Archimedes

Archimedes (287-212 BCE) was a Greek Inventor, Mathematician and Physicist born in Syracuse in Sicily. His father was Phidias the Astronomer. He is considered the greatest mathematician of Classical Antiquity.
He died during the Siege of Syracuse (214-212 BCE), during the Second Punic War when he was killed by a Roman soldier who did not recognise him, despite instructions to spare him. The following Inventions were created by Archimedes:

His inventions include the Archimedes Screw, which was a simple method of raising water to a higher level.

Vitruvius tells the story of how Archimedes invented a method to determine the volume of a body having an irregular shape.
This enabled the calculation of the Density of an irregular Mass by measuring its Volume through the displacement of Water, thus avoiding having to melt down the King’s Crown to see if it really was made of pure Gold.
King Hiero II had commissioned a crown to be made out of solid Gold. However, He distrusted the Goldsmith and asked Archimedes to find a way of calculating whether he had used cheaper silver instead, but without Archimedes melting down the crown.
Whilst getting into his bath, Archimedes observed that as he got in, the water level rose.
He realised that the crown would displace a quantity of water equal to its own volume.
Archimedes concluded that the Mass of the crown divided by its Volume would equal its Density, and he could then calculate whether it was the density of Gold or of a lighter metal.
In his excitement he leapt out of his bath and ran down the street shouting Eureka! (I have found it!).
The test was carried out and it showed that silver had been added.

Mirrors of polished Bronze or copper to focus the sun and set fire to attacking ships.

This was a claw extended from a boom and dropped onto ships approaching the harbour of Syracuse, which then hooked them up out of the water.

Archimedes is known as the greatest mathematician of Classical Antiquity. He made great advances in geometrical calculations.

After the Siege of Syracuse in 212 BCE, the Roman Commander Marcus Claudius Marcellus is reported by Cicero to have returned to Rome with two devices designed by Archimedes. The description says that they demonstrated the movements of the Sun, Moon and Five Planets, which could only mean an Orrery.
This can only work by a system of differential gears, which was thought to have been too advanced for the time, until the Antikythera Mechanism was discovered dating to c.100 BCE.

Syracuse, Sicily