Editing Euclidean geometry (section)

===17th century: Descartes===
[[René Descartes]] (1596–1650) developed [[analytic geometry]], an alternative method for formalizing geometry which focused on turning geometry into algebra.<ref>Ball, pp. 268ff.</ref>

In this approach, a point on a plane is represented by its [[Cartesian coordinate system|Cartesian]] (''x'', ''y'') coordinates, a line is represented by its equation, and so on.

In Euclid's original approach, the [[Pythagorean theorem]] follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

The equation
:<math>|PQ|=\sqrt{(p_x-q_x)^2+(p_y-q_y)^2} \, </math>
defining the distance between two points ''P'' = (''p<sub>x</sub>'', ''p<sub>y</sub>'') and ''Q'' = (''q<sub>x</sub>'', ''q<sub>y</sub>'') is then known as the ''Euclidean [[metric space|metric]]'', and other metrics define [[non-Euclidean geometry|non-Euclidean geometries]].

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., ''y'' = 2''x'' + 1 (a line), or ''x''<sup>2</sup> + ''y''<sup>2</sup> = 7 (a circle).

Also in the 17th century, [[Girard Desargues]], motivated by the theory of [[Perspective (graphical)|perspective]], introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, [[projective geometry]], but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.<ref>Eves (1963).</ref>

[[File:Squaring the circle.svg|right|thumb|Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized [[compass and straightedge]].]]