《Famous Problems and Other Monographs Second Edition》

INTRODUCTION1

PRACTICAL AND THEORETICAL CONSTRUCTIONS2

STATEMENT OF THE PROBLEM IN ALGEBRAIC FORM3

PART Ⅰ．The Possibility of the Construction of Algebraic Expressions5

CHAPTER Ⅰ．ALGEBRAIC EQUATIONS SOLVABLE BY SQUARE ROOTS5

1-4．Structure of the expression x to be constructed5

5，6．Normal form of x6

7，8．Conjugate values7

9．The corresponding equation F(x)=o8

10．Other rational equations f(x)=o8

11，12．The irreducible equation φ(x)=o10

13，14．The degree of the irreducible equation a power of 211

CHAPTER Ⅱ．THE DELIAN PROBLEM AND THE TRISECTION OF THE ANGLE13

1．The impossibility of solving the Delian problem with straight edge and compasses13

2．The general equation x3=λ13

3．The impossibility of trisecting an angle with straight edge and compasses14

CHAPTER Ⅲ．THE DIVISION OF THE CIRCLE INTO EQUAL PARTS16

1．History of the problem16

2-4．Gauss＇s prime numbers17

5．The cyclotomic equation19

6．Gauss＇s Lemma19

7，8．The irreducibility of the cyclotomic equation21

CHAPTER Ⅳ．THE CONSTRUCTION OF THE REGULAR POLYGON OF 17 SIDES24

1．Algebraic statement of the problem24

2-4．The periods formed from the roots25

5，6．The quadratic equations satisfied by the periods27

7．Historical account of constructions with straight edge and compasses32

8，9．Von Staudt＇s construction of the regular polygon of 17 sides34

CHAPTER Ⅴ．GENERAL CONSIDERATIONS ON ALGEBRAIC CONSTRUCTIONS42

1．Paper folding42

2．The conic sections42

3．The Cissoid of Diocles44

4．The Conchoid of Nicomedes45

5．Mechanical devices47

PART Ⅱ．Transcendental Numbers and the Quadrature of the Circle49

CHAPTER Ⅰ．CANTOR＇S DEMONSTRATION OF THE EXISTENCE OF TRANSCENDENTAL NUMBERS49

1．Definition of algebraic and of transcendental numbers49

2．Arrangement of algebraic numbers according to height50

3．Demonstration of the existence of transcendental numbers53

CHAPTER Ⅱ．HISTORICAL SURVEY OF THE ATTEMPTS AT THE COMPUTATION AND CONSTRUCTION OF π55

1．The empirical stage56

2．The Greek mathematicians56

3．Modern analysis from 1670 to 177058

4，5．Revival of critical rigor since 177059

CHAPTER Ⅲ．THE TRANSCENDENCE OF THE NUMBER e61

1．Outline of the demonstration61

2．The symbol hr and the function φ(x)62

3．Hermite＇s Theorem65

CHAPTER Ⅳ．THE TRANSCENDENCE OF THE NUMBER π68

1．Outline of the demonstration68

2．The function ψ(x)70

3．Lindemann＇s Theorem73

4．Lindemann＇s Corollary74

5．The transcendence of π76

6．The transcendence of y=ex77

7．The transcendence of y=sin-1x77

CHAPTER Ⅴ．THE INTEGRAPH AND THE GEOMETRIC CONSTRUCTION OF π78

1．The impossibility of the quadrature of the circle with straight edge and compasses78

2．Principle of the integraph78

3．Geometric construction of π79

NOTES81

INTRODUCTION99

DETERMINANTS101

Ⅰ．ORIGIN OF DETERMINANTS103

Ⅱ．PROPERTIES OF DETERMINANTS112

Ⅲ．SOLUTION OF SIMULTANEOUS EQUATIONS121

Ⅳ．PROPERTIES OF DETERMINANTS（continued）123

Ⅴ．THE TENSOR NOTATION131

SETS147

Ⅵ．SETS OF QUANTITIES149

Ⅶ．RELATED SETS OF VARIABLES164

Ⅷ．DIFFERENTIAL RELATIONS OF SETS177

Ⅸ．EXAMPLES FROM THE THEORY OF STATISTICS184

Ⅹ．TENSORS IN THEORY OF RELATIVITY207

APPENDIX：Product of Determinants214

INDEX OF SYMBOLS216

GENERAL INDEX217

Ⅰ305

Ⅱ314

Ⅲ330