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INTRODUCTION 1

PRACTICAL AND THEORETICAL CONSTRUCTIONS 2

STATEMENT OF THE PROBLEM IN ALGEBRAIC FORM 3

PART Ⅰ．The Possibility of the Construction of Algebraic Expressions 5

CHAPTER Ⅰ．ALGEBRAIC EQUATIONS SOLVABLE BY SQUARE ROOTS 5

1-4．Structure of the expression x to be constructed 5

5，6．Normal form of x 6

7，8．Conjugate values 7

9．The corresponding equation F(x)=o 8

10．Other rational equations f(x)=o 8

11，12．The irreducible equation φ(x)=o 10

13，14．The degree of the irreducible equation a power of 2 11

CHAPTER Ⅱ．THE DELIAN PROBLEM AND THE TRISECTION OF THE ANGLE 13

1．The impossibility of solving the Delian problem with straight edge and compasses 13

2．The general equation x3=λ 13

3．The impossibility of trisecting an angle with straight edge and compasses 14

CHAPTER Ⅲ．THE DIVISION OF THE CIRCLE INTO EQUAL PARTS 16

1．History of the problem 16

2-4．Gauss＇s prime numbers 17

5．The cyclotomic equation 19

6．Gauss＇s Lemma 19

7，8．The irreducibility of the cyclotomic equation 21

CHAPTER Ⅳ．THE CONSTRUCTION OF THE REGULAR POLYGON OF 17 SIDES 24

1．Algebraic statement of the problem 24

2-4．The periods formed from the roots 25

5，6．The quadratic equations satisfied by the periods 27

7．Historical account of constructions with straight edge and compasses 32

8，9．Von Staudt＇s construction of the regular polygon of 17 sides 34

CHAPTER Ⅴ．GENERAL CONSIDERATIONS ON ALGEBRAIC CONSTRUCTIONS 42

1．Paper folding 42

2．The conic sections 42

3．The Cissoid of Diocles 44

4．The Conchoid of Nicomedes 45

5．Mechanical devices 47

PART Ⅱ．Transcendental Numbers and the Quadrature of the Circle 49

CHAPTER Ⅰ．CANTOR＇S DEMONSTRATION OF THE EXISTENCE OF TRANSCENDENTAL NUMBERS 49

1．Definition of algebraic and of transcendental numbers 49

2．Arrangement of algebraic numbers according to height 50

3．Demonstration of the existence of transcendental numbers 53

CHAPTER Ⅱ．HISTORICAL SURVEY OF THE ATTEMPTS AT THE COMPUTATION AND CONSTRUCTION OF π 55

1．The empirical stage 56

2．The Greek mathematicians 56

3．Modern analysis from 1670 to 1770 58

4，5．Revival of critical rigor since 1770 59

CHAPTER Ⅲ．THE TRANSCENDENCE OF THE NUMBER e 61

1．Outline of the demonstration 61

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

3．Hermite＇s Theorem 65

CHAPTER Ⅳ．THE TRANSCENDENCE OF THE NUMBER π 68

1．Outline of the demonstration 68

2．The function ψ(x) 70

3．Lindemann＇s Theorem 73

4．Lindemann＇s Corollary 74

5．The transcendence of π 76

6．The transcendence of y=ex 77

7．The transcendence of y=sin-1x 77

CHAPTER Ⅴ．THE INTEGRAPH AND THE GEOMETRIC CONSTRUCTION OF π 78

1．The impossibility of the quadrature of the circle with straight edge and compasses 78

2．Principle of the integraph 78

3．Geometric construction of π 79

NOTES 81

INTRODUCTION 99

DETERMINANTS 101

Ⅰ．ORIGIN OF DETERMINANTS 103

Ⅱ．PROPERTIES OF DETERMINANTS 112

Ⅲ．SOLUTION OF SIMULTANEOUS EQUATIONS 121

Ⅳ．PROPERTIES OF DETERMINANTS（continued） 123

Ⅴ．THE TENSOR NOTATION 131

SETS 147

Ⅵ．SETS OF QUANTITIES 149

Ⅶ．RELATED SETS OF VARIABLES 164

Ⅷ．DIFFERENTIAL RELATIONS OF SETS 177

Ⅸ．EXAMPLES FROM THE THEORY OF STATISTICS 184

Ⅹ．TENSORS IN THEORY OF RELATIVITY 207

APPENDIX：Product of Determinants 214

INDEX OF SYMBOLS 216

GENERAL INDEX 217

Ⅰ 305

Ⅱ 314

Ⅲ 330