### ANCIENT PERIOD (3000 B.C. to 260 A.D.)

#### A. Number Systems and Arithmetic

- Development of numeration systems
- Creation of arithmetic techniques, lookup tables, the abacus and calculation tools.

#### B. Practical Measurement, Geometry and Astronomy

- Measurement devices to quantify distance, volume, area, and time
- Calendars invented to predict seasons and astronomical events
- Geometrical forms and patterns appear in and architecture

#### THE BABYLONIAN NUMERALS

The Babylonian civilization in Mesopotamia replaced the Sumerian civilization and the Akkadian civilization. Certainly in terms of their number system the Babylonians inherited ideas from the Sumerians and from the Akkadians. From the number systems of these earlier peoples came the base of 60, that is the sexagesimal system.

### GREEK PERIOD (600 B.C. to 450 a.d.)

#### A. Greek Logic and Philosophy

- Schools of Logic, Science, and Mathematics are established
- Greek Philosophers promote logical, rational explanations of natural phenomena.

#### B. Euclidean Geometry

#### MATHEMATICS AND GREEK PHILOSOPHY

Greek Philosopher viewed the universe in mathematical terms. Plato described the five elements that form the world and relate them to the five regular polyhedra.

#### ARCHIMEDES AND THE CROWN

### Hindu-Arabian Period (200 B.C. to 1250 A.D. )

#### A. Development and Spread of Hindu-Arabic Numbers

- A numeration system using base 10, positional notation, the zero symbol and powerful arithmetic techniques is developed by the Hindus, approx. 150 B.C. to 800 A.D.

#### B. Preservation of Greek Mathematics

- Arab scholars copied and studied Greek mathematical works, principally Baghdad.

#### C. Development of Algebra and Trigonometry

- Arab mathematicians find methods of solution for quadratic, cubic and higher degree polynomial equations.

#### ISLAMIC ASTRONOMY AND SCIENCE

### Period of Transmission (1000 AD – 1500 AD)

#### A. Discovery of Greek and Hindu-Arab Mathematics

- Greek mathematics texts are translated from Arabic into Latin; Greek ideas about logic, geometrical reasoning, and a rational view of the world are re-discovered.

#### B. Spread of the Hindu-Arabic Numeration System

- Pen and paper arithmetic algorithms based on Hindu-Arabic numerals replace the use the abacus.

#### LEONARDO OF PISA

### Early Modern Period (1450 A.D. – 1800 A.D.)

#### A. Trigonometry and Logarithms

- Publication of precise trigonometry tables, improvement of surveying methods using trigonometry, and mathematical analysis of trigonometric relationships.

#### B. Symbolic Algebra and Analytic Geometry

- Development of symbolic algebra, principally by the French mathematicians Viete and Descartes

#### C. Creation of the Calculus

- Calculus co-invented by Isaac Newton and Gottfried Leibniz. Major ideas of the calculus expanded and refined by others, especially the Bernoulli family and Leonhard Euler.

### Modern Period (1800 A.D. – Present)

#### A. Non-Euclidean Geometry

- Gauss, Lobachevsky, Riemann and others develop alternatives to Euclidean geometry in the 19th century

#### B. Set Theory

- Set theory used as a theoretical foundation for all of mathematics

#### C. Statistics and Probability

- Theories of probability and statistics are developed to solve numerous practical applications, such as weather prediction, polls, medical studies.

#### D. Computers

- Development of electronic computer hardware and software solves many previously unsolvable problems

#### E. Mathematics as a World-Wide Language

- The Hindu-Arabic numeration system and a common set of mathematical symbols are used and understood throughout the world.