**Placing Math on a Timeline
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The study of numbers has been both practical and sublime for millennia.

Without numbers it would be impossible to set a clock, keep score, or create a symphony. If numbers had not been needed the civilizations of ancient Mesopotamia, Egypt and China would not have felt it necessary to invent counting systems, which they then applied to their commerce and government. An early appreciation for the principles of geometry helped the Egyptians construct the pyramids and accurately record their boundaries.

By the sixth century, B.C., the Greeks took the practical math that they had learned from the Babylonians and Egyptians and ventured into more abstract investigations. The Greek philosopher Pythagoras and his disciples proposed a theorem, for instance, that showed the mathematical relationship among the three sides of a right triangle. Another Greek, Euclid, was the first to suggest that geometry possessed a single set of logical rules. Archimedes laid the conceptual groundwork for integral calculus in the third century, B.C., and the celebrated astronomer, Ptolemy, played a leading role in developing trigonometry.

The Romans largely contented themselves with the use of math in solving practical problems, but the more ethereal inquiries into the nature of numbers championed by the Greeks were taken up by Islamic thinkers in the 9th and 10th centuries. One of them, an astronomer named Muhammad ibn Musa al-Khwarizimi, laid many of the foundations for algebra.

Beginning in the 11th century, Islamic advances in mathematics gradually made their way into Europe. But it was not until the Renaissance in the 15th century that Europeans contributed to the breakthrough, with astronomers such as Nicolaus Copernicus, Galileo Galilei, and Johannes Kepler making major contributions.

Working independently, Sir Isaac Newton, an Englishman, and Baron Gotfried Wilhem von Leibniz invented calculus in the 1680s, which effectively ushered in the modern age of mathematics.

Following is a ready reference to many of the most commonly used mathematical concepts and operations.

*An early appreciation for the principles of geometry helped the Egyptians construct the pyramids and accurately record their boundaries. Photo: © Megan Jorgensen*