# Euler’s beautiful formula – 03/02/2021 – Marcelo Viana

Leonhard Euler (1707 – 1783) wrote about mathematics, physics, astronomy, mechanics, logic, philosophy and music in Latin, German and French. His works were published by the academies of the most advanced nations at the time, particularly Switzerland, Russia, Prussia and France. More than a century after the Swiss Academy of Sciences began collecting its works with 85 volumes already published, the task is not yet done.

His most famous public discovery, made around 1740, is the formula that bears his name: eiπ + 1 = 0. This beautiful relationship uses the most notable symbols in mathematics (π = 3.1415 …; Euler’s constant e = 2 , 7182…; the imaginary unit i = √-1; and the numbers 0 and 1 with which they are all constructed) in a single equality that combines arithmetic, algebra, geometry and analysis.

In 1758 Euler observed that the numbers F of the surfaces, A of the edges and V of the surfaces of a convex polyhedron (geometric body) always satisfy the equation FA + V = 2. For example, in the cube, F = 6, A = 12, and V = 8, and we see that 6-12 + 8 = 2. Alternative evidence of this fact came from mathematicians of Legendre and Cauchy caliber, who also pointed out that equality fails can if the polyhedron is not convex.

Despite these advances, Euler’s discovery remained little more than a curiosity until the 1890s, when Poincaré revealed its profound meaning and made it the basis of a new mathematical discipline: algebraic topology.

In 1760 Euler received an important general accuracy criterion for differential equations of any order. He submitted the work to the St. Petersburg Academy of Sciences, but it was not published until six years later. Meanwhile, Euler mentioned the discovery without evidence in his correspondence with D’Alembert, who informed Lagrange, Condorcet, and others. In 1765 the young Condorcet published evidence, not to mention Euler.

Annoyed but embarrassed to intervene directly, Euler urged Lagrange, who was close to Condorcet, to pressure him to see the source of the result. But fearful of ruining his friend’s career, Lagrange “rolled it up”. The truth about the “Condorcet Theorem” was not revealed until 1980 when these letters were published.

In June 1741 Euler left St. Petersburg for Berlin, where he would stay for almost two decades. It was a very productive time in his scientific life. However, since the mathematician has no sense of humor and is only interested in numbers and numbers, he never suits the brilliant and refined court of Frederick the Great, King of Prussia.

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