Pythagoras (ca. 569??-475?? B.C.E.) Born in Samos, Ionia, Greek mathematician and mystic Pythagoras is remembered as founder of the Pythagorean School, which claimed to have found universal truth through the study of number. As such, the Pythagoreans are credited as the first to have taken mathematics seriously as a study in its own right without regard to possible application and practical need. What precisely Pythagoras contributed himself is no longer clear, but the school is acknowledged to have discovered the role of ratios of whole numbers in the musical scale, the properties of
figurate numbers,
amicable numbers,
perfect numbers, and the existence of
irrational numbers. Although Pythagoras's Theorem was known to the Babylonians over 1,000 years earlier, the Pythagoreans are the first to have provided a general proof of the result.
Very little is known of Pythagoras's life. The sect he founded was half scientific and half religious and followed a code of secrecy that certainly promoted great mystery about the man himself. Many historical writings attribute godlike qualities to Pythagoras and are generally not regarded as accurate portrayals. It is understood that Pythagoras visited with Thales of Miletus (ca. 625-547 B.C.E.) as a young man who, most likely, contributed to Pythagoras' interest in mathematics. Around 535 B.C.E., Pythagoras traveled to Egypt and was certainly influenced by the secret sects of the Egyptian priests. (Many of the practices the Pythagoreans followed were the same as those practiced in Egypt refusal to eat beans or to wear cloth made from animal skins, for instance.)
In 525 B.C.E., Egypt was invaded by Persian forces, and Pythagoras was captured and taken to Babylon. Eventually, after being released, Pythagoras settled in southern Italy, where he founded his famous school around 518 B.C.E. Both men and women were welcomed as members.
Pythagoras believed that all physical (and metaphysical) phenomena could be understood through numbers. This belief is said to have stemmed from his observation that two vibrating strings produce a harmonious combination of tones only when the ratios of their lengths can be expressed in terms of whole numbers. (Pythagoras was an accomplished musician and made significant contributions to the theory of music.) Pythagoras, and his followers, studied whole numbers and their ratios, and even went as far as to assign mystical properties to numbers. For instance, they believed that the first natural number, 1, acted as the divine source of all numbers and so was of different stature than an ordinary number. (The number 2 was deemed the first number.) All even numbers were assumed feminine and all odd numbers masculine, and all odd numbers (except 13) represented good luck. The number 4 stood for justice, being the first perfect square, and 5 marriage, as the union (sum) of the first even number (2) and the first odd number (3). The number 6 was perfect since it equals the sum of its factors different from itself.
In Geometry, the Pythagoreans knew that the interior angles of a triangle always sum to 180° (and, more generally, that the interior angle of an n-sided figure sum to (n - 2)180°), and knew how to construct three of the five Platonic Solids. They could solve equations of the form x
2 = a(a-x) using geometrical methods, and they developed a number of techniques for constructing figures of a given area. They attributed great mystical significance to the pentagram because of the geometric ratios it contains. In astronomy, the Pythagoreans were aware that the Earth is round, but believed it lay at the center of the universe. They were aware that the orbit of the Moon is inclined to the equator of the Earth and were the first to realize that the evening star and the morning star were the same heavenly object (namely, the planet Venus).
The exact date, location, and circumstance of Pythagoras's death are not known. Despite his passing, the factions of the original Pythagorean order endured for more than 200 years. Members of the Pythagorean brotherhood so revered their founder that it was considered impious for any individual to claim a discovery for his own glory without referring back to Pythagoras himself. It is said, for instance, that member Hippasus of Metapontum (ca. 470 B.C.E.) was put to death by drowning for announcing his own discovery of the regular dodecahedroan, the fifth Platonic solid. (Other versions of this popular story submit that he was executed for claiming to have discovered that the square root of 2 is an irrational number.)
To explore more see
Cornelius Lanczos Space Through the Ages Geometrical Ideas From Pythagoras to Hilbert & Einstein, Academic Press, 1970.
James Tanton Encyclopedia Of Mathematics, Facts on File, 2005.
Carl B. Boyer A History of Mathematics, 2nd ed. revised by
Uta C. Merzbach and foreword by
Isaac Asimov, John Wiley and Sons Inc., 1991.
George F. Simmons Calculus Gems Brief Lives and Memorable Mathematics, McGraw Hill 1992.
