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本卷收錄了吳文俊的Mechanical Theorem Provingin Geometries：Basic Principles一書。《吳文俊全集·數學機械化II》論述初等幾何機器證明的基本原理，證明了奠基於各種公理系統的各種初等幾何，只需相當於乘法交換律的某一公理成立，大都可以機械化。因此在理論上，這些幾何的定理證明可以借肋於電腦來實施。可以機械化的幾何包括了多種有序或無序的常用幾何、投影幾何、非歐幾何與圓幾何等。
《吳文俊全集·數學機械化II》共分六章。前兩章是關於幾何機械化的預備知識，集中介紹了常用幾何；後四章致力於幾何的機械化問題。第3章為幾何定理證明的機械化與Hilbert機械化定理，第4，5章分別為（常用）無序幾何的機械化定理和(常用)有序幾何的機械化定理，第6章闡述各種幾何的機械化定理。
Contents
Author’s note to the English-language edition 1
1 Desarguesian geometry and the Desarguesian number system 13
1.1 Hilbert’s axiom system of ordinary geometry 13
1.2 The axiom of infinity and Desargues’ axioms 18
1.3 Rational points in a Desarguesian plane 25
1.4 The Desarguesian number system and rational number subsystem 30
1.5 The Desarguesian number system on a line 37
1.6 The Desarguesian number system associated with a Desarguesian plane 42
1.7 The coordinate system of Desarguesian plane geometry 55
2 Orthogonal geometry， metric geometry and ordinary geometry 63
2.1 The Pascalian axiom and commutative axiom of multiplication-(unordered) Pascalian geometry 63
2.2 Orthogonal axioms and (unordered) orthogonal geometry 70
2.3 The orthogonal coordinate system of (unordered) orthogonal geometry 80
2.4 (Unordered) metric geometry 91
2.5 The axioms of order and ordered metric geometry 102
2.6 Ordinary geometry and its subordinate geometries 109
3 Mechanization of theorem proving in geometry and Hilbert’s mechanization theorem 115
3.1 Comments on Euclidean proof method 115
3.2 The standardization of coordinate representation of geometric concepts 118
3.3 The mechanization of theorem proving and Hilbert’s mechanization theorem about pure point of intersection theorems in Pascalian geometry 124
3.4 Examples for Hilbert’s mechanical method 128
3.5 Proof of Hilbert’s mechanization theorem 139
4 The mechanization theorem of (ordinary) unordered geometry 149
4.1 Introduction 149
4.2 Factorization of polynomials 152
4.3 Well-ordering of polynomial sets 159
4.4 A constructive theory of algebraic varieties-irreducible ascending sets and irreducible algebraic varieties 169
4.5 A constructive theory of algebraic varieties-irreducible decomposition of algebraic varieties 178
4.6 A constructive theory of algebraic varieties-the notion of dimension and the dimension theorem 183
4.7 Proof of the mechanization theorem of unordered geometry 187
4.8 Examples for the mechanical method of unordered geometry 195
5 Mechanization theorems of (ordinary) ordered geometries 213
5.1 Introduction 213
5.2 Tarski’s theorem and Seidenberg’s method 220
5.3 Examples for the mechanical method of ordered geometries 228
6 Mechanization theorems of various geometries 235
6.1 Introduction 235
6.2 The mechanization of theorem proving in projective geometry 236
6.3 The mechanization of theorem proving in Bolyai-Lobachevsky’s hyperbolic non-Euclidean geometry 246
6.4 The mechanization of theorem proving in Riemann’s elliptic non-Euclidean geometry 258
6.5 The mechanization of theorem proving in two circle geometries 264
6.6 The mechanization of formula proving with transcendental functions 267
References 281
Subject index 285
