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that are called variational analysis. They involve the natural (or miraculous) selection of maxima or minima in quantifiable physical processes. Of all possible two-dimensional shapes with the same perimeter, the circle contains the greatest area; in three dimensions, it’s the sphere. In his Principia, Newton reports his work determining the optimal shape of round solids, with circles of revolution having the same effective cross section, in order to minimize frictional resistance to gravity in a medium.

The principle of least action says that imparting energy; say by a kick, to a physical body on a rigid two-dimensional surface like the earth, results in it taking the shortest route possible from its initial to final position. The related 1650 Fermat’s “principle of least time” is about light. As Feynman explains in his Lectures in Physics, “...out of all possible paths that light might take from one point another, light takes the path that requires the shortest time.” Feynman, using elementary relations from high school geometry, proved that the least time principle could lead directly to Snell’s law of the refraction of light at the interface of two different conducting media such as air and water. His analogy was the optimal choice of the path to take in order to rescue a pretty girl drowning in the ocean. Whereas the shortest distance to the girl leads directly into the water, faster running along the beach to the point that minimizes the distance required for the intrinsically slower rate of swimming increases the distance traveled but reduces the time required to reach her.

Euler attributed the optimization principle to an expression of the meaning and purpose of a loving God. Infused with this spirit, he developed mathematical methods describing smooth variations in position of an object in motion, the Euler differential equation, in which differential coefficients are varied to prove the principle of least action for mechanical motion. He gave the law Maupertuis’s name. Mach quoted Euler’s conclusion, “As the construction of the universe is the most perfect possible, being the handiwork of an all-wise Maker, nothing can be met with in the world in which some maximal or minimal property is not displayed.” Such faith based mathematical formalisms were rejected by Joseph Lagrange, an early 19th Century mathematician, who, among many other things, proved that every natural

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