Mathematics covers a broad range of operations from simple counting to complex theories and calculations. Among other things, it includes algebra, a system of examining the relationships among numbers; and geometry, which deals with shapes, areas, and volumes of space. The Greeks knew about all of these aspects of mathematics. The works of Greek mathematicians are the oldest known writings on mathematical subjects.

The Greeks, however, were not the first people to develop a sophisticated understanding of mathematics. That honor goes to the Egyptians and to the Babylonians, who developed numerical systems early in their history. The Egyptians created the decimal system (the counting system based on 10) and were pioneers of geometry. By about 1700 B.C., the Babylonians had created their own counting system (based on groups of 60) and had surpassed the Egyptians in algebra and basic geometry. Many modern scholars believe that much of the Babylonians’ mathematical knowledge made its way to the Greek world, although they do not know exactly when or how it did so.

We know very little about the origins of mathematics among the Greeks. According to ancient Greek historians, mathematics arose as a branch of philosophy* concerned with speculations about the meaning and relationships of numbers and forms. Tradition suggests that two early Greek mathematician-philosophers of the 600s and 500s B.C., Thales of Miletus and Pythagoras, were said to have introduced geometry to the Greeks. Neither of them left any writings, however, and modern researchers are unable to determine the extent of their mathematical knowledge, or what and whom they taught. The first person to write a book about mathematics was Hippocrates of Chios, who was active in Athens in the mid-400s B.C. Several generations of mathematicians perpetuated his work. Only fragments of their work have survived, mostly in the form of references in later writings.

Around 300 B.C., Euclid summarized Greek knowledge of mathematics in a volume called Elements of Geometry, which is the oldest surviving mathematical textbook. Euclid’s work—and Greece’s single greatest contribution to mathematics—was based on the proof. In mathematics, a proof is a series of logical steps that prove, or demonstrate, that a statement is true. The statement to be proven is called an axiom, or premise. Euclid’s mathematics, and Greek mathematics in general, introduced deductive reasoning, which became one of the principal Greek contributions to philosophy and science. Deductive reasoning is an orderly system of thought in which each step in a particular proof is firmly based on previously proven conclusions.

Greek mathematicians of the 200s B.C. produced several significant works on mathematics, especially geometry. In the centuries that followed, Greek thinkers applied Euclid’s method of deductive reasoning to various scientific challenges, such as measuring the size of the earth, creating more accurate sundials, and drawing maps that accurately represented the surface of the earth. Scholars such as Aristotle, Archimedes, Eratosthenes, and Ptolemy applied mathematical principles to astronomy, geography, and practical mechanics.

* philosophy study of ideas, including science

**A ROUND WORLD**

**The ancient Greeks knew that the world is round. In a work called On the Heavens, Aristotle listed reasons to support the idea that the earth is a sphere. For example, he pointed out that the earth's shadow, cast across the face of the moon during lunar eclipses, is dearly the shadow of a round object. Some Greek mathematicians tried to measure the size of the spherical earth. Eratosthenes may have come within a few hundred miles of an accurate measurement, but there is no way to know for sure because he gave his result in stadia, units of distance that had at least three different values in the ancient world.**

The works of Greek mathematicians had little influence on the early Christian world. Translated into Arabic, though, they helped fuel a great burst of intellectual activity in the Islamic world after the A.D. 800s. During the Renaissance*, when Europeans “rediscovered” the ancient Greek and Roman civilizations, the works of Euclid and Aristotle formed the basis for mathematical study for many years. (See also Philosophy, Greek and Hellenistic; Science.)

* Renaissance period of the rebirth of interest in classical art, literature, and learning that occurred in Europe from the late 1300s through the 1500s