Greek mathematics

Summary: Greece provided the deductive foundation for many mathematical concepts.

Historians of mathematics and ethnomathematicians have noted that we do not know what all early civilizations did in mathematics. From the evidence that is available, however, it seems that ancient Greece in the late half of the first millennium b.c.e. was the first known civilization to specifically study pure mathematics—mathematics for its own sake, mathematics as aesthetically beautiful. There are occasional examples of pure mathematics in earlier civilizations, notably mathematical proportions in art and design in Egypt and elsewhere, but the earlier peoples used mathematics mostly for practical applications, even if those applications related to religion and art.

Most of the earlier civilizations had subsistence economies, where successful life depended on success in producing food and shelter, so mathematical thinking was used to contribute to these ends. Life was difficult for most people and required full-time concentration, so there was little time for the relaxation that would allow contemplation of mathematical relationships as beauty. However, by 600 and 500 b.c.e., Greece had become prosperous, with strong markets and trade ties around the eastern Mediterranean. There was subsistence work to be done, but the upper-class elite did not have these responsibilities and could devote time to philosophy and learning for its own sake. The trade also brought ideas from other areas, and the open marketplaces encouraged the exchange of ideas and the defense of one’s own. These encounters set the stage for studying mathematics beyond the everyday uses and also for the idea of deduction to prove statements.

Early Greek Mathematicians

One of the earliest mathematicians known by name was Thales of Miletus (624–547 b.c.e.), (in modern Turkey). He was an early user of formal deduction in geometry and was known for demonstrating several basic geometric properties: that a diameter bisects a circle, that base angles of an isosceles triangle are equal, and that vertical angles formed by the intersection of lines are equal. He also used angle-side-angle and angle-angle-side triangle congruences and showed that an angle inscribed in a semicircle is always a right angle. In practical geometry, he recognized that the North Star (Polaris) could be used for navigation, and, most impressively, he is said to have predicted a solar eclipse in 585 b.c.e. (though some doubt this). He was also a businessman and bought oil-press mills when his predictions showed a good year for olives.

Pythagoras (572–497 b.c.e.) is more famous, and, for many, more interesting. After traveling as a young man, he settled in Crotona (in what is now southeastern Italy) and gathered followers in a secretive cultlike organization of number worshippers. They believed that whole numbers and ratios of whole numbers are central to everything—numbers rule the universe! They studied geometry, astronomy, and music, but linked all to numbers (including noticing how a plucked string sounds an octave higher when it is half as long, and that other common fractions of the length also make harmonic tones). Their worship led them to the beginnings of number theory as they studied odds and evens, prime numbers, and figurate numbers (numbers of objects arranged into squares, triangles, or other shapes). Some of the questions of number theory that they investigated remain as unsolved problems even in the early twenty-first century.

The most famous mathematics connected with Pythagoras and his group is the theorem of the relationship of the lengths of the sides and hypotenuse of right triangles. Others, notably the Egyptians and the Babylonians, also recognized this relationship, at least in simple cases such as the 3-4-5 triangle for the Egyptians, more such triples for the Babylonians, and, independently, the Chinese. However, the Pythagoreans were probably the first to prove the relationship in general, and hence, in Western mathematics, it is called the Pythagorean Theorem, a²+b²=h², where a and b are the lengths of the right triangle legs with the right angle between them, and h is the length of the hypotenuse across from the right angle. This theorem has been described as the first nonobvious theorem of mathematics.

The simplest example of the Pythagorean Theorem is a right triangle with each leg one unit long. This triangle has a hypotenuse of the square root of 2. Unfortunately for the whole-number-worshipping Pythagoreans, the square root of 2 can never be expressed as the ratio of any two whole numbers. Today, it is called an “irrational number,” with an infinite, nonrepeating decimal expansion. An irrational number is contrary to the beliefs of the Pythagoreans—such a serious discrepancy that they kept this result secret. More broadly, the issue of irrational numbers caused a crisis in Greek mathematics. Some have even credited this problem to the general shift of Greek mathematics from numbers to a basic geometry that does not use measurement. The geometry of the Greeks became one that allowed figures to be constructed using only a compass and an unmarked straightedge.

Three Construction Problems

Three construction problems challenged the Greeks and many others in later centuries. One was the task of constructing a square with exactly the same area as a given circle—the hope was that this would aid in finding areas of round shapes. This would require finding a way to construct a line √π units long. Another was to construct a cube of volume double that of a given cube, which would need a line of length the cube root of 2. The third problem asked for a trisection of a given angle—bisecting an angle was easy, but this asked for the angle to be cut into thirds. The problems were never solved by the Greeks, but their efforts led to interesting insights in geometry. The Greek mathematicians were redeemed in the nineteenth century when all three constructions were proved to be impossible, but there are still some skeptics who erroneously claim to have produced proofs for these constructions.

Deductive Reasoning and Euclid

This geometry and the use of deductive arguments became the standard not only of mathematics but also of clear thinking and logic. Plato’s Academy posted a sign that said only those with a knowledge of geometry could enter—deductive geometry was the prerequisite knowledge for philosophy, government, and critical thinking in all areas. Greek civilization greatly expanded under Alexander the Great late in the fourth century b.c.e., reaching as far east as modern Afghanistan and south into Egypt. The city of Alexandria was established at the mouth of the Nile and became a center of trade—and a scholarly center with the construction of the library (also called museum) of Alexandria.

One of the early leaders of the library was Euclid (c. 300 b.c.e.), a mathematician whose life is little known, but his work is one of the most published works in all of mathematics. Probably drawing on the work of earlier scholars, he set up an axiomatic, deductive structure of geometry that became the basis for much future mathematical research. He began with five postulates that mostly drew upon the rules of geometric construction, plus some fundamental obvious truths and some basic definitions. From these, he developed deductive proofs of more geometric properties.

From these early theorems, further deductions eventually led to a “tree” of proven statements, each traceable back to the original theorems. His book, The Elements, is said to have been published more than any book except the Bible, and remains the framework for the introductory study of formal geometry even today. His fifth postulate did not come from constructions and defined parallel lines, leading to the difficult use of infinity—noting that parallel lines would not even meet no matter how far they were extended. It seems Euclid himself was worried about the issue of infinity and hesitated using this postulate as long as possible. Two thousand years later, challenges and changes to the fifth postulate would lead to the development of non-Euclidean geometries in the nineteenth century.

Archimedes

Archimedes (287–212 b.c.e.) is often considered the greatest of the ancient Greek mathematicians and one of the greatest in all of history. Unlike many mathematicians, he was recognized even in his lifetime. His achievements are especially notable in that he worked in both pure and applied areas of mathematics. In pure mathematics, Archimedes came close to developing integral calculus more than 1800 years before Newton and Leibniz. He wanted to find ways to calculate areas and volumes of round shapes and used the idea of dividing the shapes into very small slices, much like the similar slices used to integrate areas and volumes in calculus. He found volumes of spheres, cones, and cylinders and discovered an interesting relationship when these shapes have the same diameter and height: the volumes of these special cones, spheres, and cylinders form a 1:2:3 ratio.

Also using calculus-like techniques, he found the value of π by inscribing and circumscribing regular polygons inside and outside a circle and then increasing the number of sides on the polygons so they would close in and estimate the circumference of the circle. He calculated the value of π to be between 3 1/7 and 3 10/71. To help handle large numbers, he greatly expanded the numeration system.

Archimedes lived in Syracuse on the island of Sicily, and his applied work often was related to his life there. He studied the mechanics of simple machines such as levers, pulleys, and screws. He was reputed to have used some of this knowledge to help the king repulse an invasion from the Romans. Once the king asked him to check the authenticity of gold in a crown. He knew he could compare densities of pure gold and an alloy, but to do so, he needed to know the volume of the very irregularly shaped crown. As he entered his bath, he noticed the water level rise to compensate for his own volume; from that he recognized that he could measure the volume of the crown from the amount of water it would displace. The story says he jumped out of the bath and ran through town naked shouting “Eureka!” (I have found it!) in his excitement at the discovery.

Although Archimedes had helped fight off the Romans, they returned when he was an old man. Legend says he refused to leave the geometry he was writing in the sand when a Roman soldier told him to go. At the refusal, the soldier killed him. In some sense, this is symbolic, in that not only did Archimedes die at the hand of a Roman soldier but much of the Greek civilization fell to the expanding Roman Empire. The Romans were good engineers and built a network of roads and aqueducts, but they mostly used existing mathematics and contributed little beyond the work of the Greeks.

Other Greek Mathematicians

However, across the Mediterranean Sea, Alexandria and its library did not fall. Following from Euclid, the Alexandria library continued to be a center for Greek mathematics that would continue even several centuries after the decline of the overall Greek civilization. Some of the work was in astronomy. As early as 200 b.c.e., Eratosthenes calculated the circumference of Earth fairly accurately (incidentally, also indicating that he knew the Earth was round) by comparing the angle of the sun at noon in Alexandria and at Cyrene and using geometrical comparisons to do the calculation.

Later, other Greek astronomers, notably Ptolemy (100–178 c.e.), found more measurements of the movements of the planets. Some of their work led to the erroneous belief that Earth was the center of the solar system, but other studies provided a sound mathematical basis for early astronomical research.

Three other names of mathematicians bring the story of ancient Greek mathematics to a close in the early centuries of the Common Era. Hero (also called Heron) in the first century designed a device that, if constructed, could have been the first steam engine, but it did not get built. He also found a remarkable formula for the area of a random triangle when only the lengths of the three sides (a, b, and c) are given:

the semiperimeter. Like the Pythagorean Theorem, this formula is considered one of the important early non-obvious theorems and is also useful in practical applications.

Diophantes, who lived in the mid-third century, has sometimes been called the “Father of Algebra.” He broke from the Greek interest in geometry and studied numerical problems with techniques that resemble later algebraic methods. He was especially interested in problems whose statements and results were all whole numbers, thus restricting the range of solutions but offering challenges that led to creative work.

Hypatia (370–415) was famous as a mathematics researcher and teacher in Alexandria. Notably, Hypatia is one of the earliest important women mathematicians known in history. Originally taught by her father, who was also a mathematician, Hypatia wrote commentaries and expansions on earlier Greek work, a common type of mathematical research of the time. She was also especially noted as a teacher. However, she inadvertently was caught up in the religious politics of her time and was captured and killed by a mob. Thus, two phases of Greek mathematics ended in tragic deaths: Archimedes at the hands of Roman soldiers approximately marked the end of Greece’s Golden Age in mathematics, while the mob killing of Hypatia came near the very end of Greek mathematical work.

Overall, Greek mathematics had continued for nearly 1000 years, providing an unequaled example for future mathematical work. The Greeks did important work in the applied areas but are especially recognized for laying the foundations for pure mathematics.

Bibliography

Boyer, Carl. A History of Mathematics. Hoboken, NJ: Wiley, 1991.

Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2005.

Eves, Howard. An Introduction to the History of Mathematics. New York: Saunders College Publishing, 1990.

Katz, Victor. A History of Mathematics: An Introduction. New York: Addison-Wesley, 2008.