Editing Singularity theory (section)

===Algebraic curve singularities===

[[File:Cubic with double point.svg|thumb|right|upright=0.75|A curve with double point]]
[[File:Cusp.svg|thumb|upright=0.75|right|A curve with a cusp]]

Historically, singularities were first noticed in the study of [[algebraic curve]]s. The ''double point'' at (0,&nbsp;0) of the curve

:<math>y^2 = x^2 + x^3 </math>

and the [[cusp (singularity)|cusp]] there of

:<math>y^2 = x^3\ </math>

are qualitatively different, as is seen just by sketching. [[Isaac Newton]] carried out a detailed study of all [[cubic curve]]s, the general family to which these examples belong. It was noticed in the formulation of [[Bézout's theorem]] that such ''singular points'' must be counted with [[Multiplicity (mathematics)|multiplicity]] (2 for a double point, 3 for a cusp), in accounting for intersections of curves.

It was then a short step to define the general notion of a [[singular point of an algebraic variety]]; that is, to allow higher dimensions.