7 - Knocking on Heaven's Door

Summary

Sunday – January 12, 1794

I am most happy when I am in my own metaphysical space of solitude and sanctity, disconnected from daily realities, a place where I endeavor and reside. I dream here, too. But most of all I am in heaven when I am in this special realm where new fascinating discoveries are illuminated by the bright light of knowledge.

For the past two weeks I've researched a most intriguing topic. In one of Euler's memoirs I found the term calculi variationum, which I understood to be different from integral or differential calculus. Because the memoir is written in Latin, I began to translate it slowly and soon it became clear that this is another branch of mathematics that deals solely with problems of maxima and minima of definite integrals. I am not sure if I translated it correctly but I will refer to this branch of mathematics as “variational calculus.”

Because translation is tedious, sometimes I find solace reading about well-known isoperimetric problems, those where one seeks to answer “Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region?” This is equivalent to the problem:Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter?

Isoperimetrics, an area of study that examines the largest area surrounded by a fixed perimeter, can be considered the roots for the development of the variational methodology, that Euler addressed starting with the observation made by Aristotle that most motion appears to be in either straight lines or circles.