P l a n e G e o m e t r y
An Adventure in Language and Logic
THE PYTHAGOREAN THEOREM
Book I. Propositions 47 and 48
PYTHAGORAS was a teacher and philosopher who lived some 250 years before Euclid, in the 6th century B.C. The theorem that bears his name is about an equality of non-congruent areas; namely the squares that are drawn on each side of a right triangle.
Thus if ABC is a right triangle with the right angle at A, then the square drawn on BC opposite the right angle, is equal to the two squares together drawn on BA, AC.
That is, if it takes one can of paint to paint the square on BC, then it will also take exactly one can to paint the other two squares.
That is how the sides of a right triangle are related -- by the squares drawn on them -- and we can illustrate it with numbers.
The square drawn on the side opposite the right angle, 25, is equal to the squares on the sides that make the right angle: 9 + 16. Thus a triangle whose sides are 3-4-5 is right-angled.
That and other facts were known to many cultures long before Pythagoras, but credit has gone to him for being the first to prove the theorem. It is the culmination of Euclid's first Book.
PROPOSITION 47. THEOREM
(The proof now shows that the square GB is equal to the parallelogram BL, and the square HC is equal to the remaining parallelogram CL.
To show that the square GB is equal to the parallelogram BL,
The side opposite the right angle is called the hypotenuse
The above proof is Euclid's, not Pythagoras's. His proof is believed to have been based on the theory of proportions; Proposition VI. 31.
Now it is also a theorem that if BC is the diameter of a circle
and A any point on the circumference, then angle BAC is always a right angle. (Proposition III. 31.) Therefore if we imagine the point A moving along the circumference, then at each point we will have a right angle. The square on BA will grow larger while the square on AC grows smaller, yet they continually adjust themselves so that together they equal the constant square on BC! That equality of areas is what makes Pythagoras's theorem so remarkable.
Moreover, actually seeing that equality -- feeling it, almost -- rather than just proving it, is the essence of geometry. For however easy or difficult it might be to prove, actually seeing the fact -- that those areas really are equal -- remains elusive. Over the centuries therefore there have been many, many proofs of this famous theorem, each one offering a different insight. For example, see this one.
The hypothesis of Proposition 47 is that the triangle is right-angled; hence the converse, which is Proposition 48 and the last theorem of Book I, has for its conclusion that the triangle is right-angled. Here is the enunciation.
PROPOSITION 48. THEOREM
It is this proposition that informs us that if the sides of a triangle are 3-4-5 -- so that the squares on them are 9-16-25 -- then the triangle is right-angled. Whole-number sides such as those are called Pythagorean triples.
For the proof, see Problem 4.
Pythagoras and the Pythagoreans
Pythagoras is remembered as the first to take mathematics seriously in relation to the world order. He taught that geometry and numbers should be studied with reverence because we are entering into knowledge of the Divine. Mathematics is therefore more than just intellectual stimulation, and its relationship to the Universe is more than just coincidence.
Pythagoras was born on the Greek island of Samos. He left to found what we might call a religious sect in the city of Crotona in southern Italy, which was then part of greater Greece. His devotees were called Pythagoreans ("Pi-thag-o-REE-ans"), and many of them, both men and women, lived communally. They had their rituals and dietary laws, and they made important contributions to the medicine and astronomy of their time. They were among the first to teach that the earth is round and that it revolves about the sun.
They also taught the continuity and reincarnation of life; and that through philosophy (literally, love of wisdom) there is purification and thus escape from the cycle of births. Hence they practiced equality between one another, and they showed compassion to all creatures because we are all forms of One.
The Pythagoreans were the first to systematically investigate both arithmetic and geometry. Not only did they discover many theorems, but they gave an ethical and spiritual significance to each of the figures they drew. As a secret society, it must be said that they were extremely successful. Not a word has survived. Nevertheless, Pythagoras is credited with discovering that the diagonal and side of a square have no common meausre. (See Topic 11 of The Evolution of the Real Numbers.) They considered their doctrine of incommensurables their most esoteric teaching -- one of their members was treated as dead for having even spoken of it to an outsider. The significance they gave to it has never been revealed.
As a scientific researcher, Pythagoras discovered how musical harmonies depend on ratios of whole numbers. He found that we
hear the interval called an octave when the length of a vibrating string is halved, that is to say, when the length of the plucked string is to the whole string in the ratio 1:2 (Fig 1). We hear the interval called a perfect fifth (G above C) when two thirds of a string is plucked (Fig. 2), that is, when the stopped string is to the whole string in the ratio 2:3. And we hear a perfect fourth (F above C) when three fourths of the string is sounded (Fig. 3); and so on, for each interval in the musical scale.
For the Pythagoreans, it was not just that they understood that. They believed that those numerical ratios created the harmonious sounds.
"All things take their form from number." (Not symbols for numbers.) All things are therefore intelligible.
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