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PREFACE TO SECOND EDITION
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
HOW TO USE THE BOOK
WHAT IS MATHEMATICS?
CHAPTER I. THE NATURAL NUMBERS  1
  Introduction  1
    1. Calculation with Integers    1
  I. Laws  of Arithmetic 2. The  Representation  of Integers 3. Computation in Systems Other than the Decimal.  1   
    2. The Infinitude of the Number System. Mathematical Induction    9
  1. The Principle of Mathematical Induction  2. The Arithmetical Progression  3. The Geometrical Progression4. The Sum of the First n Squares 5. An  Important  Inequality  6. The  Binomial  Theorem  7. Further Remarks on Mathematical Induction. SUPPLEMENT TO CHAPTER I THE THEORY OF NUMBERS    21
  Introduction 21
    1. The Prime Numbers  21
  1. Fundamental Facts2. The Distribution of the Primesa. Formulas
  Producing Primesb. Primes in Arithmetical Progressionsc. The Prime
  Nmnber Theoremd. Two Unsolved Problems Concerning Prime Num  bers.      2. Congruences      31
  1. General Concepts2. Fermat's Theorem3. Quadratic Residues.      3. Pythagorean Numbers and Fermat's  t Theorem 40
    4. The Euclidean Algorithm    42
  1. General Theory2. Application to the Fundamental Theorem of Arith metic3. Euler's  0 Function. Fermat's Theorem Again  4. Continued
  FractionsDiophantine Equations. CHAFFER II. THE NUMBER SYSTEM OF MATHEMATICS 52
  Introduction 52
    1. The Rational Numbers  52
  1. Rational Numbers as a Device for Measuring2. Intrinsic Need for the
Rational NumbersPrincipal of Generalization3. Geometrical Interpre tation of Rational Numbers.    2. Incommensurable Segments, Irrational Numbers, and the Concept ofLimit    58
1. Introduction2Decimal FractionsInfinite Decimals3. Lim itsInfinite Geometrical Series4. Rational Numbers and Periodic DeciCONTENTS
mals5. General  Definition  of  Irrational  Numbers  by  Nested
Intervals6. Alternative  Methods  of  Defining  Irrational  Num bersDedekind Cuts.    3. Remarks on Analytic Geometry    72
1. The Basic Principle2. Equations of Lines and Curves.      77
    4. The Mathematical Analysis of Infinity 
1. Fundamental Concepts. 2. The Denumerability  of ihl Raiiona] N't 
bers and the Non-Denumerability of the Continuum3. Cantor's  Cardinal
Numbers.  4. The Indirect Method of Proof5. The Paradoxes of the In finite6. The Foundations of Mathematics.      88
    5. ComplexNumbers   
1. The Ongm of Complex Numbers)  2. The  Geomeinc  inie r  iion  of
Complex  Numbers3. De  Moivre's  Formula  and  the  Roots  of
Unity4. The Fundamental Theorem of Algebra.    6. Algebraic and Transcendental Numbers  103
  1. Definition and Existence2. Liouville's Theorem and the Construction
of Transcendental Numbers. SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS 108
  1. General Theory2. Application to Mathematical Logic3. An Appli  cation to the Theory of Probability. CHAFFER III. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS    117
  Introduction        120
    Part I. Impossibility Proofs and Algebra 120
  1. Fundamental Geometrical Constructions Extraciiln'  Reguiar  y
  1. Construction of Fields and Square Root      2    Pol
  gons3. Apollonius' Problem.      2. Constructible Numbers and Number Fields  127
  1. General Theory2. All Constructible Numbers are Algebraic.      134
    3. The Unsolvability of the Three Greek Problems 
  1. Doubling the Cube. 2. A Theorem on Cubic Equations( 3 . Tns'e'c't mg
  the Angle4. The Regular Heptagon5. Remarks on the Problem of
  Squaring the Circle.    Part II. Various Methods for Performing Constructions 140
    4. GeometricalTransformationsInversion 140
  1. General Remarks2. Properties of Inversion3. Geometrical Con  struction of Inverse Points4. How to Bisect a Segment and Find the Cen  ter of a Circle with the Compass Alone.      5. Constructions with Other Tools. Mascheroni Constructions with Compass
    Alone  146
1. A Classical Construction for Doubling the Cube2. Restriction to the
  Use  of the  Compass  Alone3. Drawing with Mechanical Instru mentsMechanical CurvesCycloids4. LinkagesPeaucellier's and
Hart's Inversors.  158
      6. More About Inversions and its Applications ippiicaiio; il the  Prob
1. Invanance of AnglesFamilies of Circles.  2.
lem of Apollonius3. Repeated Reflections.  CHAFFER IV. PROJECTIVE GEOMETRY. AXIoMATICS. NON-EUCLIDEAN GEOMETRIES  165
      1. Introduction .' ''      166
1. Classification of Geometrical PropertiesInvariance under Transfor  mations2. Projective Transformations.      2. Fundamental Concepts 168
  1. The Group of Projective Transformations2. Desargues's Theorem.      3. Cross-Ratio      172
  1. Definition and Proof of Invariance2. Application to the Complete
  Quadrilateral.      4. Parallelism and Infinity    180
  1. Points at Infinity as  Ideal Points.  2. Ideal Elements and Projec  tion3. Cross-Ratio with Elements at Infinity.      5. Applications      185
  1. Preliminary  Remarks2. Proof of Desargues's  Theorem in  the
  Plane3. Pascal's Theorem4. Brianchon's Theorem5. Remark on
  Duality.      6. Analytic Representation    191
  1. Introductory Remarks2. Homogeneous Co6rdinatesThe Algebraic
  Basis of Duality.      7. Problems on Constructions with the Straightedge Alone 196
    8. Conics and Quadric Surfaces  198
  1. Elementary Metric Geometry of Conics2. Projective Properties of
  Conics3. Conics as Line Curves4. Pascal's and Brianchon's General
  Theorems for Conics5. The Hyperboloid.      9. Axiomatics and Non-Euclidean Geometry___.___. __  214
  1. The Axiomatic  Method. 2. Hyperbolic  Non-Euchdean Geome
  try3. Geometry and Reality4. Poincar6's Model5. Elliptic or Rie  mannian Geometry.      APPENDIX. GEOMETRY IN MORE THAN THREE DIMENSIONS    227
  1. Introduction2. Analytic Approach3. Geometrical or Combinatorial
  Approach. CHAFFER V. TOPOLOGY  235
  Introduction  235
    1. Euler's Formula for Polyhedra  236
    2. Topological Properties of Figures  241
  1. Topological Properties2. Connectivity.      3. Other Examples of Topological Theorems 244
  1. The Jordan Curve Theorem2. The Four Color Problem3. The Con  cept of Dimension4. A Fixed Point Theorem5. Knots.      4. The Topological Classification of Surfaces    256
  1. The Genus of a Surface2. The Euler Characteristic of a Sur  face3. One-Sided Surfaces.    APPENDIX  264
  1. The Five Color Theorem2. The Jordan Curve Theorem for Poly gons3. The Fundamental Theorem of Algebra. CHAPTER VI. FUNCTIONS AND LIMITS 272
  Introduction      272
    1. Variable and Function  273
  1. Definitions and Examples2. Radian Measure of Angles3. The
Graph  of  a  FunctionInverse  Functions4. Compound  Func tions5. Continuity6. Functions of Several Variables7. Functions
and Transformations.      289
    2. Limits    1. The    I ii of'a Sequen'il a  21 M'onoilnl Sequ'inlll. 3.  Euilr s N'um
ber e4. The Number n5. Continued Fractions.      303
    3. Limits by Continuous Approach .  1. Introduction. General Definition. 21 Rem kl on  ihl  I im;t Con cept3. The Limit of sin x/x4. Limits as x --*  .    4. Precise DefimtionofContinuity    '  ' i  iiiiii i 310312
    5. Two Fundamental Theorems on Continuous Functions     
  1. Bolzano's Theorem2. Proof of Bolzano's Theorem .  Theorem on Extreme Values4. A Theorem on Sequences. Compact Sets.      6. Some Apphcations of Bolzano s Thoerem      317
    -  1. Geometrical Applications2. Application to a Problem in Mechanics. SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY  322
    1. Examples ofLimits . _-  ' -  322
  1. General Remarks2. The Limit of q 3. The Limit of  !p  4. Discon
  tinuous Functions as Limits of Continuous Functmns5. L m ts by Itera  tion.      2. Example on Continuity    -'    327
CHAPTER VII. MAXIMA AND MINIMA    329
  Introduction    329
      330
    1. Problems in Elementary Geometry    o  S des'G ven    2.     
  1. Maximum Area of a Triangle with      i    i  Heron's Tho  eremExtremum Property of Light Rays3. Applications to Problems on
  Triangles4. Tangent  Properties    of  Ellipse  and  Hyper  bolaCorresponding Extremum Properties5. Extreme Distances to a
  Given Curve.      2. A General Principal Underlying Extreme Value Problems 338
1. The Prinople  2. Examples.      3. Stationary Points and the Differential Calculus 341
1Extrema and Stationary Points2. Maxima and Minima  Functions
  of Several VariablesSaddle Points3. Minimax Points and Topol  ogy4. The Distance from a Point to a Surface.      346
      4. Schwarz's Triangle Problemother Proof.  3    Obtuse    Tri gles
1. Schwarz's  Proof2.  4. Triangles Formed by Light Rays5. Remarks Concerning Problems of
Reflection and Ergodic Motion.      354
      5. Steiner's Problem   
1. Problem and Solution. 2    ysis of ihe  ternativel. 3.'A Cimple
mentary Problem4. Remarks and Exercises5. Generalization to the
Street Network Problem.  361
      6. Extrema and Inequalities    '      Me '  of      o     
1The  Arithmetical  and  Geometrical    Positive
Quantities2. Generalization to n Variables3. The Method of Least
Squares. 7. The Existence of an ExtremumDirichlet's Principle      366
  1. General  Remarks2. Examples3. Elementary  Extremum  Prob  lems4. Difficulties in Higher Cases.      8. The Isoperimetric Problem  373
    9. Extremum Problems with Boundary Conditions. Connection Between Stei    ner's Problem and the Isoperimetric Problem 376
    10. The Calculus of Variations  379
  1. Introduction2. The Calculus of VariationsFermat's Principle in Op  tics3. BernouUi's  Treatment  of  the  Brachistochrone  Prob  lem4. Geodesics on a SphereGeodesics and Maxi-Minima. 11. Experimental Solutions of Minimum Problems. Soap Film Experiments  385
  1. Introduction2. Soap Film Experiments3. New Experiments onPla  teau's Problem4. Experimental Solutions of Other Mathematical Prob  lems. CHAPTER VIII. THE CALCULUS    398
  Introduction      398
    1. The Integral    399
  1. Area as a Limit2. The Integral3. General Remarks on the Integral
  ConceptGeneral Definition4. Examples of IntegrationIntegration of
  x 5. Rules for the  Integral Calculus
    2. The Derivative    414
  1. The  Derivative  as  a  Slope2. The  Derivative  as  a
  Limit3. Examples4. Derivatives    of    Trigonometrical    Func  tions5. Differentiation  and  Continuity6. Derivative  and  Veloc  itySecond Derivative and Acceleration7. Geometrical Meaning of the
  Second Derivative8. Maxima and Minima.      3. The Technique of Differentiation  427
    4. Leibniz' Notation and the  Infinitely Small 433
    5. The Fundamental Theorem of the Calculus 436
  1. The Fundamental Theorem2. First ApplicationsIntegration of x ,
  cos x, sin x. Arc tan x3. Leibniz' Formula for x
    6. The Exponential Function and the Logarithm  442
  1. Definition and Properties of the LogarithmEuler's Number e2. The
  Exponential  Function3. Formulas  for  Differentiation  of  e ,  a ,
x 4. Explicit Expressions for e, e , and log x as Limits5. Infinite Series
  for the Logarithm. Numerical Calculation.      7. Differential Equations  453
  1. Defimtion2. The Differential Equation of the Exponential Func  tionRadioactive DisintegrationLaw of GrowthCompound Inter  est3. Other  ExamplesSimplest Vibrations4. Newton's Law of
  Dynamics. SUPPLEMENT TO CHAttER VIII    462
      1. Matters of Principle  462
  1. Differentiability2. The Integral3. Other Applications of the Con cept of IntegralWorkLength.      2. Orders of Magnitude    469
  1. The Exponential Function and Powers of x2. Order of Magnitude of
log (n!).    3. Infinite Series and Infinite Products  472
1. Infinite Series of Functions2. Euler's Formula, cos x + i sin x =
e%  3. The Harmonic Series and the Zeta FunctionEuler's Product for
      the Sine.    4. The Prime Number Theorem Obtained by Statistical Methods    482
CHAPTER IX. RECENT DEVELOPMENTS    487
    1. A Formula for P es i ii  ! ! ! i
    2. The GoldbachConjecture andTwin     
    3. Fermat's last Theorem 491
    4. The Continuum Hypothesis  493
    5 Set-Theoretic Notation  494
    6. The Four Color Theorem -  495
    7. Hausdorff Dimension and Fractals  499
    8. Knots    .'.  501
    9. A Problem in Mechanics    505
    10 Steiner's Problem    507
      11. Soap Films and Minimal Surfaces  513
    12. Nonstandard Analysis      518
APPENDIX  SUPPLEMENTARY REMARKS, PROBLEMS,  ND  EXERCISES        525
    Arithmetic and Algebra      525
    Analytic Geometry    526
    Geometrical Constructions  532
    Projective and Non-Euclidean Geometry  533
      }A
    Topology      oo
      db  FIw
    Functions, Limits, and Continuity 537
    Maxima and Minima      538
    The Calculus    540
    Technique of Integration 542
SUGGESTIONS FOR FURTHER READING      549
SUGGESTIONS FOR ADDITIONAL READING 553
INDEX  559