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Pythagoras

Pythagoras is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pythagoras rather than just read about it. In short: Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy.

Pythagoras — main illustration
Pythagoras — illustration

Key takeaways

  • Pythagoras belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Pythagoras to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Pythagoras from memory before moving on to harder problems.

Reference excerpt

Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Modern scholars disagree regarding Pythagoras's education and influences, but most agree that he travelled to Croton in southern Italy around 530 BC, where he founded a school in which initiates were allegedly sworn to secrecy and lived a communal, ascetic lifestyle. In antiquity, Pythagoras was credited with mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, the sphericity of the Earth, the identity of the morning and evening stars as the planet Venus, and the division of the globe into five climatic zones. He was reputedly the first man to call himself a philosopher ("lover of wisdom"). Historians debate whether Pythagoras made these discoveries and pronouncements, as some of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors, such as Hippasus and Philolaus. The teaching most securely identified with Pythagoras is the "transmigration of souls" or metempsychosis, which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematical ratios and thus resonate to produce an inaudible symphony of music. Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy, and their meeting houses were burned. Pythagoras may have been killed during this persecution, or he may have escaped to Metapontum and died there. Pythagoras influenced Plato whose dialogues (especially Timaeus) exhibit Pythagorean ideas. A major revival of his teachings occurred in the first century BC among Middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the Middle Ages and Pythagoreanism had an influence on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagorean symbolism was also used throughout early modern European esotericism, and his teachings as portrayed in Ovid's Metamorphoses would later influence the modern vegetarian movement.

Life No authentic writings of Pythagoras have survived, and almost nothing is known for certain about his life. The earliest sources on Pythagoras's life, from Xenophanes, Heraclitus, Empedocles, Ion of Chios, and Herodotus are brief, ambiguous, and often satirical. The major sources on Pythagoras's life are three biographies from late antiquity written by Diogenes Laërtius, Porphyry, and Iamblichus, all of which are filled primarily with myths and legends and which become longer and more fantastic in their descriptions of Pythagoras's achievements the more removed they are from Pythagoras's times. However, Porphyry and Iamblichus also used some material taken from earlier writings in the 4th century BC by Aristotle's students Dicaearchus, Aristoxenus, and Heraclides Ponticus, which, when it can be identified, is generally considered to be the most reliable.

Early life There is not a single detail in the life of Pythagoras that stands uncontradicted. But it is possible, from a more or less critical selection of the data, to construct a plausible account. Herodotus and Isocrates agree that Pythagoras was the son of Mnesarchus and that he was born on the Greek island of Samos in the eastern Aegean. Mnesarchus is said to have been a gem-engraver or a wealthy merchant, but his ancestry is disputed and unclear. Apollonius of Tyana writes that Pythagoras's mother was Pythaïs, a woman who was said to be a descendant of Ancaeus, the mythical founder of Samos. Iamblichus tells the story that the Pythia prophesied to her while she was pregnant with him that she would give birth to a man supremely beautiful, wise, and beneficial to humankind, and Iamblichus also held to the belief that his mother descended from Ancaeus. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, which would give a date of birth around 570 BC. Pythagoras's name led him to be associated with Pythian Apollo (Pūthíā); Aristippus of Cyrene in the 4th century BC explained his name by saying, "He spoke [ἀγορεύω, agoreúō] the truth no less than did the Pythian [πυθικός, puthikós]". During Pythagoras's formative years, Samos was a thriving cultural hub known for its feats of advanced architectural engineering, including the building of the Tunnel of Eupalinos, and for its riotous festival culture. It was a major center of trade in the Aegean where traders brought goods from the Near East. According to Christiane L. Joost-Gaugier, these traders almost certainly brought with them Near Eastern ideas and traditions. Pythagoras's early life also coincided with the flowering of early Ionian natural philosophy. He was a contemporary of the philosophers Anaximander and Anaximenes and of the historian Hecataeus, all of whom lived in Miletus, across the sea from Samos.

Reputed travels

… excerpt ends here. Continue reading the full article.

Illustrations

Pythagoras illustration
Pythagoras: Pythagoras is credited with having devised the tetractys,[87] an important sacred symbol in later Pythagoreanism.[88][89]
Pythagoras is credited with having devised the tetractys,[87] an important sacred symbol in later Pythagoreanism.[88][89]
Pythagoras: Pythagoreans Celebrate the Sunrise (1869) by Fyodor Bronnikov
Pythagoreans Celebrate the Sunrise (1869) by Fyodor Bronnikov
Pythagoras: French manuscript from 1512/1514, showing Pythagoras turning his face away from fava beans in revulsion
French manuscript from 1512/1514, showing Pythagoras turning his face away from fava beans in revulsion
Pythagoras: The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Worked examples

Example 1 — a first encounter with Pythagoras

Start with the simplest possible case. Write down what Pythagoras claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pythagoras before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pythagoras ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pythagoras

In research
Pythagoras appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Pythagoras in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Pythagoras is common in secondary-school and first-year university syllabi. It links to neighbouring topics 490s BC deaths, 570s BC births, 5th-century BC Greek mathematicians, so understanding it makes those chapters shorter.
In everyday life
Look for Pythagoras outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pythagoras in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Pythagoras means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Pythagoras out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pythagoras in simple terms?

Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through…

Why does Pythagoras matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pythagoras?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pythagoras.

Tags

  • 490s BC deaths
  • 570s BC births
  • 5th-century BC Greek mathematicians
  • 5th-century BC Greek philosophers
  • 5th-century BC religious leaders
  • 6th-century BC Greek philosophers
  • 6th-century BC mathematicians
  • 6th-century BC religious leaders
  • Ancient Greek geometers
  • Ancient Greek metaphysicians
  • Ancient Greek music theorists
  • Ancient Greek philosophers of mind

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