Archimedes (physicist)

(c. 287-212 b.c.) Greek Theoretician and Experimentalist (Mechanics, Hydrostatics), Mathematician, Astronomer

Archimedes is widely viewed as the greatest scientist of the ancient world. As a physicist, he is credited with establishing the fields of statics, a branch of mechanics dealing with the forces on an object or in a system in equilibrium, and hydrostatics, the study of fluids (liquids and gases) in equilibrium. He is most famous for Archimedes’ principle, which offered the first scientific explanation of what makes solid objects float.

Only a handful of facts about Archimedes’ life have been established with certainty. The biography of him written by his friend Heraclei-des has been lost. What remains for historians to draw on are Archimedes’ nine surviving mathematical treatises, which he published in the form of correspondence with the leading mathematicians of his time (including the Alexandrian scholars Conan of Samos and Eratosthenes of Cyrene); the accounts of his life left by his Greek contemporaries; and stories from Plutarch, Livy, and others. He was widely known during his lifetime, mainly because of his inventions that were used in war. The impression his mechanical genius made on the popular imagination gave rise to numerous legends, which today are viewed by most historians as apocryphal.

Archimedes is believed to have been born in 287 b.c., in Syracuse, Sicily, then a Greek colony. The date has been based on the claim of a 12th-century historian that he died at age 75, and then working backward from the date of his death in 212 b.c., which is reliably established. His father was Phidias, an astronomer; the family was a noble one, possibly related to that of King Hieron II of Syracuse. As a young man, he traveled to Alexandria, then a great capital of learning for mathematics, and studied under Conon and other mathematicians, who had been students of Euclid. Most disciples would remain in Alexandria, but Archimedes returned to Syracuse, where he spent the rest of his life studying mathematics and physics, diverting himself by designing the numerous mechanical devices that earned him widespread renown.

The boldness and originality of Archimedes’ mathematical work are tempered by its extreme rigor and adherence to the highest standards of the geometry of his time. Among his most important results was the determination of the value of n. He made the most accurate prediction of his time about the value of n, bracketing its value within the upper and lower limits given by 223/71 = 3.14085 > n > 3.1429 = 220/70.

What is more amazing is the fact that the average of Archimedes’ upper and lower limits on the value of n is 3.1419, less than three parts in ten thousand different from the modern approximation given: n = 3.1416. In an attempt to improve Greek numerical notation, he devised an ingenious system for the expression of very large numbers. In his treatise The Sandreckoner, he proposed a number system capable of expressing very large numbers. He then used this number system to estimate the number of grains of sand in the universe as given by the number 1063. In this manner he showed that large numbers could be considered and handled effectively. He also invented methods to solve cubic equations and to determine square roots by approximation. Most impressively, he devised formulas to determine the surface areas and volume of curved surfaces and solids, a topic that anticipated the development of the integral calculus 2,000 years later by Newton and Leibniz.

As a physicist Archimedes is credited with establishing the fields of statics of objects and hydrostatics of fluids. In statics he worked out the rigorous mathematical proofs behind the principle of the lever and the compound pulley—mechanical devices that can multiply the effects of forces. Although the scientists of his time were familiar with the use of the lever, Archimedes was the first to show that the ratio of the effort applied to the load raised by a lever is equal to the inverse ratio of the distances of the effort and load from the pivot or fulcrum about which the lever rotates. He is said to have claimed that if he could stand at a great enough distance, he could use a lever to move the world. In response to this, King Hieron purportedly issued him the lesser challenge of showing that he could move a very heavy object with ease. Archimedes allegedly responded by easily moving a ship, laden with passengers, crew, and cargo, which a number of men had struggled mightily to lift out of the harbor onto dry land. Sitting at a distance from the ship, he is said to have used a compound pulley to pull it over the land as if it were gliding through water.

His two-volume treatise on hydrostatics, On Floating Bodies, is the first known work on the topic and survives only partly in Greek, the rest in medieval Latin translation from the Greek. The first book contains his most famous result, the Archimedes’ principle, which states that the upward force on an object totally or partly submerged in a fluid is equal to the weight of fluid displaced by the object. He is said to have become engrossed in the problem of floating bodies when King Hieron ordered that his new crown be evaluated to see whether it was pure gold, without damaging the object. In what is probably the most famous Archimedes story, the great scientist is said to have been watching water overflow from the bath he was immersed in when the idea now known as Archimedes’ principle dawned on him. So jubilant was he, legend relates, that he ran through the town naked, crying, “Eureka!” (“I’ve got it!”). What he had grasped was that if the gold of the king’s crown had been mixed with silver, which is less dense, then in order to have an equal weight to that of a purely golden crown, the king’s crown would have to have a greater volume and therefore would displace more water than that of a purely golden crown. Unfortunately, as the story goes, this method proved that the crown contained silver and the unlucky goldsmith was put to death by the king.

In antiquity Archimedes was also known as an astounding astronomer, although little is known of this side of his activities. According to the Greek biographer Plutarch, Archimedes chose to publish only the results of his theoretical researches because he deemed only these worthy of serious consideration. But his interest in mechanics deeply influenced his mathematical thinking. In this context he wrote works on theoretical mechanics and hydrostatics, and his treatise Method Concerning Mathematical Theorems shows that his intuitive mechanical reasoning was an essential tool leading to his discovery of new mathematical theorems. He wrote:

Certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish actual proof. But it is of course easier, when we have previously acquired, by the [mechanical] method, some knowledge of the questions, to supply the proof than it is to find without any previous knowledge.

Inventions attributed to him include a design for a model planetarium able to show the movement of the Sun, Moon, planets, and possibly constellations across the sky. Cicero reports that when Marcellus sacked Syracuse, he took this planetarium as booty. He is also credited with the Archimedes screw, an augur pump used to raise water for irrigation, which is still used in many parts of the world. He reportedly invented it during his days in Egypt, although it is also possible that he borrowed the idea from others in Egypt.

The most dramatic of his inventions, however, were instruments of war. At the urging of King Hieron, he transformed his playful mechanical diagrams into viable machines. Some of these proved invaluable during the Roman siege of Syracuse from 212 to 215 b.c., when Archimedes’ weapons allegedly set fire to the ships of the Roman fleet under Marcellus and made them capsize. This held the Romans at bay for a long time, although they eventually succeeded in sacking the city, an operation in which Archimedes met his end. Plutarch gives three different versions of his death, all of which picture him killed while absorbed in scientific pursuits. In one version, despite orders to spare him, Archimedes was killed on the spot by a Roman soldier when he was ordered to leave his study, where he was contemplating a mathematics problem. He left instructions for his tomb to be marked with a sphere inscribed in a cylinder, together with the formula for the ratio of their volumes—since he considered this discovery his greatest achievement. Cicero found the tomb, overgrown with vegetation, a century and a half after Archimedes’ death.

Given the magnitude and originality of Archimedes’ achievements, it is ironic that his influence remained so small and undeveloped in ancient times. His work was not widely known in antiquity and did not lead directly to other advances at that time. However, his legacy was preserved by Byzantium and Islam, inspiring important work by medieval Islamic mathematicians, and from there it spread to Europe from the 12 th century onward. The greatest impact of his work on later mathematicians occurred in the late 16th and early 17th centuries, after which it had a profound impact on the history of science. In particular, Archimedes’ method of finding mathematical proof to substantiate experiment and observation became the method of modern science introduced by galileo galilei. Galileo published a study of the behavior of bodies in water, Bodies That Stay Atop Water or Move Within It, in which he championed Archimedes’ law of buoyancy, which states that the buoyant force on a body in water is equal to the weight of the water displaced.

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