Scott Phillips, age 14, of Louisville, Ky., for his question:
WHEN WERE GEOMETRY AND TRIGONOMETRY FIRST TAUGHT?
Both geometry and trigonometry have been taught to students since ancient times. Geometry, which is Greek, means "earth" and "to measure." It is a brand of mathematics that deals with the properties of space. Trigonometry is the branch of math that deals with the relationships between the sides and angles of triangles and with the properties and applications of the trigonometric functions of angles.
Geometry dates back before the dawn of history. Excavations of ancient cities show that land and buildings were carefully laid out. Egyptian architects and engineers used geometry to design and build huge temple and pyramids.
Historians believed that early Egyptian and Babylonian geometry dealt only with practical problems. It was probably not until 600 B.C. that Greek mathematicians started to develop and teach the reasoning and logic needed to prove the truth of mathematical statements.
Plato (427 347 B.C.) taught the method of forming a proof. Aristotle (384 322 B.C.) taught the difference between axioms and postulates. About 300 B.C. Euclid organized geometry as a single logical system.
Trigonometry grew out of attempts to study and describe the spheres within which the sun, moon, planets and stars were supposed to move. Two Greek astronomers, Hipparchus (100 B.C.) and Ptolemy (A.D. 100), made the first important advances in trigonometry.
Modern trigonometry developed after the Middle Ages. A German mathematician named Johann Muller (1436 1476) published one of the first systematic works on trigonometry. Then another German mathematician, Bartholomaus Pitiscus (1561 1613), published the first textbook of trigonometry.
In geometry, an angle consists of two lines extending from a point. Trigonometry uses the same definition, but adds that the angle formed by two lines can be measured.
Geometry is important as a school subject because it shows students how to develop ideas by logical reasoning, rather than by observation, description and measurement. Geometry is practical and already arranged in logical order.
We use geometry whenever we ask questions about the size, shape, volume or position of anything. Engineers and architects use geometric drawings in the plans of buildings and bridges. Auto designers use it when they figure out how to make pistons fit engine cylinders.
In daily life, we find hundreds of uses for geometry.
On an elementary level, trigonometry deals with triangles lying in a flat surface, or "plane," and on the surface of a sphere. We can speak of "plane trigonometry" and "spherical trigonometry."
Mathematicians have found that the values of trigonometric ratios have periods, or repeat again and again. On an advanced level, trigonometry involves the application of these periods of describing such things as the vibration of a violin string or the motion of a pendulum.