When the Egyptian demi-god and Pharaoh Ptolemy I Soter asked his tutor Euclid of Alexandria if there was an easier way to learn geometry, Euclid is said to have replied that there was no royal road to knowledge. While this story is almost certainly apocryphal and not written down for almost 700 years by Proclus in the 4th century AD, there is wisdom in what Euclid was reported to have said. His Elements stands as the first comprehensive mathematical work, a giant work that continues to define the basis of modern (ahem, non-Euclidean) geometry. The first main obstacle in Book 1 of the Elements is the Fifth proposition, known since antiquity in Latin as the Pons Asinorum, or in English as the ‘bridge of asses’. Perhaps Ptolemy stumbled when confronted with the difficulty of understanding the theorem of isosceles triangles. Perhaps the larger and thornier issue suggested by the Fifth postulate (note that the Fifth proposition and Fifth postulate are not the same, though both lead mathematicians to non-Euclidean geometries) gave him pause.
Today the pons asinorum is used metaphorically to mean any barrier between knowledge and how it is acquired. It also has special use in logic for the perils inherent in discovering the middle term of a syllogism as illustrated above. It has also been long suggested that the name comes from the shape of the proposition when drawn, as it resembles a bridge.
Today Euclid’s reprimand of Ptolemy still holds: do your homework!
Images, left to right: Ptolemy I Soter, a 17th century engraving of the pons asinorum in logic, and Euclid’s fifth proposition.