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The study of lines, angles, shapes, and patterns is the very essence of what we mean when we say “geometry.” The word itself comes to us from the Greek words meaning “the measure of the earth” because the ancient Greeks used it to calculate the measurements of shapes derived from the natural world, such as circles, squares, triangles, and solids.
But the story of geometry goes back even farther than ancient Greece. The earliest known geometers were the ancient Egyptians and Babylonians (c. 3000 BCE), who worked out many geometric concepts, including how to find the volume of a pyramid. Half a world away, the early Indian civilization of the Indus Valley (c. 3000 BCE) used geometry to lay out cities like Mohenjo-Daro in a perfect right-angled grid (orthogonal plan), a type of street layout still used in many cities around the world. Ancient India (c. 1500 BCE) also discovered the properties of geometric shapes, learned about the Pythagorean Theorem (before the Greek Pythagoras), and even calculated pi to two decimal places.
The ancient Greeks developed geometry largely independently of the other ancient cultures, laying down the principles we still use today in our studies of geometry. Probably the most famous of the early Greek geometers was Pythagoras (582-496 BCE), who was the first to provide a deductive proof for why the squares of the two shorter sides of a triangle equals the square of its hypotenuse, that is, a2 + b2 = c2. This became known as the Pythagorean Theorem in his honor, and it is a staple of every high school math textbook. Many centuries later, the geometer Euclid (c. 325-265 BCE) laid down the five postulates, or axioms, on which all later geometry builds:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All Right Angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough. [1]
Interestingly, the fifth postulate is the only one that cannot be conclusively proved, and a several fields of geometry called “non-Euclidean” are based on the proposition that postulate five is not true. Examples include spherical, elliptical, and hyperbolic geometries.
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[1] “Euclid’s Postulates,” Harvard University, 1997, <http://www.math.harvard.edu/~ctm/home/text/class/harvard/113/97/html/euclid.html> (17 September 2006).
