Pons asinorum

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The pons asinorum in Byrne's edition of the Elements showing part of Euclid's proof.

Pons asinorum (Latin for "bridge of donkeys") is the name given to Euclid's fifth proposition in Book 1 of his Elements of geometry, also known as the theorem on isosceles triangles. It states that the angles opposite the equal sides of an isosceles triangle are equal. Another medieval term for the theorem was Elefuga which, according to Roger Bacon, comes from Greek elegia misery, and fuga Latin for flight, that is "flight of the wretches". Though this etymology is dubious, it is echoed in Chaucer's use of the term "flemyng of wreches" for the theorem.[1]

There are two possible explanations for the name pons asinorum, the simplest being that the diagram used resembles an actual bridge. But the more popular explanation is that it is the first real test in the Elements of the intelligence of the reader and functions as a "bridge" to the harder propositions that follow.[2]

Whatever its origin, the term is also used as a metaphor for a problem or challenge which will separate the sure of mind from the simple, the fleet thinker from the slow, the determined from the dallier; to represent a critical test of ability or understanding.[3]

Metaphorical use[edit]

Uses of the term or the theorem itself as a metaphor include:

  • Richard Aungerville's Philobiblon contains the passage "Quot Euclidis discipulos retrojecit Elefuga quasi scopulos eminens et abruptus, qui nullo scalarum suffragio scandi posset! Durus, inquiunt, est hie sermo; quis potest eum audire?", which compares the theorem to a steep cliff that no ladder may help scale and asks how many would-be geometers have been turned away.[1]
  • The term pons asinorum, in both its meanings as a bridge and as a test, is used as a metaphor for finding the middle term of a syllogism.[1]
  • The 18th-century poet Thomas Campbell wrote a humorous poem called "Pons asinorum" where a geometry class assails the theorem as a company of soldiers might charge a fortress; the battle was not without casualties.[4]
  • Economist John Stuart Mill called Ricardo's Law of Rent the Pons Asinorum of economics.[5]
  • Pons Asinorum is the name given to a particular configuration of a Rubik's Cube.
  • The Finnish aasinsilta and Swedish åsnebrygga is a literary technique where a tenuous, even contrived connection between two arguments or topics, which is almost but not quite a non sequitur, is used as an awkward transition between them.[6] In serious text, it is considered a stylistic error, since it belongs properly to the stream of consciousness- or causerie-style writing. Typical examples are ending a section by telling what the next section is about, without bothering to explain why the topics are related, expanding a casual mention into a detailed treatment, or finding a contrived connection between the topics (e.g. "We bought some red wine; speaking of red liquids, tomorrow is the World Blood Donor Day").
  • In Dutch, ezelsbruggetje ('little bridge of asses') is the word for a mnemonic. The same is true for the German Eselsbrücke.
  • In Czech, oslí můstek has two meanings – it can describe either a contrived connection between two topics or a mnemonic.


Similarly, the name Dulcarnon is given to the 47th proposition of Book I of Euclid, better known as the Pythagorean theorem, after the Arabic Dhū 'l qarnain ذُو ٱلْقَرْنَيْن, meaning "the owner of the two horns", because diagrams of the theorem showed two smaller squares like horns at the top of the figure. The term is also used as a metaphor for a dilemma.[1] The theorem was also sometimes called "the Windmill" for similar reasons.[7]

Euler's identity[edit]

Gauss supposedly once espoused a similar belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.[8]

See also[edit]


  1. ^ a b c d A. F. West & H. D. Thompson "On Dulcarnon, Elefuga And Pons Asinorum as Fanciful Names For Geometrical Propositions" The Princeton University bulletin Vol. 3 No. 4 (1891) p. 84
  2. ^ D.E. Smith History of Mathematics (1958 Dover) p. 284
  3. ^ Pons asinorum - Definition and More from the Free Merriam
  4. ^ W.E. Aytoun (Ed.) The poetical works of Thomas Campbell (1864, Little, Brown) p. 385 Google Books
  5. ^ H.D. Macleod The Elements of Economics (1886 D. Appleton) Vol. 2 p. 96
  6. ^ Aasinsilta on laiskurin apuneuvo | Yle Uutiset | yle.fi
  7. ^ Charles Lutwidge Dodgson, Euclid and his Modern Rivals Act I Scene II §1
  8. ^ Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. 500 Fifth Street, NW, Washington D.C. 20001: Joseph Henry Press. p. 202. ISBN 0-309-08549-7. 

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