Area of the Q - Numberphile

TL;DR

Hippocrates discovered that any similar shape on the hypotenuse of a right-angled triangle will have the same area as the shapes on the other two sides.

Transcript

Today I'm going to talk about Hippocrates, who was the great mathematician who came after Pythagoras. Some call him the loony mathematician and I'll explain why today. What I've started with is a triangle ABC and I've echoed the triangle by producing point E down here. He knew Pythagoras' theorem, which says the square on the hippopotamus - sorry t... Read More

Key Insights

• 🔺 Hippocrates built upon Pythagoras' theorem to establish a broader understanding of the relationship between shapes in right-angled triangles.
• 👻 His concept of lunes allowed for the exact determination of the area of figures bounded purely by curves.
• 💅 The equal areas of lunes and triangles showcased the beauty and mathematical significance of Hippocrates' discoveries.
• ❓ The different examples of lunes demonstrated the versatility and applicability of his findings.
• 🤗 The mathematical relationship between lunes and triangles opened up new possibilities for geometric calculations.
• 🖐️ Hippocrates' contributions advanced mathematical knowledge and laid the foundation for further exploration in geometry.
• 🤔 His unconventional approach challenged traditional mathematical thinking and expanded the boundaries of the discipline.

Questions & Answers

Q: What did Hippocrates contribute to mathematics?

Hippocrates expanded on Pythagoras' theorem by proving that any similar shape on the hypotenuse of a right-angled triangle will have the same area as the shapes on the other two sides.

Q: What is a lune?

A lune is a curve that represents the area of the shapes on the sides of a right-angled triangle. Hippocrates used lunes to demonstrate the equal areas of different shapes within triangles.

Q: How did Hippocrates prove the equal areas of lunes and triangles?

He drew various examples of triangles and their corresponding lunes, showing that the areas of the lunes matched the areas of the triangles. This demonstrated the mathematical relationship between the shapes.

Q: Why did Hippocrates call himself the "loony mathematician"?

Hippocrates referred to himself as the "loony mathematician" due to his unconventional approach to mathematics and his discovery of the equal areas of lunes and triangles, which was groundbreaking at the time.

Summary & Key Takeaways

• Hippocrates expanded on Pythagoras' theorem and discovered that any similar shape on the hypotenuse of a right-angled triangle would have an equal area to the shapes on the other two sides.

• He introduced the concept of "lunes," which are curves that represent the areas of the shapes on the sides of the triangle.

• Hippocrates demonstrated various examples of lunes and their equal areas to different triangles, showcasing the beauty and mathematical significance of his discovery.