Archimedes’ Gravestone

Archimedes’ gravestone is said to have a sphere inside a cylinder etched into the stone as well as the symbol π. How are the two 3-D objects related if they have the same radius? And why are they on his gravestone?

Answer

Archimedes (287?–212 B.C.E.), perhaps the greatest mathematician of ancient times, was the first to calculate the volume ratio of the sphere inside the cylinder. With a sphere and a cone inside the cylinder touching top, bottom, and sides, Archimedes determined that their volumes are in the ratios 1:2:3!

The Roman general, Marcellus, tells of how he searched for and found Archimedes’ gravesite with this headstone. Archimedes was killed in 212 B.C.E. during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed. Plutarch recounts three versions of the story of his killing that had come down to him:

1. “Archimedes was, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.”

2. “A Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him.”

3. “As Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the Sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him.”

Archimedes was buried in Syracuse, where he was born, where he grew up, where he worked, and where he died. On his grave there is an inscription of π, his most famous discovery. Also placed on his tombstone is the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement. His nicknames were, “the Wise One,” “the Master,” and “the Great Geometer.”