Who invented projective space $\mathbb{P}^n$ as an extension of the usual [affine space](http://en.wikipedia.org/wiki/Affine_space%20affine%20space) $\mathbb{A}^n$?


Who was the first person to consider projective closure of
[plane affine algebraic curves](http://en.wikipedia.org/wiki/Algebraic_curve%20plane%20affine%20algebraic%20curves) (curves in $\mathbb{A}^2$)? Was it the same person?



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


