“The Pythagoreans discovered the existence of incommensurable lines, or of irrationals.”

— Thomas Little Heath

This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.
Achimedes (1920)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The Pythagoreans discovered the existence of incommensurable lines, or of irrationals." by Thomas Little Heath?

Thomas Little Heath46

British civil servant and academic 1861–1940

Related quotes

“The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable… showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established…. The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

“From experience. That something is irrational is no argument against its existence, but rather a condition for it.”

Friedrich Nietzsche book Human, All Too Human

Section IX, "Man Alone with Himself" / aphorism 515
Human, All Too Human (1878), Helen Zimmern translation

“Two girls discover
the secret of life
in a sudden line of
poetry. I who don't know the
secret wrote
the line.”

Denise Levertov (1923–1997) Poet

O Taste and See : New Poems (1964), The Secret
Source: Poems, 1960-1967

“There is no foundational mathematical or physical reason the relationship between Pythagorean and tempered western music should exist. It just does. The rich flexibility of the tempered scale and the … bountiful archives of western music are a testimonial to this wonderful coincidence provided by nature.”

Robert J. Marks II (1950) American electrical engineering researcher and intelligent design advocate

"Handbook of Fourier Analysis and Its Applications" (Oxford University Press, 2009), p. 623, Robert J. Marks II, 2009, 2011-04-29 http://books.google.com/books?id=Sp7O4bocjPAC&printsec=frontcover&dq=Handbook+of+Fourier+Analysis+and+Its+Applications&hl=en&ei=wcm5TaPvJYba0QHYi7nRDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q&f=false,

“Theism is irrational by definition, because it is not based on reason and is not reasonable. There was no evidence indicating this conclusion, yet it is a firm conviction anyway, and believers are determined not to let any evidence change their minds either. The very purpose and existence of apologetics all by itself demonstrates how theism is irrational.”

Aron Ra (1962) Aron Ra is an atheist activist and the host of the Ra-Men Podcast

Patheos, Philosophistry http://www.patheos.com/blogs/reasonadvocates/2017/04/12/philosophistry/ (April 12, 2017)

“We discovered we are both pleasantly furious half of the time,
When we're not just toeing the line.”

Ani DiFranco (1970) musician and activist

Brief Bus Stop
Song lyrics

“The Grid exists even if you don't see the lines.”

John Twelve Hawks American writer

Fourth Realm Trilogy (2005-2009), The Traveler (2005)

“The integers, the rationals, and the irrationals, taken together, make up the continuum of real numbers. It's called a continuum because the numbers are packed together along the real number line with no empty spaces between them.”

Brian Hayes (scientist) (1900) American scientist, columnist and author

Source: Group Theory in the Bedroom (2008), Chapter 11, Identity Crisis, p. 206 (See also: George Cantor)

“There is a line of poetry, a sentence in a fable, a word in an essay, by which my existence is justified; find that line, and immortality is assured.”

Alberto Manguel (1948) writer

Source: The Library at Night

“Because truths of different kinds exhibit the characteristics of incommensurability (their difference is such that they cannot be measured by a single standard or reduced to members of one series) and incompossibility (their difference is such that they cannot co-exist within the same system), such openness introduces both multiplicity and contradiction, and the creature in question stands “divided against himself.””

David L. Norton (1930–1995) American philosopher

Source: Personal Destinies: A Philosophy of Ethical Individualism (1976), pp. 9-10

“The Pythagoreans discovered the existence of incommensurable lines, or of irrationals.”

Help us to complete the source, original and additional information

Thomas Little Heath46

Related quotes

Related topics