The Pythagorean Theorem

Pythagoras, a famous Greek Scholar, Mathematician, and Philosopher, formulated a proof for a theorem that is named for him – The Pythagorean Theorem. This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The Pythagorean Theorem for right-angled triangles likely was known long before the time of Pythagoras.  It was probably used by the ancient Egyptians to construct the pyramids.

The theorem is quite believable without rigorous proof to anyone willing to expend a modest effort in some experimentation. One method is to draw a number of right-angled triangles in as wide a variety as practicable and measure all of the sides. It will be determined that, for each triangle drawn, the square of the length of the side opposite the right angle is about equal to the sum of the lengths of the squares of the other two sides.

Another method requires the availability of a balance. For this more interesting experiment, construct a right-angled triangle and a square on each side using a piece of sheet metal or cardboard. Then cut out the three squares and weigh them on the balance. The square on the hypotenuse should balance the other two.

Contained within this book are some rigorous proofs, and some interesting perspectives regarding right angles and right-angled triangles. Doubtless, this theorem is one of the most useful concepts in mathematics.

Mathematical language is precise, so precise that it is often confusing to people unaccustomed to its forms. This precision of mathematics appears as pedantic or stilted to one who does not yet appreciate that it is essential to exact thinking, for exact thinking and exact language are efficient and complementary.

Mathematical style aims at brevity and formal perfection. It sometimes succeeds too well and sacrifices the clarity its precision seeks to guarantee. Suppose we wish to express in general terms the fact illustrated on the cover of this book. We might be tempted to express:

“We have a right triangle. If we construct two squares so that each square has a side equal to a side of the triangle, and if we construct a square that has a side equal to the hypotenuse of the triangle, then the area of the third square is equal to the sum of the areas of the first two.”

However, a mathematician would not consider expressing himself in that manner. He prefers:

“The square of the hypotenuse is equal to the sum of the squares of the other two sides.”

Mathematics is more than a method, an art, and a language. It is a body of knowledge with content that serves the physical and social scientist, the philosopher, the logician, and the artist; content that influences the presentation of statesmen and theologians; content that satisfies the curiosity of the person who surveys the heavens; and content that has shaped the course of modern history. Because of its usefulness, the interest in the concept expressed by the Pythagorean Theorem is perhaps greater than any other theorem in mathematics.