While Newton made a secret of his discovery of fluxions, Leibniz publicized his calculus, and by the year of 1695 he and his student John Bernoulli developed calculus into a magnificent tool for solving a variety of problems.

To find out how much Newton really knew, Leibniz and Bernoulli devised the following test. According to the custom of that time, John Bernoulli, published in June 1696 a challenging problem, which he addressed "to acutest mathematicians of the world''

`To find the curve connecting two points, at different heights and not on the same vertical line, along which a body acted upon only by gravity will fall in the shortest time'.

Leibniz and Bernoulli were confident that only a person who knows calculus could solve this problem. Bernoulli allowed six months for the solutions but no solutions were received during this period. At the request of Leibniz, the time was publicly extended for a year in order that all contestants should have an equal chance. On 29th of January 1697 the challenge was received by Newton from France and on the next day (according to his nephew's memoirs) he sent to Montague, who was then President of the Royal society, his solution. The only other solutions were sent by Leibniz and l'Hôpital. (The latter, another student of Leibniz, was the author of the first calculus textbook). Following Bernoulli's suggestion the curve which solves the problem is called the `brachistochrone', which is the Greek for `the shortest time'.

Can you solve the problem? If you are curious to see Bernoulli's solution, click here for pdf or ps format. A prerequisite to this solution is the Fermat's explanation of Snell's Law of refraction. And of course, what you already learned in MA 366.

Bernoulli's problem was an early example of a class of problems called Calculus of Variations now. These are extremal problems (finding maxima and minima), where the independent variable is not a number, not even several numbers, but a curve or a function. A rule which assigns a number to each curve of a given collection is called a "functional". It is like an ordinary function, except that a collection of curves instead of numbers serves as an independent variable.

A general approach to this class of problems, based on differential equations, was first found by Euler. In the end of XVIII century, Lagrange discovered that the fundamental laws of mechanics can be formulated as Variational Principles. About optics a similar discovery was made much earlier, by Fermat. Thus ALL fundamental laws of nature, known by the XIX century could be formulated in terms of Calculus of Variations. Amazingly, this also applies to all new fundamental laws discovered in XIX and XX century. (For quantum mechanics this was shown by R. Feynman in 1942; his beautiful lecture for undergraduates about variational principles is mentioned below).

Thus it turns out that Calculus of Variations is a kind of universal language of physics. In XVIII century this curious fact was even considered as a proof of the existence of God. (The Nature achieves its goals in "best possible" ways, that is by minimizing some functional depending of "all possible ways").

Sources:

L. T. Moore, Isaac Newton. A Biography, Dover, NY, 1934,
Feynman Lectures on Physics, vol. 2, ch. 19.

French translation of this text.