A TREATISE ON ARITHMETIC

By E. NONID

FIRST PART: ARITHMETIC I CALCULATING WITH NATURAL NUMBERS 1 INTRODUCTION OF NUMBERS IN THE SCHOOLS 2 THE FUNDAMENTAL LAWS OF RECKONING 3 THE LOGICAL FOUNDATIONS OF OPERATIONS WITH INTEGERS 4 PRACTICE IN CALCULATING WITH INTEGERS DESCRIPTION OF THE CALCULATING MACHINE "BRUNSVIGA'' II THE FIRST EXTENSION OF THE NOTION OF NUMBER 1 NEGATIVE NUMBERS ON THE HISTORY OF NEGATIVE NUMBERS 2 FRACTIONS 3 IRRATIONAL NUMBERS ON THE NATURE OF SPACE INTUITION (PRECISION AND APPROXIMATION MATHEMATICS) III CONCERNING SPECIAL PROPERTIES OF INTEGERS NUMBER THEORY IN SCHOOLS AND IN UNIVERSITIES SPECIAL DISCUSSIONS OF NUMBER-THEORETIC ISSUES PRIME NUMBERS, DECOMPOSITION INTO PRIME FACTORS TRANSFORMATION OF ORDINARY FRACTIONS INTO DECIMAL NUMBERS CONTINUED FRACTIONS PYTHAGOREAN NUMBERS, FERMAT'S LAST THEOREM THE PROBLEM OF THE DIVISION OF THE CIRCLE IV COMPLEX NUMBERS 1 ORDINARY COMPLEX NUMBERS 2 HIGHER COMPLEX NUMBERS, ESPECIALLY QUATERNIONS REMARKS ON VECTOR CALCULUS 3 QUATERNION MULTIPLICATION – ROTATION-DILATION INTERPRETATION IN THREE-DIMENSIONAL SPACE 4 COMPLEX NUMBERS IN SCHOOL TEACHING CONCERNING THE MODERN DEVELOPMENT AND THE GENERAL STRUCTURE OF MATHEMATICS THE STRUCTURE OF ELEMENTARY ANALYSIS ACCORDING TO TWO PARALLEL PROCESSES OF DEVELOPMENT OF DIFFERING CHARACTER OVERVIEW OF THE HISTORY OF MATHEMATICS SECOND PART: ALGEBRA TEXTBOOKS OUR SPECIFIC GOAL: APPLICATION OF GEOMETRICALLY INTUITIVE METHODS TO THE SOLUTION OF EQUATIONS I REAL EQUATIONS WITH REAL UNKNOWNS 1 EQUATIONS WITH ONE PARAMETER 2 EQUATIONS WITH TWO PARAMETERS CLASSIFICATION ACCORDING TO THE NUMBER OF REAL ROOTS 3 EQUATIONS WITH THREE PARAMETERS AN APPARATUS FOR THE NUMERICAL SOLUTION OF EQUATIONS THE DISCRIMINANT SURFACE OF THE BIQUADRATIC EQUATION II EQUATIONS IN THE FIELD OF COMPLEX QUANTITIES A THE FUNDAMENTAL THEOREM OF ALGEBRA B EQUATIONS WITH A COMPLEX PARAMETER 1 THE "PURE'' EQUATION IRREDUCIBILITY; "IMPOSSIBILITY'' OF TRISECTING THE ANGLE 2 THE DIHEDRAL EQUATION 3 THE TETRAHEDRAL, THE OCTAHEDRAL, AND THE ICOSAHEDRAL EQUATIONS 4 CONTINUATION: SETTING UP THE NORMAL EQUATION 5 CONCERNING THE SOLUTION OF THE NORMAL EQUATIONS 6 UNIFORMISATION OF THE NORMAL IRRATIONALITIES BY MEANS OF TRANSCENDENTAL FUNCTIONS THE "CASUS IRREDUCIBILIS'' AND THE TRIGONOMETRIC SOLUTION OF THE CUBIC EQUATION 7 SOLVABILITY IN TERMS OF RADICAL SIGNS 8 REDUCTION OF GENERAL EQUATIONS TO NORMAL EQUATIONS THIRD PART: ANALYSIS I LOGARITHMIC AND EXPONENTIAL FUNCTIONS 1 SYSTEMATIC ACCOUNT OF ALGEBRAIC ANALYSIS 2 THE HISTORICAL DEVELOPMENT OF THE THEORY NEPER AND BÜRGI THE 17TH CENTURY: THE AREA OF THE HYPERBOLA EULER AND LAGRANGE: ALGEBRAIC ANALYSIS THE 19TH CENTURY: FUNCTIONS OF COMPLEX VARIABLES 3 REMARKS ABOUT TEACHING IN SCHOOLS 4 THE STANDPOINT OF FUNCTION THEORY THE PASSAGE TO THE LIMIT FROM THE POWER TO THE EXPONENTIAL FUNCTION II THE GONIOMETRIC FUNCTIONS 1 THEORY OF THE GONIOMETRIC FUNCTIONS EXACT COMPARISON WITH THE THEORY OF LOGARITHMS 2 TRIGONOMETRIC TABLES A PURELY TRIGONOMETRIC TABLES B LOGARITHMIC-TRIGONOMETRIC TABLES 3 APPLICATIONS OF GONIOMETRIC FUNCTIONS A TRIGONOMETRY, IN PARTICULAR, SPHERICAL TRIGONOMETRY FORMULAS OF SECOND KIND: TRIANGLES OF FIRST AND SECOND KIND THE AREA OF SPHERICAL TRIANGLES B THEORY OF SMALL OSCILLATIONS, ESPECIALLY THOSE OF THE PENDULUM TEACHING IN SCHOOLS (HIDDEN INFINITESIMAL CALCULUS) C REPRESENTATION OF PERIODIC FUNCTIONS BY MEANS OF SERIES OF GONIOMETRIC FUNCTIONS (TRIGONOMETRIC SERIES) APPROXIMATION BY SERIES WITH A FINITE NUMBER OF TERMS ESTIMATION OF ERRORS; CONVERGENCE OF INFINITE SERIES THE GIBBS PHENOMENON EXCURSIONS CONCERNING THE GENERAL NOTION OF FUNCTION HISTORICAL IMPORTANCE OF TRIGONOMETRIC SERIES; THE ROLE OF FOURIER III CONCERNING INFINITESIMAL CALCULUS PROPER 1 GENERAL CONSIDERATIONS IN INFINITESIMAL CALCULUS EMERGENCE OF THE INFINITESIMAL CALCULUS BY THE SPECIFICITY OF OUR SENSE INTUITION THE LOGICAL FOUNDATION OF DIFFERENTIAL AND INTEGRAL CALCULUS BY MEANS OF THE LIMIT CONCEPT (NEWTON AND HIS SUCCESSORS UP TO CAUCHY) CONSTRUCTION OF THE INFINITESIMAL CALCULUS BASED ON ``DIFFERENTIALS'' (LEIBNIZ AND HIS FOLLOWERS) THE ACTUAL INFINITELY SMALL QUANTITIES IN THE AXIOMATICS OF GEOMETRY THE REACTION: THE DERIVATIVE CALCULUS OF LAGRANGE FORM AND IMPORTANCE OF THE INFINITESIMAL CALCULUS IN THE PRESENT STATE OF TEACHING 2 TAYLOR'S THEOREM THE FIRST PARABOLAS OF OSCULATION INCREASING THE ORDER: QUESTIONS OF CONVERGENCE GENERALISING TAYLOR'S THEOREM TO A THEOREM OF FINITE DIFFERENCES CAUCHY'S ESTIMATE OF THE ERROR HISTORICAL REMARKS ABOUT TAYLOR AND MACLAURIN 3 HISTORICAL AND PEDAGOGICAL CONSIDERATIONS REMARKS ABOUT TEXTBOOKS FOR THE INFINITESIMAL CALCULUS CHARACTERISING OUR PROPER PRESENTATION IV SUPPLEMENT IVA TRANSCENDENCE OF THE NUMBERS E AND PHI HISTORICAL ASPECTS PROOF OF THE TRANSCENDENCE OF E PROOF OF THE TRANSCENDENCE OF PHI MORE ON TRANSCENDENT AND ALGEBRAIC NUMBERS IVB SET THEORY 1 THE POTENCY OF SETS DENUMERABILITY OF RATIONAL AND ALGEBRAIC NUMBERS THE CONTINUUM NOT BEING DENUMERABLE POTENCY OF CONTINUA OF MORE DIMENSIONS SETS WITH HIGHER POTENCIES 2 ARRANGEMENT OF THE ELEMENTS OF A SET TYPES OF ARRANGEMENT OF DENUMERABLE SETS THE CONTINUITY OF THE CONTINUUM INVARIANCE OF DIMENSION FOR CONTINUOUS ONE-TO-ONE REPRESENTATIONS CLOSING REMARKS IMPORTANCE AND GOALS OF SET THEORY APPENDIX 1: ON THE EFFORTS TO REFORM MATHEMATICS TEACHING APPENDIX 2: ABOUT MATHEMATICAL AND PEDAGOGICAL LITERATURE NAME INDEX SUBJECT INDEX BLANK PAGE