Verfügbarkeit: Mathematics emerging

Mathematics emerging: a sourcebook 1540 - 1900

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Bibliographische Detailangaben
1. Verfasser:	Stedall, Jacqueline A. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Oxford [u.a.] Oxford Univ. Press 2008
Ausgabe:	1. publ.
Schlagworte:	Geschichte 1800 v. Chr.-1902 Mathematics / History Mathematics / Sources Geschichte Mathematik Mathematics > History Quelle
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Includes bibliographical references (p. [636]-648) and index
Beschreibung:	xxi, 653 S. graph. Darst.
ISBN:	9780199226900

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Datensatz im Suchindex

_version_	1804138043155677184
adam_text	CONTENTS ACKNOWLEDGEMENTS XIII INTRODUCTION XV BEGINNINGS I 1.1 BEGINNINGS OF ARITHMETIC 2 1.1.1 LARGE NUMBER CALCULATIONS, C. 1800 BC 2 1.1.2 SACROBOSCO S ALGORISMUS, C. 1230 AD 4 1.2 BEGINNINGS OF GEOMETRY 8 1.2.1 EUCUED S DEFINITIONS, C. 250 BC 9 1.2.2 EUCLID S CONSTRUCTION OF PROPORTIONALS, C. 250 BC 12 1.2.3 ARCHIMEDES ON CIRCLE MEASUREMENT, C. 250 BC 14 1.2.4 APOLLONIUS CONICS, C. 185 BC 16 1.3 BEGINNINGS OFA THEORY OF NUMBERS 19 1.3.1 EUCLID S DEFINITIONS OF NUMBER, C. 250 BC 19 1.3.2 EUCLID S PROOF OF THE INFINITY OF PRIMES, C. 250 BC 23 1.3.3 THE ARITHMETICA OF DIOPHANTUS, (AFTER 150 AD) 24 1 .4 BEGINNINGS OF ALGEBRA 26 1.4.1 COMPLETING THE SQUARE, C. 1800 BC 26 1.4.2 AL-KHWAERIZML S A!-JABR, C. 825 AD 29 FRESHIDEAS 33 2.1 IMPROVEMENTS IN CALCULATION 34 2.1.1 STEVIN S DECIMAL FRACTIONS, 1585 34 2.1.2 NAPIER S LOGARITHMS, 1614 39 2.2 IMPROVEMENTS IN NOTATION 43 2.2.1 HARRIOT S NOTATION, C. 1600 44 2.2.2 DESCARTES NOTATION, 1637 46 2.3 ANALYTIC GEOMETRY 47 2.3.1 VIETE S INTRODUCTION TO THE ANALYTIC ART, 1591 47 2.3.2 FERMAT AND ANALYTIC GEOMETRY, 1636 50 2.3.3 DESCARTES AND ANALYTIC GEOMETRY, 1637 54 2.4 INDIVISIBLES 62 2.4.1 CAVALIERI S THEORY OF INDIVISIBLES, 1635 62 2.4.2 WALLIS AND HOBBES ON INDIVISIBLES, 1656 66 VLLL CONTENTS 3 FORESHADOWINGS OF CALCULUS 71 3.1 METHODS FOR TANGENTS 72 3.1.1 FERMAT S TANGENT METHOD, 1629 72 3.1.2 DESCARTES TANGENT METHOD, 1637 74 3.2 METHODS OF QUADRATURE 78 3.2.1 FERMAT S QUADRATURE OF HIGHER HYPERBOLAS, EARLY 1640S 78 3.2.2 BROUNCKER AND THE RECTANGULAR HYPERBOLA, C. 1655 84 3.2.3 WALLIS USE OF INDIVISIBLES, 1656 89 3.2.4 MERCATOR AND THE RECTANGULAR HYPERBOLA, 1668 95 3.3 A METHOD OFCUBATURE 100 3.3.1 TORRICELLI S INFINITE SOLID, 1644 100 3.4 A METHOD OF RECTIFICATION 102 3.4.1 NEUE AND THE SEMICUBICAL PARABOLA, 1657 102 4 THE CALCULUS OF NEWTON AND OF LEIBNIZ 105 4.1 THE CALCULUS OF NEWTON 105 4.1.1 THE CHRONOLOGY OF NEWTON S CALCULUS 105 4.1.2 NEWTON S TREATISE ON FLUXIONS AND SERIES, 1671 107 4.1.3 NEWTON S FIRST PUBLICATIONOF HIS CALCULUS, 1704 114 4.2 THE CALCULUS OF LEIBNIZ 119 4.2.1 LEIBNIZ S FIRST PUBLICATION OF HIS CALCULUS, 1684 120 5 THE MATHEMATICS OF NATURE: NEWTON S PRINCIPIA 133 5.1 NEWTON S PRINCIPIA, BOOK I 135 5.1.1 THEAXIOMS 135 5.1.2 ULTIMATE RATIOS 139 5.1.3 PROPERTIES OF SMALL ANGLES 142 5.1.4 MOTION UNDER A CENTRIPETAL FORCE 144 5.1.5 QUANTITATIVE MEASURES OF CENTRIPETAL FORCE 147 5.1.6 THE INVERSE SQUARE LAW FOR A PARABOLA 150 6 EARLY NUMBER THEORY 155 6.1 PERFECT NUMBERS 155 6.1.1 EUCLID S THEOREM ON PERFECT NUMBERS, C. 250 BC 155 6.1.2 MERSENNE PRIMES, 1644 157 6.1.3 FERMAT S LITTLE THEOREM, 1640 159 6.2 PELLV EQUATION 162 6.2.1 FERMAT S CHALLENGE AND BROUNCKER S RESPONSE, 1657 162 6.3 FERMAT S FINAL CHALLENGE 165 7 EARLY PROBABILITY 167 7.1 THE MATHEMATICS OF GAMBLING 167 7.1.1 PASCAL S CORRESPONDENCE WITH FERMAT, 1654 167 7.1.2 JACOB BERNOULLI S ARS CONJECTANDI, 1713 170 7.1.3 DE MOIVRE S CALCULATION OF CONFIDENCE, 1738 176 7.2 MATHEMATICAL PROBABILITY THEORY 180 7.2.1 BAYES THEOREM, 1763 180 7.2.2 LAPLACE AND AN APPLICATION OF PROBABILITY, 1812 182 8 POWER SERIES 189 8.1 DISCOVERIES OF POWER SERIES 190 8.1.1 NEWTON AND THE GENERAL BINOMIAL THEOREM, 1664-1665 190 8.1.2 NEWTON S DE ANALYSI , 1669 193 8.1.3 NEWTON S SETTERS TO LEIBNIZ, 1676 196 8.1.4 GREGORY S BINOMIAL EXPANSION, 1670 198 8.2 TAYLOR SERIES 201 8.2.1 TAYLOR S INCREMENT METHOD, 1715 201 8.2.2 MACLAURIN S SERIES, 1742 206 8.2.3 FUNCTIONS AS INFINITE SERIES, 1748 208 8.3 CONVERGENCE OF SERIES 209 8.3.1 D ALEMBERT S RATIO TEST, 1761 210 8.3.2 LAGRANGE AND THE REMAINDER TERM, 1 797 215 8.4 FOURIER SERIES 223 8.4.1 FOURIER S DERIVATION OF HIS COEFFICIENTS, 1822 223 9 FUNCTIONS 229 9.1 EARLY DEFINITIONS OF FUNCTIONS 229 9.1.1 JOHANN BERNOULLI S DEFINITION OF FUNCTION, 1718 229 9.1.2 EULER S DEFINITION OF A FUNCTION (1), 1748 233 9.1.3 EULER S DEFINITION OF A FUNCTION (2), 1755 238 9.2 LOGARITHMIC AND CIRCULAR FUNCTIONS 244 9.2.1 A NEW DEFINITION OF LOGARITHMS, 1748 244 9.2.2 SERIES FOR SINE AND COSINE, 1748 250 9.2.3 EULER S UNINCATION OF ELEMENTARY FUNCTIONS, 1748 253 9.3 NINETEENTH-CENTURY DEFINITIONS OF FUNCTION 254 9.3.1 A DEFINITION FROM DEDEKIND, 1888 256 10 MAKING CALCULUS WORK 259 10.1 USES OF CALCULUS 260 10.1.1 [ACOB BERNOULLI S CURVEOF UNIFORM DESCENT, 1690 260 10.1.2 D ALEMBERT AND THE WAVE EQUATION, 1747 264 10.2 FOUNDATIONS OF THE CALCULUS 27 1 10.2.1 BERKELEY AND THE ANALYST, 1734 272 10.2.2 MACLAURIN S RESPONSE TO BERKELEY, 1742 274 10.2.3 EULER AND INFINITELY SMALL QUANTITIES, 1755 276 10.2.4 LAGRANGE S ATTEMPT TO AVOID THE INFINITELY SMALL, 1797 281 11 LIMITS AND CONTINUITY 291 11.1 LIMITS 291 11.1.1 WALLIS S LESS THAN ANY ASSIGNABLE , 1656 291 11.1.2 NEWTON S FIRST AND LAST RATIOS, 1687 292 1 1.1.3 MACLAURIN S DEFINITION OF A LIMIT, 1742 295 11.1.4 D ALEMBERT S DEFINITION OF A LIMIT, 1765 297 11.1.5 CAUCHY S DEFINITION OF A LIMIT, 1821 298 11.2 CONTINUITY 301 11.2.1 WALLIS AND SMOOTH CURVES, 1656 301 11.2.2 EULER S DEFINITION OF CONTINUITY, 1748 302 CONTENTS 11.2.3 LAGRANGE S ARBITRARILY SMALL INTERVALS, 1797 304 11.2.4 BOLZANO S DEFINITION OF CONTINUITY, 1817 306 11.2.5 CAUCHY S DEFINITION OF CONTINUITY, 1821 312 11.2.6 CAUCHV AND THE INTERMEDIATE VALUE THEOREM, 1821 316 12 SOLVING EQUATIONS 325 12.1 CUBICS AND QUARTICS 325 12.1.1 CARDANO AND THE ARS MAGNA, 1545 325 12.2 FROM CARDANO TO LAGRANGE 330 12.2.1 HARRIOT AND THE STRUCTURE OF POLYNOMIALS, C. 1600 331 12.2.2 HUDDE S RULE, 1657 333 12.2.3 TSCHIRNHAUS TRANSFORMATIONS, 1683 335 12.2.4 LAGRANGE AND REDUCED EQUATIONS, 1771 339 12.3 HIGHER DEGREE EQUATIONS 348 12.3.1 LAGRANGE S THEOREM, 1771 348 12.3.2 AFTERMATH: THE UNSOLVABILITY OF QUINTICS 350 13 GROUPS, FIELDS, IDEALS, AND RINGS 353 13.1 GROUPS 353 13.1.1 CAUCHY S EARLY WORK ON PERMUTATIONS, 1815 354 13.1.2 THE PREMIER MEMOIRE OF GALOIS, 1831 361 13.1.3 CAUCHY S RETURN TO PERMUTATIONS, 1845 366 13.1.4 CAYLEY S CONTRIBUTION TO GROUP THEORY, 1854 374 13.2 FIELDS, IDEALS, AND RINGS 378 13.2.1 GALOIS FIELDS , 1830 378 13.2.2 KUMMERAND IDEA! NUMBERS, 1847 381 13.2.3 DEDEKINDON FIELDS OFFINITE DEGREE, 1877 386 13.2.4 DEDEKIND S DEFINITION OF IDEALS, 1877 391 14 DERIVATIVES AND INTEGRALS 397 14.1 DERIVATIVES 397 14.1.1 LANDEN S ALGEBRAIC PRINCIPLE, 1758 397 14.1.2 LAGRANGE S DERIVED FUNCTIONS, 1797 401 14.1.3 AMPERE S THEORY OF DERIVED FUNCTIONS, 1806 406 14.1.4 CAUCHY ON DERIVED FUNCTIONS, 1823 415 14.1.5 THE MEAN VALUE THEOREM, ANDE,5 NOTATION, 1823 421 14.2 INTEGRATION OF REAL-VALUED FUNCTIONS 425 14.2.1 EULER S INLRODUCTION TO INTEGRATION, 1768 425 14.2.2 CAUCHY S DEFINITE INTEGRAL, 1823 434 14.2.3 CAUCHY AND THE FUNDAMENTAL THEOREM OF CALCULUS, 1823 444 14.2.4 RIEMANN INTEGRATION, 1854 446 14.2.5 LEBESGUEINTEGRATION, 1902 453 15 COMPLEX ANALYSIS 459 15.1 THE COMPLEX PLANE 459 15.1.1 WALLIS S REPRESENTATIONS, 1685 459 15.1.2 ARGAND S REPRESENTATION, 1806 462 CONTENTS XI 15.2 INTEGRATION OF COMPLEX FUNCTIONS 468 15.2.1 JOHANN BERNOULLI S TRANSFORMATIONS, 1702 468 15.2.2 CAUCHYON DEFINITE COMPLEX INTEGRALS, 1814 472 15.2.3 THE CALCULUS OF RESIDUES, 1826 480 15.2.4 CAUCHY S INTEGRAL FORMULAS, 1831 486 15.2.5 THE CAUCHY-RIEMANN EQUATIONS, 1851 491 16 CONVERGENCE AND COMPLETENESS 495 16.1 CAUCHY SEQUENCES 495 16.1.1 BOLZANO AND CAUCHY SEQUENCES , 1817 495 16.1.2 CAUCHY S TREATMENT OF SEQUENCES AND SERIES, 1821 500 16.1.3 ABEL SPROOF OF THEBINOMIAL THEOREM, 1826 515 16.2 UNIFORM CONVERGENCE 524 16.2.1 CAUCHY S ERRONEOUS THEOREM, 1821 524 16.2.2 STOKES AND INFINITELY SLOW CONVERGENCE, 1847 527 16.3 COMPLETENESS OF THE REAL NUMBERS 529 16.3.1 BOLZANO AND GREATEST LOWER BOUNDS, 1817 529 16.3.2 DEDEKIND S DEHNITION OF REAL NUMBERS, 1858 534 16.3.3 CANTOR S DEFINITION OF REAL NUMBERS, 1872 542 17 LINEAR ALGEBRA 547 17.1 LINEAR EQUATIONS AND DETERMINANTS 548 17.1.1 AN EARLY EUROPEAN EXAMPLE, 1559 548 17.1.2 RULES FOR SOLVING THREE OR FOUR EQUATIONS, 1748 552 17.1.3 VANDERMONDE S ELIMINATION THEORY, 1772 554 17.1.4 CAUCHY S DEFINITION OF DETERMINANT, 1815 563 17.2 EIGENVALUE PROBLEMS 570 17.2.1 EULER S QUADRATIC SURFACES, 1748 570 17.2.2 LAPLACE S SYMMETRIE SYSTEM, 1787 573 17.2.3 CAUCHY S THEOREMS OF 1829 579 17.3 MATRICES 586 17.3.1 GAUSS AND LINEAR TRANSFORMATIONS, 1801 586 17.3.2 CAYLEY S THEORY OF MATRICES, 1858 588 17.3.3 FROBENIUS AND BILINEAR FORMS, 1878 592 17.4 VECTORS AND VECTOR SPACES 598 1 7.4. 1 GRASSMANN AND VECTOR SPACES, 1862 598 18 FOUNDATIONS 605 18.1 FOUNDATIONS OF GEOMETRY 605 18.1.1 HILBERT S AXIOMATIZATION OF GEOMETRY, 1899 607 18.2 FOUNDATIONS OFARITHMETIC 613 18.2.1 CANTOR S COUNTABILITY PROOFS, 1874 614 18.2.2 DEDEKIND S DEFINITION OF NATURAL NUMBERS, 1888 622 PEOPLE, INSTITUTIONS, AND JOURNALS 627 BIBLIOGRAPHY 635 INDEX 649
adam_txt	CONTENTS ACKNOWLEDGEMENTS XIII INTRODUCTION XV BEGINNINGS I 1.1 BEGINNINGS OF ARITHMETIC 2 1.1.1 LARGE NUMBER CALCULATIONS, C. 1800 BC 2 1.1.2 SACROBOSCO'S ALGORISMUS, C. 1230 AD 4 1.2 BEGINNINGS OF GEOMETRY 8 1.2.1 EUCUED'S DEFINITIONS, C. 250 BC 9 1.2.2 EUCLID'S CONSTRUCTION OF PROPORTIONALS, C. 250 BC 12 1.2.3 ARCHIMEDES ON CIRCLE MEASUREMENT, C. 250 BC 14 1.2.4 APOLLONIUS' CONICS, C. 185 BC 16 1.3 BEGINNINGS OFA THEORY OF NUMBERS 19 1.3.1 EUCLID'S DEFINITIONS OF NUMBER, C. 250 BC 19 1.3.2 EUCLID'S PROOF OF THE INFINITY OF PRIMES, C. 250 BC 23 1.3.3 THE ARITHMETICA OF DIOPHANTUS, (AFTER 150 AD) 24 1 .4 BEGINNINGS OF ALGEBRA 26 1.4.1 COMPLETING THE SQUARE, C. 1800 BC 26 1.4.2 AL-KHWAERIZML'S A!-JABR, C. 825 AD 29 FRESHIDEAS 33 2.1 IMPROVEMENTS IN CALCULATION 34 2.1.1 STEVIN'S DECIMAL FRACTIONS, 1585 34 2.1.2 NAPIER'S LOGARITHMS, 1614 39 2.2 IMPROVEMENTS IN NOTATION 43 2.2.1 HARRIOT'S NOTATION, C. 1600 44 2.2.2 DESCARTES'NOTATION, 1637 46 2.3 ANALYTIC GEOMETRY 47 2.3.1 VIETE'S INTRODUCTION TO THE ANALYTIC ART, 1591 47 2.3.2 FERMAT AND ANALYTIC GEOMETRY, 1636 50 2.3.3 DESCARTES AND ANALYTIC GEOMETRY, 1637 54 2.4 INDIVISIBLES 62 2.4.1 CAVALIERI'S THEORY OF INDIVISIBLES, 1635 62 2.4.2 WALLIS AND HOBBES ON INDIVISIBLES, 1656 66 VLLL CONTENTS 3 FORESHADOWINGS OF CALCULUS 71 3.1 METHODS FOR TANGENTS 72 3.1.1 FERMAT'S TANGENT METHOD, 1629 72 3.1.2 DESCARTES'TANGENT METHOD, 1637 74 3.2 METHODS OF QUADRATURE 78 3.2.1 FERMAT'S QUADRATURE OF HIGHER HYPERBOLAS, EARLY 1640S 78 3.2.2 BROUNCKER AND THE RECTANGULAR HYPERBOLA, C. 1655 84 3.2.3 WALLIS' USE OF INDIVISIBLES, 1656 89 3.2.4 MERCATOR AND THE RECTANGULAR HYPERBOLA, 1668 95 3.3 A METHOD OFCUBATURE 100 3.3.1 TORRICELLI'S INFINITE SOLID, 1644 100 3.4 A METHOD OF RECTIFICATION 102 3.4.1 NEUE AND THE SEMICUBICAL PARABOLA, 1657 102 4 THE CALCULUS OF NEWTON AND OF LEIBNIZ 105 4.1 THE CALCULUS OF NEWTON 105 4.1.1 THE CHRONOLOGY OF NEWTON'S CALCULUS 105 4.1.2 NEWTON'S TREATISE ON FLUXIONS AND SERIES, 1671 107 4.1.3 NEWTON'S FIRST PUBLICATIONOF HIS CALCULUS, 1704 114 4.2 THE CALCULUS OF LEIBNIZ 119 4.2.1 LEIBNIZ'S FIRST PUBLICATION OF HIS CALCULUS, 1684 120 5 THE MATHEMATICS OF NATURE: NEWTON'S PRINCIPIA 133 5.1 NEWTON'S PRINCIPIA, BOOK I 135 5.1.1 THEAXIOMS 135 5.1.2 ULTIMATE RATIOS 139 5.1.3 PROPERTIES OF SMALL ANGLES 142 5.1.4 MOTION UNDER A CENTRIPETAL FORCE 144 5.1.5 QUANTITATIVE MEASURES OF CENTRIPETAL FORCE 147 5.1.6 THE INVERSE SQUARE LAW FOR A PARABOLA 150 6 EARLY NUMBER THEORY 155 6.1 PERFECT NUMBERS 155 6.1.1 EUCLID'S THEOREM ON PERFECT NUMBERS, C. 250 BC 155 6.1.2 MERSENNE PRIMES, 1644 157 6.1.3 FERMAT'S LITTLE THEOREM, 1640 159 6.2 'PELLV EQUATION 162 6.2.1 FERMAT'S CHALLENGE AND BROUNCKER'S RESPONSE, 1657 162 6.3 FERMAT'S FINAL CHALLENGE 165 7 EARLY PROBABILITY 167 7.1 THE MATHEMATICS OF GAMBLING 167 7.1.1 PASCAL'S CORRESPONDENCE WITH FERMAT, 1654 167 7.1.2 JACOB BERNOULLI'S ARS CONJECTANDI, 1713 170 7.1.3 DE MOIVRE'S CALCULATION OF CONFIDENCE, 1738 176 7.2 MATHEMATICAL PROBABILITY THEORY 180 7.2.1 BAYES'THEOREM, 1763 180 7.2.2 LAPLACE AND AN APPLICATION OF PROBABILITY, 1812 182 8 POWER SERIES 189 8.1 DISCOVERIES OF POWER SERIES 190 8.1.1 NEWTON AND THE GENERAL BINOMIAL THEOREM, 1664-1665 190 8.1.2 NEWTON'S 'DE ANALYSI', 1669 193 8.1.3 NEWTON'S SETTERS TO LEIBNIZ, 1676 196 8.1.4 GREGORY'S BINOMIAL EXPANSION, 1670 198 8.2 TAYLOR SERIES 201 8.2.1 TAYLOR'S INCREMENT METHOD, 1715 201 8.2.2 MACLAURIN'S SERIES, 1742 206 8.2.3 FUNCTIONS AS INFINITE SERIES, 1748 208 8.3 CONVERGENCE OF SERIES 209 8.3.1 D'ALEMBERT'S RATIO TEST, 1761 210 8.3.2 LAGRANGE AND THE REMAINDER TERM, 1 797 215 8.4 FOURIER SERIES 223 8.4.1 FOURIER'S DERIVATION OF HIS COEFFICIENTS, 1822 223 9 FUNCTIONS 229 9.1 EARLY DEFINITIONS OF FUNCTIONS 229 9.1.1 JOHANN BERNOULLI'S DEFINITION OF FUNCTION, 1718 229 9.1.2 EULER'S DEFINITION OF A FUNCTION (1), 1748 233 9.1.3 EULER'S DEFINITION OF A FUNCTION (2), 1755 238 9.2 LOGARITHMIC AND CIRCULAR FUNCTIONS 244 9.2.1 A NEW DEFINITION OF LOGARITHMS, 1748 244 9.2.2 SERIES FOR SINE AND COSINE, 1748 250 9.2.3 EULER'S UNINCATION OF ELEMENTARY FUNCTIONS, 1748 253 9.3 NINETEENTH-CENTURY DEFINITIONS OF FUNCTION 254 9.3.1 A DEFINITION FROM DEDEKIND, 1888 256 10 MAKING CALCULUS WORK 259 10.1 USES OF CALCULUS 260 10.1.1 [ACOB BERNOULLI'S CURVEOF UNIFORM DESCENT, 1690 260 10.1.2 D'ALEMBERT AND THE WAVE EQUATION, 1747 264 10.2 FOUNDATIONS OF THE CALCULUS 27 1 10.2.1 BERKELEY AND THE ANALYST, 1734 272 10.2.2 MACLAURIN'S RESPONSE TO BERKELEY, 1742 274 10.2.3 EULER AND INFINITELY SMALL QUANTITIES, 1755 276 10.2.4 LAGRANGE'S ATTEMPT TO AVOID THE INFINITELY SMALL, 1797 281 11 LIMITS AND CONTINUITY 291 11.1 LIMITS 291 11.1.1 WALLIS'S'LESS THAN ANY ASSIGNABLE', 1656 291 11.1.2 NEWTON'S FIRST AND LAST RATIOS, 1687 292 1 1.1.3 MACLAURIN'S DEFINITION OF A LIMIT, 1742 295 11.1.4 D'ALEMBERT'S DEFINITION OF A LIMIT, 1765 297 11.1.5 CAUCHY'S DEFINITION OF A LIMIT, 1821 298 11.2 CONTINUITY 301 11.2.1 WALLIS AND SMOOTH CURVES, 1656 301 11.2.2 EULER'S DEFINITION OF CONTINUITY, 1748 302 CONTENTS 11.2.3 LAGRANGE'S ARBITRARILY SMALL INTERVALS, 1797 304 11.2.4 BOLZANO'S DEFINITION OF CONTINUITY, 1817 306 11.2.5 CAUCHY'S DEFINITION OF CONTINUITY, 1821 312 11.2.6 CAUCHV AND THE INTERMEDIATE VALUE THEOREM, 1821 316 12 SOLVING EQUATIONS 325 12.1 CUBICS AND QUARTICS 325 12.1.1 CARDANO AND THE ARS MAGNA, 1545 325 12.2 FROM CARDANO TO LAGRANGE 330 12.2.1 HARRIOT AND THE STRUCTURE OF POLYNOMIALS, C. 1600 331 12.2.2 HUDDE'S RULE, 1657 333 12.2.3 TSCHIRNHAUS TRANSFORMATIONS, 1683 335 12.2.4 LAGRANGE AND REDUCED EQUATIONS, 1771 339 12.3 HIGHER DEGREE EQUATIONS 348 12.3.1 LAGRANGE'S THEOREM, 1771 348 12.3.2 AFTERMATH: THE UNSOLVABILITY OF QUINTICS 350 13 GROUPS, FIELDS, IDEALS, AND RINGS 353 13.1 GROUPS 353 13.1.1 CAUCHY'S EARLY WORK ON PERMUTATIONS, 1815 354 13.1.2 THE PREMIER MEMOIRE OF GALOIS, 1831 361 13.1.3 CAUCHY'S RETURN TO PERMUTATIONS, 1845 366 13.1.4 CAYLEY'S CONTRIBUTION TO GROUP THEORY, 1854 374 13.2 FIELDS, IDEALS, AND RINGS 378 13.2.1 'GALOIS FIELDS', 1830 378 13.2.2 KUMMERAND IDEA! NUMBERS, 1847 381 13.2.3 DEDEKINDON FIELDS OFFINITE DEGREE, 1877 386 13.2.4 DEDEKIND'S DEFINITION OF IDEALS, 1877 391 14 DERIVATIVES AND INTEGRALS 397 14.1 DERIVATIVES 397 14.1.1 LANDEN'S ALGEBRAIC PRINCIPLE, 1758 397 14.1.2 LAGRANGE'S DERIVED FUNCTIONS, 1797 401 14.1.3 AMPERE'S THEORY OF DERIVED FUNCTIONS, 1806 406 14.1.4 CAUCHY ON DERIVED FUNCTIONS, 1823 415 14.1.5 THE MEAN VALUE THEOREM, ANDE,5 NOTATION, 1823 421 14.2 INTEGRATION OF REAL-VALUED FUNCTIONS 425 14.2.1 EULER'S INLRODUCTION TO INTEGRATION, 1768 425 14.2.2 CAUCHY'S DEFINITE INTEGRAL, 1823 434 14.2.3 CAUCHY AND THE FUNDAMENTAL THEOREM OF CALCULUS, 1823 444 14.2.4 RIEMANN INTEGRATION, 1854 446 14.2.5 LEBESGUEINTEGRATION, 1902 453 15 COMPLEX ANALYSIS 459 15.1 THE COMPLEX PLANE 459 15.1.1 WALLIS'S REPRESENTATIONS, 1685 459 15.1.2 ARGAND'S REPRESENTATION, 1806 462 CONTENTS XI 15.2 INTEGRATION OF COMPLEX FUNCTIONS 468 15.2.1 JOHANN BERNOULLI'S TRANSFORMATIONS, 1702 468 15.2.2 CAUCHYON DEFINITE COMPLEX INTEGRALS, 1814 472 15.2.3 THE CALCULUS OF RESIDUES, 1826 480 15.2.4 CAUCHY'S INTEGRAL FORMULAS, 1831 486 15.2.5 THE CAUCHY-RIEMANN EQUATIONS, 1851 491 16 CONVERGENCE AND COMPLETENESS 495 16.1 CAUCHY SEQUENCES 495 16.1.1 BOLZANO AND'CAUCHY SEQUENCES', 1817 495 16.1.2 CAUCHY'S TREATMENT OF SEQUENCES AND SERIES, 1821 500 16.1.3 ABEL'SPROOF OF THEBINOMIAL THEOREM, 1826 515 16.2 UNIFORM CONVERGENCE 524 16.2.1 CAUCHY'S ERRONEOUS THEOREM, 1821 524 16.2.2 STOKES AND 'INFINITELY SLOW' CONVERGENCE, 1847 527 16.3 COMPLETENESS OF THE REAL NUMBERS 529 16.3.1 BOLZANO AND GREATEST LOWER BOUNDS, 1817 529 16.3.2 DEDEKIND'S DEHNITION OF REAL NUMBERS, 1858 534 16.3.3 CANTOR'S DEFINITION OF REAL NUMBERS, 1872 542 17 LINEAR ALGEBRA 547 17.1 LINEAR EQUATIONS AND DETERMINANTS 548 17.1.1 AN EARLY EUROPEAN EXAMPLE, 1559 548 17.1.2 RULES FOR SOLVING THREE OR FOUR EQUATIONS, 1748 552 17.1.3 VANDERMONDE'S ELIMINATION THEORY, 1772 554 17.1.4 CAUCHY'S DEFINITION OF DETERMINANT, 1815 563 17.2 EIGENVALUE PROBLEMS 570 17.2.1 EULER'S QUADRATIC SURFACES, 1748 570 17.2.2 LAPLACE'S SYMMETRIE SYSTEM, 1787 573 17.2.3 CAUCHY'S THEOREMS OF 1829 579 17.3 MATRICES 586 17.3.1 GAUSS AND LINEAR TRANSFORMATIONS, 1801 586 17.3.2 CAYLEY'S THEORY OF MATRICES, 1858 588 17.3.3 FROBENIUS AND BILINEAR FORMS, 1878 592 17.4 VECTORS AND VECTOR SPACES 598 1 7.4. 1 GRASSMANN AND VECTOR SPACES, 1862 598 18 FOUNDATIONS 605 18.1 FOUNDATIONS OF GEOMETRY 605 18.1.1 HILBERT'S AXIOMATIZATION OF GEOMETRY, 1899 607 18.2 FOUNDATIONS OFARITHMETIC 613 18.2.1 CANTOR'S COUNTABILITY PROOFS, 1874 614 18.2.2 DEDEKIND'S DEFINITION OF NATURAL NUMBERS, 1888 622 PEOPLE, INSTITUTIONS, AND JOURNALS 627 BIBLIOGRAPHY 635 INDEX 649
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subject_GND	(DE-588)4037944-9 (DE-588)4135952-5
title	Mathematics emerging a sourcebook 1540 - 1900
title_auth	Mathematics emerging a sourcebook 1540 - 1900
title_exact_search	Mathematics emerging a sourcebook 1540 - 1900
title_exact_search_txtP	Mathematics emerging a sourcebook 1540 - 1900
title_full	Mathematics emerging a sourcebook 1540 - 1900 Jacqueline Stedall
title_fullStr	Mathematics emerging a sourcebook 1540 - 1900 Jacqueline Stedall
title_full_unstemmed	Mathematics emerging a sourcebook 1540 - 1900 Jacqueline Stedall
title_short	Mathematics emerging
title_sort	mathematics emerging a sourcebook 1540 1900
title_sub	a sourcebook 1540 - 1900
topic	Mathematics / History Mathematics / Sources Geschichte Mathematik Mathematics History Mathematik (DE-588)4037944-9 gnd
topic_facet	Mathematics / History Mathematics / Sources Geschichte Mathematik Mathematics History Quelle
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