In existographies, Leonhard Euler (1707-1783) (IQ:195|#17) (Cattell 1000:512) [RGM:101|1,500+] (Murray 4000:16|CS / 1|M) (GME:1) (CR:103) was Swiss mathematician, a top-ranked greatest mathematician ever, noted for his his 1739 reciprocity relation (or Euler reciprocity relation), the mathematical proof behind state functions, in particular entropy.

Overview
Euler made advances in a number of areas, such as the invention of various mathematical relations, proofs, symbols, and operators used in thermodynamics, such as the function, i.e. f(x), denoting the function f applied to the argument x, the imaginary number i, the natural log base e (such as used famously in S = k ln W), the summation operator ∑, among others, all introduced in years around circa 1755. His famous "Euler identity", shown below:

$e^{i \pi } + 1 =0\,$

where e is the mathematical constant sometimes known as Euler's number and i is the square root of minus one, is widely considered the most beautiful equation in mathematics, and second greatest equation of all time, behind Maxwell's field equations, according to Robert Crease's 2004 Physics World “The Greatest Equations Ever” reader poll.

The interaction of Euler with his best known student Italian mathematician Joseph Lagrange, between 1754 and 1756, resulted in the Euler–Lagrange equations of variational calculus.

Thermodynamics
In mathematical thermodynamics, Euler is noted for his homogenous function, which supposedly is the mathematical form which relates an extensive property of a system to its component variables. [1]

His his reciprocity relation (or Euler reciprocity relation) is the mathematical proof behind state functions, in particular entropy.

Collected works | Publications
Euler’s collected works, according to Albert-Laszlo Barabasi (2002), began to be published in 1911 under the title Opera Omnia (Basel, Switzerland: Birkhauser Verlag AG, 1913), of which, more than six-dozen volumes have appeared, but the work is still completed. [6]

Euler’s collected works, according to another estimate, are said to total 60 to 80 quarto volumes. [2] Euler is considered the most prolific mathematician of all time, publishing close to 900 books. When he went blind in his late 50s, however, his productivity in many areas increased. [3]

Education
Euler, as a child, was given math lessons by Johann Bernoulli, who is said to have quickly discovered his new pupil's talent.

At some point along the way, Euler carefully analyzed the basic concepts and methods introduced by Rene Descartes, Isaac Newton, and Gottfried Leibniz. [5]

Euler would eventually complete his dissertation on the physics of sound under Bernoulli in 1726 at the Universität Basel, Switzerland. [4]

Students
See main: Euler genealogy

 Student School Descendents Johann Euler Nicolaus Fuss St. Petersburg Academy Johann Hennert Königliche Akademie der Wissenschaften zu Berlin 3,958 Semen Kotelnikov Universität Berlin Joseph Lagrange 60,026 Stepan Rumovsky Universität Berlin

as well as many indirect students of his work. A statement attributed to French physicist Pierre Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Euler, read Euler, he is the master of us all."

Religion
A downfall of Euler, is that he often took recourse to the Bible to explain these things; as many were forced to do before the 1830s translation of the Rosetta stone.

References
1. Kestin, Joseph. (1979). A Course in Thermodynamics, Volume 1 (7.5.3: Euler’s theorem on homogeneous functions, pg. 326). Taylor & Francis.
2. Leonhard Euler – NNDB.
3. Bellos, Alex. (2010). “The 10 Best Mathematicians”, The Guardian, Apr 10.
4. Leonhard Euler – Mathematics Genealogy Project.
5. Suisky, Dieter. (2008). Euler as Physicist (pg. 8). Springer.
6. Barabasi, Albert-Laszlo. (2003). Linked: the New Science of Networks (pg. 244). Basic Books.