Biography Category

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Pythagoras

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Birth Date :

01/01/1970

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BIRTH AND EARLY CHILDHOOD

Born around 565 B.C. on the Greek island Samos off the Coast of Asia-Minor, his father Mnesarchos was a wealthy merchant and an engraver. His mother Pythais was a native of Samos. He passed his childhood in Samos, but often accompanied his father on business tours.

EDUCATION

He was educated according to the traditions of those times. He was a good musician and could play the Lair, a musical instrument

of ancient Greece, similar to Indian musical instrument ‘Veena’. He learned poetry and was fond of reciting ‘Homer’.

VISIT TO MILETUS

Around 545 B.C., he visited Miletus, a seat of learning. There he met the well-known philosopher Thales, who influenced him, a great deal.

Thales introduced him to mathematical ideas. He advised Pythagoras to visit Egypt to gain more knowledge about mathematics and astronomy. Thales’s pupil Anaximander was a philosopher. Pythagoras attended his lectures and got interested in Geometry and Cosmology. Afterwards, he returned to Samos.

VISIT TO EGYPT

Around 535 B.C., he visited Egypt. Polycrates, the ruler of Samos gave him a letter of introduction. He traveled far and wide in Egypt and beyond. He learnt astronomy, mathematics and philosophy. He also studied religious rites and social customs of Egypt. In short, he absorbed everything that he came across. Enlightened with Egyptian culture, he returned to Samos.

BANISHMENT FROM SAMOS

Polycrates wished to be regarded as a man of culture. He invited learned persons from Samos and surrounding areas. Pythagoras became his favorite courtier. When relations with Polycrates soured, Polycrates banished Pythagoras from Samos around 530 B.C.

As Pythagoras traveled further east, he visited Babylon and came across Magi in Persia and also probably met some Indians there. He was enlightened by the knowledge, philosophy and religion among other subjects of these distant lands. Having traveled thus far, Pythagoras returned home to Samos.

After returning from Egypt, Pythagoras founded a school named Semi Circle. He started describing himself as ‘philosopher’ i.e. lover of wisdom and set up as a teacher of the subjects that he had learnt about, earlier. Samos then, under the reign of Polycrates, attained the heights of prosperity. Its nautical trading was in full bloom. So neighboring states envied its progress and prosperity. They were eager to attack Samos and deprive it of its riches. To cope up with the threats of such rivals, Polycrates allied with Egypt. But after some time, Polycrates suddenly changed sides and allied with the powerful Persian Empire.

During all these political events and strategies, Pythagoras being an eminent all-round citizen and as a courtier might have played or had been compelled to play some role (politics) one way or other that made him unpopular with Polycrates. His relations with Polycrates gradually worsened. As a result Pythagoras was banished from Samos by Polycrates. Some believe that he was imprisoned at a place south of the island. There was a cave on the mountainside which was locally known as Pythagoras’s Gaol.

Pythagoras traveled westward after banishment from Samos and arrived at Magna-Graecia in southern Italy, around 529 B.C. He settled at a Greek colony named Croton. He called himself a philosopher, meaning a lover of wisdom. This way, he distinguished himself from the ‘Sophists’ meaning wise men, who according to Pythagoras, were blind followers of religion and not rational thinkers.

Pythagoras did major part of his works on mathematics, during the early period of his stay at Croton. In fact people knew very little about the life and work of Pythagoras, so it is not possible to distinguish his works on mathematics from that of his followers. Pythagoras had also worked on astronomy. His contribution to astronomy is as fundamental as his mathematical discoveries.

Croton was a peaceful and safe haven for him compared to Samos. Pythagoras lived there for about 30 years. Here, he could work on various subjects like mathematics, astronomy, music, philosophy etc. He worked on numbers and number ratios, applied it to vibrating strings, musical octave and musical harmony. He selected number ratios producing symmetric geometrical shapes. Spherical planets moving in circular orbits at different distances from the center, produced celestial harmony.

He gave three doctrines to the Pythagoreans – The doctrine of rebirth or transmigration of soul, the doctrine of recollection of previous generations, the doctrine of three types (tripartite) of souls. He worked on numbers and classified them in various types – odd and even numbers, odd numbers represented good things in day-to-day life and even numbers signifying bad things and their union producing harmony in life.

He founded a school of philosophy and mathematics. He framed rules to govern the society. The members of the society had no personal possessions and they lived together in colonies. All members were treated equally. It was compulsory for them to be loyal to society and to the leader. Secrecy about the work and way of life in their society was expected and adhered to. All this made the life in the colony quite harmonious. People outside were impressed by their way of life. Hence a number of colonies were established in Magna-Graecia around Croton and spread up to Tarentum. The aristocrats treated them well. Educated persons in the society could get jobs in various public offices. This way, they prospered and were honored by people of Croton. It is believed that Pythagoras held an important job in public office. He was allotted work to reform the currency and he did it successfully. The coinage of Croton became far more advanced, both in design and manufacture.

However, gradually the people of Croton and aristocrats became conscious about the activities of Pythagoreans. Secrecy about the activities of the society, staunch Pythagoreanism, loyalty towards the community and the leader made people suspicious about them. Moreover, projecting the leader as a mystic performer and spreading legends about his supernatural abilities, made people jealous of the society. Therefore, they started ridiculing the Pythagoreans on stage. They hated them. During that time, Croton attacked and defeated its neighbor Sybaris. It is believed that Pythagoras was involved in the dispute. Hence, Cylon a noble from Croton and his persons attacked Pythagoras. Pythagoras and his followers were forced to flee from Croton to Metaponteum which was a Greek colony city, about 100 miles north on the Gulf of Tarentum. Pythagoras died at Metaponteum (around 490 B. C.) sometime later, though some believed that he was burned to death when anti-Pythagorean demonstrators set afire the commune house where he was living.

Born around 565 B.C. on the Greek island Samos off the Coast of Asia-Minor, his father Mnesarchos was a wealthy merchant and an engraver. His mother Pythais was a native of Samos. He passed his childhood in Samos, but often accompanied his father on business tours.

EDUCATION

He was educated according to the traditions of those times. He was a good musician and could play the Lair, a musical instrument

of ancient Greece, similar to Indian musical instrument ‘Veena’. He learned poetry and was fond of reciting ‘Homer’.

VISIT TO MILETUS

Around 545 B.C., he visited Miletus, a seat of learning. There he met the well-known philosopher Thales, who influenced him, a great deal.

Thales introduced him to mathematical ideas. He advised Pythagoras to visit Egypt to gain more knowledge about mathematics and astronomy. Thales’s pupil Anaximander was a philosopher. Pythagoras attended his lectures and got interested in Geometry and Cosmology. Afterwards, he returned to Samos.

VISIT TO EGYPT

Around 535 B.C., he visited Egypt. Polycrates, the ruler of Samos gave him a letter of introduction. He traveled far and wide in Egypt and beyond. He learnt astronomy, mathematics and philosophy. He also studied religious rites and social customs of Egypt. In short, he absorbed everything that he came across. Enlightened with Egyptian culture, he returned to Samos.

BANISHMENT FROM SAMOS

Polycrates wished to be regarded as a man of culture. He invited learned persons from Samos and surrounding areas. Pythagoras became his favorite courtier. When relations with Polycrates soured, Polycrates banished Pythagoras from Samos around 530 B.C.

As Pythagoras traveled further east, he visited Babylon and came across Magi in Persia and also probably met some Indians there. He was enlightened by the knowledge, philosophy and religion among other subjects of these distant lands. Having traveled thus far, Pythagoras returned home to Samos.

After returning from Egypt, Pythagoras founded a school named Semi Circle. He started describing himself as ‘philosopher’ i.e. lover of wisdom and set up as a teacher of the subjects that he had learnt about, earlier. Samos then, under the reign of Polycrates, attained the heights of prosperity. Its nautical trading was in full bloom. So neighboring states envied its progress and prosperity. They were eager to attack Samos and deprive it of its riches. To cope up with the threats of such rivals, Polycrates allied with Egypt. But after some time, Polycrates suddenly changed sides and allied with the powerful Persian Empire.

During all these political events and strategies, Pythagoras being an eminent all-round citizen and as a courtier might have played or had been compelled to play some role (politics) one way or other that made him unpopular with Polycrates. His relations with Polycrates gradually worsened. As a result Pythagoras was banished from Samos by Polycrates. Some believe that he was imprisoned at a place south of the island. There was a cave on the mountainside which was locally known as Pythagoras’s Gaol.

Pythagoras traveled westward after banishment from Samos and arrived at Magna-Graecia in southern Italy, around 529 B.C. He settled at a Greek colony named Croton. He called himself a philosopher, meaning a lover of wisdom. This way, he distinguished himself from the ‘Sophists’ meaning wise men, who according to Pythagoras, were blind followers of religion and not rational thinkers.

Pythagoras did major part of his works on mathematics, during the early period of his stay at Croton. In fact people knew very little about the life and work of Pythagoras, so it is not possible to distinguish his works on mathematics from that of his followers. Pythagoras had also worked on astronomy. His contribution to astronomy is as fundamental as his mathematical discoveries.

Croton was a peaceful and safe haven for him compared to Samos. Pythagoras lived there for about 30 years. Here, he could work on various subjects like mathematics, astronomy, music, philosophy etc. He worked on numbers and number ratios, applied it to vibrating strings, musical octave and musical harmony. He selected number ratios producing symmetric geometrical shapes. Spherical planets moving in circular orbits at different distances from the center, produced celestial harmony.

He gave three doctrines to the Pythagoreans – The doctrine of rebirth or transmigration of soul, the doctrine of recollection of previous generations, the doctrine of three types (tripartite) of souls. He worked on numbers and classified them in various types – odd and even numbers, odd numbers represented good things in day-to-day life and even numbers signifying bad things and their union producing harmony in life.

He founded a school of philosophy and mathematics. He framed rules to govern the society. The members of the society had no personal possessions and they lived together in colonies. All members were treated equally. It was compulsory for them to be loyal to society and to the leader. Secrecy about the work and way of life in their society was expected and adhered to. All this made the life in the colony quite harmonious. People outside were impressed by their way of life. Hence a number of colonies were established in Magna-Graecia around Croton and spread up to Tarentum. The aristocrats treated them well. Educated persons in the society could get jobs in various public offices. This way, they prospered and were honored by people of Croton. It is believed that Pythagoras held an important job in public office. He was allotted work to reform the currency and he did it successfully. The coinage of Croton became far more advanced, both in design and manufacture.

However, gradually the people of Croton and aristocrats became conscious about the activities of Pythagoreans. Secrecy about the activities of the society, staunch Pythagoreanism, loyalty towards the community and the leader made people suspicious about them. Moreover, projecting the leader as a mystic performer and spreading legends about his supernatural abilities, made people jealous of the society. Therefore, they started ridiculing the Pythagoreans on stage. They hated them. During that time, Croton attacked and defeated its neighbor Sybaris. It is believed that Pythagoras was involved in the dispute. Hence, Cylon a noble from Croton and his persons attacked Pythagoras. Pythagoras and his followers were forced to flee from Croton to Metaponteum which was a Greek colony city, about 100 miles north on the Gulf of Tarentum. Pythagoras died at Metaponteum (around 490 B. C.) sometime later, though some believed that he was burned to death when anti-Pythagorean demonstrators set afire the commune house where he was living.

BIRTH :

(Around) 565 B.C. (Around) 490 B.C. in Island Samos – Aegean Sea at Metaponteum

DEATH :

Off Coast of Asia-Minor 100 miles north of Croton near Gulf of Tarantum.

Greek philosopher, religious thinker and mathematician was known for ‘Pythagoras Theorem’. Son of Mnesarchos, he established a religious society at Croton, and his followers were known as Pythagoreans. His enemies forced him to escape to Metaponteum, where he died.

He wrote nothing though numerous works are attributed to him.

Showing Italy, Greece, Samos, and Asia-Minor, Egypt etc.

1. Samos (Island in Aegean Sea) – Birthplace of Pythagoras

2. Miletus – Asia-Minor – where Thales & Anaximander lived.

3. Croton – Magna Graecia – Southern Italy – where Pythagoras settled after banishment.

4. Tarantum – Military Commander and leading Pythagorean mathematician Archytas lived there (Archytas of Tarentum).

5. Athens – Greece

6. Magna Graecia – Southern Italy

Showing Greece, Samos, Asia-Minor, Syria and Egypt.

1.Samos – Birth Place of Pythagoras.

2.Miletus – In Asia-Minor where Thales and Anaximander lived.

3.Tyre – (Syria) – Mnesarchos- Pythagoras’s father came from

4.Athens – in Greece.

5.Thebes – in Mainland Greece where Philolaus lived.

6.Babylon – Pythagoras visited and gathered knowledge.

(Around) 565 B.C. (Around) 490 B.C. in Island Samos – Aegean Sea at Metaponteum

DEATH :

Off Coast of Asia-Minor 100 miles north of Croton near Gulf of Tarantum.

Greek philosopher, religious thinker and mathematician was known for ‘Pythagoras Theorem’. Son of Mnesarchos, he established a religious society at Croton, and his followers were known as Pythagoreans. His enemies forced him to escape to Metaponteum, where he died.

He wrote nothing though numerous works are attributed to him.

Showing Italy, Greece, Samos, and Asia-Minor, Egypt etc.

1. Samos (Island in Aegean Sea) – Birthplace of Pythagoras

2. Miletus – Asia-Minor – where Thales & Anaximander lived.

3. Croton – Magna Graecia – Southern Italy – where Pythagoras settled after banishment.

4. Tarantum – Military Commander and leading Pythagorean mathematician Archytas lived there (Archytas of Tarentum).

5. Athens – Greece

6. Magna Graecia – Southern Italy

Showing Greece, Samos, Asia-Minor, Syria and Egypt.

1.Samos – Birth Place of Pythagoras.

2.Miletus – In Asia-Minor where Thales and Anaximander lived.

3.Tyre – (Syria) – Mnesarchos- Pythagoras’s father came from

4.Athens – in Greece.

5.Thebes – in Mainland Greece where Philolaus lived.

6.Babylon – Pythagoras visited and gathered knowledge.

Around 565 B.C. Birth of Pythagoras.

Around 545 B.C. Visited Miletus.

Around 535 B.C. Travelled in Egypt and Babylonia.

Around 530 B.C. Banished from Samos.

Around 529 B.C. Settled at Croton in Magna-Gracia in Southern, Italy.

Around 520 B.C. Pythagoras and his followers were compelled to flee from Croton.

Around 500 B.C. Pythagoras died at Metaponteum.

Around 490B.C. Hippasus, a leading Pythagorean of Metaponteum, did major work on mathematics.

Wave of revolution swept Magna-Graecia, resulting in dispersal of Pythagoreans.

Around 420B.C. Philolus of Thebes in Greece became the authentic source and an authority on Pythagorean theory.

Around 400B.C. Archytas of Tarentum, a military commander, a mechanical genius, a Pythagorean philosopher and mathematician and friend of Plato believed to have lived and propagated the Pythagorean theories.

Around 4th Century A.D. Pythagoreanism absorbed into Neo-Platonism

Around 545 B.C. Visited Miletus.

Around 535 B.C. Travelled in Egypt and Babylonia.

Around 530 B.C. Banished from Samos.

Around 529 B.C. Settled at Croton in Magna-Gracia in Southern, Italy.

Around 520 B.C. Pythagoras and his followers were compelled to flee from Croton.

Around 500 B.C. Pythagoras died at Metaponteum.

Around 490B.C. Hippasus, a leading Pythagorean of Metaponteum, did major work on mathematics.

Wave of revolution swept Magna-Graecia, resulting in dispersal of Pythagoreans.

Around 420B.C. Philolus of Thebes in Greece became the authentic source and an authority on Pythagorean theory.

Around 400B.C. Archytas of Tarentum, a military commander, a mechanical genius, a Pythagorean philosopher and mathematician and friend of Plato believed to have lived and propagated the Pythagorean theories.

Around 4th Century A.D. Pythagoreanism absorbed into Neo-Platonism

Pythagoras was the first genius of western culture. He had a multifaceted magnetic personality. An intelligent mathematician and a religious thinker, both co-existed in him.

Having visited several countries like Miletus (in Asia-Minor), Egypt, Tyre (in Syria), Greece and southern Italy, he gained knowledge from all walks of life. He had also visited learned personalities like Thales, Anaximander, Pherekydes of that era, discussed and learned from them, subjects like philosophy, astronomy and mathematics. He could therefore, work and contribute on several subjects. His main contributions are in geometry, numbers, music, cosmology, astronomy, philosophy and religion.

Contribution to Mathematics :

A. Geometry :

After banishment from Samos, Pythagoras settled at Croton in Magna–Graecia. During his early stay there, Pythagoras conducted major part of his mathematical research work; including his famous Pythagoras Theorem. The Egyptians and Babylonians knew that a right-angled triangle having sides of 3,4 units had a hypotenuse of 5 units, but they could not provide requisite proof. Pythagoras gave proof of the theorem.

Pythagoras had proved two specific types of right-angled triangles.

i .A right angled triangle having sides 3,4 and 5 units [fig. (a)]

ii. A triangle having two mutually perpendicular and equal sides of an isosceles right angled triangle [fig. (b)]

Pythagoras was the first genius of western culture. He had a multifaceted magnetic personality. An intelligent mathematician and a religious thinker, both co-existed in him.

Having visited several countries like Miletus (in Asia-Minor), Egypt, Tyre (in Syria), Greece and southern Italy, he gained knowledge from all walks of life. He had also visited learned personalities like Thales, Anaximander, Pherekydes of that era, discussed and learned from them, subjects like philosophy, astronomy and mathematics. He could therefore, work and contribute on several subjects. His main contributions are in geometry, numbers, music, cosmology, astronomy, philosophy and religion.

Contribution to Mathematics :

A. Geometry :

After banishment from Samos, Pythagoras settled at Croton in Magna–Graecia. During his early stay there, Pythagoras conducted major part of his mathematical research work; including his famous Pythagoras Theorem. The Egyptians and Babylonians knew that a right-angled triangle having sides of 3,4 units had a hypotenuse of 5 units, but they could not provide requisite proof. Pythagoras gave proof of the theorem.

Pythagoras had proved two specific types of right-angled triangles.

i .A right angled triangle having sides 3,4 and 5 units [fig. (a)]

ii. A triangle having two mutually perpendicular and equal sides of an isosceles right angled triangle [fig. (b)]

We can see from figure (a) that (square on 3) + (square on 4) =(square on 5)

i.e. 32 + 42=9 + 16 = 25

We can see from figure (b) that Square A = 1 + 2, Square B = 3+4,Square C = 1+2+3+4

So (Square A) + (Square B) = (1+2) + (3+4)= 1+2+3+4 = (Square C)

Pythagoras theorem states that : The square on hypotenuse is equal to the sum of the squares on the other two sides. The remarkable thing is that he was the first mathematician to provide proof of the theorem. Thus, Pythagoras introduced abstraction, proof and reasoning in geometry.

The work on Pythagoras theorem led to various properties of right-angled triangles with integral sides, which are whole numbers. For instance, the triangle with sides 3, 4 and 5 has peculiar properties not found in other Pythagorean triangles (I). The sides a, b, c of this triangle are in Arithmetic Progression (AP). If a, b, c are in A. P. then c-b = b-a or 2b = a + c.

We can see in a given triangle that, 2(4) = (3+5) = 8.

i. 3,4,5, is the only right-angled triangle whose area is equal to half its perimeter.

We can see that

The area of the Pythagorean triangle (3,4,5) = ½ (base x height)

= ½ (3 x 4)

= ½ (12)

= 6.

The perimeter of the triangle = sum of its sides = (a+b+c)

= (3+4+5)

= 12

Half the perimeter = ½ (12) = 6

Thus, area of the Pythagorean triangle (3,4,5) is equal to half its perimeter.

ii. Pythagorean triangle and Pythagorean triads :

If a,b,c are sides of a right-angled triangle ABC and AC is the hypotenuse of ABC then according to Pythagoras theorem c2 = a2 + b2. The triangle ABC is known as Pythagorean triangle. The triad of positive integers (a,b,c) satisfying the relation c2=a2 + b2 is called the Pythagorean triad of numbers. About fifteen such triads were previously known like (3,4,5), (5,12,13), (7,24,25), (9,12,15), (15,36,39). It is believed that Pythagoras himself discovered the formula for determining triads of numbers satisfying the relation c2 = a2+b2.

The formula for determining Pythagorean triad is

Here n is an odd positive integer. For example, put n=3 in the above formula,

then

that is 32 + 42=52 . Thus the triad is (3,4,5) for n=5

52 + 122=132

Hence the Pythagorean triad is (5,12,13).

The Pythagorean triads in which the numbers a,b,c do not have a common factor are called primitive Pythagorean triads. For example (3,4,5) (5,12,13) (7,24,25) etc. are primitive Pythagorean triads. But (9,12,15), (15,36,39) are not primitive triads.

Geometric method for constructing four regular solids :

Pythagoras believed that the universe is made of five solid figures. They are regular polyhedra known as platonic solids, the tetrahedron or triangular pyramids, the cube having square faces, the octahedron having eight identical triangular faces, dodecahedron having 12 regular pentagonal faces etc. It is believed that Pythagoras discovered the methods for constructing these four regular polyhedra. They are shown in figures (a) (b) (c) (d).

A Tetrahedron

A Cube

A Regular Octahedron

A Regular Icosahedron

having 12 Faces

A Regular Icosahedron

having 20Faces

(B) Irrational number :

The mathematical notion of irrational number was an outcome of Pythagoras' theorem. Consider an isosceles right-angled triangle with equal sides of unit length. Then the length of its hypotenuse will be . Now is a number, which could not be expressed as a ratio of positive integers . Also its value could not be expressed as a decimal, which either finishes or repeats. We can see that =1.4142135623……and so on …with no recurring pattern. Pythagoras called it 'incommensurate' number. Later on, a leading Pythagorean named Hippasus of Metaponteum discovered irrational numbers like . Euclid gave the proof to show that is an irrational number. He used the Reductio-Ad-Absurdum method.

(C) Number and Mysticism :

Speculation about numbers by Pythagoras is indeed surprising. He said, a number has its own personality and accordingly, he named numbers as masculine (odd) and feminine (even), perfect (complete) and incomplete numbers, beautiful and ugly numbers etc.

According to him, One (1) (though odd), was undivided so (1) was not a real number. Two (2), being a feminine number was also not real number. Pythagoras called 'three' as the first real number. It was called a complete number because it had a beginning, a middle and an end. The numbers arising from product of even and odd numbers were named even-odd numbers.

Pythagoras gave two different types of perfect numbers. According to him, 'Ten' was a perfect number because the sum of first four integral numbers is Ten that is 1+2+3+4=10. It was a perfect number being base to decimal system. The four numbers 1,2,3,4 known as 'tetractys' could be represented in pyramidal shape as shown in the figure.

The pyramid contained numbers, which formed musical harmonies 2:1, 3:2 and 4:3 which are known musical octave, a fifth and a fourth.

Second type of perfect numbers : According to Pythagoras the second type of perfect numbers were 6, 28, 496, 8126 etc. They were the numbers equal to sum of their factors, for instance.

6 has factors 1, 2, 3 and 1+2+3=6

28 has factors 1, 2, 4, 7, 14 and 1+2+4+7+14=28

496 has factors 1, 2, 4, 8, 16, 31, 62, 124, 248 and 1+2+4+8+16+31+62+124+248=496 etc.

Amicable Numbers :

From perfect numbers, Pythagoras was led to amicable numbers like 220 and 284. Amicable numbers form a pair of numbers where each number is equal to the sum of the factors of the other numbers. For instance 220 has factors 1, 2, 4, 5, 10, 11, 20, 22,

44, 55 & 110. The sum of these factors is 1+2+4+5+10+11+20+22+44+55+110=284

Moreover, 284 has factors 1, 2, 4, 71 & 142

The sum of these factors is 1+2+4+71+142=220

Triangular numbers of Pythagoras : Pythagoras called numbers : 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 as triangular numbers because the numbers can be arranged so as to form triangles (as shown in the figures).

Number triangle : Pythagoreans knew about the number triangle. Here, the arrangement of numbers forms a triangle and the sum of the numbers represents squares.

1=12

1+2+1=22

1+2+3+2+1=32

1+2+3+4+3+2+1=42

1+2+3+4+5+4+3+2+1 =52

Square numbers : Square numbers are clearly the squares of the integers 1, 2, 3, 4, 5 etc. They are represented by square of dots as shown in the figure. From the arrangement of numbers the well-known formula

1+3+5+7+-------------+(2n-1)=n2 is thus, proved.

We can see from the above formula that the sum of n odd numbers is 1+3+5+7+-------------+(2n-1) = n2.

We apply this formula for n=5

It is 1+3+5+7+9

1+3+5+7+1[2(5)-1] = 25

And n2 = 52 =25

Thus 1+3+5+7+9=25

Musical harmony and celestial harmony :

Pythagoras conducted remarkable investigation in ‘music’ as he was a musician. Harmonies correspond to most beautiful mathematical ratio, he stated. Melodious musical tunes could be produced on a stringed instrument by plucking the string at particular points, which correspond to mathematical ratios. Such beautiful mathematical ratios are 2:1 (an octave), 3:2 (a fifth), and 4:3 (a fourth).

During his meeting with Anaximander at Miletus, Pythagoras got interested in astronomy. Later, Pythagoras saw that the Babylonians regularly and continuously observed the motion of planets. They predicted solar and lunar eclipses. Knowing this, Pythagoras also started working on astronomy. The Babylonians believed that there were seven planets moving round the earth. They were Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn. Pythagoras compared these eight planets (including the earth) with the musical octave and the seven planets (excluding earth) as seven strings of the musical instrument Lair. The planets situated at different distances and moving at different speed correspond to different notes on musical octave. The planets moving with higher speed produce higher notes and those with lower speed produce lower notes. The celestial harmony (or the music of the spheres) of moving planets produces heavenly music analogous to different notes of musical octave.

On having put forth this theory, the rationalists questioned him as to why the heavenly music not heard ? The answer was that either the musical notes were very high so that they were not perceptible to human ears (like ultra-sonic sound as known today) or that, we hear the heavenly music right from our birth and are accustomed to it. (Like a blacksmith with the noise of his anvil and hammer.)

According to Pythagoras, the sphere was the most beautiful solid and the circle the most beautiful shape. Thus, a spherical planet moving round the earth in circular orbit would form a harmonious constellation. Pythagoras worked out the distances of the planets from the earth. He arranged the planets in order of increasing distances of the planets from the earth. The order given by him was the Moon, Mercury, Venus, the Sun, Mars, Jupiter and Saturn. Some Pythagoreans believed that the Earth moved round a central fire. The Earth did not always face the central fire.

because they reflected light from their surfaces received from the central fire. Perhaps the idea of central fire later on led to the heliocentric (sun at the centre of the Solar system) configuration of solar system.

Pythagoras’ observation of heavens suggested to him that the motion of the heavenly bodies was cyclic and that the heavenly bodies returned to the place from which they had started. From this, Pythagoras concluded that there must be a cycle of cycles, a greater year and on its completion the heavenly bodies returned to the original position and the same heavenly constellation would be observed again and again. He called this the eternal recurrence.

Some Pythagoreans like Philalous said that the Moon appeared like Earth, as animals and plants inhabited it. The animals there were greater and more powerful. They are 15 times more powerful. Their day is 15 times longer than that on Earth. Pythagoreans said that the Milky Way is the burning of a star that fell from its position. It set on fire the circular region through which it passed. To them a comet was a star that is periodically seen for some time.

According to Pythagoras, the universe was made of five solid figures. They were known as mathematical solids. He said the earth had arisen from a cube, fire from pyramid, air from octahedron, water from Icosahedron (24 faces) and the spherical universe from Dodecahedron with (12 faces). There is nothing (void) outside the universe.

Pythagoras’ dictum: All is Number

Pythagoras associated numbers with geometrical notions and numerical ratios with shapes. He associated number one with a point, too with a line, three with a triangle and four with a tetrahedron (as shown in the figure)

Thus, one point generates dimensions, two points generate a line of one dimension, three points generate two dimensions, and four points generate three-dimensional solid figures. In geometry, numbers represent lengths, their squares represent areas, their cubes represent volumes etc. Starting from numbers, numerical ratios and their powers, one can construct geometrical figures of different shapes and geometrical solids of different sizes. Using distance the arrangement of planets, their motion, their orbital path, their distances from the center and their inter relations with each other can be worked out. Thus, according to Pythagoras all relations could be reduced to number relations and hence, the whole cosmos is a scale and a number based phenomenon.

From his observations in music, mathematics and astronomy, Pythagoras generalized that everything could be expressed in terms of numbers and numerical ratios. Numbers are not only symbols of reality, but also substances of real things. Hence, he claimed – All is number.

PYTHAGOREANS AND PYTHAGOREANISM

Pythagoras was forced to leave Samos by Polycrates. He was highly disturbed by continuous harassment, conflicts and strife’s in Samos. From Samos, Pythagoras traveled westwards to and arrived at Magna-Gracia in Southern Italy around 529 B. C. and settled in a Greek colony named Croton, a place of peace and safety.

Pythagoras described himself as a philosopher. He wanted to set up as a teacher of philosophy, astronomy and mathematics, so began his society. He founded an ethico–mathematical school following ethical beliefs and pursuits and worked on mathematics. Many persons were attracted by the society and joined it. Pythagoras was the head of the society. It was in fact Pythagorean brotherhood observing some rules and following a particular way of life. Pythagorean School regarded men and women equally. They enjoyed a common way of life. The society had an inner circle and an outer circle of followers.

The inner circle was known as ‘mathematikoi’. The members of the inner circle had to live with the society. They had no personal possessions and vegetarianism was strictly practiced there. Pythagoras personally taught them. Both men and women could become members of the society and they were treated equally. The followers had to abstain from beans, not to pick up what had fallen and not to touch the fire with iron. They had to observe top secrecy about every activity and works of the society. They were bound by an oath not to speak of the doctrines and discoveries of their school. Loyalty towards Pythagoreans and their leader was the basic tenet of the Pythagoreans.

The outer circle of the society was known as the ‘akousmatics’. They lived in their own houses and came to the society during daytime. They were allowed to have their own possessions. They were not required to be vegetarians. But they were also bound by oath not to speak of the doctrines and discoveries of their school. Loyalty towards fellow men and the leader was expected. Some of them were well educated and well versed in mathematics. Such members held important key posts in public offices, they were influential and were honored by a society. Pythagoreans were famous for their mutual friendship, unselfishness and honesty. Besides framing rules to govern the Pythagorean brotherhood, Pythagoras taught his disciples some doctrines about rebirth, recollection and key to knowledge. He called himself semi-divine. There has hardly been the followers of any school holding their master in such high esteem and reverence as the Pythagoreans had. They often cited the phrase ipse dixit meaning the master said so; which was a watchword for them.

Pythagorean philosophy : Pythagoras gave three doctrines to his followers.

1. The doctrine of tripartite (of three kinds) souls

2. The doctrine of rebirth or transmigration of souls

3. The doctrine of recollection of previous births.

(1) Doctrine of tripartite soul :

Pythagoras compared the human life with an Olympiad gathering and classified the souls into three types : lower category, middle category and higher category. He said three types of persons gather at the Olympic games.

i. The merchants come to sell their commodities to gain profit and the customers buy it. They are lovers of gain. They are put in lower category.

ii. The participants compete, they show their skills and get rewards. They are classified as lovers of honor. They are put in middle category.

iii. The spectators come only to see performances. They are lovers of wisdom; they are put in the higher category. This classification of persons into lovers of gain, lovers of honor and lovers of wisdom can be applied to souls giving the doctrine of tripartite souls.

(2) Doctrine of rebirth or transmigration :

Perhaps Pythagoras might have borrowed this from the knowledge that he had gathered form his visits to the East. The whole spirit of the doctrine is religious and ethical. It advocates the soul as immortal and residing in the body. The body is the tomb of the soul. The soul changes from one body to another, known as the birth-death cycle. During successive incarnations (Avataras), the soul resides in bodies of various animals and according to their vices and virtues (Karma) they are punished or rewarded (Phala) after death and souls climb up the scale. Highest in the scale is the saintly human whose supreme moral effects enable the soul to break the cycle of birth and death thus becoming free and achieving salvation (Moksha). This is analogous to our Hindu philosophy of ‘Purvas– Janma’ ‘Punar – Janma’ and ‘Moksha’.

(3) The doctrine of recollection :

As the soul is immortal, it resides in the body and changes from one body to another, the human recalls about previous births. It is said that Pythagoras claimed to know about his previous births, which could be a tale or a legend. Referring to recollection of previous births Xenophanes, a critic, made fun of Pythagoras and said that he could recognize the voice of his departed friend in the howls of a beaten dog.

Later on, Pythagorean philosophy was dominated by number theory. Pythagoras had applied number theory to music and astronomical phenomena. The Pythagoreans said numbers are not only symbols of reality but also the very substance of real things. Pythagoreans attached philosophical meaning to numbers. They said, all things of the universe had numerical attributes. According to them, one was a number of reason, two a number of opinion, three of harmony (as all things are defined by three which has beginning, middle and end. All the three together form harmony). Four was a number of justice, five number of marriage, and six number of creation. Seven attributed fate. Seven dominates human life because infancy ceases at seven, maturity begins at 14, marriage takes place at 21 and 70 years is the span of life. Pythagoreans believed that 10 was a perfect number. They considered it to be holy hence, they swore by number 10. Numbers were classified into odd and even. Under odd they took, light, straight, good and right under even they took : dark, crooked, evil, and left. Such opposites are found everywhere in nature and their union constitutes harmony. The philosophy of number greatly influenced the Greeks.

Having visited several countries like Miletus (in Asia-Minor), Egypt, Tyre (in Syria), Greece and southern Italy, he gained knowledge from all walks of life. He had also visited learned personalities like Thales, Anaximander, Pherekydes of that era, discussed and learned from them, subjects like philosophy, astronomy and mathematics. He could therefore, work and contribute on several subjects. His main contributions are in geometry, numbers, music, cosmology, astronomy, philosophy and religion.

Contribution to Mathematics :

A. Geometry :

After banishment from Samos, Pythagoras settled at Croton in Magna–Graecia. During his early stay there, Pythagoras conducted major part of his mathematical research work; including his famous Pythagoras Theorem. The Egyptians and Babylonians knew that a right-angled triangle having sides of 3,4 units had a hypotenuse of 5 units, but they could not provide requisite proof. Pythagoras gave proof of the theorem.

Pythagoras had proved two specific types of right-angled triangles.

i .A right angled triangle having sides 3,4 and 5 units [fig. (a)]

ii. A triangle having two mutually perpendicular and equal sides of an isosceles right angled triangle [fig. (b)]

Pythagoras was the first genius of western culture. He had a multifaceted magnetic personality. An intelligent mathematician and a religious thinker, both co-existed in him.

Having visited several countries like Miletus (in Asia-Minor), Egypt, Tyre (in Syria), Greece and southern Italy, he gained knowledge from all walks of life. He had also visited learned personalities like Thales, Anaximander, Pherekydes of that era, discussed and learned from them, subjects like philosophy, astronomy and mathematics. He could therefore, work and contribute on several subjects. His main contributions are in geometry, numbers, music, cosmology, astronomy, philosophy and religion.

Contribution to Mathematics :

A. Geometry :

After banishment from Samos, Pythagoras settled at Croton in Magna–Graecia. During his early stay there, Pythagoras conducted major part of his mathematical research work; including his famous Pythagoras Theorem. The Egyptians and Babylonians knew that a right-angled triangle having sides of 3,4 units had a hypotenuse of 5 units, but they could not provide requisite proof. Pythagoras gave proof of the theorem.

Pythagoras had proved two specific types of right-angled triangles.

i .A right angled triangle having sides 3,4 and 5 units [fig. (a)]

ii. A triangle having two mutually perpendicular and equal sides of an isosceles right angled triangle [fig. (b)]

We can see from figure (a) that (square on 3) + (square on 4) =(square on 5)

i.e. 32 + 42=9 + 16 = 25

We can see from figure (b) that Square A = 1 + 2, Square B = 3+4,Square C = 1+2+3+4

So (Square A) + (Square B) = (1+2) + (3+4)= 1+2+3+4 = (Square C)

Pythagoras theorem states that : The square on hypotenuse is equal to the sum of the squares on the other two sides. The remarkable thing is that he was the first mathematician to provide proof of the theorem. Thus, Pythagoras introduced abstraction, proof and reasoning in geometry.

The work on Pythagoras theorem led to various properties of right-angled triangles with integral sides, which are whole numbers. For instance, the triangle with sides 3, 4 and 5 has peculiar properties not found in other Pythagorean triangles (I). The sides a, b, c of this triangle are in Arithmetic Progression (AP). If a, b, c are in A. P. then c-b = b-a or 2b = a + c.

We can see in a given triangle that, 2(4) = (3+5) = 8.

i. 3,4,5, is the only right-angled triangle whose area is equal to half its perimeter.

We can see that

The area of the Pythagorean triangle (3,4,5) = ½ (base x height)

= ½ (3 x 4)

= ½ (12)

= 6.

The perimeter of the triangle = sum of its sides = (a+b+c)

= (3+4+5)

= 12

Half the perimeter = ½ (12) = 6

Thus, area of the Pythagorean triangle (3,4,5) is equal to half its perimeter.

ii. Pythagorean triangle and Pythagorean triads :

If a,b,c are sides of a right-angled triangle ABC and AC is the hypotenuse of ABC then according to Pythagoras theorem c2 = a2 + b2. The triangle ABC is known as Pythagorean triangle. The triad of positive integers (a,b,c) satisfying the relation c2=a2 + b2 is called the Pythagorean triad of numbers. About fifteen such triads were previously known like (3,4,5), (5,12,13), (7,24,25), (9,12,15), (15,36,39). It is believed that Pythagoras himself discovered the formula for determining triads of numbers satisfying the relation c2 = a2+b2.

The formula for determining Pythagorean triad is

Here n is an odd positive integer. For example, put n=3 in the above formula,

then

that is 32 + 42=52 . Thus the triad is (3,4,5) for n=5

52 + 122=132

Hence the Pythagorean triad is (5,12,13).

The Pythagorean triads in which the numbers a,b,c do not have a common factor are called primitive Pythagorean triads. For example (3,4,5) (5,12,13) (7,24,25) etc. are primitive Pythagorean triads. But (9,12,15), (15,36,39) are not primitive triads.

Geometric method for constructing four regular solids :

Pythagoras believed that the universe is made of five solid figures. They are regular polyhedra known as platonic solids, the tetrahedron or triangular pyramids, the cube having square faces, the octahedron having eight identical triangular faces, dodecahedron having 12 regular pentagonal faces etc. It is believed that Pythagoras discovered the methods for constructing these four regular polyhedra. They are shown in figures (a) (b) (c) (d).

A Tetrahedron

A Cube

A Regular Octahedron

A Regular Icosahedron

having 12 Faces

A Regular Icosahedron

having 20Faces

(B) Irrational number :

The mathematical notion of irrational number was an outcome of Pythagoras' theorem. Consider an isosceles right-angled triangle with equal sides of unit length. Then the length of its hypotenuse will be . Now is a number, which could not be expressed as a ratio of positive integers . Also its value could not be expressed as a decimal, which either finishes or repeats. We can see that =1.4142135623……and so on …with no recurring pattern. Pythagoras called it 'incommensurate' number. Later on, a leading Pythagorean named Hippasus of Metaponteum discovered irrational numbers like . Euclid gave the proof to show that is an irrational number. He used the Reductio-Ad-Absurdum method.

(C) Number and Mysticism :

Speculation about numbers by Pythagoras is indeed surprising. He said, a number has its own personality and accordingly, he named numbers as masculine (odd) and feminine (even), perfect (complete) and incomplete numbers, beautiful and ugly numbers etc.

According to him, One (1) (though odd), was undivided so (1) was not a real number. Two (2), being a feminine number was also not real number. Pythagoras called 'three' as the first real number. It was called a complete number because it had a beginning, a middle and an end. The numbers arising from product of even and odd numbers were named even-odd numbers.

Pythagoras gave two different types of perfect numbers. According to him, 'Ten' was a perfect number because the sum of first four integral numbers is Ten that is 1+2+3+4=10. It was a perfect number being base to decimal system. The four numbers 1,2,3,4 known as 'tetractys' could be represented in pyramidal shape as shown in the figure.

The pyramid contained numbers, which formed musical harmonies 2:1, 3:2 and 4:3 which are known musical octave, a fifth and a fourth.

Second type of perfect numbers : According to Pythagoras the second type of perfect numbers were 6, 28, 496, 8126 etc. They were the numbers equal to sum of their factors, for instance.

6 has factors 1, 2, 3 and 1+2+3=6

28 has factors 1, 2, 4, 7, 14 and 1+2+4+7+14=28

496 has factors 1, 2, 4, 8, 16, 31, 62, 124, 248 and 1+2+4+8+16+31+62+124+248=496 etc.

Amicable Numbers :

From perfect numbers, Pythagoras was led to amicable numbers like 220 and 284. Amicable numbers form a pair of numbers where each number is equal to the sum of the factors of the other numbers. For instance 220 has factors 1, 2, 4, 5, 10, 11, 20, 22,

44, 55 & 110. The sum of these factors is 1+2+4+5+10+11+20+22+44+55+110=284

Moreover, 284 has factors 1, 2, 4, 71 & 142

The sum of these factors is 1+2+4+71+142=220

Triangular numbers of Pythagoras : Pythagoras called numbers : 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 as triangular numbers because the numbers can be arranged so as to form triangles (as shown in the figures).

Number triangle : Pythagoreans knew about the number triangle. Here, the arrangement of numbers forms a triangle and the sum of the numbers represents squares.

1=12

1+2+1=22

1+2+3+2+1=32

1+2+3+4+3+2+1=42

1+2+3+4+5+4+3+2+1 =52

Square numbers : Square numbers are clearly the squares of the integers 1, 2, 3, 4, 5 etc. They are represented by square of dots as shown in the figure. From the arrangement of numbers the well-known formula

1+3+5+7+-------------+(2n-1)=n2 is thus, proved.

We can see from the above formula that the sum of n odd numbers is 1+3+5+7+-------------+(2n-1) = n2.

We apply this formula for n=5

It is 1+3+5+7+9

1+3+5+7+1[2(5)-1] = 25

And n2 = 52 =25

Thus 1+3+5+7+9=25

Musical harmony and celestial harmony :

Pythagoras conducted remarkable investigation in ‘music’ as he was a musician. Harmonies correspond to most beautiful mathematical ratio, he stated. Melodious musical tunes could be produced on a stringed instrument by plucking the string at particular points, which correspond to mathematical ratios. Such beautiful mathematical ratios are 2:1 (an octave), 3:2 (a fifth), and 4:3 (a fourth).

During his meeting with Anaximander at Miletus, Pythagoras got interested in astronomy. Later, Pythagoras saw that the Babylonians regularly and continuously observed the motion of planets. They predicted solar and lunar eclipses. Knowing this, Pythagoras also started working on astronomy. The Babylonians believed that there were seven planets moving round the earth. They were Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn. Pythagoras compared these eight planets (including the earth) with the musical octave and the seven planets (excluding earth) as seven strings of the musical instrument Lair. The planets situated at different distances and moving at different speed correspond to different notes on musical octave. The planets moving with higher speed produce higher notes and those with lower speed produce lower notes. The celestial harmony (or the music of the spheres) of moving planets produces heavenly music analogous to different notes of musical octave.

On having put forth this theory, the rationalists questioned him as to why the heavenly music not heard ? The answer was that either the musical notes were very high so that they were not perceptible to human ears (like ultra-sonic sound as known today) or that, we hear the heavenly music right from our birth and are accustomed to it. (Like a blacksmith with the noise of his anvil and hammer.)

According to Pythagoras, the sphere was the most beautiful solid and the circle the most beautiful shape. Thus, a spherical planet moving round the earth in circular orbit would form a harmonious constellation. Pythagoras worked out the distances of the planets from the earth. He arranged the planets in order of increasing distances of the planets from the earth. The order given by him was the Moon, Mercury, Venus, the Sun, Mars, Jupiter and Saturn. Some Pythagoreans believed that the Earth moved round a central fire. The Earth did not always face the central fire.

because they reflected light from their surfaces received from the central fire. Perhaps the idea of central fire later on led to the heliocentric (sun at the centre of the Solar system) configuration of solar system.

Pythagoras’ observation of heavens suggested to him that the motion of the heavenly bodies was cyclic and that the heavenly bodies returned to the place from which they had started. From this, Pythagoras concluded that there must be a cycle of cycles, a greater year and on its completion the heavenly bodies returned to the original position and the same heavenly constellation would be observed again and again. He called this the eternal recurrence.

Some Pythagoreans like Philalous said that the Moon appeared like Earth, as animals and plants inhabited it. The animals there were greater and more powerful. They are 15 times more powerful. Their day is 15 times longer than that on Earth. Pythagoreans said that the Milky Way is the burning of a star that fell from its position. It set on fire the circular region through which it passed. To them a comet was a star that is periodically seen for some time.

According to Pythagoras, the universe was made of five solid figures. They were known as mathematical solids. He said the earth had arisen from a cube, fire from pyramid, air from octahedron, water from Icosahedron (24 faces) and the spherical universe from Dodecahedron with (12 faces). There is nothing (void) outside the universe.

Pythagoras’ dictum: All is Number

Pythagoras associated numbers with geometrical notions and numerical ratios with shapes. He associated number one with a point, too with a line, three with a triangle and four with a tetrahedron (as shown in the figure)

Thus, one point generates dimensions, two points generate a line of one dimension, three points generate two dimensions, and four points generate three-dimensional solid figures. In geometry, numbers represent lengths, their squares represent areas, their cubes represent volumes etc. Starting from numbers, numerical ratios and their powers, one can construct geometrical figures of different shapes and geometrical solids of different sizes. Using distance the arrangement of planets, their motion, their orbital path, their distances from the center and their inter relations with each other can be worked out. Thus, according to Pythagoras all relations could be reduced to number relations and hence, the whole cosmos is a scale and a number based phenomenon.

From his observations in music, mathematics and astronomy, Pythagoras generalized that everything could be expressed in terms of numbers and numerical ratios. Numbers are not only symbols of reality, but also substances of real things. Hence, he claimed – All is number.

PYTHAGOREANS AND PYTHAGOREANISM

Pythagoras was forced to leave Samos by Polycrates. He was highly disturbed by continuous harassment, conflicts and strife’s in Samos. From Samos, Pythagoras traveled westwards to and arrived at Magna-Gracia in Southern Italy around 529 B. C. and settled in a Greek colony named Croton, a place of peace and safety.

Pythagoras described himself as a philosopher. He wanted to set up as a teacher of philosophy, astronomy and mathematics, so began his society. He founded an ethico–mathematical school following ethical beliefs and pursuits and worked on mathematics. Many persons were attracted by the society and joined it. Pythagoras was the head of the society. It was in fact Pythagorean brotherhood observing some rules and following a particular way of life. Pythagorean School regarded men and women equally. They enjoyed a common way of life. The society had an inner circle and an outer circle of followers.

The inner circle was known as ‘mathematikoi’. The members of the inner circle had to live with the society. They had no personal possessions and vegetarianism was strictly practiced there. Pythagoras personally taught them. Both men and women could become members of the society and they were treated equally. The followers had to abstain from beans, not to pick up what had fallen and not to touch the fire with iron. They had to observe top secrecy about every activity and works of the society. They were bound by an oath not to speak of the doctrines and discoveries of their school. Loyalty towards Pythagoreans and their leader was the basic tenet of the Pythagoreans.

The outer circle of the society was known as the ‘akousmatics’. They lived in their own houses and came to the society during daytime. They were allowed to have their own possessions. They were not required to be vegetarians. But they were also bound by oath not to speak of the doctrines and discoveries of their school. Loyalty towards fellow men and the leader was expected. Some of them were well educated and well versed in mathematics. Such members held important key posts in public offices, they were influential and were honored by a society. Pythagoreans were famous for their mutual friendship, unselfishness and honesty. Besides framing rules to govern the Pythagorean brotherhood, Pythagoras taught his disciples some doctrines about rebirth, recollection and key to knowledge. He called himself semi-divine. There has hardly been the followers of any school holding their master in such high esteem and reverence as the Pythagoreans had. They often cited the phrase ipse dixit meaning the master said so; which was a watchword for them.

Pythagorean philosophy : Pythagoras gave three doctrines to his followers.

1. The doctrine of tripartite (of three kinds) souls

2. The doctrine of rebirth or transmigration of souls

3. The doctrine of recollection of previous births.

(1) Doctrine of tripartite soul :

Pythagoras compared the human life with an Olympiad gathering and classified the souls into three types : lower category, middle category and higher category. He said three types of persons gather at the Olympic games.

i. The merchants come to sell their commodities to gain profit and the customers buy it. They are lovers of gain. They are put in lower category.

ii. The participants compete, they show their skills and get rewards. They are classified as lovers of honor. They are put in middle category.

iii. The spectators come only to see performances. They are lovers of wisdom; they are put in the higher category. This classification of persons into lovers of gain, lovers of honor and lovers of wisdom can be applied to souls giving the doctrine of tripartite souls.

(2) Doctrine of rebirth or transmigration :

Perhaps Pythagoras might have borrowed this from the knowledge that he had gathered form his visits to the East. The whole spirit of the doctrine is religious and ethical. It advocates the soul as immortal and residing in the body. The body is the tomb of the soul. The soul changes from one body to another, known as the birth-death cycle. During successive incarnations (Avataras), the soul resides in bodies of various animals and according to their vices and virtues (Karma) they are punished or rewarded (Phala) after death and souls climb up the scale. Highest in the scale is the saintly human whose supreme moral effects enable the soul to break the cycle of birth and death thus becoming free and achieving salvation (Moksha). This is analogous to our Hindu philosophy of ‘Purvas– Janma’ ‘Punar – Janma’ and ‘Moksha’.

(3) The doctrine of recollection :

As the soul is immortal, it resides in the body and changes from one body to another, the human recalls about previous births. It is said that Pythagoras claimed to know about his previous births, which could be a tale or a legend. Referring to recollection of previous births Xenophanes, a critic, made fun of Pythagoras and said that he could recognize the voice of his departed friend in the howls of a beaten dog.

Later on, Pythagorean philosophy was dominated by number theory. Pythagoras had applied number theory to music and astronomical phenomena. The Pythagoreans said numbers are not only symbols of reality but also the very substance of real things. Pythagoreans attached philosophical meaning to numbers. They said, all things of the universe had numerical attributes. According to them, one was a number of reason, two a number of opinion, three of harmony (as all things are defined by three which has beginning, middle and end. All the three together form harmony). Four was a number of justice, five number of marriage, and six number of creation. Seven attributed fate. Seven dominates human life because infancy ceases at seven, maturity begins at 14, marriage takes place at 21 and 70 years is the span of life. Pythagoreans believed that 10 was a perfect number. They considered it to be holy hence, they swore by number 10. Numbers were classified into odd and even. Under odd they took, light, straight, good and right under even they took : dark, crooked, evil, and left. Such opposites are found everywhere in nature and their union constitutes harmony. The philosophy of number greatly influenced the Greeks.

SOME VIEWS AND QUOTATIONS

Many Renaissance humanists regarded Pythagoras as the father of exact sciences. The earth moved around the sun. Copernicus calls this a Pythagorean idea. Leibnitz, a well-known German mathematician admired Pythagoras and regarded himself as part of Pythagorean traditions. An ancient commentator, Aulus Gellius gave an ingenious explanation of Pythagoras’ view to abstain from beans. It did not mean what it seemed to mean literally. In earlier times, it was believed that consuming beans had heightened effect on libido, hence Pythagoras’ sought ban on beans as it was in fact related to sexual activity.

ABOUT BIRTH OF PYTHAGORAS :

He was the son of a wealthy engraver and merchant Mnesarchos, but some insists that he was the son of Apollo, the ancient Greek God of music, poetry and dance. The great philosopher and thinker of the 20th century Bertrand Russell had to say this about his birth: I leave the reader to take his choice between these alternatives.

ARISTOTLE WROTE :

‘Pythagorean having been brought up in the study of mathematics, thought that things could be represented by numbers …and that the whole cosmos consists of a scale and a number’.

Miracles and legends about Pythagoras were heard all over Magna-Graecia "He was seen at two places at the same time". "When he crossed a stream, the stream rose out of the dead and greeted Pythagoras Hail Pythagoras. The Crotonians did not like this, they ridiculed Pythagoreans, and called them superstitious, filthy vegetarians.

Xenophanes, the poet and singer said about the doctrine of transmigration of souls, "A small dog was being thrashed by some one, seeing this Pythagoras rushed there and said, ‘stop beating, for the dog lives in the soul of my friend, I know him by his voice.’

From Croton, Pythagoras escaped to Metaponteum. Most of the authors said he died there. Some said he committed suicide. But Iamblichus, an authority on Pythagoras said, "Pythagoras escaped to Metaponteum but after some time, he returned to Croton and lived there for a longer period even after 480 B.C."

Many Renaissance humanists regarded Pythagoras as the father of exact sciences. The earth moved around the sun. Copernicus calls this a Pythagorean idea. Leibnitz, a well-known German mathematician admired Pythagoras and regarded himself as part of Pythagorean traditions. An ancient commentator, Aulus Gellius gave an ingenious explanation of Pythagoras’ view to abstain from beans. It did not mean what it seemed to mean literally. In earlier times, it was believed that consuming beans had heightened effect on libido, hence Pythagoras’ sought ban on beans as it was in fact related to sexual activity.

ABOUT BIRTH OF PYTHAGORAS :

He was the son of a wealthy engraver and merchant Mnesarchos, but some insists that he was the son of Apollo, the ancient Greek God of music, poetry and dance. The great philosopher and thinker of the 20th century Bertrand Russell had to say this about his birth: I leave the reader to take his choice between these alternatives.

ARISTOTLE WROTE :

‘Pythagorean having been brought up in the study of mathematics, thought that things could be represented by numbers …and that the whole cosmos consists of a scale and a number’.

Miracles and legends about Pythagoras were heard all over Magna-Graecia "He was seen at two places at the same time". "When he crossed a stream, the stream rose out of the dead and greeted Pythagoras Hail Pythagoras. The Crotonians did not like this, they ridiculed Pythagoreans, and called them superstitious, filthy vegetarians.

Xenophanes, the poet and singer said about the doctrine of transmigration of souls, "A small dog was being thrashed by some one, seeing this Pythagoras rushed there and said, ‘stop beating, for the dog lives in the soul of my friend, I know him by his voice.’

From Croton, Pythagoras escaped to Metaponteum. Most of the authors said he died there. Some said he committed suicide. But Iamblichus, an authority on Pythagoras said, "Pythagoras escaped to Metaponteum but after some time, he returned to Croton and lived there for a longer period even after 480 B.C."

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