David and Elise Price A Mathematical Travelogue Session: S084
Mathematics as a subject of systematic inquiry originated about 600 B.C. in Miletus, an ancient Greek city-state located in the southwestern part of modern Turkey. The Egyptian and Mesopotamian civilizations made some significant mathematical progress, but their haphazard approach led only to isolated results with little theoretical coherence. After studying in Egypt and Mesopotamia, Thales, a resident of Miletus, returned there and launched Greek geometry by establishing five results:
1. A circle is bisected by a diameter; 2. The base angles of an isosceles triangle are equal; 3. Vertical angles formed by intersecting lines are equal; 4. If two angles and a side of one triangle are equal respectively to two angles and a side of another triangle, then the triangles are congruent; 5. An angle inscribed in a semicircle is a right angle.
In particular, this last assertion is known as the Theorem of Thales. Although at least some of these propositions were probably already known, Thales is credited with being the first person to provide some sort of justification for them. Because he introduced the concept of proof into mathematics, he is often hailed as the first mathematician.
In addition to his work in mathematics, Thales is even more famous as the father of Greek philosophy. The nature of change and multiplicity was the fundamental question that concerned the thinkers of the time. Prior to Thales, attempts to explain these phenomena had always appealed to mythology; his revolutionary advance was to offer an explanation in natural terms. He held that the universe was composed of one basic stuff, namely water, and that change was the process by which it became the multitude of entities we experience.
Thales had a student named Anaximander who raised an objection to his teacher’s theory. Although some things may be forms of water, he said, how can everything be? In particular, how can objects with properties opposite to those of water, such as hotness and dryness, be representations of water?
Anaximenes, another Milesian, offered an answer to Anaximander’s question. He proposed that the fundamental stuff was air, not water, and that it generates the universe in all of its aspects by undergoing quantitative variation. In particular, this could explain the appearance of apparently contradictory characteristics, as when a sober person becomes inebriated by consuming a sufficient quantity of alcohol. Although Anaximenes’ explanation is inadequate to us, it was one of the first suggestions that nature could be explained in mathematical terms.
Pythagoras was born on the island of Samos, which is located about fifty miles north of Miletus just off of the coast of Turkey. He also studied in Egypt and Mesopotamia and then established a school at Croton in southern Italy. Like Thales, he was a philosopher, and his fascination with mathematics stemmed from his philosophical beliefs. Whereas Thales sought to explain the universe in terms of water, Pythagoras’ fundamental principle was that everything is number, by which he meant natural number. By studying the properties of numbers, Pythagoras and his school thought that they were unlocking the secrets of the universe, and they inaugurated the branch of mathematics that we today call number theory.
Of course, the Pythagoreans are best known for two developments: proving the Pythagorean Theorem and consequently discovering irrational numbers. Their failure to explain incommensurability undermined their belief in the fundamentality of numbers and led to a sharp division between arithmetic and geometry, a breach that did not begin to close until the development of analytic geometry in the seventeenth century. When dealing with quantities that could be irrational, the Pythagoreans represented them geometrically as lengths of line segments without assigning numerical values to the lengths. By using geometrical arguments, an approach known as geometrical algebra, they obtained a number of formulas, such as the result of squaring a binomial, which we today establish by purely algebraic means.
Despite the Pythagoreans’ inability to provide a satisfactory explanation of irrational numbers, they had a fundamental impact on the future of mathematics. Many of their results appear in Euclid’s Elements, and their quantitative approach to the study of music, astronomy, and medicine promoted the belief that nature could be explained mathematically. When combined with the view that nature operates uniformly by mechanical law, the Pythagorean approach played a pivotal role in launching the Scientific Revolution during the Renaissance.
Hippocrates (not to be confused with the father of medicine) was born on the island of Khios, also located in the eastern Mediterranean. He is considered to be the foremost Greek mathematician of the fifth century B.C., in large part because he was the first to calculate a curvilinear area in terms of a rectilinear one. He also introduced the method of indirect proof and arranged theorems in order so that later ones could be proved on the basis of earlier ones, thereby foreshadowing the axiomatic method. Books III and IV of Euclid’s Elements are based on Hippocrates’ work.
Delos, another island in the eastern Mediterranean, lies just south of Mykonos and according to Greek mythology, was the birthplace of Apollo. When the Peloponnesian War began in 431 B.C., the Athenian leader, Pericles, moved residents of the countryside into Athens to protect them from the Spartans. Shortly thereafter, however, the crowded conditions in the city led to a plague that claimed numerous lives, including Pericles himself. There was an oracle to Apollo at Delos, and a committee of Athenian citizens traveled there to seek its guidance on how to end the plague. From the cubical altar on which it sat, the oracle proclaimed that the Athenians should build an altar twice as large. When the citizens returned to Athens, they doubled the side of the Delian altar, and thus constructed one eight times the original size. When the plague did not abate, the problem of doubling the cube, one of the three famous construction problems of classical Greek mathematics, was born.
Zeno lived at Elea in southern Italy, not far from Croton, where Pythagoras established his school. He was a student of the philosopher, Parmenides, who held that change is an illusion and only permanence exists. To support Parmenides’ thesis, Zeno constructed his famous paradoxes by exploiting the Pythagorean view that numbers are extended points. The inability of Zeno’s contemporaries to refute his reasoning, together with the discovery of incommensurable magnitudes, turned Greek mathematics away from arithmetic and toward geometry as its primary focus.
In approximately 388 B.C., Plato founded the Academy in Athens as the first university in the Western world. His strong interest in mathematics was fueled by Pythagorean doctrines and attracted the leading mathematicians of the day to study and work there. Foremost among these was Eudoxus, whose achievements included his theory of proportion, which resolved the crisis precipitated by the Pythagorean discovery of incommensurable magnitudes, the method of exhaustion, which eventually became the basis for integral calculus, and his explanation of planetary motion, which attempted to rationalize the apparently random motions of the planets. Although Eudoxus’ approach to the solar system was geocentric in nature, it was the first major attempt to apply mathematics to astronomy.
Plato himself was not a mathematician, but his philosophy of mathematics, that mathematicians study forms, not their concrete representations, has been profoundly influential. Another member of the Academy, Theaetetus, proved that there are exactly five regular polyhedra, and Plato took such a strong interest in these figures that they are frequently referred to as the Platonic solids. He associated four of them with air, earth, fire, and water, the four elements of Greek science, and he identified the fifth with the universe as a whole. To understand their properties, he advocated the study of solid geometry, which in his view had been neglected.
The leading student at the Academy was Aristotle, who later established his own school, the Lyceum, also in Athens. Like Plato, he was not a mathematician, but he had a strong interest in the subject and made fundamental contributions to its foundations. He formulated the laws of logic and identified the axiomatic method as the proper means of organizing a body of knowledge. In addition, he was the first to define the concept of a specific science. Previously, all knowledge had been considered to be philosophy; subdivisions of learning had not been recognized. Aristotle was the first to realize that each subject has fundamental truths that lie at its foundation and that provide the context for its development. Because of his insight, it became possible for scholars, using the laws of logic as their common base, to specialize in various aspects of intellectual inquiry.
In 336 B.C., Alexander the Great assumed control of the army of Macedonia and quickly conquered the known world. With his death in 323 B.C., the Hellenic period of Greek history ended, and the Hellenistic era began. The intellectual center of antiquity shifted from Athens to Alexandria when Alexander’s first two successors in Egypt, Ptolemy I and Ptolemy II, built a museum and a library respectively to attract scholars throughout his empire. One of these was Euclid, who wrote his Elements while in Alexandria and thereby demonstrated the epistemological efficacy of the axiomatic method. Euclid’s accomplishment paved the way for Archimedes, whose vast range of achievements included the systematic use of the method of exhaustion to calculate a wide variety of areas and volumes, the discovery of the law of levers, and the founding of mathematical physics by formulating his principle of buoyancy. Another major mathematical figure in Alexandria was Apollonius, whose work on conic sections provided the first unified treatment of the subject. He showed that all of the conics could be generated by passing a plane through a double-napped cone at various angles. He coined the terms, parabola, ellipse, and hyperbola, that we use today, and he was the first to recognize that a hyperbola has two branches. There are hints of analytic geometry in his work in that he seems to have understood that curves can be associated with equations. However, he did not grasp that equations can be associated with curves, and it would be incorrect to call him the true father of analytic geometry.
In the second century B.C., Hipparchus, who worked at both Alexandria and Rhodes, established trigonometry as a distinct branch of mathematics. He also did extensive work on a scheme of planetary motion based on epicycles, which superseded the Eudoxian approach. Claudius Ptolemy extended Hipparchus’ work and completed the development of the geocentric theory of the solar system that dominated astronomy until it was displaced by Copernicus’ heliocentric theory.
The school at Alexandria lasted for several more centuries and included mathematicians such as Heron, Diophantus, and Pappus. Heron’s work focused on applications of the theoretical advances of his predecessors. He used similar triangles to devise a method for constructing a straight tunnel through a mountain and to formulate an early version of the parallelogram law for the addition of vectors. The famous formula for the area of a triangle that bears his name was known and presumably proved by Archimedes, but Heron’s proof is the first one to have survived. One of its significant features is that it involves the multiplication of four factors. Following the Pythagorean discovery of irrational numbers, Greek mathematicians restricted products to two or three factors only and conceived of them as representing areas or volumes. Heron’s formula was a step toward treating arithmetic as a branch of mathematics independent of geometry. He also developed an approximation technique for the calculation of square roots. It is not certain whether he originated this method or based it on Babylonian mathematics, but it was needed to use his formula for the area of a triangle.
Just as Heron did not base his arithmetic work on geometry, so Diophantus treated algebraic questions unrelated to a geometric context. He worked with powers greater than three, and he introduced algebraic abbreviations in place of the verbal approach to problems used by earlier mathematicians. He did not develop the fully symbolic notation that we use today, but his innovation, called syncopated algebra, was an important advance toward it. His efforts focused primarily on the solving of indeterminate equations, an area known today as Diophantine analysis. Although his work had no deductive structure and did not use general methods, one of his results motivated Fermat to state his famous Last Theorem.
Pappus was the last major geometer of the Alexandrian period. Among his achievements was a generalization of the Pythagorean Theorem to parallelograms constructed on the sides of any triangle. He was the first to state the focus-directrix property of conic sections, and he considered the possibility of studying curves corresponding to equations of degree greater than three. Although he did not pursue this idea, it foreshadowed the development of analytic geometry.
Alexandria declined in importance as a center of scholarship and intellectual advancement with the rise of the Roman Empire, which placed little significance on theoretical advances in mathematics and science. Its last contribution consisted of commentaries on Diophantus, Ptolemy, and Apollonis written by Hypatia. With her brutal death in 415 A.D., its influence came to an end.