Born circa 287 B.C.; died 212 B.C. Ancient Greek scientist, mathematician, and specialist in mechanics.
Archimedes developed methods for finding the areas of surfaces and also the volumes of various figures and bodies. His mathematical works were far ahead of his time, and they were correctly evaluated only during the time of the creation of differential and integral calculus. Archimedes was a pioneer in mathematical physics. The mathematics in his works is systematically utilized for investigating problems of natural science and technology. Archimedes was one of the founders of mechanics as a science. He also invented various mechanical devices.
Archimedes was born in Syracuse on the island of Sicily and lived in that city during the First and Second Punic Wars. It is thought that he was the son of the astronomer Pheidias. He began his scientific activity as a specialist in mechanics and technician. Archimedes made a journey to Egypt, and there he became closely associated with the Alexandrian scientists, including Conon and Eratosthenes. This association served as an impetus to the development of his outstanding abilities. Archimedes was also close to the king of Syracuse, Hieron II. During the Second Punic War, Archimedes organized the engineering defenses of Syracuse against the Roman Army. His military machines compelled the Romans to abandon their attempts to take the city by storm and forced them to begin a lengthy siege. After the capture of the city by Marcellus’ army, Archimedes was killed by a Roman soldier whom, according to tradition, he had greeted with the words: “Don’t touch my drawings!” A monument with a depiction of a sphere inscribed within a cylinder was placed on Archimedes’ grave. The epitaph indicated that the volumes of these two bodies have a ratio of 2:3—a discovery of which Archimedes was particularly proud.
The works of Archimedes show that he was highly familiar with the mathematics and astronomy of his time; these works are astonishing because of Archimedes’ deep penetration into the essence of the problems. A number of his works are in the form of letters to friends and colleagues. At times Archimedes would communicate his discoveries to them beforehand, without proofs, adding several incorrect propositions with a fine sense of irony.
From the ninth to the 11th centuries the works of Archimedes were translated into Arabic, and beginning in the 13th century they appeared in Western Europe in Latin translation. Editions of Archimedes’ works began to be published from the 16th century on, and from the 17th to the 19th centuries they were translated into modern languages. The first edition of Archimedes’ individual works in Russian dates back to 1823. Certain works by Archimedes have not come down to us or are known only in fragments, and his Letter to Eratosthenes was discovered only in 1906.
The central theme running through Archimedes’ mathematical works is concerned with finding the areas of surfaces and volumes. Archimedes found a solution to many problems of this type using mechanical considerations which essentially correspond to the method of indivisibles, and then he rigorously proved it by the method of exhaustive approach, which he developed to a considerable extent. Archimedes’ examination of two-sided estimates of error in carrying through integration processes allows him to be considered a predecessor not only of I. Newton and G. Leibniz but also of G. Riemann. Archimedes calculated the area of an ellipse and a parabolic segment; he discovered the surface area of a cone and a sphere as well as the volume of a sphere, a spherical segment, and various rotating bodies and their segments. Archimedes studied the properties of the so-called Archimedean spiral. He constructed the tangent to this spiral and found the area of its pitch. Here Archimedes was a predecessor of the use of the methods of differential calculus. Archimedes also examined one problem of the isoperimetric type. In the course of his researches he discovered the sum of an infinite geometric progression with the ratio 1/4; this was the first appearance in mathematics of an infinite series. In studying one problem that led to a cubic equation, Archimedes elucidated the role of the characteristic that later came to be called the discriminant. The formula for determining the area of a triangle by means of its three sides (incorrectly called Hero’s formula) is attributed to him. Archimedes provided a theory (not completely comprehensive) of semiregular convex polyhedrons (Archimedean bodies). Especially important is the Axiom of Archimedes: of unequal segments, the smallest when repeated a sufficient number of times will surpass the largest. This axiom defines the so-called Archimedean principle of ordering, which plays an important role in modern mathematics. Archimedes constructed a number system that allowed him to note and name extremely large numbers. With great exactness he calculated the value of the number ir and demonstrated its limits of error as follows: 310/71<π<31/7.
Mechanics was one of Archimedes’ constant interests. In one of his first works he studied the distribution of loads on a beam between its supports. Archimedes defined the concept of a body’s center of gravity. In particular, using integration methods, he discovered the position of the center of gravity of various figures and bodies. Archimedes provided the mathematical derivation for the laws of the fulcrum. To him is attributed the proud phrase: “Give me a place to stand, and I will move the earth.” Archimedes laid the foundations of hydrostatics. He formulated the basic propositions of this discipline, including the well-known Law of Archimedes. Archimedes’ last work was devoted to a study of the equilibrium of floating bodies. Here he distinguished stable positions of equilibrium. Archimedes invented a water-raising mechanism, the so-called Archimedes’ screw, which was the prototype of propellers for ships and airplanes. The story is told that the solution to the problem of determining the amounts of gold and silver in a crown that had been given to Hieron was discovered by Archimedes while he was stepping into a bath, whereupon he ran home naked, shouting “Eureka!” (“I have found it!”). Archimedes also studied astronomy. He constructed a device for determining the visible (angular) diameter of the sun, and he found the value of this angle with amazing precision. Here Archimedes introduced a correction owing to the size of the pupil of the eye. He was the first to conduct studies relating to the center of the earth. Finally, Archimedes constructed a celestial sphere—a mechanical device on which it was possible to observe planetary motions, lunar phases, and solar and lunar eclipses.
S. B. STECHKIN