# Definition:Tangent/Analytic Geometry

## Definition

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

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

Let $A = \left({x, f \left({x}\right)}\right)$ be a point on $G$.

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

- $\displaystyle \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$

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

- $B = \left({x + h, f \left({x + h}\right)}\right)$

as $B$ gets closed 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 $A$.

## 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

- 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 ($1601$ – $1665$)