# Development of Geometry in Different Cultures

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Geometry is the study of shapes and how they relate to each other, and people have been trying to understand it for thousands of years. In this lesson, learn about how geometry developed independently in several ancient cultures.

## Early History of Geometry

Geometry is all around us - from the repeating pattern of the Moon's orbit to the complex shapes found in a spiderweb. Ancient people certainly saw these things and many more, and came up with rules to measure and explain what they saw. Geometry is the branch of mathematics that studies shapes and their relationships to each other.

More than 5000 years ago, in the valley of the Nile river, we know that Egyptian scholars were already using the principles of geometry to survey land and construct complex structures like the great pyramids.

Even before this time, people in various parts of the world used basic geometrical ideas to map their lands and construct their homes. In fact, the word 'geometry' comes from the Greek word geometrein, meaning Earth measuring. Although it has evolved to include many types of more abstract measurements, geometry arose from these early measurement systems.

Because the study of geometry arose from the simple observations and measurements, it developed independently in many cultures in the ancient world. Some of the most famous early forms of geometry were developed in Greece, India, and China.

## Greek Geometry

While the Egyptians and other ancient cultures developed many useful geometry rules, they did not attempt to expand their knowledge of geometry. Later, Greek philosophers and mathematicians like Thales, Pythagoras, Euclid, and Archimedes, would take on this challenge.

Thales, who lived in the 5th century B.C.E, was the first person to use deductive reasoning to prove mathematical relationships. Because of this, he made many contributions to the development of geometry.

Pythagoras, who lived at about the same time, expanded on the ideas of Thales. Among other things, he proved that the three interior angles of a triangle will always add to give 180 degrees. He also proved the famous theorem that bears his name even now, the Pythagorean theorem, which demonstrates the relationship between the sides of a right triangle and the hypotenuse.

In the 3rd century B.C.E, Euclid of Alexandria wrote a series of books known as The Elements of Geometry or just The Elements. In this revolutionary work, he laid out many of the axioms of geometry that we still use today, such as the principle that any two points can be joined by a straight line, and all right angles are equal to each other.

Euclid's books were so popular that The Elements became the most important mathematical textbook throughout the Western world for the next 2000 years. Even now, we still call the geometry of flat surfaces Euclidean geometry because it was first explained by Euclid!

Archimedes, who lived in the 2nd century B.C.E., was one of the most important scientists, inventors, and mathematicians who ever lived. Among his many contributions to mathematics, he invented an early form of coordinate geometry.

## Indian Geometry

Centuries before the axioms of Euclidean geometry were proven and recorded by the ancient Greeks, people were using geometry to construct elaborate ceremonial altars to the Hindu gods throughout the Indian subcontinent.

Instructions used to construct these alters were recorded in a series of books called the Sulba Sutras. While developing processes for more and more complex altar construction, the writers of the Sulba Sutras developed a method for calculating the mathematical constant pi, estimated the square root of two, and wrote down the earliest known statement of what would later come to be known as the Pythagorean theorem hundreds of years before Pythagoras was even born!

The Sulba Sutras also describe ways to create various geometric shapes with the same area. For example, using these geometrical principles, it was possible to make a circle, square, and rectangle that each had the same area.

## Chinese Geometry

We know that geometry had been developed in China at least by 330 B.C.E, when the oldest existing Chinese book about geometry, the Mo Jing, was written. Because the mathematical principles described in the Mo Jing were already quite advanced, many modern historians believe that there may have been earlier works that have been lost.

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