Hero Image
Reading: The fifth postulate

Further reading Leitura 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

October 11, 2026 Read
Hero Image
Exercises from chapter 5 — Parallelism

Exercises and resolutions 130 Since the line $a$ is parallel to the line $b$, determine $x$ in the cases: Ative o JavaScript para ver esta figura interativa. Ver respostaa) $50^{\circ}$ · b) $60^{\circ}$

October 11, 2026 Read
Hero Image
Consequences in the triangle

The sum of the interior angles of any triangle is 180°. It is perhaps the best-known geometry theorem after Pythagoras'. Who discovered Proclus, citing Eudemus of Rhodes, attributes to the Pythagoreans (6th–5th centuries BC) the first general demonstration that the angles of a triangle add up to two right angles. The commentator Gemino adds that the ancients had proven the fact separately for equilateral triangles, isosceles and scalenes. The version that has come down to us is the proposition I.32 of the The Elements of Euclid, which also proves that the external angle is equal to the sum of the two non-adjacent internal angles.

October 11, 2026 Read
Hero Image
Existence and uniqueness of parallel

A parallel passes through a point outside a straight line. That only one passes is the most discussed statement in the history of geometry. Who discovered The existence of parallelism is demonstrated by Euclid (proposition I.31). uniqueness depends on its fifth postulate, the one that many have tried to transform into a theorem. The form used today at school — “a single parallel passes through a point outside a straight line” — became known as Playfair’s axiom, after the Scotsman John Playfair, who popularized it in 1795, although Proclus had already stated it. In 1829, the Russian Nikolai Lobachevsky published a geometry in which infinite parallels pass.

October 11, 2026 Read
Hero Image
Parallel lines

Parallel lines never meet — and yet, in a perspective drawing, they appear to meet at the horizon. Who discovered The classic definition is from Euclid (c. 300 BC), in definition I.23 of The Elements: parallel lines are those that are in the same plane and, extended indefinitely in both directions, do not meet. The relationships between the angles formed by a transversal — alternate, corresponding, collateral — are the propositions I.27 to I.29. The philosopher Posidonius (1st century BC) proposed another definition, as straight lines always at the same distance; Proclus recorded the debate.

October 11, 2026 Read