Definition:Tangent Line

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 $A$.

Limit of Secant

Let $\CC$ be a space curve.

Let $\CC$ be embedded in a Cartesian plane.

Let $P$ be a point on $\CC$.

Let $\SS$ be a secant line to $\CC$ passing through $P$ and another point $Q$ also on $\CC$.

The tangent to $\CC$ at $P$ can be considered as the limit of $\SS$ as $Q$ approaches $P$.

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 $C$.

Point of Contact

Let $\LL$ be tangent to $f$ at $A$.

Then $A$ is known as the point of contact of $\LL$ to $f$.

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 ($\text {1601}$ – $\text {1665}$)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): tangent: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tangent: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): tangent (to a curve)