Reading: Euler, analysis incarnate

Leonhard Euler (1707–1783) was born in Basel, Switzerland. His father, a Protestant pastor, wanted him to pursue a religious career, but the young man was a student of Johann Bernoulli, then the greatest mathematician in Europe, who soon recognized his talent and convinced the family to let him study mathematics.

Leonhard Euler, pictured in 1753.Jakob Emanuel Handmann · Wikimedia Commons · Public domain

In 1727, at the age of twenty, he went to the newly created Saint Petersburg Academy of Sciences, in Russia, where Daniel and Nicolau Bernoulli, sons of his master, already worked. Around 1738, after an illness, he lost the vision in his right eye. In 1741 he accepted the invitation of Frederick the Great, and spent 25 years at the Berlin Academy; in 1766 he returned to Saint Petersburg, where he stayed until he died. Shortly after his return, a cataract took away his sight in his other eye: Euler spent the last seventeen years of his life practically blind — and it was during this period that he produced almost half of his work, dictating to assistants and making enormous calculations in his head.

Euler wrote about almost everything: analysis, number theory, geometry, mechanics, astronomy, optics, music, shipbuilding. His books were so widely read that many of the notations he used have become ours: $e$ for the base of natural logarithms, $\pi$ for the ratio of circumference to diameter, $i$ for $\sqrt{-1}$, $f(x)$ for a function, $\Sigma$ for a sum, and lowercase letters $a$, $b$, $c$ for the sides of a triangle opposite the angles $A$, $B$, $C$.

For trigonometry, the decisive work was Introductio in analysin infiniterum (“Introduction to the analysis of infinities”), by 1748. In it, sine and cosine stopped being segments of a circle of arbitrary radius and became functions of a number, measured on a circle of radius 1 — exactly the trigonometric cycle of this book. There also appears the famous formula that links trigonometry to exponential:

$$e^{ix} = \cos x + i\operatorname{sen} x.$$

Euler died in Saint Petersburg, in 1783, aged 76. The astronomer François Arago said that he “calculated without apparent effort, as men breathe and eagles sustain themselves in the air”; and his contemporaries called him analysis incarnate.

Sources: Euler (MacTutor) · L. Euler, Introductio in analysin infiniterum (1748).

Euler's Eyes

Reading: Euler, analysis incarnate

Further reading

On this page