Euler’s Homework

Even the best and most prolific of mathematicians have had to do homework assignments. Famed Swiss mathematician Leonhard Euler (1707–1783) was no exception.

Euler was only 14 years old when he was sent to the University of Basel in 1720 to study for the ministry. Not long after his arrival, he got himself introduced to Johann Bernoulli (1667–1748) and persuaded the famous scholar to serve as his mentor.

In his unpublished autobiographical writings, Euler noted, “True, [Bernoulli] was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Saturday afternoon and he kindly explained to me everything I could not understand.”

Euler obtained his Master’s degree in philosophy in 1723, then embarked on his theological studies. But his heart was set on mathematics. After hearing from Bernoulli, an acquaintance from undergraduate days, Euler’s father agreed to let his son switch to mathematics.

Euler completed his studies at the university in 1726. By that time he already had a mathematical article in print and, in the following year, submitted an entry for a prestigious competition sponsored by the Paris Academy on the best arrangement of masts on a ship.

That year, before he left Basel for St. Petersburg, Euler received the following assignment from his mentor: Find the shortest line between two given points on a surface.

A copy of the manuscript that Euler handed in to Bernoulli is in the Euler archives in Moscow and a paper based on the homework was published in the journal Commentarii academiae scientiarum Petropollitanae in 1728. Both documents are reprinted in Opera Omnia, a massive edition of Euler’s collected papers.

As part of a project to work through Euler’s papers chronologically, Ed Sandifer of Western Connecticut State University recently examined Euler’s efforts to solve the problem that Bernoulli had posed. The written evidence is illuminating. Mathematician Constantin Carathéodory (1873–1950) once remarked that “this work reads like a worksheet, and one watches Euler’s discoveries as he makes them.” Sandifer agrees. “It is a thrill to watch the Master at work,” he says.

Euler started with two easy cases. On a flat surface, the shortest distance between two given points is along a straight line. On the surface of a sphere, the shortest distance is along a segment of a great circle passing through the points.

Finding the equation for the shortest line on any given convex, concave, or lumpy surface is far more difficult, however. Indeed, in Euler’s time, there was no known general solution to the problem.

For a convex surface (like that of a sphere), Euler noted that the problem can be solved mechanically by stretching a string between the two given points–but that gives only the curve, not an equation describing the curve.

Interestingly, Euler’s consideration of a mechanical process to solve the problem suggests his willingness to use any available analytical tool to help reach an answer. “To him, anything is fair in mathematics,” Sandifer says.

To solve the general problem, Euler turned to a novel system of three-dimensional coordinates suited to his task, then step by step worked his way through to an answer expressed in terms of differentials and applicable to various types of surfaces. The details are in Sandifer’s paper “Euler and the greatest homework ever?” (available at http://www.southernct.edu/~sandifer/Ed/History/Preprints/Preprints.htm).

Toward the end of his paper, Euler returned to the special cases he had mentioned at the start. He showed how the fact that the shortest line between two points on a sphere is the arc of a great circle arises naturally from his novel formulation of the problem.

“The Celebrated Johann Bernoulli proposed this question to me and urged me to write up my solution and to investigate these three kinds of surfaces which lead to solutions that are integrable equations,” Euler concluded. “I wanted to include the solutions to these questions because they followed so easily from what I had done earlier.”

Sandifer translates Euler’s remarks into this comment: “Once I knew what I was doing, this homework was pretty easy.”

Nonetheless, Euler’s homework represents a remarkable achievement. Sandifer argues that the manuscript and the paper together laid the foundations for a branch of mathematics known as the calculus of variations, which deals with such problems as characterizing optimal forms and finding the peaks and valleys (maxima and minima) of mathematical landscapes.