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Pythagoras

Greek philosopher and mathematician (c. 570 – c. 495 BC)

Pythagoras of Samos (Πυθαγόρας; c. 570 BC – c. 496 BC) was an Ionian Greek philosopher often revered as a great mathematician, mystic, scientist, and putative founder of the traditions referred to as Pythagoreanism.

Rest satisfied with doing well, and leave others to talk of you as they please.
As long as Man continues to be the ruthless destroyer of lower living beings, he will never know health or peace. For as long as men massacre animals, they will kill each other. Indeed, he who sows the seed of murder and pain cannot reap joy and love.

Quotes

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Let no one persuade you by word or deed to do or say whatever is not best for you.
 
The soul of man is divided into three parts, intelligence, reason, and passion. Intelligence and passion are possessed by other animals, but reason by man alone.
 
Number is the ruler of forms and ideas, and the cause of gods and daemons.
 
Reason is immortal, all else mortal.
 
Truth is so great a perfection, that if God would render himself visible to men, he would choose light for his body and truth for his soul.
 
Sooner throw a pearl at hazard than an idle or useless word; and do not say a little in many words, but a great deal in a few.
 
There is no word or action but has its echo in Eternity.
 
Thought is an Idea in transit, which when once released, never can be lured back, nor the spoken word recalled.
 
Time is the soul of this world.
  • Dear youths, I warn you cherish peace divine,
    And in your hearts lay deep these words of mine.
  • Τὴν δ' ἀνθρώπου ψυχὴν διῃρῆσθαι τριχῆ, εἴς τε νοῦν καὶ φρένας καὶ θυμόν. νοῦν μὲν οὖν καὶ θυμὸν εἶναι καὶ ἐν τοῖς ἄλλοις ζῴοις, φρένας δὲ μόνον ἐν ἀνθρώπῳ.
  • ἀλλήλοις θ᾽ ὁμιλεῖν, ὡς τοὺς μὲν φίλους ἐχθροὺς μὴ ποιῆσαι, τοὺς δ᾽ ἐχθροὺς φίλους ἐργάσασθαι. ἴδιόν τε μηδὲν ἡγεῖσθαι.
  • ἐν ὀργῇ μήτε τι λέγειν μήτε πράσσειν
    • In anger we should refrain both from speech and action.
    • As quoted in Lives and Opinions of Eminent Philosophers, "Pythagoras", Sect. 23–24, as translated in Dictionary of Quotations (1906) by Thomas Benfield Harbottle, p. 370
  • Reason is immortal, all else mortal.
    • As quoted in Lives and Opinions of Eminent Philosophers, Sect. 30, as translated by Robert Drew Hicks (1925); also in The Demon and the Quantum: From the Pythagorean Mystics to Maxwell's Demon (2007) by Robert J. Scully, Marlan O. Scully, p. 11
  • The most momentous thing in human life is the art of winning the soul to good or to evil.
    • As quoted in Lives and Opinions of Eminent Philosophers, as translated by Robert Drew Hicks (1925)
    • Variant translation: The most momentous thing in human life is the art of winning the soul to good or evil.
      • As quoted in Ionia, a Quest (1954) by Freya Stark, p. 94
  • Power is the near neighbour of necessity.
    • As quoted in Aurea Carmina (8) by Hierocles of Alexandria, as translated in Dictionary of Quotations (1906) by Thomas Benfield Harbottle, p. 356
  • Number is the ruler of forms and ideas, and the cause of gods and daemons.
    • As quoted in Life of Pythagoras (c. 300) by Iamblichus of Chalcis, as translated by Thomas Taylor (1818)
    • Variants:
    • Number rules the universe.
    • As quoted in The Story of a Number‎ (1905) by E. Maor; also in Comic Sections (1993) by Desmond MacHale
  • Sobriety is the strength of the soul, for it preserves its reason unclouded by passion.
    • As quoted in The History of Philosophy: From the Earliest Times to the Beginning of the Present Century (1819) by William Enfield
    • Sobriety is the strength of the mind; for it preserves reason unclouded by passion.
      • As quoted in Bible of Reason (1831) by Benjamin F. Powell, p. 157
    • Strength of mind rests in sobriety; for this keeps your reason unclouded by passion.
      • As quoted in Dictionary of Quotations from Ancient and Modern English and Foreign Sources (1899) by James Wood
  • None but God is wise.
  • Silence is better than unmeaning words.
    • As quoted in Encyclopaedia Americana (1832) Vol. X, p. 445 edited by Francis Lieber, E. Wigglesworth, and Thomas Gamaliel Bradford
  • If there be light, then there is darkness; if cold, heat; if height, depth; if solid, fluid; if hard, soft; if rough, smooth; if calm, tempest; if prosperity, adversity; if life, death.
    • As quoted in Bibliotheca Sacra and Theological Review by ? Vol. IV, No. 8 (1847) by Dallas Theological Seminary, p. 107
  • Rest satisfied with doing well, and leave others to talk of you as they please.
    • As quoted in The World's Laconics: Or, The Best Thoughts of the Best Authors (1853) by Everard Berkeley
    • Variant: Rest satisfied with doing well, and leave others to talk of you as they will.
  • As soon as laws are necessary for men, they are no longer fit for freedom.
    • As quoted in Short Sayings of Great Men: With Historical and Explanatory Notes‎ (1882) by Samuel Arthur Bent, p. 454
  • Friends are as companions on a journey, who ought to aid each other to persevere in the road to a happier life.
    • As quoted in Gems of Thought: Being a Collection of More Than a Thousand Choice Selections, Or Aphorisms, from Nearly Four Hundred and Fifty Different Authors, and on One Hundred and Forty Different Subjects (1888). p. 97 by Charles Northend
  • Anger begins in folly, and ends in repentance.
    • As quoted in Treasury of Thought: Forming an Encyclopædia of Quotations from Ancient and Modern Authors (1894) by Maturin Murray Ballou
  • Choose always the way that seems the best, however rough it may be; custom will soon render it easy and agreeable.
    • As quoted in A Dictionary of Thoughts: Being a Cyclopedia of Laconic Quotations from the Best Authors of the World, both Ancient and Modern (1908) by Tyron Edwards, p. 101
  • It is better wither to be silent, or to say things of more value than silence. Sooner throw a pearl at hazard than an idle or useless word; and do not say a little in many words, but a great deal in a few.
    • As quoted in A Dictionary of Thoughts: Being a Cyclopedia of Laconic Quotations from the Best Authors of the World, both Ancient and Modern (1908) by Tyron Edwards, p. 525
  • Truth is so great a perfection, that if God would render himself visible to men, he would choose light for his body and truth for his soul.
    • As quoted in A Dictionary of Thoughts: Being a Cyclopedia of Laconic Quotations from the Best Authors of the World, both Ancient and Modern (1908) by Tyron Edwards, p. 592
  • There is no word or action but has its echo in Eternity.
    Thought is an Idea in transit, which when once released, never can be lured back, nor the spoken word recalled. Nor ever can the overt act be erased All that thou thinkest, sayest, or doest bears perpetual record of itself, enduring for Eternity.
    • As quoted in Pythagoron: The Religious, Moral, and Ethical Teachings of Pythagoras (1947) by Hobart Huson, p. 99
  • There is geometry in the humming of the strings, there is music in the spacing of the spheres.
    • As quoted in The Mystery of Matter‎ (1965) edited by Louise B. Young, p. 113
  • As long as Man continues to be the ruthless destroyer of lower living beings, he will never know health or peace. For as long as men massacre animals, they will kill each other. Indeed, he who sows the seed of murder and pain cannot reap joy and love.
    • Attribution to Pythagoras by Ovid, as quoted in The Extended Circle: A Dictionary of Humane Thought (1985) by Jon Wynne-Tyson, p. 260; also in Vegetarian Times, No. 168 (August 1991), p. 4
  • Time is the soul of this world.
  • Most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge.
    • As quoted in The Golden Ratio (2002) by Mario Livio
  • Man know thyself; then thou shalt know the Universe and God.
    • As quoted in Fragments of Reality: Daily Entries of Lived Life (2006) by Peter Cajander, p. 109
  • A blow from your friend is better than a kiss from your enemy.
    • As quoted in Geary's Guide to the World's Great Aphorists‎ (2007) by James Geary, p. 118
  • Write in the sand the flaws of your friend.
    • As quoted in Geary's Guide to the World's Great Aphorists‎ (2007) by James Geary
  • Educate the children and it won't be necessary to punish the men.
    • As quoted in Geary's Guide to the World's Great Aphorists‎ (2007) by James Geary

The Symbols

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English translations of the Symbols of Pythagoras recorded by Iamblichus of Chalcis from those in The Golden Verses of Pythagoras and other Pythagorean Fragments (1904); selected and arranged by Florence M. Firth
 
Disbelieve nothing wonderful concerning the gods, nor concerning divine dogmas.
 
Assist a man in raising a burden; but do not assist him in laying it down.
 
Abstain from beans.
  • When going to the temple to adore Divinity neither say nor do any thing in the interim pertaining to the common affairs of life.
    • Symbol 1
  • Sacrifice and adore unshod.
    • Symbol 3
  • Disbelieve nothing wonderful concerning the gods, nor concerning divine dogmas.
    • Symbol 4
  • Declining from the public ways, walk in unfrequented paths.
    • Symbol 5
  • Govern your tongue before all other things, following the gods.
    • Symbol 7
  • The wind is blowing, adore the wind.
    • Symbol 8
  • Cut not fire with a sword.
    • Symbol 9
    • Variant translation: Poke not the fire with a sword.
      • As quoted in Short Sayings of Great Men: With Historical and Explanatory Notes‎ (1882) by Samuel Arthur Bent, p. 455
  • Assist a man in raising a burden; but do not assist him in laying it down.
    • Symbol 11
  • Step not beyond the beam of the balance.
    • Symbol 14
  • Having departed from your house, turn not back; for the furies will be your attendants.
    • Symbol 15
  • Eat not the heart.
    • Symbol 30; explained in the edition used here: "This Symbol signifies that it is not proper to divulse the union and consent of the universe. And still further it signifies this, Be not envious, but philanthropic and communicative; and from this it exhorts us to philosophize. For philosophy alone among the sciences and arts is neither pained with the goods of others, nor rejoices in evils of neighbours, these being allied and familiar by nature, subject to the like passions, and exposed to one common fortune; and evinces that all men are equally incapable of foreseeing future events. Hence it exhorts us to sympathy and mutual love, and to be truly communicative, as it becomes rational animals.
    • Variant translation: Do not eat your heart.
  • Eat not the brain.
    • Symbol 31
  • Κυάμων ἀπέχεσθαι
    • Abstain from beans.
    • Symbol 37; This was long thought by many to be simply a dietary proscription, and often ridiculed, but many consider it to have originally been intended as advice against getting involved in politics, for voting on issues in his time was often done by using differently colored beans. Others have stated that it might signify a more general admonition against relying on the votes of people to determine truths of reality.
  • Abstain from animals.
    • Symbol 39; explained in the edition used here: "This Symbol exhorts to justice, to all the honour of kindred, to the reception of similar life, and to many other things of a like kind."

The Golden Verses

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Quotes cited as from the Golden Verses, but drawn from various translations.
 
Above all things reverence thy Self.
 
Work at these things, practice them... they are what will put you on the path of divine virtue — yes, by the one who entrusted our soul with the tetraktys, source of ever-flowing nature.
 
Meditate upon my counsels; love them; follow them; To the divine virtues will they know how to lead thee...
 
Holding fast to these things, you will know the worlds of gods and mortals which permeates and governs everything.
 
You will know, as is right, nature similar in all respects, so that you will neither entertain unreasonable hopes nor be neglectful of anything.
  • Above and before all things, worship GOD!
    • As quoted in The Sayings of the Wise: Or, Food for Thought: A Book of Moral Wisdom, Gathered from the Ancient Philosophers (1555) by William Baldwin [1908 edition]
    • Variant translation: Honor first the immortal gods, in the manner prescribed, and respect the oath.
      Next, honor the reverent heroes and the spirits of the dead by making the traditional sacrifices.
      Honor your parents and your relatives. As for others, befriend whoever excels in virtue.
      Yield to kind words and helpful deeds, and do not hate your friend for a trifling fault as you are able. For ability is near to necessity.
      • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999) ISBN 0-9653774-5-8
  • Above all things reverence thy Self.
    • Variant translations:
    • Respect yourself above all.
    • Above all things reverence thy self.
      Above all things, respect yourself.
      Above the cloud with its shadow is the star with its light. Above all things reverence thyself.
  • Work at these things, practice them, these are the things you ought to desire; they are what will put you on the path of divine virtue — yes, by the one who entrusted our soul with the tetraktys, source of ever-flowing nature. Pray to the gods for success and get to work.
    • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999)
  • Practice justice in word and deed, and do not get in the habit of acting thoughtlessly about anything.
    • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999)
  • Know that death comes to everyone, and that wealth will sometimes be acquired, sometimes lost. Whatever griefs mortals suffer by divine chance, whatever destiny you have, endure it and do not complain. But it is right to improve it as much as you can, and remember this: Fate does not give very many of these griefs to good people.
    • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999)
  • Many words befall men, mean and noble alike; do not be astonished by them, nor allow yourself to be constrained.
    If a lie is told, bear with it gently.
    But whatever I tell you, let it be done completely.
    Let no one persuade you by word or deed to do or say whatever is not best for you.
    • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999)
  • Let not sleep fall upon thy eyes till thou has thrice reviewed the transactions of the past day. Where have I turned aside from rectitude? What have I been doing? What have I left undone, which I ought to have done? Begin thus from the first act, and proceed; and, in conclusion, at the ill which thou hast done, be troubled, and rejoice for the good.
    • As translated in The Rambler No. 8 (14 April 1750) by Samuel Johnson
    • Let not sleep e'er close thy eyes
      Without thou ask thyself: What have I omitted and what done?
      Abstain thou if 'tis evil; persevere if good.
    • Do not let sleep close your tired eyes until you have three times gone over the events of the day. 'What did I do wrong? What did I accomplish? What did I fail to do that I should have done?' Starting from the beginning, go through to the end. Then, reproach yourself for the things you did wrong, and take pleasure in the good things you did.
      • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999)
  • Meditate upon my counsels; love them; follow them;
    To the divine virtues will they know how to lead thee.
    I swear it by the One who in our hearts engraved
    The sacred Tetrad, symbol immense and pure,
    Source of Nature and model of the Gods.
  • Holding fast to these things, you will know the worlds of gods and mortals which permeates and governs everything. And you will know, as is right, nature similar in all respects, so that you will neither entertain unreasonable hopes nor be neglectful of anything.
    • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999)
  • You will know that wretched men are the cause of their own suffering, who neither see nor hear the good that is near them, and few are the ones who know how to secure release from their troubles. Such is the fate that harms their minds; like pebbles they are tossed about from one thing to another with cares unceasing. For the dread companion Strife harms them unawares, whom one must not walk behind, but withdraw from and flee.
    • As quoted in Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook (1999)
  • There is geometry in the humming of the strings. There is music in the spacings of the spheres.
    • As quoted in the preface of the book entitled Music of the Spheres by Guy Murchie (1961)

Florilegium

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Remind yourself that all men assert that wisdom is the greatest good, but that there are few who strenuously seek out that greatest good.
Quotes of Pythagoras from the Florilegium of Stobaeus, using various translations, including those from "Pythagorean Ethical Sentences From Stobæus" in The Golden Verses of Pythagoras and other Pythagorean Fragments (1904); selected and arranged by Florence M. Firth
  • Do not even think of doing what ought not to be done.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • Choose rather to be strong in soul than in body.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
    • Choose rather to be strong of soul than strong of body.
      • As quoted in Florilegium, I.22, as translated in Dictionary of Quotations (1906) by Thomas Benfield Harbottle, p. 396
  • It is difficult to walk at one and the same time many paths of life.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • It is requisite to choose the most excellent life; for custom will make it pleasant. Wealth is an infirm anchor, glory is still more infirm; and in a similar manner, the body, dominion, and honour. For all these are imbecile and powerless. What then are powerful anchors. Prudence, magnanimity, fortitude. These no tempest can shake. This is the Law of God, that virtue is the only thing that is strong; and that every thing else is a trifle.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • It is requisite to defend those who are unjustly accused of having acted injuriously, but to praise those who excel in a certain good.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • Neither will the horse be adjudged to be generous, that is sumptuously adorned, but the horse whose nature is illustrious; nor is the man worthy who possesses great wealth, but he whose soul is generous.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • When the wise man opens his mouth, the beauties of his soul present themselves to the view, like the statues in a temple
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • Remind yourself that all men assert that wisdom is the greatest good, but that there are few who strenuously seek out that greatest good.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • Despise all those things which when liberated from the body you will not want; invoke the Gods to become your helpers.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • Wind indeed increases fire, but custom love.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • Those alone are dear to Divinity who are hostile to injustice.
    • "Pythagorean Ethical Sentences From Stobæus" (1904)
  • None can be free who is a slave to, and ruled by, his passions.
    • As quoted in Florilegium, XVIII, 23, as translated in Dictionary of Quotations (1906) by Thomas Benfield Harbottle, p. 368
    • No one is free who has not obtained the empire of himself.
      • As translated by Nicholas Rowe(1732)
    • No man is free who cannot command himself.
      • As quoted in Moral Encyclopaedia, Or, Varlé's Self-instructor, No. 3 (1831) by by Charles Varle
    • No man is free who cannot control himself.
      • As quoted in 25 Days to Better Thinking and Better Living: A Guide for Improving Every Aspect of Your Life (2006) by Linda Elder and Richard Paul
  • It is not proper either to have a blunt sword or to use freedom of speech ineffectually.
    Neither is the sun to be taken from the world, nor freedom of speech from erudition.
  • Not frequently man from man.
    • As quoted in the translation of Thomas Taylor (1818); This has been interpreted as being an exhortation to moderation in homosexual liaisons.

The Sayings of the Wise (1555)

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Dispose thy Soul to all good and necessary things!
Quotes of Pythagoras as translated in The Sayings of the Wise: Or, Food for Thought: A Book of Moral Wisdom, Gathered from the Ancient Philosophers (1555) by William Baldwin [1908 edition]
 
True and perfect Friendship is, to make one heart and mind of many hearts and bodies.
 
The best and greatest winning is a true friend; and the greatest loss is the loss of time.
  • When a reasonable Soul forsaketh his divine nature, and becometh beast-like, it dieth. For though the substance of the Soul be incorruptible: yet, lacking the use of Reason, it is reputed dead; for it loseth the Intellective Life.
  • A good Soul hath neither too great joy, nor too great sorrow: for it rejoiceth in goodness; and it sorroweth in wickedness. By the means whereof, when it beholdeth all things, and seeth the good and bad so mingled together, it can neither rejoice greatly; nor be grieved with over much sorrow.
  • Order thyself so, that thy Soul may always be in good estate; whatsoever become of thy body.
  • Dispose thy Soul to all good and necessary things!
  • Patience cometh by the grace of the Soul.
  • True and perfect Friendship is, to make one heart and mind of many hearts and bodies.
  • He is not rich, that enjoyeth not his own goods.
  • By Silence, the discretion of a man is known: and a fool, keeping Silence, seemeth to be wise.
  • A fool is known by his Speech; and a wise man by Silence.
  • The King that followeth Truth, and ruleth according to Justice, shall reign quietly: but he that doth the contrary, seeketh another to reign for him.
  • Tell not abroad what thou intendest to do; for if thou speed not, thou shalt be mocked!
  • If thy fellows hurt thee in small things, suffer it! and be as bold with them!
  • Take not thine enemy for thy friend; nor thy friend for thine enemy!
  • Rejoice not in another man's misfortune!
  • Let thy mind rule thy tongue!
  • Hear gladly!
  • Attempt nothing above thy strength!
  • Be not hasty to speak; nor slow to hear!
  • Wish not the thing, which thou mayest not obtain!
  • If thou intend to do any good; tarry not till to-morrow! for thou knowest not what may chance thee this night.
  • Use examples; that such as thou teachest may understand thee the better!
  • Reason not with him, that will deny the principal truths!
  • Honor Wisdom; and deny it not to them that would learn; and shew it unto them that dispraise it! Sow not the sea fields!
  • Wisdom thoroughly learned, will never be forgotten.
    Science is got by diligence; but Discretion and Wisdom cometh of GOD.
  • Without Justice, no realm may prosper.
  • Happy is that City that hath a wise man to govern it.
  • To use Virtue is perfect blessedness.
  • Envy has been, is, and shall be, the destruction of many. What is there, that Envy hath not defamed, or Malice left undefiled? Truly, no good thing.
  • A solitary man is a God, or a beast.
  • None but a Craftsman can judge of a craft.
  • Repentance deserveth Pardon.
  • The best and greatest winning is a true friend; and the greatest loss is the loss of time.
  • It is better to suffer, than to do, wrong.
  • He is worst of all, that is malicious against his friends.
  • Evil destroyeth itself.
  • Better be mute, than dispute with the Ignorant.


Disputed

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Virtue is harmony.
  • Virtue is harmony.
    • This is often published as a direct quote of Pythagoras, but seems to be derived from the account of Diogenes Laertius of Pythagorean doctrines, where he simply describes the statement as a precept of his followers. In the translation of C. D. Yonge (1853) it is rendered, in regard to Pythagoreans:
They also say, that the most important privilege in man is, the being able to persuade his soul to either good or bad. And that men are happy when they have a good soul; yet, that they are never quiet, and that they never retain the same mind long. Also, that an oath is justice; and that on that account, Jupiter is called Jupiter of Oaths. Also, that virtue is harmony, and health, and universal good, and God; on which account everything owes its existence and consistency to harmony. Also, that friendship is a harmonious equality.
  • The oldest, shortest words— "yes" and "no"— are those which require the most thought.
    • Quoted in Grist: The National Conference on State Parks magazine, 1965 link but not found earlier. Appears to be a misconstrual of a story in Ausonius:
est etenim comis brevitas. sic fama renatum
Pythagoram docuisse refert : cum multa loquaces
ambiguis sererent verbis, contra omnia solum
“est” respondebat vel “non”. o certa loquendi
regula!
(“For brevity is indeed polite. Thus, legend has it that Pythagoras, once reborn, taught the following: when babblers would string together mouthfuls with indecisive words, to everything he used to respond simply ‘Yes’ or ‘No’. O what a steadfast rule of speech!”) link


Misattributed

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  • There is nothing so easy but that it becomes difficult when you do it reluctantly.
    • Terence, in Heauton Timoroumenos [The Self-Tormentor]
  • Concern should drive us into action and not into a depression.
    • The Collected Works of Karen Horney‎ (1957) by Karen Horney, p. 154: "We may feel genuinely concerned about world conditions, though such a concern should drive us into action and not into a depression."
  • In this theater of man's life it is reserved only for God and angels to be lookers on.
    • Francis Bacon, in The Advancement of Learning (1605) Book II, xx, 8.

Quotes about Pythagoras

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It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle of unity beneath the forms, changes, and other phenomena of the universe. ~ H.P. Blavatsky,
 
I have often admired the mystical way of Pythagoras, and the secret Magic of numbers. ~ Thomas Browne
 
It was Pythagoras who first called heaven kosmos, because it is perfect, and "adorned" with infinite beauty and living beings.
 
It was through philosophy, he said, that he had come to be surprised at nothing. ~ Plutarch
 
I do not know of any other man who has been as influential as he was in the sphere of thought. ~ Bertrand Russell
 
Even the seeming remoteness of Pythagorean teaching helps one to realize that the current world view, while it seems destined to dominate the planet, is fleeting and temporary and, like others before it, will pass. ~ John Strohmeier and Peter Westbrook
  • Ten is the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unity. And again, Pythagoras maintains, the power of the number 10 lies in the number 4, the tetrad. This is the reason: if one starts at the unit (1) and adds the successive number up to 4, one will make up the number 10 (1+2+3+4 = 10). And if one exceeds the tetrad, one will exceed 10 too.... So that the number by the unit resides in the number 10, but potentially in the number 4. And so the Pythagoreans used to invoke the Tetrad as their most binding oath: "By him that gave to our generation the Tetractys, which contains the fount and root of eternal nature..."
  • Pythagoras... assumed as first principles the numbers and symmetries existing among them, which he calls harmonies, and the elements compounded of both, that are called geometrical. ...he says that the nature of Number is the Decad.
  • Let us not link ourselves with the vilifiers of Plato and the persecutors of Confucius. They were oppressed by citizens who were considered the pride of the country. Thus has the world raised its hand against the great Servitors. Be assured that the Brotherhood formed by Pythagoras appeared dangerous in the eyes of the city guard. Paracelsus was a target for mockery and malignance. Thomas Vaughan seemed to be an outcast, and few wished to meet with him. Thus was the reign of darkness manifested.
  • If one examines the reasons for the persecution of the best minds of different nations, and compares the reasons for the persecution and banishment of Pythagoras, Anaxagoras, Socrates, Plato, and others, one can observe that in each case the accusations and reasons for banishment were almost identical and unfounded. But in the following centuries full exoneration came, as if there had never been any defamation. It would be correct to conclude that such workers were too exalted for the consciousness of their contemporaries, and the sword of the executioner was ever ready to cut off a head held high. Pericles was recognized in his time only after people had reduced him to a sorry state. Only in that state could his fellow citizens accept him as an equal! A book should be written about the causes of the persecution of great individuals. By comparing the causes is it possible to trace the evil will.
  • Pythagoras, it seems, did not only call the supreme Deity a monad, but also a tetrad, or tetractys... It is, in the golden verses, said to be the fountain of the eternal nature; and by Hierocles, the maker of all things, the intelligent god, the cause of the heavenly and sensible god, that is, of the animated world or heaven. The later Pythogoreans endeavour to give reasons why God should be called Tetractys, from certain mysteries in the number four; but... much more probable... this name was really nothing else but the tetragrammaton, or that proper name of the supreme God amongst the Hebrews, consisting of four letters; nor is it strange Pythagoras should be so well acquainted with the name Jehovah, since, besides travelling into other parts of the East, he is by Josephus, Porphyry, and others, to have conversed with the Hebrews also.
  • These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent states.
  • They thought they found in numbers, more than in fire, earth, or water, many resemblances to things which are and become; thus such and such an attribute of numbers is justice, another is soul and mind, another is opportunity, and so on; and again they saw in numbers the attributes and ratios of the musical scales. Since, then, all other things seemed in their whole nature to be assimilated to numbers, while numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.
  • Whenever he heard a person who was making use of his symbols, he immediately took him into his circle, and made him a friend.
    • Aristoxenus, as quoted in Birth of the Symbol: Ancient Readers at the Limits of Their Texts (2004) by Peter Struck
  • But it is, however, in the teachings of Pythagoras that we find the closest and most frequent identities of teachings and argumentation [about as to the nature and position of the Gods], explained as due to Pythagoras himself having visited India and learned his philosophy there, as tradition asserts. In later centuries we find some peculiarly Sânkhyan and Buddhist ideas playing a prominent part in Gnostic thought.
  • The school of the Pythagoras and those of the Neo-Platonists kept up the tradition for Greece, and we know that Pythagoras gained some of his learning in India, while Plato studied, and was initiated in the schools of Egypt. More precise information has been published of the Grecian schools than of others; the Pythagorean had pledged disciples as well as an outer discipline, the inner circle passing through three degrees during five years of probation.
  • It is further interesting to remark that the finest characters among women with which ancient Greece presents us were formed in the school of Pythagoras, and the same is true of the men.
  • The world is always ungrateful to its great men. Florence has built a statue to Galileo, but hardly even mentions Pythagoras. The former had a ready guide in the treatises of Copernicus, who had been obliged to contend against the universally established Ptolemaic system. But neither Galileo nor modern astronomy discovered the emplacement of the planetary bodies. Thousands of ages before, it was taught by the sages of Middle Asia, and brought thence by Pythagoras, not as a speculation, but as a demonstrated science. "The numerals of Pythagoras," says Porphyry, "were hieroglyphical symbols, by means whereof he explained all ideas concerning the nature of all things." Verily, then, to antiquity alone have we to look for the origin of all things. p. 31
  • Pythagoras called his Gnosis “the knowledge of things that are,” or ἡ γνῶσις τῶν ὄντων, and preserved that knowledge for his pledged disciples only: for those who could digest such mental food and feel satisfied; and he pledged them to silence and secrecy. Occult alphabets and secret ciphers are the development of the old Egyptian hieratic writings, the secret of which was, in the days of old, in the possession only of the Hierogrammatists, or initiated Egyptian priests.
  • Pythagoras repeats our archaic doctrine when stating that the Ego (Nous) is eternal with Deity; that the soul only passed through various stages to arrive at divine excellence; while thumos returned to the earth, and even the phren, the lower Manas, was eliminated.
    • H.P. Blavatsky in The Key to Theosophy p. 91, (1889)
  • Pythagoras was said to have been the first man to call himself philosopher; in fact, the world is indebted to him for the word philosopher. Before that time the wise men called themselves sages, which was interpreted to mean those who know. Pythagoras was more modest. He coined the word philosopher, which he defined as one who is attempting to find out.
    • Grover W. Brunton, Pythagoras: The First Philosopher and Discoverer of the Forty-seventh Problem of Euclid (2005)
  • There remains no firm basis for the belief that Pythagoras was a geometer and in any case no attestation of his having written anything.
  • The apparently ancient reports of the importance of Pythagoras and his pupils in laying the foundations of mathematics crumble on touch, and what we can get hold of is not authentic testimony by the efforts latecomers to paper over a crack, which they obviously found surprising, by the use of various kinds of reconstruction and reinterpretation. On the other hand, there are ancient and unassailable indications of a Greek mathematics antedating Pythagoras and quite outside his sphere of influence.
  • I wished to show that Pythagoras, the first founder of the vegetable regimen, was at once a very great physicist and a very great physician; that there has been no one of a more cultured and discriminating humanity; that he was a man of wisdom and of experience; that his motive in commending and introducing the new mode of living was derived not from any extravagant superstition, but from the desire to improve the health and the manners of men.
  • Koyré's exaltation of the "Platonic and Pythagorean" elements of the Scientific Revolution... was based on a demonstrably false understanding of how Galileo reached his conclusions. Koyré asserted that Galileo merely used experiments as a check on the theories he devised by mathematical reasoning. But later research has definitively established that Galileo's experiments preceded his attempts to give a mathematical account of their results.
    • Clifford D. Conner, A People's History of Science (2005)
  • Pythagoras could not have been the discoverer of the relation, because... this property was known and used by scholars and artisans of Oriental lands thousands of years before Pythagoras... While deductive geometry is barely more than twenty-five hundred years old, empirical geometry is probably as old as civilization itself.
  • Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy.
  • Let us suppose that we have set the problem of finding a solution to the equation   This is a problem for which the Babylonians around 1700 BC found the excellent approximation   ...This is the identical problem which Pythagoras asserted had no fractional solution and in whose honor he was supposed to have sacrificed a hecatomb of oxen—the problem which caused the existentialist crisis in ancient Greek mathematics. The   exists (as the diagonal of the unit square); yet it does not exist (as a fraction)!
  • A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines. With consistency a great soul has simply nothing to do. He may as well concern himself with his shadow on the wall. Speak what you think now in hard words, and to-morrow speak what to-morrow thinks in hard words again, though it contradict every thing you said to-day. — "Ah, so you shall be sure to be misunderstood." — Is it so bad, then, to be misunderstood? Pythagoras was misunderstood, and Socrates, and Jesus, and Luther, and Copernicus, and Galileo, and Newton, and every pure and wise spirit that ever took flesh. To be great is to be misunderstood.
  • But there was among them a man of prodigious knowledge who acquired the profoundest wealth of understanding and was the greatest master of skilled arts of every kind; for, whenever he willed with his whole heart, he could with ease discern each and every truth in his ten—nay, twenty—men's lives.
    • Empedocles (c. 460 BC) as quoted by Sir Thomas Little Heath, A History of Greek Mathematics, Vol. 1, p.65, citing Diog. L. viii. 54 and Porph. V. Pyth. 30 (Fr. 129 in Vorsokratiker i3, p. 272. 15-20).
  • Pythagoras is the founder of European culture in the Western Mediterranean sphere.
  • The games which can be built up from the simple idea of dots and lines... can be a productive source of teaching material. After all, figurate numbers provided the Pythagoreans and neo-Pythagoreans with important theorems about the summing of series.
    • Graham Flegg, Numbers: Their History and Meaning (1983)
  • Nor need you question but that Pythagoras a long time be­fore he found the demonstration for which he offered the Heca­tomb, had been certain, that the square of the side subtending the right angle in a rectangle triangle, was equal to the square of the other two sides: and the certainty of the conclusion condu­ced not a little to the investigating of the demonstration, un­derstanding me alwayes to mean in demonstrative Sciences.
  • Some fundamental unity was surely to be discerned either in the matter or the structure of things. The Ionic philosophers chose the former field: Pythagoras took the latter. ...The geometry which he had learnt in Egypt was merely practical. ...It was natural to nascent philosophy to draw, by false analogies, and the use of a brief and deceptive vocabulary,2 enormous conclusions from a very few observed facts: and it is not surprising if Pythagoras, having learnt in Egypt that number was essential to the exact description of forms and of the relations of forms, concluded that number was the cause of form and so of every other quality. Number, he inferred, is quantity and quantity is form and form is quality.
    Footnote2 Primitive men, on seeing a new thing, look out especially for some resemblance in it to a known thing, so that they may call both by the same name. This developes a habit of pressing small and partial analogies. It also causes many meanings to be at attached to the same word. Hasty and confused theories are the inevitable result.
  • Pythagoras was indeed the first man to call himself a philosopher. Others before had called themselves wise (sophos), but Pythagoras was the first to call himself a philosopher, literally a lover of wisdom.
    More importantly, for Pythagoras and his followers philosophy was not merely an intellectual pursuit, but a way of life, the aim of which was the assimilation to God.
    • Kenneth Sylvan Guthrie and David R. Fideler, The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy (1919)
  • Pythagoras, the son of Mnesarchus, was the most learned of all men of history; and having selected from these writings, he thus formed his own wisdom and extensive learning, and mischievous art.
  • No one will deny that the soul of Pythagoras was sent to mankind from Apollo's domain, having either been one of his attendants, or more intimate associates, which may be inferred both from his birth, and his versatile wisdom.
  • After his father's death, though he was still but a youth, his aspect was so venerable, and his habits so temperate that he was honored and even reverenced by elderly men, attracting the attention of all who saw and heard him speak, creating the most profound impression. That is the reason that many plausibly asserted that he was a child of the divinity. Enjoying the privilege of such a renown, of an education so thorough from infancy, and of so impressive a natural appearance he showed that he deserved all these advantages by deserving them, by the adornment of piety and discipline, by exquisite habits, by firmness of soul, and by a body duly subjected to the mandates of reason. An inimitable quiet and serenity marked all his words and actions, soaring above all laughter, emulation, contention, or any other irregularity or eccentricity; his influence at Samos was that of some beneficent divinity. His great renown, while yet a youth, reached not only men as illustrious for their wisdom as Thales at Miletus, and Bias at Prione, but also extended to the neighboring cities. He was celebrated everywhere as the "long-haired Samian," and by the multitude was given credit for being under divine inspiration.
    • Iamblichus of Chalcis in Life of Pythagoras translated by Thomas Taylor; Ch. 2: Youth, Education, Travels
  • Pythagoras conceived that the first attention that should be given to men should be addressed to the senses, as when one perceives beautiful figures and forms, or hears beautiful rhythms and melodies. Consequently he laid down that the first erudition was that which subsists through music's melodies and rhythms, and from these he obtained remedies of human manners and passions, and restored the pristine harmony of the faculties of the soul.
  • Pythagoras taught, accordingly, that he had himself been originally Euphorbus, and then Callides, thirdly Hermotimus, fourthly Pyrrhus, and lastly Pythagoras; and that those things which had existed, after certain revolutions of time, came into being again; so that nothing in the world should be thought of as new. He said that true philosophy was a meditation on death; that its daily struggle was to draw forth the soul from the prison of the body into liberty: that our learning was recollection, and many other things which Plato works out in his dialogues, especially in the Phaedo and Timæus. For Plato, after having formed the Academy and gained innumerable disciples, felt that his philosophy was deficient on many points, and therefore went to Magna Græcia, and there learned the doctrines of Pythagoras from Archytas of Tarentum and Timæus of Locris: and this system he embodied in the elegant form and style which he had learned from Socrates. The whole of this, as we can prove, Origen carried over into his book Περὶ ᾿Αρχῶν, only changing the name.
    • St. Jerome, Apology Against Rufinus, Book III, sec. 40
  • Nicomachus concludes his first book with a theorem that indicates that mathematics was not yet free from ethical and æsthetic mixture. From Pythagoras onward two ideas were widespread in Greek, especially Platonic, philosophy. These are that the beautiful and the definite are prior to the ugly and the indefinite, and that from them are formed all the parts and classes of the infinite and indefinite. Nicomachus aims to show that in mathematics the same principle holds good in that from equality may be derived all the species of inequality.
  • The Ionians were optimistic, heathenly materialists... Every philosopher of the period seems to have had his own theory regarding the nature of the universe around him. ...The sixth century scene evokes the image of an orchestra expectantly tuning up, each player absorbed in his own instrument only, deaf to the caterwaulings of the others. Then there is a dramatic silence, the conductor enters the stage, raps three times with his baton, and harmony emerges from the chaos. The maestro is Pythagoras of Samos, whose influence on the ideas, and thereby on the destiny, of the human race was probably greater than that of any single man before or after him.
    • Arthur Koestler, The Sleepwalkers: A History of Man's Changing Vision of the Universe (1959, 1963)
  • It is impossible to decide whether a particular detail of the Pythagorean universe was the work of the master, or filled in by a pupil—a remark which equally applies to Leonardo or Michelangelo. But there can be no doubt that the basic features were conceived by a single mind; that Pythagoras of Samos was both the founder of a new religious philosophy, and the founder of Science, as the word is understood today.
    • Arthur Koestler, The Sleepwalkers: A History of Man's Changing Vision of the Universe (1959, 1963)
  • It may perhaps help us to realize the human side of our Masters if we remember that many of Them in comparatively recent times have been known as historical characters. The Master K.H. for example, appeared in Europe as the philosopher Pythagoras. Before that He was the Egyptian priest Sarthon, and on yet another occasion chief-priest of a temple at Agade, in Asia Minor, where He was killed in a general massacre of the inhabitants by a host of invading barbarians who swooped down upon them from the hills
  • Ah, Pythagoras' metempsychosis, were that true,
    This soul should fly from me, and I be changed
    Unto some brutish beast!
    All beasts are happy, for when they die,
    Their souls are soon dissolved in elements;
    But mine must live still to be plagued in hell.
  • Pythagoras and Plato and Boehme and Paracelsus and Thomas Vaughan were men who bore their lamps amidst their fellowmen in life under a hail of nonunderstanding and abuse. Anyone could approach them, but only a few were able to discern the superearthly radiance behind the earthly face. It is possible to name great Servitors of East and West, North and South. It is possible to peruse their biographies; yet everywhere we feel that the superearthly radiance appears rarely in the course of centuries. One should learn from reality. Let us not link ourselves with the vilifiers of Plato and the persecutors of Confucius. They were oppressed by citizens who were considered the pride of the country. Thus has the world raised its hand against the great Servitors. Be assured that the Brotherhood formed by Pythagoras appeared dangerous in the eyes of the city guard. (175)
    • Morya, Brotherhood (1937)
  • What appeared here, at the center of the Pythagorean tradition in philosophy, is another view of psyche that seems to owe little or nothing to the pan-vitalism or pan-deism (see theion) that is the legacy of the Milesians.
    • Francis E. Peters, in Greek Philosophical Terms: A Historical Lexicon (NYU Press 1967), p. 169 ISBN: 0814765521
  • The votaries of Pythagoras of Samos have this story to tell of him, that he was not an Ionian at all, but that, once on a time in Troy, he had been Euphorbus, and that he had come to life after death, but had died as the songs of Homer relate. And they say that he declined to wear apparel made from dead animal products and, to guard his purity, abstained from all flesh diet, and from the offering of animals in sacrifice. For that he would not stain the altars with blood; nay, rather the honey-cake and frankincense and the hymn of praise, these they say were the offerings made to the Gods by this man, who realized that they welcome such tribute more than they do the hecatombs note and the knife laid upon the sacrificial basket. For they say that he had of a certainty social intercourse with the gods, and learnt from them the conditions under which they take pleasure in men or are disgusted, and on this intercourse he based his account of nature.
  • It was through philosophy, he said, that he had come to be surprised at nothing.
  • The following became universally known: first, that he maintains that the soul is immortal; second, that it changes into other kinds of living things; third, that events recur in certain cycles and that nothing is ever absolutely new; and fourth, that all living things should be regarded as akin. Pythagoras seems to have been the first to bring these beliefs into Greece.
  • He ordained that his disciples should speak well and think reverently of the Gods, muses and heroes, and likewise of parents and benefactors; that they should obey the laws; that they should not relegate the worship of the Gods to a secondary position, performing it eagerly, even at home; that to the celestial divinities they should sacrifice uncommon offerings; and ordinary ones to the inferior deities. (The world he Divided into) opposite powers; the "one" was a better monad, light, right, equal, stable and straight; while the "other" was an inferior duad, darkness, left, unequal, unstable and movable.
    • Porphyry of Tyre, in "The Life of Pythagoras" as translated by Kenneth Sylvan Guthrie in The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy (1919); also quoted in The Golden Chain: An Anthology of Pythagorean and Platonic Philosophy (2004) by Algis Uzdavinys
  • Such things taught he, though advising above all things to speak the truth, for this alone deifies men. For as he had learned from the Magi, who call God Oremasdes, God's body is light, and his soul is truth. He taught much else, which he claimed to have learned from Aristoclea at Delphi.
    • Porphyry of Tyre, as translated by Kenneth Sylvan Guthrie in The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy (1919); also quoted in The Golden Chain: An Anthology of Pythagorean and Platonic Philosophy (2004) by Algis Uzdavinys
    • Unsourced variant: Speak the truth in all situations.
  • According to the account of Proclus (Book II. c. 4 ), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles. ...The traditionary account, that Pythagoras was the founder of scientific mathematics, is in some degree, supported by the statement of Diogenes Laertius, that he was chiefly occupied with the consideration of the properties of number, weight, and extension, besides music and astronomy. The passage of Cicero (De Nat. Deor. III. 36) may be referred to as evidence that later writers were unable to give any precise account of the mathematical discoveries of Pythagoras. To Pythagoras, however, is attributed the discovery of some of the most important elementary properties contained in the first book of Euclid's Elements. The very important truth contained in Prop. 47, Book I. is also ascribed to Pythagoras. ...Proclus attributes to him the discovery of that right-angled triangle, the three sides of which are respectively 3, 4, and 5 units. To Pythagoras also belongs the discovery, that there are only three kinds of regular polygons which can be placed so as to fill up the space round a point; namely, six equilateral triangles, four squares, and three regular hexagons. Proclus attributes to him the doctrine of incommensurables, and the discovery of the five regular solids, which, if not due to Pythagoras, originated in his school. In Astronomy he is reputed to have held, that the Sun is the centre of the system, and that the planets revolve round it. This has been called, from his name, the Pythagorean System, which was revived by Copernicus, A.D.1541, and proved by Newton.
  • Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrationals [or 'proportions'] and the construction of the cosmic figures.
    • Proclus A Commentary on the First Book of Eudlid's Elements (c. 450 AD), as quoted by Thomas Little Heath, A History of Greek Mathematics (1921) Vol. 1, p. 128, citing Proclus on Eucl. I, p. 65. 15-21.
  • Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles [or calculi] (or as we would more naturally say, shot) required to make the shapes in question.
  • Personal religion is derived from ecstasy, theology from mathematics, and both are to be found in Pythagoras.
    • Bertrand Russell, in A History of Western Philosophy (1945), Book One, Part I, Chapter III, Pythagoras, p. 36
  • The combination of mathematics and theology, which began with Pythagoras, characterized religious philosophy in Greece, in the Middle Ages, and in modern times down to Kant. Orphism before Pythagoras was analogous to Asiatic mystery religions. But in Plato, Saint Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant there is an intimate blending of religion and reasoning, of moral aspiration with logical admiration of what is timeless, which comes from Pythagoras, and distinguishes the intellectualized theology of Europe from the more straightforward mysticism of Asia. It is only in quite recent times that it has been possible to say clearly that Pythagoras was wrong. I do not know of any other man who has been as influential as he was in the sphere of thought. I say this because what appears as Platonism is, when analyzed, found to be in essence Pythagoreanism. The whole conception of an eternal world, revealed to the intellect but not to the senses, is derived from him. But for him, Christians would not have thought of Christ as the Word; but for him, theologians would not have sought logical proofs of God and immortality.
  • Inasmuch as I have begun to explain to you how much greater was my impulse to approach philosophy in my youth than to continue it in my old age, I shall not be ashamed to tell you what ardent zeal Pythagoras inspired in me. Sotion used to tell me why Pythagoras abstained from animal food, and why, in later times, Sextius did also. In each case, the reason was different, but it was in each case a noble reason. … Pythagoras … held that all beings were inter-related, and that there was a system of exchange between souls which transmigrated from one bodily shape into another. If one may believe him, no soul perishes or ceases from its functions at all, except for a tiny interval – when it is being poured from one body into another.
  • What seems certain is that Pythagoras developed the idea of mathematical logic... He realized that numbers exist independently of the tangible world and therefore their study was untainted by inaccuracies of perception. This meant he could discover truths which were independent of opinion of prejudice and which were more absolute then any previous knowledge.
  • Number, its kinds; the first kind, intellectual in the divine mind.
    Number is of two kinds, the Intellectual (or immateriall) and the Scientiall. The intellectuall is that eternal substance of number, which Pythagoras in his discourse concerning the Gods asserted to be the principle most providentiall of all Heaven and Earth, and the nature that is betwixt them. Moreover, it is the root of divine Beings, and of gods, & of Dæmons. This is that which he termed the principle, fountain,and root of all things, and defined it to be that which before all things exists in the divine mind; from which and out of which all things are digested into order, and remain numbred by an indissolube series.
    For all things which are ordered in the world by nature according to an artificiall course in part and in whole appear to be distinguished and adorn'd by Providence and the All-creating Mind, according to Number; the exemplar being established by applying (as the reason of the principle before the impression of things) the number præxistent in the Intellect of God, maker of the world. This only in intellectual, & wholly immaterial, really a substance according to which as being the most exact artificiall reason, all things are perfected, Time, Heaven, Motion, the Stars and their various revolutions.
    ...The other kind of number, Scientiall; its principles.
    Scientiall Number is that which Pythagoras defines the extension and production into act of the seminall reasons which are in the Monad, or a heap of Monads, or a progressian of multitude beginning from Monad, and a regression ending in Monad.
  • In spite of the dominance of mechanistic thought in the contemporary world, a perplexing residue of the magical tradition still survives in the form of several issues, solutions to which do not appear possible within the context of a purely mechanical view of the world.... It is important to recognize that the materialist, scientific paradigm that dominates the late twentieth century world and provides the basis for its dominant institutions, has its basis in the life and work of Pythagoras, one of the most significant representatives of the perennial philosophy and a founder of the magical tradition. This spirit, which gave rise to our world view, is a spirit that must be recaptured if our civilization is to flourish. The choice is a clear one to many, and was summed up in a book title by the late Pythagorean and futurist Buckminster Fuller, Utopia or Oblivion.
    • John Strohmeier and Peter Westbrook, in Divine Harmony : The Life and Teachings of Pythagoras (1999).
  • Pythagoras was a teacher of the purest system of morals ever propounded to man.
  • Pythagoras was a man; and with all his imperfections on his head, we shall look among the race of men, for his better, in yain, yea, for his equal, or his second, but in vain. Pythagoras was entirely a Deist, a steady maintainer of the unity of God, and of the eternal obligations of moral virtue. No Christian writings, even to this day, can compete in sublimity and grandeur with what this illustrious philosopher has laid down concerning God, and the end of all our actions; and it is likely, says Bayle, that he would have carried his orthodoxy much farther, had he had the courage to expose himself to martyrdom.
  • It was Pythagoras who first called heaven kosmos, because it is perfect, and "adorned" with infinite beauty and living beings.
    • Algis Uzdavinys in The Golden Chain: An Anthology of Pythagorean and Platonic Philosophy (2004) p. 4; also in The Life of Pythagoras by an unknown ancient author, as quoted in The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy (1919)
  • From of old, amid the rage of robbery and blood-lust, it came to wise men's consciousness that the human race was suffering from a malady which necessarily kept it in progressive deterioration. Many a hint from observation of the natural man, as also dim half-legendary memories, had made them guess the primal nature of this man, and that his present state is therefore a degeneration. A mystery enwrapped Pythagoras, the preacher of vegetarianism; no philosopher since him has pondered on the essence of the world, without recurring to his teaching. Silent fellowships were founded, remote from turmoil of the world, to carry out this doctrine as a sanctification from sin and misery.
  • It was a maxim of Pythagoras that the two most excellent things for man were to speak the truth, and to render benefits to each other.
    • Joseph Dame Weeks, History of the Knights of Pythias, with an Account of the Life and Times of Damon and Pythias (1874) Note: The bolded portion of this has sometimes been presented as a quote of Pythagoras, but has not been found in this form in any existing translations of his statements.
  • The association between religion and mathematically based science has its origins in the mists of history. ...the very dawn of Western culture in sixth-century B.C. Greece. ...[W]hen the Greeks were turning away from the mythological picture immortalized by Homer and Hesiod, the Ionian philosopher Pythagoras of Samos pioneered a worldview in which mathematics was seen as the key to reality. In place of the mythological gods, Pythagoras painted a picture in which the universe was conceived as a great musical instrument resonating with divine mathematical harmonies. ...[inspiring] mystics, theologians, and physicists ever since. ...But to Pythagoras and his followers, mathematics was the key not simply to the physical world, but more importantly to the spiritual world—for they believed that numbers were literally gods. By contemplating numbers and their relationships, the Pythagoreans sought union with the "divine." For them, mathematics was first and foremost a religious activity.
    • Margaret Wertheim, Pythagoras' Trousers: God, Physics, and the Gender Wars (1995) Introduction, pp. 9-10.
  • Around 600 BCE, Pythagoras observed that the tones of a lyre sound most harmonious when the ratio of string lengths forms a simple whole-number fraction. Inspired by such hints, Pythagoras and his followers made a remarkable intuitive leap. They foresaw the possibility of a different kind of world-model, less dependent on the accident of our senses but more in tune with Nature's hidden harmonies, and ultimately more faithful to reality. This is the meaning of... "All things are number."

"Pythagoras" in Lives of the Necromancers (1835)

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He appears to have spent the latter half of his life in that part of Italy, called Magna Graecia, so denominated in some degree from the numerous colonies of Grecians by whom it was planted, and partly perhaps from the memory of the illustrious things which Pythagoras achieved there.
Statements on Pythagoras in Lives of the Necromancers (1835) by William Godwin, p. 77-92
 
He professed to have already in different ages appeared in the likeness of man: first as Aethalides, the son of Mercury … he became Euphorbus … then appeared as Hermotimus, then Pyrrhus, a fisherman of Delos, and finally Pythagoras.
 
There was nothing that Pythagoras was more fixed to discountenance, than the communication of the truths upon which he placed the highest value, to the uninitiated. It is not probable therefore that he wrote any thing: all was communicated orally, by such gradations, and with such discretion, as he might think fit to adopt and to exercise.
  • Pythagoras was a man of the most various accomplishments, and appears to have penetrated in different directions into the depths of human knowledge. He sought wisdom in its retreats of fairest promise, in Egypt and other distant countries. In this investigation he employed the earlier period of his life, probably till he was forty, and devoted the remainder to such modes of proceeding, as appeared to him the most likely to secure the advantage of what he had acquired to a late posterity.
    He founded a school, and delivered his acquisitions by oral communication to a numerous body of followers. He divided his pupils into two classes, the one neophytes, to whom was explained only the most obvious and general truths, the other who were admitted into the entire confidence of the master. These last he caused to throw their property into a common stock, and to live together in the same place of resort. He appears to have spent the latter half of his life in that part of Italy, called Magna Graecia, so denominated in some degree from the numerous colonies of Grecians by whom it was planted, and partly perhaps from the memory of the illustrious things which Pythagoras achieved there.
  • But this marvellous man in some way, whether from the knowlege he received, or from his own proper discoveries, has secured to his species benefits of a more permanent nature, and which shall outlive the revolutions of ages, and the instability of political institutions. He was a profound geometrician. The two theorems, that the internal angles of every right-line triangle are equal to two right angles, and that the square of the hypothenuse of every right angled triangle is equal to the sum of the squares of the other two sides, are ascribed to him. In memory of the latter of these discoveries he is said to have offered a public sacrifice to the Gods; and the theorem is still known by the name of the Pythagorean theorem.
  • To inculcate a pure and a simple mode of subsistence was also an express object of pursuit to Pythagoras. He taught a total abstinence from every thing having had the property of animal life. … He taught temperance in all its branches, and a resolute subjection of the appetites of the body to contemplation and the exercises of the mind; and, by the unremitted discipline and authority he exerted over his followers, he caused his lessons to be constantly observed. There was therefore an edifying and an exemplary simplicity that prevailed as far as the influence of Pythagoras extended, that won golden opinions to his adherents at all times that they appeared, and in all places.
  • One revolution that Pythagoras worked, was that, whereas, immediately before, those who were most conspicuous among the Greeks as instructors of mankind in understanding and virtue, styled themselves sophists, professors of wisdom, this illustrious man desired to be known only by the appellation of a philosopher, a lover of wisdom. The sophists had previously brought their denomination into discredit and reproach, by the arrogance of their pretensions, and the imperious way in which they attempted to lay down the law to the world.
    The modesty of this appellation however did not altogether suit with the deep designs of Pythagoras, the ascendancy he resolved to acquire, and the oracular subjection in which he deemed it necessary to hold those who placed themselves under his instruction. This wonderful man set out with making himself a model of the passive and unscrupulous docility which he afterwards required from others. He did not begin to teach till he was forty years of age, and from eighteen to that period he studied in foreign countries, with the resolution to submit to all his teachers enjoined, and to make himself master of their least communicated and most secret wisdom.
  • What however seems to be agreed in by all his biographers, is that he professed to have already in different ages appeared in the likeness of man: first as Aethalides, the son of Mercury; and, when his father expressed himself ready to invest him with any gift short of immortality, he prayed that, as the human soul is destined successively to dwell in various forms, he might have the privilege in each to remember his former state of being, which was granted him. From Aethalides he became Euphorbus, who slew Patroclus at the siege of Troy. He then appeared as Hermotimus, then Pyrrhus, a fisherman of Delos, and finally Pythagoras. He said that a period of time was interposed between each transmigration, during which he visited the seat of departed souls; and he professed to relate a part of the wonders he had seen. He is said to have eaten sparingly and in secret, and in all respects to have given himself out for a being not subject to the ordinary laws of nature.
    Pythagoras therefore pretended to miraculous endowments. Happening to be on the sea-shore when certain fishermen drew to land an enormous multitude of fishes, he desired them to allow him to dispose of the capture, which they consented to, provided he would name the precise number they had caught. He did so, and required that they should throw their prize into the sea again, at the same time paying them the value of the fish.
  • The close of the life of Pythagoras was, according to every statement, in the midst of misfortune and violence. Some particulars are related by Iamblichus, which, though he is not an authority beyond all exception, are so characteristic as seem to entitle them to the being transcribed. … Cylon, the richest man, or, as he is in one place styled, the prince, of Crotona, had manifested the greatest partiality to Pythagoras. He was at the same time a man of rude, impatient and boisterous character. He, together with Perialus of Thurium, submitted to all the severities of the Pythagorean school. They passed the three years of probation, and the five years of silence. They were received into the familiarity of the master. They were then initiated, and delivered all their wealth into the common stock. They were however ultimately pronounced deficient in intellectual power, or for some other reason were not judged worthy to continue among the confidential pupils of Pythagoras. They were expelled. The double of the property they had contributed was paid back to them. A monument was set up in memory of what they had been; and they were pronounced dead to the school. … Cylon, from feelings of the deepest reverence and awe for Pythagoras, which he had cherished for years, was filled even to bursting with inextinguishable hatred and revenge. The unparalleled merits, the venerable age of the master whom he had so long followed, had no power to control his violence. His paramount influence in the city insured him the command of a great body of followers. He excited them to a frame of turbulence and riot. He represented to them how intolerable was the despotism of this pretended philosopher. They surrounded the school in which the pupils were accustomed to assemble, and set it on fire. Forty persons perished in the flames. According to some accounts Pythagoras was absent at the time. According to others he and two of his pupils escaped. He retired from Crotona to Metapontum. But the hostility which had broken out in the former city, followed him there. He took refuge in the Temple of the Muses. But he was held so closely besieged that no provisions could be conveyed to him; and he finally perished with hunger, after, according to Laertius, forty days' abstinence.
  • It is difficult to imagine any thing more instructive, and more pregnant with matter for salutary reflection, than the contrast presented to us by the character and system of action of Pythagoras on the one hand, and those of the great enquirers of the last two centuries, for example, Bacon, Newton and Locke, on the other. Pythagoras probably does not yield to any one of these in the evidences of true intellectual greatness. In his school, in the followers he trained resembling himself, and in the salutary effects he produced on the institutions of the various republics of Magna Graecia and Sicily, he must be allowed greatly to have excelled them. His discoveries of various propositions in geometry, of the earth as a planet, and of the solar system as now universally recognised, clearly stamp him a genius of the highest order.
    Yet this man, thus enlightened and philanthropical, established his system of proceeding upon narrow and exclusive principles, and conducted it by methods of artifice, quackery and delusion. One of his leading maxims was, that the great and fundamental truths to the establishment of which he devoted himself, were studiously to be concealed from the vulgar, and only to be imparted to a select few, and after years of the severest noviciate and trial. He learned his earliest lessons of wisdom in Egypt after this method, and he conformed through life to the example which had thus been delivered to him. The severe examination that he made of the candidates previously to their being admitted into his school, and the years of silence that were then prescribed to them, testify this. He instructed them by symbols, obscure and enigmatical propositions, which they were first to exercise their ingenuity to expound. The authority and dogmatical assertions of the master were to remain unquestioned; and the pupils were to fashion themselves to obsequious and implicit submission, and were the furthest in the world from being encouraged to the independent exercise of their own understandings. There was nothing that Pythagoras was more fixed to discountenance, than the communication of the truths upon which he placed the highest value, to the uninitiated. It is not probable therefore that he wrote any thing: all was communicated orally, by such gradations, and with such discretion, as he might think fit to adopt and to exercise.
  • All was oracular and dogmatic in the school of Pythagoras. He prized and justly prized the greatness of his attainments and discoveries, and had no conception that any thing could go beyond them. He did not encourage, nay, he resolutely opposed, all true independence of mind, and that undaunted spirit of enterprise which is the atmosphere in which the sublimest thoughts are most naturally generated. He therefore did not throw open the gates of science and wisdom, and invite every comer; but on the contrary narrowed the entrance, and carefully reduced the number of aspirants. He thought not of the most likely methods to give strength and permanence and an extensive sphere to the progress of the human mind. For these reasons he wrote nothing; but consigned all to the frail and uncertain custody of tradition. And distant posterity has amply avenged itself upon the narrowness of his policy; and the name of Pythagoras, which would otherwise have been ranked with the first luminaries of mankind, and consigned to everlasting gratitude, has in consequence of a few radical and fatal mistakes, been often loaded with obloquy, and the hero who bore it been indiscriminately classed among the votaries of imposture and artifice.

Isis Unveiled (1877)

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: A Master-Key to the Mysteries of Ancient and Modern Science and Theology by H. P. Blavatsky, A source of Vol. 1.
  • It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle 'of unity beneath the forms, changes, and other phenomena of the universe. Aristotle asserted that he taught that “ numbers are the first principles of all entities.” Ritter has expressed the opinion that the formula of Pythagoras should be taken symbolically, which is doubtless correct. (p. XV Before the Veil)
    • Before the Veil, xv.
  • Aristotle was no trustworthy witness. He misrepresented Plato, and he almost caricatured the doctrines of Pythagoras. There is a canon of interpretation, which should guide us in our examinations of every philosophical opinion: The human mind has, under the necessary operation of its own laws, been compelled to entertain the same fundamental ideas, and the human heart to cherish the same feelings in all ages. It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle of unity beneath the forms, changes, and other phenomena of the universe. Aristotle asserted that he taught that numbers are the first principles of all entities. Ritter has expressed the opinion that the formula of Pythagoras should be taken symbolically, which is doubtless correct.
    • Vol. I, Before the Veil, xv.
  • If the Pythagorean metempsychosis should be thoroughly explained and compared with the modern theory of evolution, it would be found to supply every "missing link" in the chain of the latter. But who of our scientists would consent to lose his precious time over the vagaries of the ancients. Notwithstanding proofs to the contrary, they not only deny that the nations of the archaic periods, but even the ancient philosophers had any positive knowledge of the Heliocentric system. The "Venerable Bedes," the Augustines and Lactantii appear to have smothered, with their dogmatic ignorance, all faith in the more ancient theologists of the pre-Christian centuries. But now philology and a closer acquaintance with Sanskrit literature have partially enabled us to vindicate them from these unmerited imputations. In the Vedas, for instance, we find positive proof that so long ago as 2000 B.C., the Hindu sages and scholars must have been acquainted with the rotundity of our globe and the Heliocentric system. Hence, Pythagoras and Plato knew well this astronomical truth; for Pythagoras obtained his knowledge in India, or from men who had been there, and Plato faithfully echoed his teachings.
  • If we carefully trace the terms nazar, and nazaret, throughout the best known works of ancient writers, we will meet them in connection with “Pagan” as well as Jewish adepts. Thus, Alexander Polyhistor says of Pythagoras that he was a disciple of the Assyrian Nazarrt, whom some suppose to be Ezekiel. Diogenes Laertius states most positively that Pythagoras, after being initiated into all the Mysteries of the Greeks and barbarians, “went into Egypt and afterward visited the Chaldeans and Magi;” and Apuleius maintains that it was Zoroaster who instructed Pythagoras. (p. 140)
    • The Veil of Isis, Part 1, Science.
  • Zoroaster, Pythagoras, Epicharmus, Empedocles, Kebes, Euripides, Plato, Euclid, Philo, Boethius, Virgil, Marcus Cicero, Plotinus, Iamblichus, Proclus, Psellus, Synesius, Origen, and, finally, Aristotle himself, far 'from denying our immortality, support it most emphatically. (p. 251)
    • The Veil of Isis, Part 1, Science.
  • The laws of Manu are the doctrines of Plato, Philo, Zoroaster, Pythagoras, and of the Kabala. The esoterism of every religion may be solved by the latter. The kabalistic doctrine of the allegorical Father and Son, or Xlarrjp and Aoyos is identical with the groundwork of Buddhism. Moses could not reveal to the multitude the sublime secrets of religious speculation, nor the cosmogony of the universe ; the whole resting upon the Hindu Illusion, a clever mask veiling the Sanctum Sanctorum, and which has misled so many theological commentators. (p. 271)
    • The Veil of Isis, Part 1, Science.
  • Scholastic skeptics, as well as ignorant materialists, have greatly amused themselves for the last two centuries over the absurdities attributed to Pythagoras by his biographer, Iamblichus. The Samian philosopher is said to have persuaded a she-bear to give up eating human flesh; to have forced a white eagle to descend to him from the clouds, and to have subdued him by stroking him gently with the hand, and by talking to him. On another occasion, Pythagoras actually persuaded an ox to renounce eating beans, by merely whispering in the animal's ear! (Iamblichus: "De Vita Pythag.") Oh, ignorance and superstition of our forefathers, how ridiculous they appear in the eyes of our enlightened generations! Let us, however, analyze this absurdity. Every day we see unlettered men, proprietors of strolling menageries, taming and completely subduing the most ferocious animals, merely by the power of their irresistible will... Every one has either witnessed or heard of the seemingly magical power of some mesmerizers and psychologists. They are able to subjugate their patients for any length of time. Regazzoni, the mesmerist who excited such wonder in France and London, has achieved far more extraordinary feats than what is above attributed to Pythagoras. Why, then, accuse the ancient biographers of such men as Pythagoras and Apollonius of Tyana of either wilful misrepresentation or absurd superstition?
    • p.284.

Divine Harmony (1999)

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Quotes on Pythagoras from Divine Harmony: The Life and Teachings of Pythagoras by John Strohmeier and Peter Westbrook. (1999) ISBN 0-9653774-5-8
 
The concept of a harmonious universe ordered according to "the Great Chain of Being" — a chain that connects the continuum of matter, body, mind, soul and spirit — stands as one of the most fundamental ideas of western thought.
  • Pythagoras stands at the fountainhead of our culture. The ideas he set in motion were, according to Daniel Boorstin, "among the most potent in modern history," resulting directly in many of the pillars upon which the modern world is built. In particular, the very existence of science becomes possible only when it is realized that inner, purely subjective, mathematical forms have a resonance with the form and behavior of the external world — a Pythagorean perception. And a world at peace — that is to say, in a nuclear age, the survival of our planet — is predicated upon ideas of universal brotherhood to which Pythagoras, while not the sole author, made an enormous contribution. Even the seeming remoteness of Pythagorean teaching helps one to realize that the current world view, while it seems destined to dominate the planet, is fleeting and temporary and, like others before it, will pass.
  • Pythagoras' teachings have enormous relevance in understanding both the sources of our culture and, perhaps more importantly, where it may be heading or may need to head. But to appreciate this we have to understand him in modern terms.
  • At the dawn of our century, scientists were proclaiming that our understanding of the world was almost complete. Only one or two small problems in physics remained to be solved. One of these problems had to do with black body radiation and was solved by Max Planck. His solution, however, formed the foundation for quantum mechanics which was to sweep aside almost the whole edifice of fundamental assumptions in physics, and with it our understanding of the world.
    A hundred years later we are faced with a similar situation. The mechanistic viewpoint that began to dominate our world view in the seventeenth century has almost completed its hegemony. This paradigm, as historian Hugh Kearney points out, stems from only one of three main systems of thought that flowed from Greek thought into the modern world, each of which has dominated our world view at different points in our history. … In spite of the dominance of mechanistic thought in the contemporary world, a perplexing residue of the magical tradition still survives in the form of several issues, solutions to which do not appear possible within the context of a purely mechanical view of the world.
  • It is important to recognize that the materialist, scientific paradigm that dominates the late twentieth century world and provides the basis for its dominant institutions, has its basis in the life and work of Pythagoras, one of the most significant representatives of the perennial philosophy and a founder of the magical tradition. This spirit, which gave rise to our world view, is a spirit that must be recaptured if our civilization is to flourish. The choice is a clear one to many, and was summed up in a book title by the late Pythagorean and futurist Buckminster Fuller, Utopia or Oblivion.
  • The concept of a harmonious universe ordered according to "the Great Chain of Being" — a chain that connects the continuum of matter, body, mind, soul and spirit — stands as one of the most fundamental ideas of western thought. … It continues to be a profound influence upon the deepest strata of our thought. And yet a major rift has appeared in the consciousness of our time because the theme of harmonia has not been translated into the realm of human conduct. The challenge of our time may be to revive it, and make divine harmony "the great theme" of the next millennium. Any success we have in accomplishing this will be based, in large part, on the achievements of Pythagoras.

Quotes about Pythagoreanism

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They seem to have been the first who reckoned the elements to be five in number, on the supposition of their derivation from the five regular solids. They made the cube, earth; the pyramid, fire; the octohedron, air; the icosahedron, water; and the dodecahedron, aether. The analogy of the five senses and the five elements was another favourite notion of the Pythagoreans. ~ Robert Potts
  • If someone associates with a true Pythagorean, what will he will get from him, and in what quantity? I would say: statesmanship, geometry, astronomy, arithmetic, harmonics, music, medicine, complete and god-given prophecy, and also the higher rewards — greatness of mind, of soul, and of manner, steadiness, piety, knowledge of the gods and not just supposition, familiarity with blessed spirits and not just faith, friendship with both gods and spirits, self-sufficiency, persistence, frugality, reduction of essential needs, ease of perception, of movement, and of breath, good color, health, cheerfulness, and immortality.
  • It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed us down a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being.
  • It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity.
    Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master.
  • The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction.
    • Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1949).
  • Ionian philosophers... had sought to identify a first principle for all things. Thales had thought to find this in water, but others preferred to think of air or fire as the basic element. The Pythagoreans had taken a more abstract direction, postulating that number... was the basic stuff behind phenomena; this numerical atomism... had come under attack by the followers of Parmenides of Elea... The fundamental tenet of the Eleatics was the unity and permanence of being... contrasted with the Pythagorean ideas of multiplicity and change. Of Parmenides' disciples the best known was Zeno the Eleatic... who propounded arguments to prove the inconsistency in the concepts of multiplicity and divisibility.
  • We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As apeiron in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself.
    • W. K. C. Guthrie, A History of Greek Philosophy Vol. 1, "The Earlier Presocratics and the Pythagoreans" (1962)
  • Those who dwelt in the common auditorium adopted this oath:
    "I swear by the discoverer of the Tetraktys,
    which is the spring of all our wisdom;
    The perennial fount and root of Nature.    "
  • The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter forms the symphony diapente; 4 to 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason.
  • Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the quadrivium of the Pythagoreans. With the trivium (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire.
  • The Neo-Pythagoreans treated all the divisions of philosophy. In Metaphysics they held that the Unit and the (indeterminate) Two are the basis of all things. the Unit being the form, and the Two the matter. ...The Unit being the prior principle may be identified with Deity, and, as such, was thought of either as the former [creator] of indefinite matter into individual things, or, as in Neo-Platonism, as the transcendent origin of the derivative Unit and Two. Another mode of conception was to identify the numbers with the Platonic Ideas and then to think of the Unit as comprehending them in the same manner as the mind comprehends its thoughts and gives them form. In Logic the Neo-Pythagoreans were for the most part imitators of Aristotle. Their Physics was Aristotelian and Stoic. Their Anthropology was Platonic. In Ethics and Politics they merely reechoed the Academy and the Lyceum with Stoic additions. In all this Neo-Pythagoreanism has little originality.
  • Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry — the springboard to the Greeks' epochal revolution in thought — the number 1 represented a point... 2 represented a line... 3 represented a surface... and 4 represented a three-dimensional tetrahedral solid... The Tetraktys, therefore appeared to encompass all the perceived dimensions of space.
  • On the question whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt — mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician — mathematics was God! ...By setting the stage, and to some extent the agenda, for the next generation of philosophers — Plato in particular — the Pythagoreans established a commanding position in Western thought.
  • As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space. They regarded the unit, as the point; the duad, as the line; the triad, as the surface; and the tetractys, as the geometrical volume. They assumed the pentad as the physical body with its physical qualities. They seem to have been the first who reckoned the elements to be five in number, on the supposition of their derivation from the five regular solids. They made the cube, earth; the pyramid, fire; the octohedron, air; the icosahedron, water; and the dodecahedron, aether. The analogy of the five senses and the five elements was another favourite notion of the Pythagoreans.
  • While most sophists emphasized the reality of change — in particular, the Atomists, followers of Leucippus and Democritus — the Pythagoreans stressed the study of the unchangeable elements in nature and society. In their search for the eternal laws of the universe they studied geometry, arithmetic, astronomy, and music (the quadrivium). Their most outstanding leader was Archytas of Tarentum...and to whose school, if we follow... E. [Eva] Frank, much of the Pythagorean brand of mathematics may be ascribed. ...Numbers were divided into classes: odd, even, even-times-even, odd-times-odd, prime and composite, perfect, friendly, triangular, square, pentagonal, etc. ...Of particular importance was the ratio of numbers (logos, Lat. ratio). Equality of ratio formed a proportion. They discriminated between an arithmetical  , geometrical  , and a harmonical   proportion that they interpreted philosophically and socially.
  • The Pythagoreans knew some properties of regular polygons... how a plane can be filled by... regular triangles, squares, or regular hexagons, and space by cubes... [They] may also have known the regular oktahedron and dodekahedron—the latter figure because pyrite, found in Italy, crystallizes in dodekahedra, and models... date to Etruscan times.
  • [T]he most striking result of the Greeks' faith that the world could be understood in terms of rational principles was the invention of abstract mathematics. The most grandiose ambition they conceived was to explain all the properties of Nature in arithmetical terms alone. This was the aim of the Pythagoreans... [T]hey... knew that the phenomena of the Heavens recurred in a cyclical manner; and... discovered ...that the sound of a vibrating string ...is simply related to the length ...and its 'harmonics' always go with simple fractional lengths. ...[S]ince the Pythagoreans were a religious brotherhood... they thought that this search would lead to more than explanations alone. If one discovered the mathematical harmonies in things, one should... discover how to put oneself in harmony with Nature. ...[T]hey had ...positive grounds for thinking that both astronomy and acoustics were at the bottom arithmetical; and the study of simple fractions was called 'music' right down until the late Middle Ages.

The Commentary of Pappus on Book X of Euclid's Elements (1930)

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Arabic Text and Translation by William Thomson with Introductory Remarks, Notes, and a Glossary of Technical Terms by Gustav Junge and William Thomson. A source.
  • Not one of the philosophical ideas in Part I of the commentary is peculiarly Neoplatonic. The doctrine of the Threeness of things... is found in Aristotle and goes back to the early Pythagoreans or to Homer even; paragraph 8 is mathematical in content rather than philosophical... although there is an allusion in it to the Monad as the principle of finitudes, again a very early Pythagorean doctrine; and these two paragraphs are the source of [Heinrich] Sitter's suggestion of the authorship of Proclus. As a matter of fact, the philosophical notions in Part I have been borrowed for the most part directly from Plato, with two or three exceptions that are Aristotelian... Plato's Theaetetus, Parmenides, and the Laws, are specifically mentioned. The Timaeus forms the background of much of the thought. And the Platonism of a mathematician of the turn of the third century A. D. need not surprise us, if we but recall Aristotle's accusation that the Academy tended to turn philosophy into mathematics.
    • pp. 40-41.
  • §1. The aim of Book X of Euclid's treatise on the Elements is to investigate the commensurable and incommensurable, the rational and irrational continuous quantities. This science (or knowledge) had its origin in the sect (or school) of Pythagoras, but underwent an important development at the hands of the Athenian, Theaetetus, who had a natural aptitude for this as for other branches of mathematics most worthy of admiration.
    • p. 63.
  • §2. Since this treatise (i. e. Book X of Euclid.) has the aforesaid aim and object, it will not be unprofitable for us to consolidate the good which it contains. Indeed the sect (or school) of Pythagoras was so affected by its reverence for these things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common herd, perished by drowning: which is most probably a parable by which they sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of nonidentity (i. e. lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement. This was the consideration which Pythagoreans and the Athenian Stranger held to be an incentive to particular care and concern for these things and to imply of necessity the grossest foolishness in him who imagined these things to be of no account.
    • p. 64.

See also

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Ancient Greek schools of philosophy
Pre-Socratic Anaxagoras • Anaximander • Anaximenes • Democritus • Empedocles • Heraclitus • Leucippus • Melissus • Parmenides • Protagoras • Pythagoras • Thales • Zeno of Elea
Socratic Antisthenes • Aristippus • Aristotle • Diogenes of Sinope • Euclid of Megara • Phaedo of Elis • Plato • Socrates
Hellenistic Apollonius of Tyana • Augustine • Epictetus • Epicurus • John Philoponus • Lucretius • Plotinus • Proclus • Pyrrho • Sextus Empiricus • Zeno of Citium

Mathematics
Mathematicians
(by country)

Abel • Anaxagoras • Archimedes • Aristarchus of Samos • Averroes • Arnold • Banach • Cantor • Cartan • Chern • Cohen • Descartes • Diophantus • Erdős • Euclid • Euler • Fourier • Gauss • Gödel • Grassmann • Grothendieck • Hamilton • Hilbert • Hypatia • Lagrange • Laplace • Leibniz • Milnor • Newton • von Neumann • Noether • Penrose • Perelman • Poincaré • Pólya • Pythagoras • Riemann • Russell • Schwartz • Serre • Tao • Tarski • Thales • Turing • Weil • Weyl • Wiles • Witten

Numbers

1 • 23 • 360 • e • π • Fibonacci numbers • Irrational number • Negative number • Number • Prime number • Quaternion • Octonion

Concepts

Abstraction • Algorithms • Axiomatic system • Completeness • Deductive reasoning • Differential equation • Dimension • Ellipse • Elliptic curve • Exponential growth • Infinity • Integration • Geodesic • Induction • Proof • Partial differential equation • Principle of least action • Prisoner's dilemma • Probability • Randomness • Theorem • Topological space • Wave equation

Results

Euler's identity • Fermat's Last Theorem

Pure math

Abstract algebra • Algebra • Analysis • Algebraic geometry (Sheaf theory) • Algebraic topology • Arithmetic • Calculus • Category theory • Combinatorics • Commutative algebra • Complex analysis • Differential calculus • Differential geometry • Differential topology • Ergodic theory • Foundations of mathematics • Functional analysis • Game theory • Geometry • Global analysis • Graph theory • Group theory • Harmonic analysis • Homological algebra • Invariant theory • Logic • Non-Euclidean geometry • Nonstandard analysis • Number theory • Numerical analysis • Operations research • Representation theory • Ring theory • Set theory • Sheaf theory • Statistics • Symplectic geometry • Topology

Applied math

Computational fluid dynamics • Econometrics • Fluid mechanics • Mathematical physics • Science

History of math

Ancient Greek mathematics • Euclid's Elements • History of algebra • History of calculus • History of logarithms • Indian mathematics • Principia Mathematica

Other

Mathematics and mysticism • Mathematics education • Mathematics, from the points of view of the Mathematician and of the Physicist • Philosophy of mathematics • Unification in science and mathematics


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