The Pythagoreans

About five centuries B.C. the school founded by the Greek philosopher, mathematician and astronomer Pythagoras flourished in Samos, Greece. The Pythagoreans believed (but failed to prove) that the universe could be understood in terms of whole numbers. This belief stemmed from observations in music, mathematics and astronomy. For example, they noticed that vibrating strings produce harmonious tones when the ratios of the their lengths are whole numbers. From this first attempt to express the universe in terms of numbers the idea was born that the world could be understood through mathematics, a central concept in the development of mathematics and science.

The importance of pure numbers is central to the Pythagorean view of the world. A point was associated with 1, a line with 2 a surface with 3 and a solid with 4. Their sum, 10, was sacred and omnipotent .

Pythagoras also developed a rather sophisticated cosmology. He and his followers believed the earth to be perfectly spherical and that heavenly bodies, likewise perfect spheres, moved as the Earth around a central fire invisible to human eyes (this was not the sun for it also circled this central fire) as shown in Fig. 2.8. There were 10 objects circling the central fire which included a counter-earth assumed to be there to account from some eclipses but also because they believed the number 10 to be particularly sacred. This is the first coherent system in which celestial bodies move in circles, an idea that was to survive for two thousand years.

**Figure 2.8:** The universe according to the Pythagoreans.
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It was also stated that heavenly bodies give forth musical sounds ``the harmony of the spheres'' as they move in the cosmos, a music which we cannot discern, being used to it from childhood (a sort of background noise); though we would certainly notice if anything went wrong! The Pythagoreans did not believe that music, numbers and cosmos were just related, they believed that music was number and that the cosmos was music

**Figure 2.9:** Pythagoras' theorem (the areas of the squares attached to the smaller sides of the triangle equal the area of the largest square).
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Pythagoras is best known for the mathematical result (Pythagoras' theorem) that states that the sum of the squares of the sides of a right triangle equals the square of the diagonal; see Fig. 2.9. This result, although known to the Babylonians 1000 years earlier, was first proved by Pythagoras (allegedly: no manuscript remains). Pythagoras' theorem will be particularly important when we study relativity for, as it turns out, it is not valid in the vicinity of very massive bodies! Similar statements hold for Euclid's postulate that parallel lines never meet, see Sect. 7.9.