Euler’s Function: The Birth of a Mathematical Machine

By Eben van Tonder, 26 March 25

A 3-Part Illustrated Learning Journey for Armin, Christa, and Eben

Part I: The Function as a Machine

Visual Metaphor: A Factory Machine with an Input Pipe (x) and Output Chute (f(x))

1. What Did Euler See?

In the early 1700s, people were still writing math in words. “Let y be equal to the square of x plus three times x.” Clumsy. Limited.

Euler asked a brilliant question:

“What if the relationship itself could become the object?”

And with that, he invented:

f(x) — a function that acts on x

He called it a function because it functions on x. You feed it x, and it gives you an output:

  • Input x = 2 → Output f(x) = 10
  • Input x = 5 → Output f(x) = 40
    (if f(x) = x² + 3x)

It’s just like a factory machine.

Part II: The Function as a Graph Machine

Visual Metaphor: A Big Chart Plotter Drawing Curves

2. The Power of This New Idea

Euler realized that with a function, we could do amazing things:

  • Describe how two things are connected
  • Test what happens when x changes
  • Draw graphs and see the shape of change

“A function of a variable quantity is an analytical expression composed… of the variable and constants.” — Euler

He allowed for any kind of expression:

  • Polynomials:
  • Trigonometric:
  • Rational:

– Before Euler: Math was about static things.
– After Euler: Math became about movement and relationships.

The function graph is like a map of motion across the plane.

Part III: The Function as a Crystal Ball

Visual Metaphor: A Wizard’s Ball that Predicts Future Values

3. The Power to Predict

The function gave scientists the ability to predict.

If you model speed as a function of time:
Then you can say:

  • “How fast will the car be going in 5 seconds?”
  • “What happens if we double the weight?”

Euler gave us a way to model reality:

  • Physics
  • Engineering
  • Biology
  • Economics
  • Nutrition
  • Weather
  • Meat science

Anything that changes can be modeled with functions.

4. A Philosophical Revolution

Euler wasn’t just writing symbols. He changed the meaning of mathematics.

  • From counting to mapping change
  • From static forms to dynamic flows
  • From isolated numbers to systems of prediction

Carl Boyer wrote:

“The concept of function is the most important single concept in mathematics.”

Euler made it beautiful, useful, and universal.

Closing Thought: What Is a Function, Really?

It’s not just math.

It’s a language for change, a machine for thought, and a tool for vision.

Euler gave us a way to see the future—one value at a time.

In this short series on Beautiful Mathematics:

Euler’s Function: The Birth of a Mathematical Machine

A Deep Dive into f(x) = mx + b

Parent (Index) Page: The Power of One Variable