# Method of normals

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In [calculus](/source/Calculus), the **method of normals** was a technique invented by [Descartes](/source/Descartes) for finding [normal](/source/Surface_normal) and [tangent](/source/Tangent) lines to [curves](/source/Curve). It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that the [radius](/source/Radius) of a [circle](/source/Circle) is always normal to the circle itself. With this in mind Descartes would construct a circle that was tangent to a given curve. He could then use the radius at the point of intersection to find the slope of a normal line, and from this one can easily find the slope of a tangent line.

This was discovered about the same time as [Fermat](/source/Fermat)'s method of [adequality](/source/Adequality). While Fermat's method had more in common with the [infinitesimal](/source/Infinitesimal) techniques that were to be used later, Descartes' method was more influential in the early history of calculus. ([Katz 2008](#CITEREFKatz2008))

One reason Descartes' method fell from favor was the algebraic complexity it involved. On the other hand, this method can be used to rigorously define the [derivative](/source/Derivative_(mathematics)) for a wide class of functions using neither infinitesimal nor [limit](/source/Limit_(mathematics)) techniques. It is also related to a completely general definition of differentiability given by [Carathéodory](/source/Carath%C3%A9odory) ([Range 2011](#CITEREFRange2011)).

## References

- [Katz, V.](/source/Victor_J._Katz) (2008), *A History of Mathematics: An Introduction*, Addison Wesley

- Range, R. Michael (May 2011), "Where Are Limits Needed in Calculus?", *American Mathematical Monthly*, **118** (5): 404–417, [doi](/source/Doi_(identifier)):[10.4169/amer.math.monthly.118.05.404](https://doi.org/10.4169%2Famer.math.monthly.118.05.404)

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