The Greek mathematician Pythagoras has high name recognition, not only in the history of mathematics. Everyone who has studied geometry can recall, well after the high school years, some aspect of the Pythagorean Theorem. However, the story of Pythagoras and his famous theorem is not well known. Some story plot points are: the famous theorem goes by several names grounded in the behavior of the day (discussed later in the text), including the Pythagorean Theorem, Pythagoras’ Theorem and notably Euclid I 47. The Pythagorean Theorem is arguably the most famous statement in mathematics, and the fourth most beautiful equation.1, 2 There are well over 371 Pythagorean Theorem proofs originally collected by an eccentric mathematics teacher, who put them in a 1927 book, which includes those by a 12-year-old Einstein, Leonardo da Vinci (a master of all disciplines) and President of the United States James A. Garfield.3, 4, 5

Pythagoras’ likeness in pictures and sculptures, as shown in Figure 1, appears in all geometry textbooks, and books about the history of mathematics. However, ironically, not much is really known about him – not even his likeness. What is known about Pythagoras is generally considered more fiction than fact, as historians who lived hundreds of years later provided the facts about his life. There are definite details of Pythagoras’ life from early biographies that use original sources, yet are written by authors who attribute divine powers to him, and present him as a deity figure. Consequently, most historians treat this information as legend. Few historians view the information with any degree of historical importance because it is obtained from rare original sources.6

Historians generally agree that Pythagoras of Samos (born circa 569 BC in Samos, Ionia and died circa 475 BC) was the first mathematician. He is an extremely important figure in the development of mathematics, yet relatively little is known about his mathematical achievements. Unlike many later Greek mathematicians, who wrote a number of books, there are no writings by Pythagoras. However, there is evidence that Pythagoras founded a school (in what is now Crotone, to the east of the heel of southern Italy) named the Semicircle of Pythagoras – half-religious and half-scientific, which followed a code of secrecy. Because secrecy is often controversial, Pythagoras is a mysterious figure. The members of the Semicircle of Pythagoras – the Pythagoreans – were bound by an allegiance that was strictly enforced. However, the Semicircle was more than just a school that studied intellectual disciplines, including in particular philosophy, mathematics and astronomy.7 The scientific dimension of the school treated numbers in ways similar to the Jewish mysticism of Kaballah, where each number has divine meaning and combined numbers reveal the mystical worth of life. The latter is reflected in the Pythagorean motto: Number Rules the Universe.6 The religious dimension of the school included diverse lectures held by Pythagoras attended by men and women, even though the law in those days forbade women from being in the company of men. Pythagoreans consumed vegetarian dried and condensed food and unleavened bread (as matzos, used by the Biblical Jewish priestly class (the Kohanim), and used today during the Jewish holiday of Passover).8

One reason for the rarity of Pythagoras original sources was that Pythagorean knowledge was passed on from one generation to the next by word of mouth, as writing material was scarce. Moreover, out of respect for their leader, many of the discoveries made by the Pythagoreans were attributed to Pythagoras himself; this would account for the term ‘Pythagoras’ Theorem’. Consequently, of Pythagoras’ actual work nothing is known. On the other hand, his school practiced collectivism, making it hard to distinguish between the work of Pythagoras and that of his followers; this would account for the term ‘Pythagorean Theorem’. Therefore, the true discovery of a particular Pythagorean result may never be known. Regardless of the uncertainty of Pythagoras’ actual contributions, however, his school made outstanding contributions to mathematics.

Is seems that Pythagoras was the first person to define the consonant acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension of proportional lengths create tones of proportional frequencies when plucked. For example, a string that is 2 feet long will vibrate x times per second (that is, hertz, a unit of frequency equal to one cycle per second), while a string that is 1 foot long will vibrate twice as fast: 2x. Furthermore, those two frequencies create a perfect octave.9

The most important discovery of Pythagoras’ school was the fact that the diagonal of a square is not a rational multiple of its side.10 This result proved the existence of irrational numbers.11 This finding greatly disturbed the Pythagoreans, as it was inconsistent with their divine belief in numbers: whole numbers and their ratios, which account for geometrical properties, were challenged by their own result. The Pythagoreans were so troubled over the finding of irrational numbers that they swore each other to secrecy about its existence. It is known that one Pythagorean did tell someone outside the school, and he was never to be found thereafter, that is, he was murdered, as Pythagoras himself was murdered by oppressors of the Semicircle of Pythagoras.12