Biography – Who was Pythagoras
The attribution of pure mathematics to Pythagoras, often dubbed the first “true” mathematician, remains a subject of debate. Pythagoras himself did not leave behind any mathematical writings, and much of our understanding of Pythagorean thought comes from the writings of later Pythagorean scholars like Philolaus. It is unclear whether many, if any, of the theorems credited to Pythagoras were actually solved by him or his followers.
Pythagoras founded a school in Croton, southern Italy, around 530 BCE, which became the center of a unique Pythagorean sect. While mathematics was a dominant aspect of Pythagorean thought, it was deeply intertwined with mysticism. Pythagoras imposed quasi-religious philosophies, strict vegetarianism, communal living, secret rituals, and peculiar rules on all members of his school. These rules included eccentric directives such as not urinating toward the sun, not marrying women wearing gold jewelry, avoiding passing an ass lying in the street, and refraining from consuming or even touching black fava beans.
The school’s members were divided into two groups: the “mathematikoi” or “learners,” who further developed Pythagoras’s mathematical and scientific work, and the “akousmatikoi” or “listeners,” who focused on the religious and ritualistic aspects of his teachings. Tensions often arose between these groups, and the sect eventually became embroiled in local conflicts, leading to its dispersion. The Pythagoreans’ secrecy and exclusivity fueled resentment, and in 460 BCE, their meeting places were burned and destroyed, with at least 50 members killed in Croton alone.
The fundamental tenet of Pythagoras’s school was “All is number” or “God is number.” The Pythagoreans practiced a form of numerology or number-worship, assigning distinct characteristics and meanings to each number. For instance, one represented the origin of all numbers, two symbolized opinion, three represented harmony, four stood for justice, five denoted marriage, six signified creation, and seven corresponded to the seven planets or “wandering stars.” Odd numbers were considered female, while even numbers were regarded as male.
The most sacred number was “Tetractys,” representing ten, a triangular number obtained by adding one, two, three, and four. Notably, the Pythagoreans deduced the significance of the number 10 through abstract mathematical reasoning rather than simply counting the fingers on two hands.
Pythagoras and his school, alongside a few other ancient Greek mathematicians, played a crucial role in establishing a more rigorous form of mathematics than what preceded it. They constructed a comprehensive mathematical system based on axioms and logic, wherein geometric elements corresponded to numbers. This approach showed that integers and their ratios were sufficient to build an entire system of logical and truthful mathematics. Before Pythagoras, geometry had largely relied on empirical measurements, lacking the foundational rigor that he and his followers introduced.
The Pythagorean Theorem
Pythagoras is primarily renowned for his discovery of what we now call Pythagoras’ Theorem, also known as the Pythagorean Theorem. This theorem states that in any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides, often referred to as the “legs.”
Expressed as an equation, it is written as: a^2 + b^2 = c^2, where ‘a’ and ‘b’ are the lengths of the two legs, and ‘c’ is the length of the hypotenuse.
What Pythagoras and his followers may not have realized at the time is that this theorem extends beyond triangles; it holds true for any shape. This means that, for instance, the area of a pentagon constructed on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the pentagons constructed on the other two sides. This principle applies to various shapes, whether they are regular or irregular, including semi-circles and beyond.
The Pythagorean theorem is famously illustrated by the simplest Pythagorean triangle with sides of 3, 4, and 5 units, where 3^2 + 4^2 = 5^2. This relationship is easily visualized by drawing unit squares on each side, demonstrating the theorem’s validity. However, an infinite number of integer “Pythagorean triples” exists, including (5, 12, 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), and many more. Notably, (6, 8, 10) is not considered a “primitive” Pythagorean triple, as it’s a multiple of (3, 4, 5).
The principles of Pythagoras’ theorem and the properties of right-angled triangles were explored in ancient mathematical texts from Babylon and Egypt over a thousand years earlier. One of the simplest proofs, dating back to ancient China and likely preceding Pythagoras, was discovered. While Pythagoras gave the theorem its definitive form, it remains uncertain whether he personally proved it or merely described it. Regardless, it has become one of the most well-known mathematical theorems, with over 400 different proofs, including geometrical, algebraic, and involving advanced differential equations.
It became evident that non-integer solutions were possible, such as an isosceles triangle with sides 1, 1, and √2, a concept known to the Babylonians centuries earlier. When Pythagoras’ student Hippasus attempted to calculate the value of √2, he found it couldn’t be expressed as a fraction. This discovery hinted at the existence of irrational numbers, which could not be represented as simple integer fractions. This revelation challenged the Pythagoreans’ belief in the divinity of integers, leading to a significant shift in mathematical thinking.
Hippasus paid a heavy price for sharing this discovery with the world, as the secretive Pythagoreans reportedly drowned him. However, this shift from the concept of divine integers to the richer concept of the continuum marked a pivotal development in mathematics, giving birth to Greek geometry, which deals with continuous elements like lines, planes, and angles.
Pythagoras and his followers, the Pythagoreans, also made important contributions to geometry, including the understanding that the sum of angles in a triangle equals 180°. They likely generalized the concept that the sum of interior angles in a polygon with n sides equals (2n – 4) right angles, with the sum of exterior angles totaling 4 right angles. They developed methods to construct figures with specific areas and employed geometrical algebra to solve equations like a(a – x) = x^2.
Furthermore, the Pythagoreans laid the foundation for number theory through their investigations of triangular, square, and perfect numbers (numbers equal to the sum of their divisors). They uncovered properties of square numbers, such as the square of a number n being equal to the sum of the first n odd numbers (e.g., 4^2 = 16 = 1 + 3 + 5 + 7). They also discovered at least one pair of amicable numbers, 220 and 284, where the sum of the divisors of one number equals the other (e.g., the sum of the divisors of 220 is 284, and vice versa).
Music Theory
Pythagoras is renowned for his profound insight into the world of harmonious music, where he uncovered the fundamental connection between musical intervals and whole number ratios. This discovery revolutionized our understanding of music. For example, when you pluck half the length of a guitar string, it produces the same note as the open string but an octave higher. Similarly, dividing the string into thirds creates a different yet harmonious note, and so on.
In contrast, intervals based on non-whole number ratios tend to result in dissonant or unpleasant sounds. Pythagoras described the first four overtones that form the essential building blocks of musical harmony: the octave (1:1), the perfect fifth (3:2), the perfect fourth (4:3), and the major third (5:4). Pythagorean tuning, the oldest method for tuning the 12-note chromatic scale, relies on stacking perfect fifths, each tuned in the ratio of 3:2.
Pythagoras’s mystical nature led him to believe that the entire universe was inherently rooted in numbers. He postulated that the planets and stars moved in accordance with mathematical equations that corresponded to musical notes. This cosmic interplay generated what he called the “Musical Universalis” or the “Music of the Spheres,” a harmonious symphony that governed the celestial bodies in their orbits.
