Pons asinorum
Encyclopedia
Pons asinorum is the name given to Euclid
's fifth proposition in Book 1 of his Elements
of geometry
, also known as the theorem on isosceles triangle
s. 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.
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 as a bridge to the harder propositions that follow.
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.
, 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. The theorem was also sometimes called "the Windmill" for similar reasons.
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's fifth proposition in Book 1 of his Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, also known as the theorem on isosceles triangle
Isosceles triangle theorem
In Euclidean geometry, the isosceles triangle theorem, also known as the pons asinorum, states that the angles opposite the two equal sides of an isosceles triangle are equal...
s. 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
Roger Bacon
Roger Bacon, O.F.M. , also known as Doctor Mirabilis , was an English philosopher and Franciscan friar who placed considerable emphasis on the study of nature through empirical methods...
, 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
Geoffrey Chaucer
Geoffrey Chaucer , known as the Father of English literature, is widely considered the greatest English poet of the Middle Ages and was the first poet to have been buried in Poet's Corner of Westminster Abbey...
use of the term "flemyng of wreches" for the theorem.
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 as a bridge to the harder propositions that follow.
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.
Metaphorical use
Uses of the term or the theorem itself as a metaphor include:- Richard AungervilleRichard AungervilleRichard de Bury , also known as Richard Aungerville , was an English writer, bibliophile, Benedictine monk and bishop. He was a patron of learning, and one of the first English collectors of books. He is chiefly remembered for his Philobiblon, written to inculcate in the clergy the pursuit of...
'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. - 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 syllogismSyllogismA syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
. - 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.
- Economist John Stuart MillJohn Stuart MillJohn Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
called Ricardo'sDavid RicardoDavid Ricardo was an English political economist, often credited with systematising economics, and was one of the most influential of the classical economists, along with Thomas Malthus, Adam Smith, and John Stuart Mill. He was also a member of Parliament, businessman, financier and speculator,...
Law of RentLaw of RentThe law of rent was formulated by David Ricardo around 1809, and this is the origin of the term Ricardian rent. Ricardo's formulation of the law was the first clear exposition of the source and magnitude of land rents, and is among the most important and firmly established principles of economics. ...
the Pons Asinorum of economics. - Pons Asinorum is the name given to a particular configuration of a Rubik's CubeRubik's CubeRubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...
. - The FinnishFinnish languageFinnish is the language spoken by the majority of the population in Finland Primarily for use by restaurant menus and by ethnic Finns outside Finland. It is one of the two official languages of Finland and an official minority language in Sweden. In Sweden, both standard Finnish and Meänkieli, a...
aasinsilta and SwedishSwedish languageSwedish is a North Germanic language, spoken by approximately 10 million people, predominantly in Sweden and parts of Finland, especially along its coast and on the Åland islands. It is largely mutually intelligible with Norwegian and Danish...
åsnebrygga is a literary technique where a tenuous, even contrived connection between two arguments or topics, which is almost but not quite a non sequiturNon sequiturNon sequitur is Latin for "it does not follow." It is most often used as a noun to describe illogical statements.Non sequitur may refer to:* Non sequitur , an irrelevant, often humorous comment to a preceding topic or statement....
, 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 causerieCauserieCauserie is a literary style of short informal essays mostly unknown in the English-speaking world. A causerie is generally short, light and humorous and is often published as a newspaper column . Often causerie is a current opinion piece, but it contains more verbal acrobatics and humor than a...
-style writing.
Dulcarnon
Similarly, the name Dulcarnon is given to the 47th proposition of Book I of Euclid, better known as the Pythagorean theoremPythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
, 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. The theorem was also sometimes called "the Windmill" for similar reasons.