calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...

, the method of normals was a technique invented by Descartes for finding normal and

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mor ...

lines to

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s. It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that the

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

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 Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...

's method of

adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''

. While Fermat's method had more in common with the

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally ref ...

techniques that were to be used later, Descartes' method was more influential in the early history of calculus. 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 In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

for a wide class of functions using neither infinitesimal nor limit techniques. It is also related to a completely general definition of differentiability given by Carathéodory .

References

* * {{citation, title=Where Are Limits Needed in Calculus?, last1=Range, first1=R. Michael, journal=American Mathematical Monthly, volume=118, number=5, date=May 2011, pages=404–417, doi=10.4169/amer.math.monthly.118.05.404 History of calculus