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This Dover edition, published in 2005, is an unabridged version of the work first issued by Academic Press in 1952. At that time, the majority of texts on projective geometry based their approach on synthetic methods, so this book would then have been out of the ordinary due to its heavy reliance on algebraic methods.
It certainly isn't a book for those new to the subject and it is aimed at upperlevel undergraduates and graduate students (more likely the latter, I feel). In span of 300 pages, there are only 19 diagrams and, although the ideas from linear algebra are invoked throughout the book, matrices are hardly used at all. The main thrust is a series of theorems on the representation of projective geometries by linear manifolds and of collineations by linear transformations and of dualities by semilinear forms. Such theorems are used to present the geometry within algebraic structures such as the endomorphism of the underlying manifold or the full linear group.
This, however, is one of those books in which the author states that Little actual knowledge is presupposed, but those without good prior knowledge of linear algebra, groups, rings and transfinite set theory and elementary projective geometry will find it extremely heavy going.
Peter Ruane (ruane.p@blueyonder.co.uk) is retired from university teaching, where his interests lay predominantly within the field of mathematics education
Preface


I.  Motivation  
I.1  The ThreeDimensional Affine Space as Prototype of Linear Manifolds  
I.2  The Real Projective Plane as Prototype of the Lattice of Subspaces of a Linear Manifold  
II.  The Basic Properties of a Linear Manifold  
II.1  Dedekind's Law and the Principle of Complementation  
II.2  Linear Dependence and Independence; Rank  
II.3  The Adjoint Space  
Appendix I.  Application to Systems of Linear Homogeneous Equations  
Appendix II.  Paired Spaces  
II.4  The Adjunct Space  
Appendix III.  Fano's Postulate  
III.  Projectivities  
III.1  Representation of Projectivities by Semilinear Transformations  
Appendix I.  Projective Construction of the Homothetic Group  
III.2  The Group of Collineations  
III.3  The Second Fundamental Theorem of Projective Geometry  
Appendix II.  The Theorem of Pappus  
III.4  The Projective Geometry of a Line in Space; Cross Ratios  
Appendix III.  Projective Ordering of a Space  
IV.  Dualities  
IV.1  Existence of Dualities; Semibilinear Forms  
IV.2  Null Systems  
IV.3  Representation of Polarities  
IV.4  Isotropic and Nonisotropic Subspaces of a Polarity; Index and Nullity  
Appendix I.  Sylvester's Theorem of Inertia  
Appendix II.  Projective Relations between Lines Induced by Polarities  
Appendix III.  The Theorem of Pascal  
IV.5  The Group of a Polarity  
Appendix IV.  The Polarities with Transitive Group  
IV.6  The Nonisotropic Subspaces of a Polarity  
V.  The Ring of a Linear Manifold  
V.1  Definition of the Endomorphism Ring  
V.2  The Three Cornered Galois Theory  
V.3  The Finitely Generated Ideals  
V.4  The Isomorphisms of the Endomorphism Ring  
V.5  The Antiisomorphisms of the Endomorphism Ring  
Appendix I.  The Twosided Ideals of the Endomorphism Ring  
VI.  The Groups of a Linear Manifold  
VI.1  The Center of the Full Linear Group  
VI.2  First and Second Centralizer of an Involution  
VI.3  Transformations of Class 2  
VI.4  Cosets of Involutions  
VI.5  The Isomorphisms of the Full Linear Group  
Appendix I.  Groups of Involutions  
VI.6  Characterization of the Full Linear Group within the Group of Semilinear Transformations  
VI.7  The Isomorphisms of the Group of Semilinear Transformations  
VII.  Internal Characterization of the System of Subspaces  
A Short Bibliography of the Principles of Geometry  
VII.1  Basic Concepts, Postulates and Elementary Properties  
VII.2  Dependent and Independent Points  
VII.3  The Theorem of Desargues  
VII.4  The Imbedding Theorem  
VII.5  The Group of a Hyperplane  
VII.6  The Representation Theorem  
VII.7  The Principles of Affine Geometry  
Appendix S.  A Survey of the Basic Concepts and Principles of the Theory of Sets  
A Selection of Suitable Introductions into the Theory of Sets  
Sets and Subsets  
Mappings  
Partially Ordered Sets  
Well Ordering  
Ordinal Numbers  
Cardinal Numbers  
Bibliography  
Index 