Reading: The fifth postulate

The parallel postulate is the fifth postulate of The Elements. Since Antiquity, many mathematicians have distrusted it: it seemed more like a theorem than an evident truth, and several generations tried to demonstrate it from the other four. All attempts failed — and in the 19th century, Gauss, Bolyai and Lobachevsky realized why: by changing the fifth postulate to “through a point several parallels pass”, one obtains a geometry as consistent as Euclid’s, hyperbolic geometry. In it, the sum of the angles of a triangle is less than 180°. Euclid’s postulate is not a logical necessity; It is a choice that defines the geometry of our daily lives.

Carl Friedrich Gauss.Christian Albrecht Jensen (1792–1870) · Wikimedia Commons · Public domain

János Bolyai, imaginary portrait by Ferenc Márkos (2012).Ferenc Márkos · Wikimedia Commons · CC BY-SA 3.0

Nikolai Lobachevsky, portrait by Lev Kryukov.Lev Kryukov · Wikimedia Commons · Public domain

Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain

Euler's Eyes

Reading: The fifth postulate

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