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Euclid’s fifth proposition in the first book of his Elements (that the base angles in an isosceles triangle are equal) may have been named the Bridge of Asses (Latin: Pons Asinorum) for medieval students who, clearly not destined to cross over into more abstract mathematics, had difficulty understanding the proof—or even the need for the proof. An alternative name for this famous theorem was Elefuga, which Roger Bacon, writing circa AD 1250, derived from Greek words indicating “escape from misery.” Medieval schoolboys did not usually go beyond the Bridge of Asses, which thus marked their last obstruction before liberation from the Elements.
We are given that ΔABC is an isosceles triangle—that is, that AB = AC.
Extend sides AB and AC indefinitely away from A.
With a compass centred on A and open to a distance larger than AB, mark off AD on AB extended and AE on AC extended so that AD = AE.
∠DAC = ∠EAB, because it is the same angle.
Therefore, ΔDAC ≅ ΔEAB; that is, all the corresponding sides and angles of the two triangles are equal. By imagining one triangle to be superimposed on another, Euclid argued that the two are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of the other triangle (known as the side-angle-side theorem).
Therefore, ∠ADC = ∠AEB and DC = EB, by step 5.
Now BD = CE because BD = AD − AB, CE = AE − AC, AB = AC, and AD = AE, all by construction.
ΔBDC ≅ ΔCEB, by the side-angle-side theorem of step 5.
Therefore, ∠DBC = ∠ECB, by step 8.
Hence, ∠ABC = ∠ACB because ∠ABC = 180° − ∠DBC and ∠ACB = 180° − ∠ECB.
Bridge of Asses or Pons Asinorum (Latin for "Bridge of Asses") is a term used to refer to a problem that severely tests the ability of an inexperienced person, and therefore separates the serious and dedicated students from the “asses.” It is said that students are as reluctant to tackle these problems as donkeys (asses) are to cross over a bridge. Once a student is experienced in his field, however, the problem appears relatively simple. The term can be used to refer to a problem that is a stumbling block in any field, or to a problem whose solution seems pointless.
The term “Bridge of Asses” first came into use during the Middle Ages, and is most commonly applied to a diagram used to help students of logic identify the middle term in a syllogism, or to Euclid's fifth proposition in Book 1 of his Elements of geometry. As early as the sixth century, the Greek philosopher Philoponus used a diagram to show what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises.
“Pons Asinorum” in Logic
The sixth century Greek philosopher Philoponus, presented a diagram showing what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises, to enable students of logic to construct valid syllogisms more easily.[1]
The French philosopher Jean Buridan (Joannes Buridanus, c. 1297 – 1358), professor of philosophy in the University of Paris, is credited with devising a set of rules to help slow-witted students in the discovery of syllogistic middle terms, which later became known as the pons asinorum.
In 1480, Petrus Tartaretus applied the Latin expression “pons asinorum” to a diagram illustrating these rules, whose purpose was to help the student of logic find the middle term of a syllogism and disclose its relations to the other terms.[2]
The “asses’ bridge” was usually presented with the predicate, or major term, of the syllogism on the left, and the subject on the right. The three possible relations of the middle term to either the subject or the predicate (consequent, antecedent and extraneous) were represented by six points arranged in two rows of three in the middle of the diagram, between the subject and the predicate. The student was then asked to identify the nineteen valid combinations of the three figures of the syllogism and evaluate the strength of each premise.[3][4]
Fifth Proposition of Euclid
Euclid's Fifth Proposition reads:
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
Pappus provided the shortest proof of the first part, that if the triangle is ABC with AB being the same length as AC, then comparing it with the triangle ACB (the mirror image of triangle ABC) will show that two sides and the included angle at A of one are equal to the corresponding parts of the other, so by the fourth proposition (on congruent triangles) the angles at B and C are equal. The difficulty lies in treating one triangle as two, or in making a correspondence, but not the correspondence of identity, between a triangle and itself. Euclid's proof was longer and involved the construction of additional triangles:
Proposition 5
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC. (Book I.Definition 20; Postulate 2)
I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE. Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB. (Book I. Proposition 3.; Postulate.1)
Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG.
Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB. (Book I.Proposition 4)
Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG. (Common Notion 3)
But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them. Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG. (Book I. Proposition 4)
Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB, and they are under the base.(Common Notion 3)
Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
==It is the ass's pitfall, not his bridge.
If this be rightly called the “Bridge of Asses,”
He's not the fool who sticks, but he that passes.[6]==
In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (Latin pronunciation: [ˈpons asiˈnoːrʊm]; English: /ˈpɒnz ˌæsɪˈnɔːrəm/ PONZ ass-i-NOR-əm), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Euclid's Elements, and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal.
The name of this statement is also used metaphorically 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.[1]
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 involves drawing auxiliary lines to these extensions. But, as Euclid's commentator Proclus 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, given that it makes the proof more complicated, Euclid added the second conclusion to the theorem. 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.[2] 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:[3] 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 angle A is equal to itself, so by side-angle-side, the triangles are congruent and corresponding sides and angles are equal. Therefore angle ABE = angle ACD, angle ADC = angle AEB, 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 angle DBE = angle ECD have just been shown, so applying side-angle-side again, the triangles are congruent. Therefore angle BDE = angle CED and angle BED = angle CDE. Since angle BDE = angle CED and angle CDE = angle BED, angle BDC = angle CEB by subtraction of equal parts. Consider a third pair of triangles, BDC and CEB; DB=EC, DC=EB, and angle BDC = angle CEB, so applying side-angle-side a third time, the triangles are congruent. In particular, angle CBD = BCE, which was to be proved.
Proclus gives a much shorter proof attributed to Pappus of Alexandria. 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.[4] This method is lampooned by Charles Lutwidge Dodgson in Euclid and his Modern Rivals, calling it an "Irish bull" because it apparently requires the triangle to be in two places at once.[5]
The proof is as follows:[6] 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. Angle A is equal to itself, AB=AC and AC=AB, so by side-angle-side, triangles ABC and ACB are congruent. In particular angle B = angle C.[7]
A standard textbook method is to construct the bisector of the angle at A.[8] 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:[9] As before, let the triangle be ABC with AB = AC. Construct the angle bisector of angle BAC and extend it to meet BC at X. AB = AC and AX is equal to itself. Furthermore angle BAX = angle CAX, 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.[10] 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.
Moo moo piggies of Baal
Neck yourself immediately
The isosceles triangle theorem holds in inner product spaces over the real or complex numbers. In such spaces, it takes a form that says of vectors x, y, and z that if[11]
x
+
y
+
z
=
0
and
‖
x
‖
=
‖
y
‖
,
{\displaystyle x+y+z=0{\text{ and }}\|x\|=\|y\|,}
then
‖
x
−
z
‖
=
‖
y
−
z
‖
.
{\displaystyle \|x-z\|=\|y-z\|.}
Since
‖
x
−
z
‖
2
=
‖
x
‖
2
−
2
x
⋅
z
+
‖
z
‖
2
,
{\displaystyle \|x-z\|^{2}=\|x\|^{2}-2x\cdot z+\|z\|^{2},}
and
x
⋅
z
=
‖
x
‖
‖
z
‖
cos
θ
{\displaystyle x\cdot z=\|x\|\|z\|\cos \theta }
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
Edit
Another medieval term for the pons asinorum 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.[12]
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.[13] 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.[14]
Similarly, 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 , 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.[12] The theorem was also sometimes called "the Windmill" for similar reasons.[15]
Uses of the pons asinorum 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 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.[12]
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.[12]
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.[16]
Economist John Stuart Mill called Ricardo's Law of Rent the pons asinorum of economics.[17]
Pons Asinorum is the name given to a particular configuration of a Rubik's Cube.
Eric Raymond referred to the issue of syntactically-significant whitespace in the Python programming language as its pons asinorum.[18]
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.[19] 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.
>In Czech, oslí můstek has two meanings – it can describe either a contrived connection between two topics or a mnemonic.
Around 1250 a man named Roger Bacon gave an alternate name to Euclid’s fifth proposition in the first book of his elements, which I will from here on out refer to as “the Bridge of Asses” or the fifth proposition. The name he gave it was Elefuga, another word I will use freely to refer to the fifth proposition. Elefuga, derived from Greek, means, “escape from misery.” Medieval boys were presented with the Elefuga shortly before their “escape from misery.” That is to say most medieval young men’s experience in geometry ended shortly after they encountered the fifth element, because it proved they simply did not want to go on or their mentor felt they should not. They, like a donkey fears crossing a bridge, had a hard time grasping the fifth proposition or refused to grasp it. I personally believe they refused to try to grasp it or the mentor did not want to walk them through it well enough. This is because I think with time and patience people can overcome most barriers, but again I am digressing.
To better explain this, “the Bridge of Asses,” also known as the isosceles triangle theorem, is Proposition 5 of Book 1 of Euclid’s Elements. But, also, pons asinorum, the Latin translation of “the Bridge of Asses,” became a metaphorical statement for a problem that will separate the confident from the unconfident. In other words it is a critical test, of the ability and understanding, of an individual. You see things like this all the time in movies. Usually someone has a sensei or master and they are trying to prove themselves. Eventually they come to the test that decides if they will continue with their training or not. For Bruce Wayne in Batman Begins it is, possibly, when he brings the flower to the League of Shadows high up in the mountain so that he can begin training with them. Now we want to pass “the Bridge of Asses” for math, or proposition 5. Let’s see if you and I can manage to cross the bridge of elements together.
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