Thread: Euler reciprocity relation (origin?)

Anyone know what publication Leonhard Euler first derived the reciprocity relation?

Re: “Even if you were clear on what you were talking about, one would have to study the works of Liebnitz, Gauss and newton (and several others) in particular besides Euler to know whether whatever contribution you are talking about is unique and no one else was involved in essentially similar work.” (by Mr Lonny) (28 Dec 2011), the Euler reciprocity relation is said to determine (or prove mathematically) whether or not a function is an exact differential equation (as contrasted with an inexact differential equation), and is the mathematical justification behind the concept of the “state function” (in thermodynamics), in particular the second law of thermodynamics, that which governs the direction of time and the nature of the universe.

Re: ”Even if you were clear about what you were talking about”, the subject was first outlined in German physicist Rudolf Clausius’ 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable” and then expanded on as the first chapter (Mathematical Introduction) to his 1865 and 1875 textbook The Mechanical Theory of Heat:

http://www.eoht.info/page/Mathematical+Introduction

Clausius called it the “condition for an exact differential”:

http://www.eoht.info/page/condition+...t+differential

It has since come to be known as the “Euler reciprocity relation” in textbooks on higher mathematics in chemistry in physics. Hence, if anyone has a clue as to where in Euler’s works this relation was first used, that is what I am looking for?