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Archimedes of Syracuse

Ancient Greek Scientist, Mathematician, and Inventor (c. 287 BCE – 212 BCE)

Archimedes of Syracuse was one of the most brilliant scientists, mathematicians, and inventors of the ancient world. Born in the Greek city of Syracuse on the island of Sicily, his contributions laid the foundation for many modern scientific and mathematical principles. Archimedes made discoveries in fields like geometry, mechanics, hydrostatics (fluids), and invented devices far ahead of his time.

Archimedes’ Principle (Buoyancy)

One of Archimedes’ most famous discoveries relates to the behavior of objects submerged in fluids. Legend says he discovered this principle while taking a bath and noticing the water level rise when he got in. This led to the understanding that any object submerged in a fluid is pushed up by a force equal to the weight of the fluid it displaces. This force is called the buoyant force.

Formula:
\[ F_b = \rho \times g \times V \]

• \(F_b\) = buoyant force
• \(\rho\) = density of the fluid
• \(g\) = acceleration due to gravity
• \(V\) = volume of the displaced fluid

This principle remains essential in designing ships, submarines, and fluid mechanics.

Volume of Sphere and Cylinder

Archimedes made significant progress in geometry by proving that the volume of a sphere is two-thirds the volume of the cylinder in which it fits exactly (same height and diameter).

Volume of a sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

Volume of a cylinder with same height and diameter as sphere:
\[ V_{\text{cylinder}} = \pi r^2 h \]

Since \(h = 2r\) for a cylinder that perfectly contains the sphere:
\[ V_{\text{cylinder}} = \pi r^2 \times 2r = 2 \pi r^3 \]

Ratio of the volumes:
\[ \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{\frac{4}{3} \pi r^3}{2 \pi r^3} = \frac{2}{3} \]

Archimedes was so proud of this discovery that he requested a sphere and cylinder be carved on his tombstone.

Law of the Lever

Archimedes studied simple machines and explained how levers work. He discovered that a small force can move a large weight if applied at the right distance. His famous quote is, "Give me a place to stand, and I will move the Earth."

Formula:
\[ F_1 \times d_1 = F_2 \times d_2 \]

• \(F_1\) and \(F_2\) are the forces applied on each side of the lever
• \(d_1\) and \(d_2\) are the distances from the fulcrum (pivot point)

This principle is still taught in physics and used in designing seesaws, balance scales, and many tools.

The Archimedean Screw

Archimedes invented a machine to move water from a low level to a higher one. It is a screw-shaped device inside a tube. As it turns, it scoops up water and pushes it upward. This invention, called the Archimedean screw, is still used today in irrigation and wastewater management systems.

Approximation of Pi (\(\pi\))

Archimedes developed a method to estimate the value of pi (\(\pi\)), the ratio of the circumference of a circle to its diameter. Using polygons with many sides inscribed and circumscribed around a circle, he calculated bounds for \(\pi\).

He concluded that:
\[ 3.1408 < \pi < 3.1429 \]

This estimate remained one of the most accurate for over a thousand years.

His Death and Legacy

Archimedes was killed in 212 BCE during the Roman invasion of Syracuse, despite orders from the Roman general to capture him alive. According to legend, Archimedes was so absorbed in a mathematical drawing in the sand that he ignored a Roman soldier, who then killed him. His ideas profoundly influenced later scientists like Galileo, Newton, and many others. He is remembered as one of history's greatest minds.