Describing the plane and space requires a way of locating points. Once an origin has been chosen, every point A can be associated with the vector . But reaching this viewpoint first required the concept of a coordinate system, which emerged only in the 17th century with René Descartes and Pierre de Fermat. In this way, these scholars reduced geometric problems to the manipulation of algebraic equations.
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Visionary scholars
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In the closing years of that same century, Gottfried Wilhelm Leibniz criticized this approach: analytic geometry, he argued, reduced everything to numbers and was therefore certainly not the shortest route to solving a problem. He then put forward the visionary—but at the time impracticable—idea of creating what he called a "geometry of position": an algebra acting on geometric entities such as angles or motions.
Around the middle of the 18th century, Gabriel Cramer began studying the intersections of algebraic curves—that is, curves whose coordinates satisfy polynomial equations. This led him to solve systems of linear equations and introduce the concept of a determinant. The Swiss mathematician's name remains associated with systems having as many equations as unknowns (see Tangente 136). To resolve an apparent contradiction in the work of his compatriot, Leonhard Euler realized that n equations in n unknowns do not necessarily have exactly one solution. Euler observed that one equation may "be implied by the others." In other words, he was the first to recognize linear dependence.
