Pythagoras was a Greek mathematician and philosopher who lived around 570 BC in Samos, which is now an island in Greek. Around 530 BC, Pythagoras and his family moved to Croton, Southern Italy, where he started his school. He taught his student how to save the human soul through mathematical principles. His followers were known as Pythagorean scholars. When Croton got victory over Sybaris, then some prominent citizens of Croton wanted democracy, but Pythagoreans refused to adopt that. This resulted in conflicts, and Pythagoras, along with his followers, had to move to Metapontum, where he died. There is no evidence of his writings. Whatever facts and principles are known today in his name are the work done by his scholars or Philolaus. It is still a mystery that whether his most famous theorem Pythagorean Theorem was also developed by him or by his followers. Besides Pythagoras theorem, there are many more significant contributions made by him. He worked more on the mystical study of mathematics rather than its practical application. We will discuss them next. It is said that he was a ‘true’ mathematician and we owe ‘pure’ mathematics to him.

## 1. Metempsychosis

This is among one of the teachings of Pythagoras. Metempsychosis means the transmission of the soul from one body to another. Pythagoras mentioned that after death, the soul gets transferred into another body. As said by other philosophers, Pythagoras used to claim that he could remember his previous four lives in detail. He also mentioned in one of his poems that he could recollect his former incarnation. According to him, his first incarnation was Aethalides as the son of Hermes, the second was a minor hero of the Trojan war, Euphorbus. His third incarnation was as a philosopher of Hermotimus and last was as Pyrrhus, a fisherman from Delos.

## 2. Numerology

The principle of Pythagoras school was “All is number” or “God is number.” As mentioned by Aristotle, Pythagoreans mostly used their knowledge of science in spiritualism. They believed that each number has its meaning and character. They used to worship numbers and very strongly practice numerology. For example, number 1 was called a generator as it can generate all other numbers, number 2 (dyad) represents matter, number 3 represents harmony, number 4 signifies four seasons and four elements, number 5 represents marriage because it is the sum of two and three, number 6 represents creation; number 7 was sacred as it was the number of 7 planets. Odd numbers were considered as female and even numbers as male.

## 3. Music and Lifestyle

Pythagoras and his followers believed that music produces the same effect on the body as medicine does. They found music to be the best source for purifying and saving one’s soul. Pythagoras mentioned in one of his stories that how he saved a lady from being robbed by some drunk youths who forcefully tried to enter her house. According to Pythagoras, when he saw those drunk youths entering the lady’s house, he started singing a song, and the song was so effective that it changed the mind of the drunk youths, and they escaped from the scene. Pythagoras was the first person to link music to self-improvement and peace of mind. According to historians, one day, he was passing by a blacksmith, and suddenly, he noticed the sound of a hammer and found it to be musical. He rushed inside the shop and started beating hammers of different sizes on the table. He observed that the tune played by the hammer was directly proportional to the size of the hammer, hence he proved that music was mathematical. Pythagoras and his followers also found that a string of a particular thickness and length makes a particular sound, whereas a string of half of its length will make the same sound but on a higher note. They used ratio 1:2 for octave, 2:3 for perfect fifth, and 3:4 for perfect fourth.

## 4. Cosmology

Before Pythagoras, it was believed that Earth was the focal point of the universe with everything revolving around it. He was the first person who discovered planet Earth to be a sphere. He also suggested that the morning star and the evening star are the same, i.e., Venus. Pythagoreans believed that all planets move in a large circle and when they come in contact with each other, they produce melodious sounds. These sounds were known as the music of spheres. Because of a large amount of background noise, this music was not audible. He also discovered that the orbit of the Moon was leaning towards the Equator of the Earth. Pythagoreans had a belief that the Earth, stars and planets all move in a circle around a central flame, and nights and days are the results of this movement. They further added that there existed a counter Earth on the other side of the flame.

## 5. Pythagoras Theorem

He is mainly credited for the theorem that is named after him, i.e, the ‘Pythagoras Theorem’. This theorem states that

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of length of other two sides that contains the right angle.’

In equation form, it is written as {a}^{2}={b}^{2}+{c}^{2}. The most common example quoted for the theorem is {5}^{2}={3}^{2}+{4}^{2}. One very easy proof of this theorem was given by China, even before the birth of Pythagoras. But Pythagoras gave a perfect geometric explanation of this theorem. However, it is not known whether this theorem was given by him or by his followers.

## 6. Pythagoras Reasonings

Aristotle mentions that most of Plato’s reasonings were based on the teachings of Pythagoras. There are chances that both Plato and Pythagoras believed that the science of mathematics can change the way we look at the world. It is believed that Plato might have learned that mathematical ideas are behind logic and morality from Pythagoras.

## 7. Five Regular Solids

Proclus credited Pythagoras with the discovery of five regular solids, i.e., cube, triangle, octahedron, dodecahedron, icosahedron. Properties of these solids were further elaborated by Plato, hence these are named after him like Plato’s regular solids. The properties of these solids were extensively discussed in the books published in the 16th and 17th centuries.

## 8. Properties of Triangle

Adding further to his achievements, Pythagoras and his followers gave the idea about the angles of the triangle. They stated that the sum of all angles of a triangle is equal to two right angles. They also proposed a generalization that states,

the sum of interior angles of a polygon with n sides is equal to (2n-4) right angles and the sum of its exterior angles is equal to four right angles.”

They were also able to compute the solution of equations like a(a-x)={x}^{2} with the help of geometry.

## 9. Tetractys

The Tetractys is a triangular structure also known as an equilateral triangle that consists of 10 dots arranged in four rows. One point in the first row, two points in the second, three points in the third, and four points in the fourth row. The four rows add up to number 10. Pythagoreans gave a very special place to the number 10. According to them, 10 is a holy number. Tetractys represents the four classical elements – fire, air, water, and earth.

## 10. Irrational numbers

Inspired by the inventions of Pythagoras, one of his students named Hippasus discovered a new set of numbers, i.e., Irrational numbers. One day, Hippasus tried to calculate the value of √2 and found out that it was impossible to express this number in terms of fractions. So, he proposed a new set of numbers in mathematics. This discovery shattered the Pythagoreans as it was against their belief that ‘All is number,’ and every number can be expressed as the ratio of God’s creation. Enraged by Hippasus’ findings, Pythagoreans sentenced him to death by drowning and threw him into the sea.

## 11. Amicable Numbers

Pythagoreans are credited for the discovery of the first pair of amicable numbers. Amicable numbers are those pairs of numbers for which the sum of divisors of one number is equal to the other number and vice versa. For example, (220,284), here, sum of perfect divisors of 220 is 1+2+4+5+10+11+20+22+44+55+110=284 and sum of perfect divisors of 284 is 1+2+4+71+142=220.

## 12. Perfect number

Pythagoreans were the first to identify the perfect numbers. Perfect numbers are those numbers that are equal to the sum of their proper divisors. For Example, 6, 28, 496, and 8128. It is believed that Pythagoreans studied the ‘mystical’ properties of these numbers. An actual explanation about perfect numbers was later given by Euclid.