The work done by Archimedes (ca. 287-212 B.C.), a Greek mathematician, was wide ranging, some of it leading to what has become integral calculus. He is considered one of the greatest mathematicians of all time.

# Archimedes

Archimedes probably was born in the seaport city of Syracuse, a Greek colony on the island of Sicily. He was the son of an astronomer, Phidias, and may have been related to Hieron, King of Syracuse, and his son Gelon. Archimedes studied in Alexandria at the school established by Euclid and then settled in his native city.

To the Greeks of this time, mathematics was considered one of the fine arts—something without practical application but pleasing to the intellect and to be enjoyed by those with the requisite talent and leisure. Archimedes did not record the many mechanical inventions he made at the request of King Hieron or simply for his own amusement, presumably because he considered them of little importance compared with his purely mathematical work. These inventions did, however, make him famous during his life.

## Fact and Fancy

The many stories that are told of Archimedes are the prototype of the absentminded-professor stories. A famous one tells how Archimedes uncovered a fraud attempted on Hieron. The King ordered a golden crown and gave the goldsmith the exact amount of gold needed. The goldsmith delivered a crown of the required weight, but Hieron suspected that some silver had been used instead of gold. He asked Archimedes to consider the matter. Once Archimedes was pondering it while he was getting into a bathtub full of water. He noticed that the amount of water overflowing the tub was proportional to the amount of his body that was being immersed. This gave him an idea for solving the problem of the crown, and he was so elated he ran naked through the streets repeatedly shouting *"Heureμka, heureμka!"* (I have discovered it!)

There are several ways Archimedes may have determined the proportion of silver in the crown. One likely method relies on a proposition which Archimedes later wrote in a treatise, *On Floating Bodies,* and which is equivalent to what is now called Archimedes' principle: a body immersed in a fluid is buoyed up by a force equal to the weight of fluid displaced by the body. Using this method, he would have first taken two equal weights of gold and silver and compared their weights when immersed in water. Next he would have compared the weight of the crown and an equal weight of pure silver in water in the same way. The difference between these two comparisons would indicate that the crown was not pure gold.

On another occasion Archimedes told Hieron that with a given force he could move any given weight. Archimedes had investigated properties of the lever and pulley, and it is on the basis of these that he is said to have asserted, "Give me a place to stand and I can move the earth." Hieron, amazed at this, asked for some physical demonstration. In the harbor was a new ship which the combined strength of all the Syracusans could not launch. Archimedes used a mechanical device that enabled him, standing some distance away, to move the ship. The device may have been a simple compound pulley or a machine in which a cogwheel with oblique teeth moves on a cylindrical helix turned by a handle.

Hieron saw that Archimedes had a most inventive mind in such practical matters as constructing mechanical aids. At this time one use for such inventions was in the military field. Hieron persuaded Archimedes to construct machines for possible use in warfare, both defensive and offensive.

## A Time of War

Plutarch in his biography of the Roman general Marcellus describes the following incident. After the death of Hieron, Marcellus attacked Syracuse by land and sea. Now the instruments of warfare made at Hieron's request were put to use. "The Syracusans were struck dumb with fear, thinking that nothing would avail against such violence and power. But Archimedes began to work his engines and hurled against the land forces all sorts of missiles and huge masses of stones, which came down with incredible noise and speed; nothing at all could ward off their weight, but they knocked down in heaps those who stood in the way and threw the ranks into disorder. Furthermore, beams were suddenly thrown over the ships from the walls, and some of the ships were sent to the bottom by means of weights fixed to the beams and plunging down from above; others were drawn up by iron claws, or crane-like beaks, attached to the prow and were plunged down on their sterns, or were twisted round and turned about by means of ropes within the city, and dashed against the cliffs. … Often there was the fearful sight of a ship lifted out of the sea into mid-air and whirled about as it hung there, until the men had been thrown out and shot in all directions, when it would fall empty upon the walls or slip from the grip that had held it."

Later writers tell how Archimedes set the Roman ships on fire by focusing an arrangement of concave mirrors on them he basic idea is that the mirror reflects to one point all the sun's light entering parallel to the mirror axis.

Marcellus, according to Plutarch, gave up trying to take the city by force and relied on a siege. The city surrendered after 8 months. Marcellus gave orders that the Syracusan citizens were not to be killed, taken as slaves, or mistreated. But some Roman soldier did kill Archimedes. There are different accounts of his death. One version is that Archimedes, now 75 years old, was alone and so absorbed in examining a diagram that he was unaware of the capture of the city. A soldier ordered him to go to Marcellus, but Archimedes would not leave until he had worked out his problem to the end. The soldier was so enraged, he killed Archimedes. Another version is that Archimedes was bringing Marcellus a box of his mathematical instruments, such as sundials, spheres, and angles adjusted to the apparent size of the sun, when he was killed by soldiers who thought he was carrying valuables in the box. "What is, however, agreed," Plutarch says, "is that Marcellus was distressed, and turned away from the slayer as from a polluted person, and sought out the relatives of Archimedes to do them honor."

Archimedes had requested his relatives to place upon his tomb a drawing of a sphere inscribed within a cylinder with a notation giving the ratio of the volume of the cylinder to that of the sphere—an indication of what Archimedes considered to be his greatest achievement. The Roman statesman and writer Cicero tells of finding this tomb much later in a state of neglect.

## Other Inventions

Perhaps while in Egypt, Archimedes invented the water screw, a machine for raising water to irrigate fields. Another invention was a miniature planetarium, a sphere whose motion imitated that of the earth, sun, moon, and the five other planets then known (Saturn, Jupiter, Mars, Venus, and Mercury); the model may have been kept in motion by a flow of water. Cicero tells of seeing it over a century later and claimed that it actually represented the periods of the moon and the apparent motion of the sun with such accuracy that it would, over a short period, show the eclipses of the sun and moon. Since astronomy was a branch of mathematics in Archimedes' time, he undoubtedly considered this and his other astronomical inventions much more important than those which could be put to practical use.

Archimedes is said to have made observations of the solstices to determine the length of the year and to have discovered the distances of the planets. In *The sand Reckoner* he describes a simple device for measuring the angle subtended by the sun at an observer's eye.

## Contributions to Mathematics

Euclid's *Elements* had catalogued practically all the results of Greek geometry up to Archimedes' time. Archimedes adopted Euclid's uniform and rigorously logical form: axioms followed by theorems and their proofs. But the problems Archimedes set himself and his solutions were on another level from any that preceded him.

In geometry Archimedes continued the work in Book XII of Euclid's *Elements.* In Book XII the method of exhaustion, discovered by Eudoxus, is used to prove theorems on areas of circles and volumes of spheres, pyramids, and cones. Two of the theorems are mentioned by Archimedes in the preface to *On the Sphere and Cylinder.* After stating the result concerning the ratio of the volumes of a cylinder and an inscribed sphere, he says that this result can be put side by side with his previous investigations and with those theorems of Eudoxus on solids, namely: the volume of a pyramid is one-third the volume of a prism with the same base and height; and the volume of a cone is one-third the volume of a cylinder with the same base and height.

There was no direct computation of areas and volumes enclosed by various curved lines and surfaces, but rather a comparison of these with each other or with the areas and volumes enclosed by rectilinear figures such as rectangles and prisms. The reason for this is that the area, for a simple example, of a circle with radius of length one cannot be expressed exactly by any fraction or integer. It is possible, however, to say as is done in Proposition 2 of Book XII of the *Elements* that the ratio of the area of one circle to another is exactly equal to the ratio of the squares of their diameters, or, in a more concise form closer to the Greek, circles are to one another as the squares of the diameters. The proof of this theorem relies on (theoretically) being able to "exhaust" the circle by inscribing in it successively polygons whose sides increase in number and hence which fit closer to the circle. Thus the curved line, the circle, can be closely approximated by a rectilinear figure, a polygon.

Recognizing this, it would be easy to conclude that the circle itself is a polygon with "infinitely" many "infinitesimal" sides. Even by Euclid's time this concept had a long history of philosophic controversy beginning with the well-known Zeno's paradoxes discussed by Aristotle. Archimedes, aware of the logical problems involved in making such a facile statement, avoids it and proceeds in his proofs in an invulnerable manner. However, a student with a knowledge of integral calculus today would find Archimedes' method very cumbersome. It should nevertheless be remembered that the theorems which make the work almost trivial to any modern mathematician were obtained only in the 17th, 18th, and 19th centuries, about 2000 years after Archimedes.

In modern terminology, the area of a circle with radius of length one is the irrational number denoted by π, and although Archimedes knew it could not be calculated exactly, he knew how to approximate it as closely as desired. In his treatise *Measurement of a Circle,* using the method of exhaustion, Archimedes proves that π is between 3 1/7 and 3 10/71 (it is actually 3.14159).

Large numbers seem to have some fascination of their own. A common Greek proverb was to the effect that the quantity of sand eludes number, that is, is infinite. To the Greeks this might seem especially true since their numeral system did not include a zero. Numbers were represented by letters of the alphabet, and for large numbers this notation becomes clumsy. In *The Sand Reckoner* Archimedes refutes the idea expressed by the proverb by inventing a notation which enables him to calculate in a reasonably concise way the number of grains of sand required to fill the "universe." He takes the universe to be the size of a sphere centered at the earth and having as radius the distance from the earth to the sun. After saying this he also points out an alternative view of the universe that had been expressed by a contemporary astronomer, Aristarchus of Samos, namely, that the sun is fixed, the earth revolves about the sun, and the stars are fixed a long distance beyond the earth. Astronomical data, together with the assumption that there are no more than 10,000 grains of sand in a volume the size of a poppyseed, are the basis of calculations leading up to the conclusion that the number of grains of sand which could be contained in a sphere the size of the universe is less than 10
51, in modern notation.

Other known works by Archimedes that are purely geometrical are *On Conoids and Spheroids, On Spirals,* and *Quadrature of the Parabola.* The first is concerned with volumes of segments of such figures as the hyperboloid of revolution. The second describes what is now known as Archimedes' spiral and contains area computations. The third is on finding areas of segments of the parabola.

Another of Archimedes' works in mechanics, besides *On Floating Bodies* mentioned previously, is *On the Equilibrium of Planes.* From such simple postulates as "Equal weights at equal distances balance," positions of centers of gravity are determined for parabolic segments.

As is true of all other mathematicians of antiquity, Archimedes usually wrote in a way which left no indication of how he arrived at the theorems; all the reader sees is a theorem followed by a proof. But in 1906 a hitherto-lost treatise by Archimedes, *The Method,* was found. In it Archimedes explains a certain method by which it is possible to get a start in investigating some of the problems in mathematics by means of mechanics. "For," Archimedes writes, "certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration." Thus Archimedes is careful to distinguish between a heuristic approach to verifying a theorem and the proof of the theorem. *The Method* utilizes theorems from his mechanical treatise *On the Equilibrium of Planes* and provides an excellent example of the interplay between pure and applied mathematics.

## Further Reading on Archimedes

The standard English translation of Archimedes is Thomas L. Health, ed., *The Works of Archimedes* (1897), which includes a supplement, *The Method of Archimedes* (1912). For biographical information see E. J. Dijksterhuis, *Archimedes* (1938; trans. 1956). Archimedes' place in the development of integral calculus is described in Carl B. Boyer, *The History of the Calculus and Its Conceptual Development* (1949). Works on mathematics for the general reader are Thomas L. Heath, *A Manual of Greek Mathematics* (1931); Bartel L. van der Waerden, *Science Awakening* (1950; trans. 1954); and James R. Newman, ed., *The World of Mathematics* (4 vols., 1956). See also Robert S. Brumbaugh, *Ancient Greek Gadgets and Machines* (1966).