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Mathematician David Joyce knows that the study of geometry has a lot to offer students who will never become mathematicians. He has used the internet to make available to a wide audience a new translation of the classic text on geometry, Euclid's Elements. |
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Angles, lines and logic
Professor David Joyce's research
Star Trek's science officer Mr. Spock is greatly admired by fans throughout the world for his ability to solve problems using logic. But according to Clark mathematics professor David Joyce, U.S. high school students are at a disadvantage when it comes to learning deductive reasoning, an important branch of logic. He maintains that geometry, one of the most important subjects for teaching deductive reasoning, is often poorly taught in U.S. high schools.
Using the web to make geometry accessible
In an effort to make the principles and beauty of geometry more accessible to a wide audience, Joyce is creating an online version of one of the most important books ever written about geometry, the Elements. The Elements was written about 300 B.C.E.* by the Greek mathematician Euclid to set forth the principles of geometry and other branches of mathematics. Joyce has complemented his new translation of the Elements with Java applets. By interacting with the applets on screen, viewers are able to demonstrate the principles of Euclidean geometry for themselves in a dynamic environment. Joyce hopes that this approach will encourage more people to study geometry.
What is geometry?
Webster's dictionary defines geometry as that branch of mathematics concerned with the measurement of and relationships between points, lines, angles, surfaces (such as triangles and rectangles) and solids (such as cubes and cylinders). In the Elements, Euclid sets forth a series of definitions (for example, parallel lines never intersect) and statements assumed to be true (which he calls common notions and postulates). An example of a postulate is the statement that all right angles are equal. Building on these definitions and accepted statements of fact, Euclid uses deductive reasoning to derive new knowledge, or propositions. For example, he shows how the sum of the internal angles of any triangle equals two right angles. (Another type of geometry, termed Non-Euclidean, was developed in the 19th century.)
Why is the study of geometry important?
Geometry is important for two reasons. First, a thorough understanding of the principles of geometry is important to a wide range of scientific and technological fields of study such as engineering, physics and cartography. Since ancient times surveyors have used geometric principles to map the earth's surface. Global positioning systems (GPS) incorporate geometric principles for a similar purpose. And you wouldn't want to drive over a bridge designed by an engineer with no understanding of geometry! In addition, an understanding of geometry can help improve your game of billiards or calculate the square footage of your dream house.
Second, the study of geometry sharpens thinking skills by teaching the process of deductive reasoning. We often use deductive reasoning unconsciously in our daily lives to make decisions and judgments about the world around us. In mathematics and the sciences, where nothing is considered true until it is proved, deductive reasoning is an important tool.
How does deductive reasoning work?
Deductive reasoning consists of statements accepted as fact (premises), and conclusions that can be drawn from them.
Here's a non-geometry example (also called a syllogism):
- Premise: U.S. Presidents must be at least 35 years old.
- Premise: Jane Smith is president of the U.S.
- Conclusion: Therefore Jane Smith must be at least 35.
Proofs in geometry are structured using deductive reasoning. For example:
- Premise: Figure ABC is a triangle
- Premise: The sum of the angles of a triangle equals 180 degrees.
- Premise: Angle A = 20 and Angle B = 40
- Conclusion: Angle C = 120 (180 - (20 + 40) = 120)
*BCE refers to "Before the Common Era". The "Common Era" starts with the birth of Christ.
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Additional Resources
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Euclid demonstrating principles of geometry. Detail from The School of Athens by Raphael. In the Vatican.
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Location of Alexandria, where Euclid lived and worked.
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