Archimedes made early advances in approximating Pi through drawing polygons inside and outside circles and using the formula for the circumference of a circle. He established fundamental principles of density that are still used today, such as in determining if objects will float based on their density compared to the fluid around them. Archimedes also discovered the principle of buoyancy when bathing and noticing how much water his submerged body displaced. This principle is critical for ship design and submarine operations.