One of the first details we read about Archimedes (287-212 BCE) in almost every account of his life is the famous scene where he runs wet and naked through the streets of Syracuse shouting “Eureka!, Eureka!” (“I have found it!”). This nudist episode, however, fails to capture the respect that the life of the greatest Greek mathematician and mechanical engineer of antiquity deserves. Archimedes was a pioneer in mathematics and engineering, many centuries ahead of his contemporaries. He was the son of an astronomer named Phidias, lived in the Greek city of Syracuse, studied in Alexandria under the successors of Euclid, and was on intimate terms with King Hieron II, the ruler of Syracuse.
Like all important figures in antiquity who were supremely talented, his story became filled throughout the centuries with many myths and other non-historical accounts to sustain his specialness. It is, therefore, a challenge to clearly distinguish between actual historical facts and legends that were added to decorate his story.
Archimedes' success in applying his mathematical knowledge to weapons of war played a major role during the war between Rome and Syracuse during the Second Punic War. The development of this conflict can be traced back to around 290 BCE, when the Romans became the new rulers of central Italy and began to conquer the Greek cities on the Italian coast. In 270 BCE Hieron II (308-215 BCE) became king of Syracuse, located on the island of Sicily, and the city enjoyed a last period of prosperity. In Sicily, Romans and Carthaginians were brought face to face and in 264 BCE, the First Punic War started. The Carthaginians were the masters of the sea, so the Romans relied on help from the Greek cities in the south in order to build their own ships and so were able to fight the Carthaginians at sea. In 241 BCE Rome defeated Carthage and took over Sicily. During his reign, Hieron II remained on peaceful terms with the Romans and when Rome took over Sicily after the First Punic War, Syracuse remained independent.
In 218 BCE the Second Punic War started; this was the second major war between Carthage and Rome. In 215 BCE, Hieron II died and his successor Hieronymus made a very poor decision by switching sides and supporting Carthage: He felt the Romans would lose the war. The Romans were not happy about this decision, and they made it clear by besieging the city of Syracuse from 214 to 212 BCE. In the end, the Romans entered the city, slaughtered and enslaved its citizens, and sacked it.
During the time of Archimedes, the centre of Greek culture was Alexandria, the greatest centre of scholarship at this time. Here Archimedes received the finest training available in several disciplines, including mathematics. Archimedes’ devotion to mathematics has been compared with that of Newton’s: Both often neglected food, drink, and even the basic care of their bodies in order to continue studying mathematics. Plutarch wrote on Archimedes some three centuries later:
It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unlabored results.
(Durant, p. 629)
Archimedes’ works on mathematics can be categorized into three groups:
1. Works that prove theorems related to solids and areas bounded by curves and surfaces.
2. Works that analyse problems in statics and hydrostatics from a geometrical viewpoint.
3. Miscellaneous works, including some that emphasize counting, such as The Sand Reckoner.
In his work On the Measurement of the Circle, Archimedes arrives at the logical conclusion that the ratio of a circle’s circumference to its diameter, the mathematical constant we today call “pi” (π), is greater than 3 1/7 but less than 3 10/71, a very good approximation.
The famous incident where Archimedes runs naked started with a crown made for Hieron II: The king suspected that the artisan might have kept for himself some of the gold provided for the task and replaced it with a mixture of gold and materials of lower quality. The king wanted to know whether the artisan replaced the gold, but he wanted to find out without damaging the crown, so he requested that many experts test the crown without damaging it. We are told that Archimedes was among those experts and after several weeks thinking about the matter, he found the answer while stepping into a tub at the public baths. He noticed two things; first, that the water overflowed in accordance to the depth of his immersion, and second, that his body appeared to weigh less the deeper it was submerged. Upon this revelation, if we are to believe the legend, Archimedes rushed off down the streets of Syracuse, presumably naked and wet, shouting in excitement that he had found the answer to the king’s question. He formulated the “Principle of Archimedes”, also known as the law of buoyancy, which states that any object fully or partially immersed in a fluid will experience an upward force equal to the weight of the displaced fluid. This principle offered Archimedes a test for the material make-up of the crown. Back home he discovered that a given weight of silver, when immersed, displaced more water than an equal weight of gold. The reason for this is that silver has more volume per weight in comparison to gold. He then proceeded to submerge the crown and compared the water displaced by it with a quantity of gold equal to the crown in weight. Archimedes concluded that the crown was not made entirely of gold, confirming the king’s suspicions, and so he was able to tell exactly how much gold was missing.
In a lost treatise which we know only through summaries, Archimedes formulated the Law of the Lever and Balance. He did it so accurately that no advancement was made until the 16th century CE. He also discovered the benefits of the pulley for lifting large weights. He was so amazed by the mechanical advantages provided by both the lever and the pulley that he famously stated, “give me a place to stand, and I will move the Earth”. King Hieron challenged Archimedes to put his claim to the test, so Archimedes arranged a cleverly designed series of cogs and pulleys in such a manner that he alone, sitting on one end of the mechanism, managed to draw a fully loaded vessel out of the water and place it onto the land, a task that a hundred men could barely accomplish.
Despite all of the physical laws he discovered, Archimedes never actually referred to them as laws, nor did he describe them in reference to observation and measurement; he instead treated them as pure mathematical theorems, within the logic of a system similar to the one Euclid developed for geometry. Greek science during Archimedes’ day had a tendency to undervalue observations and favour logical arguments: Greeks believed that the highest knowledge was based on deductive reasoning. This, however, did not prevent Archimedes from experimenting; in fact, he stands out from his contemporaries because he successfully applied his theoretical knowledge into practice. But the way he presents his discoveries is always from a mathematical perspective, and he never attempted to offer a systematic description from an engineering viewpoint. Moreover, when he refers to mechanical experiments he is actually using them to help the understanding of mathematics: This shows a key difference in approach between ancient science, where experimentation was used to help theoretical understanding, and modern science, where theory is used to pursue practical results.
Death & Legacy
After the death of Hieron II, war began between Syracuse and the Romans. The city was attacked by both land and sea. Seventy-five years of age were no obstacle for Archimedes in playing a central role defending the city. Applying his skills as an engineer, he developed and arranged catapults that hurled heavy stones to a great distance, pierced holes in the city walls for bowmen to shoot their arrows, and set up cranes that were able to release a large weight of stones on the Roman ships when they came within reach. These inventions were so effective that Marcus Claudius Marcellus, the Roman commander, abandoned the idea of attacking Syracuse and decided that a siege was the only way of breaking the city. In 212 BCE, the starving city surrendered and the Romans captured Syracuse.
Marcellus was so impressed by the genius of Archimedes that he ordered that the talented Greek should be captured alive. Nonetheless, when the Roman soldiers located Archimedes, he was on the beach drawing geometrical figures in the sand and working on one of his many theorems. He ignored the soldiers’ orders and requested some extra time to finish his work. The furious soldiers, probably feeling a little insulted, immediately killed one of the greatest minds of all history.
Archimedes died, but his ideas could not be killed, and Archimedes’ works, after many adventures and translations during the Middle Ages, have survived in an accessible form. During the Renaissance, the work of Archimedes gained a wide interest in the developing scientific movement. Galileo was very interested in Archimedes due to the application of mathematics to physics and many of his clever experiments. The western world would have to wait until Leonardo Da Vinci to see a greater mechanical genius.