Projective Geometry - History

History

The first geometrical properties of a projective nature were discovered in the third century by Pappus of Alexandria. Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective in 1425 (see the history of perspective for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the pivotal concept of the "point at infinity". Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy in 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.

This early 19th century projective geometry was a stepping stone from analytic geometry to algebraic geometry. When treated in terms of homogeneous coordinates, projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems to algebra, an extension reducing the number of special cases. The detailed study of quadrics and the "line geometry" of Julius Plücker still form a rich set of examples for geometers working with more general concepts.

The work of Poncelet, Steiner and others was not intended to extend analytic geometry. Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra.

This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. Towards the end of the century the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.

In the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now seen as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians.

Paul Dirac studied projective geometry and used it as a basis for developing his concepts of Quantum Mechanics, although his published results were always in algebraic form. See a blog article referring to an article and a book on this subject, also to a talk Dirac gave to a general audience in 1972 in Boston about projective geometry, without specifics as to its application in his physics.