# who invented projective space $\mathbb{P}^n$?

Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$?

Who was the first person to consider projective closure of plane affine algebraic curves (curves in $\mathbb{A}^2$)? Was it the same person?

• Start here? en.wikipedia.org/wiki/Projective_geometry – Michael Joyce Apr 3 '13 at 14:46
• I do not know. Anyway, it seems that homogeneous coordinates were introduced by Moebius en.wikipedia.org/wiki/Homogeneous_coordinates – Francesco Polizzi Apr 3 '13 at 14:48
• Francesko, thank you. Very helpful. Interestingly, the article says that "Möbius' original formulation of homogeneous coordinates specified the position of a point as the center of mass (or barycenter) of a system of three point masses placed at the vertices of a fixed triangle". -- I.e., his goal was different rather than the completion of $\mathbb{A}^n$! – Maxim Leyenson Apr 3 '13 at 15:15
• Michael, the current version of the Wikipedia page on Projective geometry does not answer these questions, I checked before posting... – Maxim Leyenson Apr 3 '13 at 17:37
• The idea of the projective plane is at least implicit in the work of the ancient Greek Pappus. It's not clear to me that one can say when projective space was developed. Anyway, it's still an interesting question, but I think it might be helpful to indicate exactly which aspect of projective space you are interested in. (I realize your second question does that, but I didn't know how to interpret your first question. Are you specifically interested in when people understood the generalization to $n$-dimensional space, or the first incarnations when $n = 2, 3$?) – Michael Joyce Apr 3 '13 at 19:08

## 3 Answers

The idea of projective space goes back to the study of perspective in painting. The first formalization known is due to G. Desargues, with the book Brouillon Projet d'une atteinte aux événements des rencontres du Cône avec un Plan (Rough draft for an essay on the results of taking plane sections of a cone) published in 1639. There it was developed a geometry of incidence without parallel lines. It was very dense and difficult to read.

Until XIX century the topic did not developed in full. Monge and Gergonne redeveloped it. Möbius introduced the homogeneous coordinates and Plücker also worked in these early developments. Steiner gave the first axiomatic (or synthetic) treatment. from there on, it playe a central role specially in the study of sets of solutions of polynomial equations. Today it makes one of the fundamental traits in modern algebraic geometry. But projective space considerations are present more or less implicitly also in topology, differential geometry, certain kind of differential equations and some descriptions of particle behavior in quantum mechanics.

If the question concerns projective space specifically rather than projective geometry in a broader sense, then the answer would have to be Jean-Victor Poncelet (1788 – 1867). Desargues already introduced the notion of a point at infinity, but I believe Poncelet was the first to consider a LINE at infinity.

One should also mention Karl Georg Christian von Staudt (1798 – 1867), a German mathematician.

His book "Geometrie der Lage (1847)" was a landmark in projective geometry.

Staudt was the first to adopt a fully rigorous approach. Without exception his predecessors still spoke of distances, perpendiculars, angles and other entities that play no role in projective geometry.