions ................................................................. , 107 6. Percentages, interest and annuities ...•..............•..........•...........•.. , 139 7. Plane geometry .........•.............•••............•••••.•...•...•.......• 146 8. Solid geometry .................•...........•..••.......•....••.......•.•.•. , 184 9. Descriptive geometry ............•••••.•.•..............•.........•.....•...• 203 10. Trigonometry ...............•.......................................••..•... 220 11. Plane trigonometry ............•..........•......•.....•.........•...•...... , 241 12. Spherical trigonometry .....................................•.•..........•..• 261 13. Analytic geometry of the plane ...•.••....•.....•...•.......••••..••..••....... 282 ll. Steps towards higher mathematics 14. Set theory ..................••.....••.......••..•.........••.•........•..... 320 15. The elements of mathematical logic ...............••................•..•...... 332 16. Groups and fields ...............••.....•........•.......................... 343 17. Linear algebra . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . .. 356 18. Sequences, series, limits. . . . . . . . . . . . . • • . . . • • . • . . • . . • . . . . . • . . • . . . . • • . . • • . . . • . .. 381 19. Differential calculus .•.........•............................................. 406 20. Integral calculus .•.........................•.....•.....•.......•••••.••..... 443 21. Series of functions. . . . . . . . . . • . . . . . • . . . • . . . • . . • . • . . . . . . . . • . . • . . . . . . • • . • . . . . . .• 479 22. Ordinary differential equations. . . • • . . . . . . . . . . • . . . • • . . . . . . . . • • . • • • . . . . • • • • • . . .. 500 23. Complex analysis .................•.....•••••••......•...•.•...•..••.......• 517 24. Analytic geometry of space ..........•..•.................•..........••...... 530 25. Projective geometry ......••..•...........•...••..•....••..•.•..•..••.•.•..•. 547 26. Differential geometry, convex bodies, integral geometry •....••.•.••.•..•..•...... 561 27. Probability theory and statistics ..•....••.......•..•..............•.•...•..... 575 28. Calculus of errors, adjustment of data, approximation theory . • • . . . • . . . . . . . . . . • . •. 607 29. Numerical analysis •...••...........•...••.••.•.............••.....•......... 630 30. Mathematical optimization. • . . . • . . . • . . . . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . . . . .. 653 m. Brief reports on selected topics 31. Number theory •.•...............•••••.••............•...•....••.......•...• 669 32. Algebraic geometry .•....•.•...........•...••••••.........•••............... 675 33. Further algebraic structures ..••...••......•••..•.•.................•......•.. 678 34. Topology .•........•...••.....•.......•...•....••.......................... 680 35. Measure theory. . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . • . • . . . . . . . . . . . . . . . . .. 687 36. Graph theory. . . . . . . . • . . . . . . . . . • • . • • . . . . . • • . . • • . . . . . . . . . . . . . . . . . . . . . . • . . . . .• 688 37. Potential theory and partial differential equations ..•.....•.......•.............. 693 38. Calculus of variations. . . . • . • • • • . . . • • • . . . . . . . . . . . • . . . . . • . . . . . • . • • . • . . . . . . . . . .. 698 39. Integral equations .......•..•••....••......................•....•........... 703 40. Functional analysis. . . . . . . . . . . . . • . . . . . • . . . • . . . • . . . . . . . . . . . . . . • . . . . . . . . . . . . . .. 705 41. Foundation of geometry - Euclidean and non-Euclidean geometry ................ 711 42. Foundations of mathematics .............•................................... 717 43. Game theory ......•........................................................ 723 44. Perturbation theory ......................................................... 731 45. The pocket calculator .•............• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 732 46. Microcomputers .....•.............•........................................ 745 Preface It is commonplace that in our time sc:iem:e and technology cannot be mastered without the tools of mathematics; but the same applies to an ever growing extent to many domains of everyday life, not least owing to the spread of cybernetic methods and arguments. As a consequence, there is a wide demand for a survey of the results of mathematics. for an unconventional approach that would also make it possible to fill gaps in one's knowledge. We do not think that a mere juxtaposition of theorems or a collection of formulae would be suitable for this purpose, because this would over· emphasize the symbolic language of signs and letters rather than the mathematical idea, the only thing that really matters. Our task was to describe mathematical interrelations as briefly and precisely as possible. In view of the overwhelming amount of material it goes without saying that we did not just compile details from the numerous text-books for individual branches: what we were aiming at is to smooth out the access to the specialist literature for as many readers as possible. Since well over 700000 copies of the German edition of this book have been sold, we hope to have achieved our difficult goal. Colours are used extensively to help the reader. Important definitions and groups of formulae are on a yellow background, examples on blue, and theorems on red. The course of more complic: ated calculations is indicated by red arrows. Also, in the illustrations in the text colours show up the essential features. Ample examples help to make general statements understandable. Frequently the numerical calculations have been arranged separately so that a problem can be read as ali explanatory text, without reference to calculations, while the latter can be regarded as worked exam· pies with explicit details. Physical units, which occur in some examples, are given in the SI-system, which is coming more and more into legal and practical use. Eveayday examples are given in everyday units, both metric and others. A systematic subdivision of the material, many brief section headings, and tables are meant to provide the reader with quick and reliable orientation. The detailed index to the book gives an easy access to specific questions. In the plates at the end numerous photographs and colour plates help to make the material more vivid and provide interesting glimpses of the history of mathematics. We thank the authors of the various chapters, specially to acceding to our request for generally understandable diction even at the risk of deviating from the usual terminology. Above all in the brief reports on special topics many an author has found it difficult to be content with mere in· dications about a topic in which he is an expert. Our particular thanks are due to our advisors, Professor K. A. Hirsch, Queen Mary College, University of London, and Professor H. Reichardt, Section for Mathematics, Humboldt University of Berlin. They have worked untiringly for the improvement of the book and have helped to create a work which is a reliable source of information for every user and should convince everyone that mathematics is essentially a simple and learnable discipline. The Editors and the Publishers Plates Archimedes 17 Mathematics and the visual arts I Poster of the town of Syracuse (Italy) Ancient Egyptian mural: catching fish and 2/3 Mathematics in sdtool l/ll hunting birds in a papyrus thicket Introduction of the number seven and Painting by Melozzo da Forli (1438-1494): revision exercises Pope Sixtus IV appoints Platina as Prefect 4 Mathematics in industrial arts of the Vatican Surfaces of revolution in the design of a 18 Mathematics and the visual arts II pottery set Proportions of the human body 5 Drawing instruments I Drawings by Leonardo da Vinci and sketch Geometry sets by Albrecht DOrer 6 Drawing instruments II 19 Mathematics and the visual arts ill Slide rules Melancholia, copper engraving by Albrecht 7 Drawing instruments ill Viirer Rulers, protractors, and French curves 20 Geometric: forms in architecture and 8 Graph papers technology I Millimetre paper - doubly logarithmic Egyptian pyramids near Giza paper - simply logarithmic paper - polar Tower of city walls coordinate paper - triangular net paper - The old town hall of Leipzig probability paper 21 Geometric forms in architecture and tecb 9 From the earliest period of mathematics nologyll Clay vessels of the new stone age Modern water tower Early Egyptian surveying Cooling towers of a generating plant 10 Ancient Egyptian mathematics 22 Geometric: forms in architecture and tech Original text of the Hau problem in Demotic nologyill writing and transcription of the same text Obelisk in the great temple of Amun at into hieroglyphics Karnak Calculation of a frustum of a pyramid Wedge as a cleaving tool 11 Babylonian mathematics Hyperbolic paraboloid shells as roofs of Cuneiform tablet with calculations of areas an exhibition hall Section of the tablet above 23 Famous mathematicians of the 15th/16th 12 Graeco-Roman mathematics century The Elements of Euclid, first printed edition Regiomontanus - Simon Stevin - Albrecht 1482 Diirer - Niccolo Tartaglia - Geronimo Roman hand abacus Cardano - Jost Biirgi - Luca Pacioli 13 Andent Chinese mathematics 24 Famous mathematicians of the 16th century From a manuscript dated 1303 Title page of Robert Recorde's ' Algebra' Bamboo sticks to represent numbers Title page of Adam Ries's 'Rechnung auff Chinese slide rule (about 1600) der Linihen und Federn ...' 14 Andent Hindu mathematics A problem out of this book concerning the Mathematical-astronomical buildings of the purchase of livestock 17th century 25 From old arithmetic books Mathematical manuscript of the 16th Conclusion of a business deal at a calculat century ing desk 15 Arabic: mathematics Calculation of the capacity of a cask Theorem of Pythagoras in an Arabic 26 Two libraries mathematical manuscript of the 14th The mathematics room of the National and century University Library in Prague Arabic astrolabe Entrance to the Science Library of Erfurt 16 Mathematics in Europe, 15th to 17th century (Boyneburg portal) Triumph of the modem algorithm (digital 27 Old matbematkal aids I calculation) over the ancient counter reckon Pedometer, 1741 ing (abacus) Slit bamboo as counting stick The use of Jacob's staff Tally stick 8 28 Old ..........t icaI aids II 42 F~ matbelllatfdus of tile 1M Counters or markers for arithmetic and an c:eDtIIrJ IV elaborate box, 16th century Friedrich Wilhelm Bessel - Augustin Louis 29 Old matbelllatical aids m Cauchy - Jakob Steiner - Niels Henrik Surveyor's compass, about 1600 Abel - Peter Gustav Lejeune Dirichlet - 30 0 ........ 1 :avariste Galois - Pafnuti Lvovich Cheby- Illustration of a rod, by juxtaposition of sheY 16 feet 43 F~ matbelllatfdus of tile 1M 16th century measuring rods with various ceatIIr)' V graduations Carl Gustav Jacob Jacobi - Bernhard 31 01d ...... 11 Riemann - Leopold Kronecker - Karl Set of weights, Nuremberg 1S88 WeierstraB - Arthur Cayley - Sophus Lie- Hinged sun dial, ivory Sonya KovaJevskay 32 F~ matbelllatidaDs of tile 17th 44 Matbelllatical iIIstruDalts 1 ceaDryl Instrument for drawing an integra) curve Title page of Descartes' 'Discours de Ja of a given function or differential equation mabode' Instrument to evaluate the integra) of a Ren6Descartes function whose graph is given 33 FIUIIOUS matbematiclans of tile 17th 45 Mathematical instrumeats II ceaturyll Compensating polar planimeter with polar Fran~is Vieta - John Napier - Galileo arm Galilei - Johannes Kepler - Buonaventura Compensating polar planimeter with polar Cavalieri - Pierre de Fermat - James carriage Gregory 46 Mathematical instrumeats m 34 F~ matbematidaDs of tile 17th/18th Precision pantograph ceatury 1 Instrument for the measurement of reel- Blaise Pascal angular coordinates or the drawing of Gottfried Wilhelm Leibniz points with given coordinates Isaac Newton 47 Matbematical instruments IV 35 FIUIIOUS matliematiciaDs of tile 17th/18th Harmonic analyser eent.yll Instrument to determine the tangent or Extract from a manuscript of Leibniz with normal to a curve whose graph is given the integral sign The mechanical calculator constructed by 48 FamollS matllematidans of the 1M/20th century 1 Pascal in 1642 George Stokes - Richard Dedekind - Georg 36 F~ matbematidans of tile 17th/18th Frobenius - Georg Cantor - Henri Poin- centurym care - Felix Klein - Emmy Noether Jakob Bernoulli 49 Famous matbematicians of the 19th/20th Johann Bernoulli century II Daniel Bernoulli 37 Famous matbematiclans of the 18th David Hilbert - :alie Joseph Cartan - Henri ceaturyl L60n Lebesgue - John von Neumann - Page from a manuscript by Euler Hermann Weyl - Jacques Hadamard - Leonhard Euler Stefan Banach 38 FaDlOlIS mathematicians of the 18th 50 Surveying ceaturyll Signals for the observation of trigonometric Brook Taylor - Moreau Maupertuis - nets Johann Heinrich Lambert - Joseph Louis Trigonometric point (TP) Lagrange - Gaspard Monge - Adrien Marie 51 Matbematical education I Legendre - Jean Baptiste Joseph de Fourier Work on a wall board 39 FaDlOlIS mathematidaDs of the 1M Determination of an angle with a hand-made ceaturyl apparatus Drawing by Janos B6lyai on non-Euclidean Giant slide rule for instructional purposes. geometry 52 Matbematical education II Nikolai lvanovich Lobachevskii Computations on part of an exhaust system 40 Famous mathematicians of the 19th Geometrical constructions on the black- ceaturyll board Portrait of the young Gauss Application of Pythagoras' theorem m Gauss in his old age 53 Mathematical education Gauss's signature Models for pupils: Cube with surface and The University in Gottingen space diagonals - Prism decomposable into 41 Famous matbematicians of the 19th three pyramids of equal volume - Cylinder ceatury III with sections - Sphere with plane sections - A page from Gauss's scientific diary Sections of a right circular cone 9 S4 Mirror images The patli of-thelight ray is the solution of Negative and positive of a photograph a minimal problem Reflection in water S6 Mathematical models Ship's Diesel engine in a left- and right Moebius strip hand version A closed surface of genus 1 SS Variational problems Formation of a minimal surface in a lobster Pseudosphere pot Surface representing the modulus of the Formation of a minimal surface by a soap film function w = exp (l/z) Index of mathematicians Abel, Niels Henrik, 1802-1829 Cavalieri, Bonaventura, c. 1598-1647 d'Alembert, Jean Ie Rond, 1717-1783 Cayley, Arthur, 1821-1895 Apollonius of Perga, c. 262-190 ? B. C. Ceva, Giovanni, 1647-1734 Archimedes, 287?-212 B. C. Chebyshev, Pafnuti Lvovich, 1821-1894 Argand, Jean Robert 1768-1832 Clavius, Christoph, 1537-1612 Aristotle, 384-322 B. C. Cramer, Gabriel, 1704-1752 Banach, Stefan, 1892-1945 Cusanus, Nicolaus, 1401-1464 Beltrami, Eugenio, 1835-1900 Dandelin, Pierre, 1794-1847 Bernoulli, Daniel, 1700-1782 Dedekind, Richard, 1831-1916 Bernoulli, Jakob, 1654-1705 de la Vallee-Poussin, Charles, 1966-1962 Bernoulli, Johann, 1667-1748 Descartes, Rene, 1596-1650 Bessel, Friedrich Wilhelm, 1784-1846 Diphantos of Alexandria, c. 250 A. D. Bezout, Etienne, 1730-1783 Dirichlet, Peter Gustav Lejeune, IW5-1859 Bhaskara, 1114-1185? Durer, Albrecht, 1471-1528 Birkhoff, George David, 1884-1944 Eisenhart, Luther Pfahler, 1876-1965 Blaschke, Wilhelm, 1885-1962 Enriques, Federigo, 1871-1946 B6lyai, Farkas, 1775-1856 Eratosthenes of Kyrene, c. 276-194 B. C. B6lyai, Janos, 1802-1860 Euclid of Alexandria, c. 450-380 B. C. Bolzano, Bernard, 1781-1848 Eudoxus, c. 408-355 B. C. Bombelli, Rafael, 16. century Euler, Leonhard, 1707-1783 Bahmagupta, born 598 Fermat, Pierre de, 1601-1665 Briggs, Henry, 1561-1630 Ferrari, Ludovico, 1522-1565 Brouwer, Luitzen Egbertus Jan, 1881-1966 Ferro, Scipione del, c. 1465-1526 Buffon, Georges Louis de, 1707-1788 Fibonacci t Leonardo of Pisa Burgi, Jost, 1552-1632 Fisher, Ronald Aylmer, 1890-1962 Burnside, William, 1852-1927 Fourier, Jean Baptiste Joseph de, Cantor, Georg, 1845-1918 1768-1830 Caratheodory, Constantin, 1873-1950 Fraenkel, Abraham, 1891-1965 Cardano, Geronimo, 1501-1576 Fredholm, Erik Ivar, 1866-1927 Cartan, Elie Joseph, 1869-1951 Frege, Gottlob, 1848-1925 Cartesius t Descartes Frobenius, Ferdinand Georg, 18491917 Cauchy, Augustin Louis, 1789-1857 Galilei, Galiko, 1564-1642 10 Index of mathematicians Galois, Evariste, 1811-1832 Ostrogradskii, Michail Wassilyevich,1801-1862 GauB, Carl Friedrich, 1777-1855 Ought red, William, 1574-1660 Girard, Albert, 1595-1632 Pacioli, luca, 1445?-1514 Goldbach, Christian, 1690-1764 Partridge, Seth, 1603-1686 Green, George, 1793-1841 Pappus of Alexandria, 4. century Gregory, James, 1638-1675 Pascal, Blaise, 1623-1662 Guldin, Paul, 1577-1643 Peano, Giuseppe, 1858-1932 Gunter, Edmund, 1561-1626 Pearson, Karl, 1857-1936 Hadamard, Jaques Salomon, 1865-1963 Pell, John, 1610-1685 Hamilton, Sir William Rowan, 1805-1865 Plato, 427-347? B. C. Hankel, Hermann, 1839-1874 Plucker, Julius, 1801-1868 Herbrand, Jacques, 1908-1931 Poincare, Henri, 1854-1912 Hermite, Charles, 1822-1901 Poisson, Simeon Denis, 1781-1840 Heron of Alexandria, c. 75 A. D. Poncelet, Jean Victor, 1788-1867 Hesse, Ludwig Otto, 1811-1874 Poseidon ius, c. 135-51 B. C. Hilbert, David, 1862-1943 Proclus,c.410-485 Hippasos of Metapontum, c. 450 B. C. Pythagoras of Samos, c. 580-496 B. C. Hippocrates of Chios, c. 440. B. C. Quetelet, Lambert Adolphe Jacques, 1796-1874 I'Hospital, Guillaume Fran~ois Antoine Mar- Recorde, Robert, 1510?-1558 quis de, 1661-1704 Regiomontanus, Johannes, 1436-1476 I'Huilier, Simon, 1750-1840 Riemann, Bernhard, 1826-1866 Huygens, Christiaan, 1629-1695 Ries, Adam, 1492-1559 Jacobi, Carl Gustav Jacob, 1804-1851 Rolle, Michel, 1652-1719 Jordan, Marie Ennemond Camille, 1838-1922 RudoltT, Christoph, c. 1500-1545 Kepler, Johannes, 1571-1630 Ruffini, Paolo, 1765-1822 Klein, Felix, 1849-1925 Russell, Bertrand, 1872-1970 Kovalevski, Sonya, 1850-1891 Rytz, David, 1801-1868 Kronecker, Leopold, 1823-1891 Saccheri, Girolarr.o, 1667-1733 Krull, Wolfgang Adolf ludwig Helmuth, Schmidt, Erhard, 1876-1959 1899-1971 Schwarz, Hermann Amandus, 1843-1921 Kummer, Ernst Eduard, 1810-1893 Segre, Corrado, 1863-1924 lagrange, Joseph Louis, 1736-1813 Severi, Francesco, 1879-1961 Lambert, Johann Heinrich, 1728-1777 Simpson, Thomas, 1710-1761 Laplace, Pierre Simon de, 1749-1827 Staudt, Carl Georg Christian von, 1798-1867 Lasker, Emmanuel, 1868-1941 Steiner, Jakob, 1796-1863 Lebesgue, Henri Leon, 1875-1941 Stevin, Simon, 1548-1620 Legendre, Adrien Marie, 1752-1833 Stifel, Michael, 1487-1567 Leibniz, Gottfried Wilhelm, 1646-1716 Stirling, James, 1696-1770 Leonardo da Vinci, 1452-1519 Stokes, George Gabriel, 1819-1903 Leonardo of Pisa, called Fibonacci, Tartaglia, Niccolb, originally Fontana Niccolb, 1180?-1250? c. 1500-1557 Lie, Sophus, 1842-1899 Taylor, Brook, 1685-1731 Lindemann, Ferdinand von, 1852-1939 Thales of Miletus, c. 624-547 B. C. Liouville, Joseph, 1809-1882 Theaitetus, 410?-368 B. C. Lipschitz, Rudolf, 1832-1903 Theodoros von Cyrene, c. 390 B. C. lobachevskii, Nikolai Iwanowich, 1792-1856 Tschirnhaus, Ehrenfried Walter Graf von, Lullus, Raimundus, Lull, Ramon, c. 1235-1315 1651-1708 Machin, John, 1685-1751 Viet a t Viete Maclaurin, Colin, 1698-1746 Viete, Fran~ois, 1540-1603 Maupertuis, Pierre Louis Moreau de, Vlacq, Adrien, c. 1600-1667 1698-1759 Wallis, John, 1616-1703 Menelaus of Alexandria, c. 98 A. D. Waring, Edward, 1734-1798 Miilkowski, Hermann, 1864-1909 WeierstraB, Karl, 1815-1897 Mobius, August Ferdinand, 1790-1868 Wessel, Caspar, 1745-1818 Moivre, Abraham de, 1667-1754 Weyl, Hermann, 1885-1955 Monge, Gaspard, 1746-1818 Whitehead, Alfred North, 1861-1947 Morgan, Augustus de, 1806-1871 Widmann, Johann, born 1460 Napier, Neper, John, 1550-1617 Wingate, Edmund, 1593-1656 Neumann, John von, 1903-1957 Wittich, Paul, 1555-1587 Newton, Isaac, 1643-1727 Wronski, Josef Maria, 1775-1853 Noether, Emmy, 1882-1935 Zenodoros, c. 180 B. C. Noether, Max, 1844-1921 Zenon of Elea, 490-430 B. C. Oresme, Nicole, 1323?-1382 Zermelo, Ernst, 1871-1953

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