Marcel Berger, the great French geometer who died in October 2016, used to say: "An affine space is a vector space whose origin has been forgotten." By this he meant that, in the former, all points play analogous roles, whereas in the latter, the zero vector is a distinguished element. Historically, however… the reverse path was actually taken!
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Describing our space
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Until the early 19th century, no real distinction was made between physical reality and its mathematical formulation. Some properties were therefore justified simply by their visual obviousness. There seemed to be no need to define a mathematical structure representing the space around us: it appeared to follow directly from experience and intuition. The introduction of coordinates in the 17th century by René Descartes and then Pierre de Fermat had shown that choosing an origin O and two points A and B in the plane (three in space) makes it possible to locate any point M—or, in modern terms, to express the coordinates of the vector in the basis . Without saying so explicitly, working with coordinates already amounted to working in a vector space; nevertheless, it took two centuries for this structure to be formalized. Reversing Marcel Berger's formulation, we might say that, historically, a vector space is an affine space in which one point has been fixed. But what is an affine space?
A set is endowed with the structure of an affine space associated with a real vector space E by specifying a map Θ from the Cartesian product × to E that satisfies the following two conditions, using the notation :
