# Archimedes

born: 287 B.C. in Syracuse, Sicily
died: 212 B.C. in Syracuse

Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.

One of the greatest mathematicians of all time. Method of Exhaustion — integration. “Eureka” — the first law of hydrostatics. Discovered the laws of levers and used pulleys. His mechanical inventions defeated the Roman fleet of Marcellus.

Archimedes was an aristocrat, the son of an astronomer, but little is known of his early life except that he studied for a time in Alexandria, Egypt. Several of his books were preserved by the Greeks and Arabs into the Middle Ages, and, fortunately, the Roman historian Plutarch described a few episodes from his life. In many areas of mathematics as well as in hydrostatics and statics, his work and results were not surpassed for over 1500 years!

He approximated the area of circles (and the value of ¼) by summing the areas of inscribed and circumscribed rectangles, and generalized this “method of exhaustion,” by taking smaller and smaller rectangular areas and summing them, to find the areas and even volumes of several other shapes. This anticipated the results of the calculus of Newton and Leibniz by almost 2000 years!

He found the area and tangents to the curve traced by a point moving with uniform speed along a straight line which is revolving with uniform angular speed about a fixed point. This curve, described by r = aƟ in polar coordinates, is now called the “spiral of Archimedes.” With calculus it is an easy problem; without calculus it is very difficult.

The king of Syracuse once asked Archimedes to find a way of determining if one of his crowns was pure gold without destroying the crown in the process. The crown weighed the correct amount but that was not a guarantee that it was pure gold. The story is told that as Archimedes lowered himself into a bath he noticed that some of the water was displaced by his body and flowed over the edge of the tub. This was just the insight he needed to realize that the crown should not only weigh the right amount but should displace the same volume as an equal weight of pure gold. He was so excited by this idea that he reportedly ran naked through the streets shouting “Eureka” (“I have found it”).

“Give me a place to stand and I will move the earth” was his boast when he discovered the laws of levers and pulleys. Since it was impossible to challenge that statement directly, he was asked to move a ship which had required a large group of laborers to put into position. Archimedes did so easily by using a compound pulley system.

During the war between Rome and Carthage, the Roman fleet decided to attack Syracuse, but Archimedes had been at work devising a few surprises. There were catapults with adjustable ranges which could throw objects which weighted over 500 pounds. The ships which survived the catapults were met with poles which reached over the city walls and dropped heavy stones onto the ships. Large grappling hooks attached to levers lifted the ships out of the water and then dropped them. During another failed assault, it is said that Archimedes had the soldiers of Syracuse use specially shaped and shined shields to focus the sunlight onto the sails to set them afire. This was more than the terrified sailors could stand, and the fleet withdrew. Unfortunately, the city began celebrating a bit early, and Marcellus captured Syracuse by attacking from the landward side during the celebration. “Archimedes, who was then, 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 upon him, commanded him to follow to Marcellus, which he declined to do before he had worked out his problem to a demonstration; the soldier, enraged, drew his sword and ran him through.” (Plutarch)

Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with the ratio of the cylinder’s volume to that of the sphere. Archimedes considered the discovery of this ratio the greatest of all his accomplishments.

The results of Archimedes are all the more remarkable when one considers the times in which he lived. Archimedes made fundamental discoveries in several fields, and he then advanced them so far that his results were not improved upon for many centuries. Archimedes certainly ranks as one of the greatest minds in recorded history.

Condensed from Men of Mathematics by E.T. Bell (1937, Simon and Schuster) and An Introduction to the History of Mathematics, 4th ed., by H. Eves (1976, Holt, Rinehart and Winston). DTH

Last Updated September 19, 2022