# Definition:Tangent to Curve

## Definition

Let $f: \R \to \R$ be a real function.

Let the graph of $f$ be depicted on a Cartesian plane.

Let $A = \tuple {x, \map f x}$ be a point on $G$.

The **tangent to $f$ at $A$** is defined as:

- $\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

Thus the **tangent to $f$ at $x$** can be considered as the secant $AB$ to $G$ where:

- $B = \tuple {x + h, \map f {x + h} }$

as $B$ gets closer and closer to $A$.

By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$.

Hence the **tangent** to $f$ is a straight line which intersects the graph of $f$ locally at a single point.

In the above diagram, the **tangent** is the straight line passing through $A$.

### Tangent to Circle

In the words of Euclid:

*A straight line is said to***touch a circle**which, meeting the circle and being produced, does not cut the circle.

(*The Elements*: Book $\text{III}$: Definition $2$)

In the above diagram, the line is **tangent** to the circle at the point $C$.

## Also see

- Results about
**tangents**can be found here.

## Historical Note

The definition of the tangent to a curve as the limit of a sequence of secants was made by Pierre de Fermat.

It first appeared in his *Introduction to Plane and Solid Loci*, but the idea most probably dates from considerably earlier.

This definition anticipates the invention of differential calculus.

In fact, Isaac Newton, in a letter that was discovered as late as $1934$, specifically states that it was the work of Fermat which inspired his own ideas about calculus.

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $2$. The tangent at a given point - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**tangent**(to a curve)