Overview: Mathematics 2000 B.c. to A.d. 699

The basic notions of number and magnitude can be seen in markings on cave walls and primitive tools. However, it took many thousands of years for the first use of quantities to evolve into the abstract concept of numbers as we use them today.

The cultures of the Mesopotamian region had developed the use of written numbers by about 1800 B.C. The Babylonians, in particular, developed a sophisticated number system, estimated π with some accuracy, used fractions, and solved complex quadratic equations. Mesopotamian mathematics was spread far and wide by the many conquerors of the region. Their ideas influenced other cultures from Europe to China.

Chinese mathematics also developed at an early date. The earliest surviving works, dating from about 300 B.C., contain detailed astronomical calculations, as well as surveying, agricultural, and other practical math problems. The Chinese were interested in number patterns, and the magic square was one of their innovations.

The oldest surviving Egyptian mathematical documents are the Moscow and Rhind Papyri. The Moscow Papyrus, dating from about 1890 B.C., contains examples of geometry that have great practical use for the calculation of areas and volumes. The Rhind (or Ahmes) Papyrus dates from around 1650 B.C., and contains copied information from earlier sources. It includes many examples of fractions, and is written in the form of arithmetic problems that relate to practical concerns. The text mainly discusses addition and subtraction, but also deals with a multiplication shortcut and proportions.

The Rhind Papyrus also contains the earliest example of an algebraic problem. Egyptian mathematics was absorbed by the Greeks, who revered the Egyptians as the fathers of mathematics. The first book on algebra was written by Diophantus of Alexandria (third century A.D.), a Roman scholar writing in the Greek tradition. Algebra was a minor field until the later work of Arab scholar Muhammad ibn-Musa (800?-847), also known as al-Khwarizmi. The term algorithm is derived from his name.

Numbers were often studied for their mystical properties, as well as their mathematical importance. Pythagoras (580?-500? B.C.) and his many followers were particularly interested in numbers that had special properties, such as the prime numbers. They also discovered the so-called perfect and amicable numbers. While Pythagoras's mystical teachings had a great impact on later mathematics, he is better remembered for the theorem regarding right-angled triangles that bears his name. Indeed geometry was the major preoccupation of ancient mathematics.

Greek mathematicians, such as Thales of Miletus (624?-548? B.C.) and Pythagoras, traveled to Egypt and Babylonia, where they learned basic geometry. The Greeks transformed the subject by insisting on deductive proofs. Thales is credited with the earliest geometrical theorems, and later thinkers used these to expand the field. Euclid (330?-260? B.C.) collected together all the theorems of previous mathematicians into his Elements, which was to remain the standard text in geometry for over 2,000 years. Later geometers, such as Archimedes (287?-212 B.C.) and Apollonius (262?-190? B.C.), continued to add new theorems.

There were three so-called Great Problems that dominated Greek mathematics and occupied the minds of many mathematicians for centuries to come. Doubling the cube was the most famous problem of its time, and one that many Greek thinkers attempted to solve. Hippocrates of Chios (fl. c. 460 B.C.) made some important early steps, and there was an elegant, if somewhat complex, solution proposed by Archytas (428?-350? B.C.). However, it was the work of Menaechmus (380?-320? B.C.) that is most remembered, for not only did he give two solutions to the problem, he also did foundational work on conic sections as a result.

However, the other two Great Problems, squaring the circle and trisecting an angle are both unsolvable problems. At first sight they seem tantalizingly possible, and squaring the circle, in particular, was to occupy the minds of many mathematicians for almost four millennia. It is impossible to construct a circle and a square of the same area because the value of π is an irrational number. Attempts to solve this, while futile, did lead to some very accurate approximations of π.

Trisecting an angle seems like an easy problem, and indeed for certain angles it is. However, there can be no method for trisecting an arbitrary angle with a ruler and compass. Even today many amateur mathematicians attempt to solve this unsolvable problem.

Practical problems in astronomy inspired the study of trigonometry. Indeed, many early mathematicians were most interested in astronomy. The first major work on trigonometric functions was by the Greek mathematician Hipparchus (180?-125? B.C.), using many Babylonian ideas. Eratosthenes (276?-194? B.C.) was able to use the rules of trigonometry to calculate the Earth's circumference with surprising accuracy. Later work by Menelaus (70?-130?), and particularly Ptolemy (100?-170?) developed the field of trigonometry and widened its practical applications.

Mathematics was also used to pose and solve logical puzzles. The Greek thinker Parmenides (515?-445? B.C.) and his pupil Zeno of Elea (490?-425? B.C.) raised a number of interesting logical puzzles with mathematical implications. The paradoxes of Zeno were concerned with ideas of infinity, something that worried many Greek philosophers. Zeno used mathematical logic to prove that motion was impossible. This was obviously not the case, but his mathematical proofs seemed unshakable, thereby causing many Greek thinkers to reevaluate their ideas of the world.

While the Roman world inherited much of Greek and Egyptian learning, they did not develop many mathematical fields. Roman numerals, which evolved from the early Greek and Egyptian number systems, represent each number by a unique collection of symbols. They remained the dominant number system of Europe for hundreds of years, and are still used in some situations today. However, the Roman system results in very long collections of symbols for some numbers, and made complex calculations very difficult.

One of the earliest mathematical tools was the counting board, a flat surface with marked lines on which counters were moved to represent numbers. Counting boards developed into the abacus, a frame with rods and sliding beads. Such devices made calculations quicker and easier, but also limited the development of mathematical theory, as the users of such devices were more interested in the results than the processes involved.

Counting boards and abacuses use a place-value system, where a small number of symbols (counters) are repeatedly used in different positions to represent all numbers. In our decimal system, based on 10, the first column represents the ones, the next the tens, the third is the hundreds, and so on. Some cultures developed written number systems using place values, such as the Babylonians, who used 60 as their base, and the Mayas of Central America, who used 20. One advantage of place-value notation is the ease with which multiplication and division can be carried out.

Place-value systems also generated a need for a symbol to represent nothing, the zero. When recording the results of a calculation from a counting board, there often arose confusion when the result had nothing in a certain place value, such as the number 203, which has no tens. In Roman notation there is no problem—it is CCIII—and so the need for the zero did not arise in the West. However, place-value notation required a placeholder in the empty spot, and so the zero was invented independently by the Babylonians, Maya, and Indians.

In India the zero became more than just a placeholder, it became a number in its own right. This meant that written calculation became just as easy to understand as the counting board, leading to a new appreciation of the rules of calculation. The Hindu zero also made it easier to deal with larger numbers. The idea of zero was to cause many philosophical headaches where ever it traveled, for the concept of nothing was frightening for many, and was related to the even more worrying notion of infinity.

The majority of ancient learning was lost to Europe after the collapse of Rome, but was studied, copied, and developed in the Arab world. Arab mathematicians absorbed the knowledge of many nearby cultures, including the Babylonians and the Hindus. This potent mix of mathematical learning was distilled, refined, and eventually transmitted to Europe from the time of the Crusades on. However, it took many centuries to break down the resistance to change encountered in Europe. When Eastern learning was at last accepted, the resulting revolution in mathematics was to redefine the field and lead to the mathematics of the modern era.