Katana VentraIP

Pons asinorum

In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ˈpɒnz ˌæsɪˈnɔːrəm/ PONZ ass-i-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem. The theorem appears as Proposition 5 of Book 1 in Euclid's Elements. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal.

Pons asinorum is also used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the "asses' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.[1]

Etymology[edit]

There are two common explanations for the name pons asinorum, the simplest being that the diagram used resembles a physical 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]


Another medieval term for the isosceles triangle theorem was Elefuga which, according to Roger Bacon, comes from Greek elegia "misery", and Latin fuga "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.[3]


The name Dulcarnon was 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. That term has similarly been used as a metaphor for a dilemma.[3] The name pons asinorum has itself occasionally been applied to the Pythagorean theorem.[4]


Gauss supposedly once suggested that understanding Euler's identity might play a similar role, as a benchmark indicating whether someone could become a first-class mathematician.[5]

's 14th century Philobiblon contains the passage "Quot Euclidis discipulos retrojecit Elefuga quasi scopulos eminens et abruptus, qui nullo scalarum suffragio scandi posset! Durus, inquiunt, est his 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.[3]

Richard Aungerville

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 .[3]

syllogism

The 18th-century poet 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.[19]

Thomas Campbell

Economist called Ricardo's Law of Rent the pons asinorum of economics.[20]

John Stuart Mill

The 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. 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").

Finnish

In , ezelsbruggetje ('little bridge of asses') is the word for a mnemonic. The same is true for the German Eselsbrücke.

Dutch

In , oslí můstek has two meanings – it can describe either a contrived connection between two topics or a mnemonic.

Czech

Uses of the pons asinorum as a metaphor for a test of critical thinking include:

Artificial intelligence proof myth[edit]

A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem.[21][22] In fact, Marvin Minsky recounts that he had rediscovered the Pappus proof (which he was not aware of) by simulating what a mechanical theorem prover might do.[23][9]

Euclid, commentary and trans. by Elements Vol. 1 (1908 Cambridge) Google Books

T. L. Heath

Euclid, commentary by , ed. and trans. by T. Taylor Elements Vol. 2 (1789) Google Books

Proclus

at PlanetMath.

Pons asinorum

D. E. Joyce's presentation of Euclid's Elements