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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.

Monday, May 28, 2012

The ancient astronomer Ptolemy (in Ancient Greek: Κλαύδιος Πτολεμαῖος, Klaudios Ptolemaios) was a Greek astronomer and Roman citizen who lived from around 90 AD to 168 AD in Egypt, at the time part of the Roman Empire. A prolific thinker and scientist, today Ptolemy is known chiefly for his work on astronomy, known today as The Almagest. It remains the only complete and comprehensive treatise on astronomy to survive from ancient times. Ptolemy also wrote books on optics, music and harmony, geography and even astrology.

The title of his key work the Almagest was not the title Ptolemy gave it. The first title attributed to this work was Μαθηματικὴ Σύνταξις or Mathematical Treatise. This was soon expaned to Ἡ Μεγάλη Σύνταξις (hey megalay zuntaxis) or The Great Treatise. When the manuscript was rediscovered by Arab mathematicians and astronomers centuries later, its importance as an astronomical treatise was clearly understood, and they named it The Almagest. In this case, the prefix al- is redundant, al- being the definite article the. The Arab astronomers who found it elevated it from the Great Treatise to the Greatest, and used the Ancient Greek superlative megiste, meaning greatest. Today, its name and importance converge-it is the most important early astronomical treatise, and therefore the greatest!

Image of Ptolemy’s triquetrum (or ‘parallactic instrument’) according to William Cunningham’s The Cosmographical Glasse, conteinyng the Pleasant Principles of Cosmographie, Geographie, Hydrographie, or Navigation (London: John Day, 1559), in the public domain.