In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the statement that the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s opposite the equal sides of an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
are themselves equal is known as the ''pons asinorum'' (, ), typically translated as "bridge of
asses". This statement is Proposition 5 of Book 1 in
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's ''
Elements'', and is also known as the isosceles triangle theorem. Its
converse is also true: if two angles of a
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
are equal, then the sides opposite them are also equal. The term is also applied to the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
.
''Pons asinorum'' is also used
metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wi ...
ically for a problem or challenge which acts as a test of
critical thinking
Critical thinking is the analysis of available facts, evidence, observations, and arguments to form a judgement. The subject is complex; several different definitions exist, which generally include the rational, skeptical, and unbiased ana ...
, referring to the "ass' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.
A persistent piece of mathematical folklore claims that an
artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech ...
program discovered an original and more elegant proof of this theorem. In fact,
Marvin Minsky
Marvin Lee Minsky (August 9, 1927 – January 24, 2016) was an American cognitive and computer scientist concerned largely with research of artificial intelligence (AI), co-founder of the Massachusetts Institute of Technology's AI laboratory ...
recounts that he had rediscovered the
Pappus proof (which he was not aware of) by simulating what a mechanical theorem prover might do.
[Michael A.B. Deakin, "From Pappus to Today: The History of a Proof", ''The Mathematical Gazette'' 74:467:6-11 (March 1990) ]
Proofs
Euclid and Proclus
Euclid's statement of the pons asinorum includes a second conclusion that if the equal sides of the triangle are extended below the base, then the angles between the extensions and the base are also equal. Euclid's
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
involves drawing auxiliary lines to these extensions. But, as Euclid's commentator
Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...
points out, Euclid never uses the second conclusion and his proof can be simplified somewhat by drawing the auxiliary lines to the sides of the triangle instead, the rest of the proof proceeding in more or less the same way.
There has been much speculation and debate as to why Euclid added the second conclusion to the theorem, given that it makes the proof more complicated. One plausible explanation, given by Proclus, is that the second conclusion can be used in possible objections to the proofs of later propositions where Euclid does not cover every case. The proof relies heavily on what is today called
side-angle-side, the previous proposition in the ''Elements''.
Proclus' variation of Euclid's proof proceeds as follows:
:Let ''ABC'' be an isosceles triangle with ''AB'' and ''AC'' being the equal sides. Pick an arbitrary point ''D'' on side ''AB'' and construct ''E'' on ''AC'' so that ''AD'' = ''AE''. Draw the lines ''BE'', ''DC'' and ''DE''.
:Consider the triangles ''BAE'' and ''CAD''; ''BA'' = ''CA'', ''AE'' = ''AD'', and
is equal to itself, so by side-angle-side, the triangles are
congruent and corresponding sides and angles are equal.
:Therefore
and
, and ''BE'' = ''CD''.
:Since ''AB'' = ''AC'' and ''AD'' = ''AE'', ''BD'' = ''CE'' by subtraction of equal parts.
:Now consider the triangles ''DBE'' and ''ECD''; ''BD'' = ''CE'', ''BE'' = ''CD'', and
have just been shown, so applying side-angle-side again, the triangles are congruent.
:Therefore
and
.
:Since
and
,
by subtraction of equal parts.
:Consider a third pair of triangles, ''BDC'' and ''CEB''; ''DB'' = ''EC'', ''DC'' = ''EB'', and
, so applying side-angle-side a third time, the triangles are congruent.
:In particular, angle ''CBD'' = ''BCE'', which was to be proved.
Pappus
Proclus gives a much shorter proof attributed to
Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself.
This method is lampooned by
Charles Lutwidge Dodgson
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ...
in ''
Euclid and his Modern Rivals'', calling it an "
Irish bull" because it apparently requires the triangle to be in two places at once.
The proof is as follows:
:Let ''ABC'' be an isosceles triangle with ''AB'' and ''AC'' being the equal sides.
:Consider the triangles ''ABC'' and ''ACB'', where ''ACB'' is considered a second triangle with vertices ''A'', ''C'' and ''B'' corresponding respectively to ''A'', ''B'' and ''C'' in the original triangle.
:
is equal to itself, ''AB'' = ''AC'' and ''AC'' = ''AB'', so by side-angle-side, triangles ''ABC'' and ''ACB'' are congruent.
:In particular,
.
Others
A standard textbook method is to construct the
bisector of the angle at ''A''.
This is simpler than Euclid's proof, but Euclid does not present the construction of an angle bisector until proposition 9. So the order of presentation of the Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.
The proof proceeds as follows:
:As before, let the triangle be ''ABC'' with ''AB'' = ''AC''.
:Construct the angle bisector of
and extend it to meet ''BC'' at ''X''.
:''AB'' = ''AC'' and ''AX'' is equal to itself.
:Furthermore,
, so, applying side-angle-side, triangle ''BAX'' and triangle ''CAX'' are congruent.
:It follows that the angles at ''B'' and ''C'' are equal.
Legendre uses a similar construction in ''Éléments de géométrie'', but taking ''X'' to be the midpoint of ''BC''. The proof is similar but
side-side-side must be used instead of side-angle-side, and side-side-side is not given by Euclid until later in the ''Elements''.
In inner product spaces
The isosceles triangle theorem holds in
inner product spaces over the
real or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. In such spaces, it takes a form that says of vectors ''x'', ''y'', and ''z'' that if
:
then
:
Since
:
and
:
where ''θ'' is the angle between the two vectors, the conclusion of this inner product space form of the theorem is equivalent to the statement about equality of angles.
Etymology and related terms
Another medieval term for the pons asinorum was Elefuga which, according to
Roger Bacon
Roger Bacon (; la, Rogerus or ', also '' Rogerus''; ), also known by the scholastic accolade ''Doctor Mirabilis'', was a medieval English philosopher and Franciscan friar who placed considerable emphasis on the study of nature through emp ...
, 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.
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.
Gauss supposedly once espoused a similar belief in the necessity of immediately understanding
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
as a benchmark pursuant to becoming a first-class mathematician.
Similarly, the name ''
Dulcarnon'' was given to the 47th proposition of Book I of Euclid, better known as the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
, 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.
[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] The theorem was also sometimes called "the Windmill" for similar reasons.
Metaphorical usage
Uses of the ''pons asinorum'' as a metaphor for a test of critical thinking include:
*
Richard Aungerville'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.
*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
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
.
*The 18th-century poet
Thomas Campbell Thomas Campbell may refer to:
Arts and entertainment
* Thomas Campbell (poet) (1777–1844), Scottish poet
* Thomas Campbell (sculptor) (1790–1858), Scottish sculptor
* Thomas Campbell (visual artist) (born 1969), California-based visual artist ...
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 Mill
John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
called
Ricardo's Law of Rent the ''pons asinorum'' of economics.
*''Pons Asinorum'' is the name given to a particular configuration of a
Rubik's Cube
The Rubik's Cube is a Three-dimensional space, 3-D combination puzzle originally invented in 1974 by Hungarians, Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik t ...
.
*
Eric Raymond
The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''Eiríkr'' (or ''Eríkr'' in Old East Norse due to monophthongization).
The first element, ''ei-'' may be derived from the older Proto-Norse ''* ain ...
referred to the issue of syntactically-significant whitespace in the
Python programming language as its ''pons asinorum.''
*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.
Aasinsilta on laiskurin apuneuvo , Yle Uutiset , yle.fi
/ref> 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
Dutch commonly refers to:
* Something of, from, or related to the Netherlands
* Dutch people ()
* Dutch language ()
Dutch may also refer to:
Places
* Dutch, West Virginia, a community in the United States
* Pennsylvania Dutch Country
People E ...
, ''ezelsbruggetje'' ('little bridge of asses') is the word for a mnemonic
A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding.
Mnemonics make use of elaborative encoding, retrieval cues, and image ...
. The same is true for the German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
''Eselsbrücke''.
*In Czech, ''oslí můstek'' has two meanings – it can describe either a contrived connection between two topics or a mnemonic.
References
External links
*
D. E. Joyce's presentation of Euclid's ''Elements''
{{Ancient Greek mathematics
History of mathematics
Elementary geometry
Latin words and phrases
Articles containing proofs
Euclidean geometry
Theorems about special triangles