Projective Geometry Vol I by Oswald Veblen and John Wesley Young (PDF)

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⭐This first volume deals with the classical axiomatic, formal deductive approach. The second explores the influence of calculus or analysis methods. Around p.11 the authors introduce the homogeneous coordinates approach which essentially constructs a model of the projective plane as a subset of Euclidean 3-space which shows this geometry to be as consistent as Euclidean geometry. This model satisfies the axioms of projective geometry. After this proofs are given as deductions from axioms in the spirit of Euclid. I tend to believe that mathematicians who were trained from these texts later seized on the homogeneous coordinates as an economical simplification, i.e., these allow Euclidean geometry methods to be extended or generalized to projective geometry theorems through analytic geometry and linear algebraic methods-these tools being more familiar to themselves and their students. For example the famous Pascal theorem which generalizes the Pappus theorem to conic sections can easily be proven for a circle in Euclidean geometry (involves Menelaus’s theorem and the secant theorem for a circle from high school). Next the circle is represented in homogeneous coordinates and with a general conic represented as a quadratic form you can show with matrix methods that there are projective transforms which transform the circle to this form-ellipse, parabola, or hyperbola. As the transform is projective the three collinear points found in the circle case remain collinear in the general case-an easier or at least more familiar proof (you can find this on the web).I believe the authors intended this choice to occur as the axiomatic approach becomes divorced from its underlying physical or pictorial foundations and the authors give extensive development to the homogeneous coordinate approach later in the text. This subject is rarely taught in schools today but still creeps up in places like algebraic topology-the Crowell and Fox text on knot theory. Application is found for example in the Pascal theorem-that line containing the three points must be somewhere-possibly the limiting line at infinity. At any rate this text is the standard for the axiomatic approach and seminal for the use of homogeneous coordinates which many of you know imply a topological model for the projective plane. This is the classic reference but there are gentler introductions like Coxeter or even Hartshorne.

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