"Natural Extremes" Series --- 5. The Story of the Brachistochrone

If the previous content of this series has not yet satisfied you, then the upcoming discussion on the Brachistochrone and Catenary problems might just meet your needs. However, before diving into the theoretical exploration of the Brachistochrone problem, let us first recount an exciting story of a mathematical competition that took place in the 17th century. I believe every friend who loves mathematics and physics will be inspired and moved by it. What is permeated within is not just an academic competition, but the unyielding spirit of generation after generation in the pursuit of truth and the exploration of new paths.

(The following content is sourced from the internet and organized by Scientific Space)

In 1630, the Italian scientist Galileo proposed a fundamental problem of analysis: "Given two points A and B in a vertical plane, what is the curve traced out by a point mass moving under the influence of gravity, which starts at A and reaches B in the shortest time, neglecting friction?" This can be considered the origin of this famous problem (Why didn’t others think of this? This shows that the quality of a great scientist lies in reflection and innovation; one must have thoughts, for without them, a person is no different from a walking corpse). Unfortunately, Galileo claimed the curve was a circular arc, which was an incorrect answer.

In 1696, the Swiss mathematician Johann Bernoulli proposed the Brachistochrone problem again (problem of brachistochrone), seeking solutions from mathematicians across Europe. Bernoulli named the problem "Brachistochrone," derived from the Greek words for "shortest" (brachistos) and "time" (chronos).

Naturally, people first thought of the straight line connecting A and B. Bernoulli said: "Although the straight line segment between A and B is the shortest distance, the time for a small ball to roll down it is not the shortest. If no one discovers this curve by the end of the year (referring to 1696), I will announce it myself." A straight line might not be the path of shortest time because, since the ball starts from zero velocity, the initial path should be steeper to accelerate and gain speed more quickly.

This was somewhat like a challenge in a martial arts novel. Clearly, Bernoulli had already derived the answer before issuing this challenge. The difficulty of this problem lies in finding a curve, which is essentially finding an unknown function that satisfies given conditions. This was unprecedented and had the potential to open a new field of study. Consequently, mathematicians took a great interest and began their research.

Bernoulli’s original deadline was the end of 1696, but he received only one solution, which was from his teacher Leibniz (an independent inventor of calculus and a great mathematician). Leibniz requested that Bernoulli extend the deadline to Easter of the following year (roughly between late March and late April) to allow European mathematicians more time to fully resolve this difficult problem.

Interestingly, in his "challenge letter," Bernoulli specifically hinted at his intended target. He wrote: "...few have been able to solve our unique problem, even among those who claim to have... not only explored the secrets of geometry deeply but also extended the field of geometry in an extraordinary way through special methods; these people think their great theorems are unknown to anyone, but in fact, others have already published them."

This was a blatant reference to the great Isaac Newton! The "theorems" Bernoulli mentioned clearly referred to the Method of Fluxions (Newton’s own name for calculus), and Newton had claimed to have discovered this theory long before Leibniz published his calculus paper in 1684. As mentioned earlier, Leibniz was Bernoulli’s teacher; since his master was fighting Newton for the priority of inventing calculus, the disciple felt duty-bound to defend the honor of his school. Johann Bernoulli personally copied the Brachistochrone problem, put it in an envelope, and sent it to England (Newton).

At this time, Newton was no longer the Newton of his youth. He admitted himself that his mind was not as sharp as it had been twenty years prior, and he was busy all day with the mundane affairs of the Mint. Regarding this event, we can look at the account by Newton’s niece, Catherine Conduitt: "One day in 1697, when the problem sent by Bernoulli arrived, Sir Isaac Newton was busy with the work of re-coining at the Mint and returned home very late, exhausted. However, he did not go to bed until he had solved the problem, which was at 4 o’clock in the morning."

Even in his later years, and after a full day of work, Newton solved a problem in a few hours that many European mathematicians could not solve! This gives a glimpse into the power of this great genius. Newton felt that his honor and reputation as a grandmaster were being challenged, and his opponents were waiting to laugh at him. Therefore, Newton took up the challenge and solved the problem in just a few hours. Newton was irritated; he is reported to have said: "I do not love to be... teased by foreigners about mathematical things." (Of course, if you don’t have the strength, you can only be teased).

By the Easter deadline of 1697, Bernoulli had received a total of five solutions. One was his own, one from his teacher Leibniz, and the third was from his brother, Jakob Bernoulli. This surely made Johann Bernoulli quite unhappy, as the two brothers were always competing and never yielded to one another. L’Hôpital was the fourth. The final solution arrived in an envelope with a British postmark; interestingly, it was anonymous, but the answer was perfectly correct! Clearly, this letter came from a supreme genius, none other than Isaac Newton. It is said that Bernoulli, half in anger and half in awe, put down the anonymous solution and remarked knowingly: "I recognize the lion by his paw."

Except for L’Hôpital’s solution, the others were published in the May 1697 issue of Acta Eruditorum. The answer was a segment of a cycloid. Pascal and Huygens had studied this important curve before, but neither of them realized it was also the brachistochrone. Because the time taken for a clock pendulum to complete one full swing is equal regardless of the amplitude (if it follows a cycloid), the cycloid is also known as the tautochrone curve.

The Brachistochrone also has beautiful applications in architecture. The "Big Roofs" of ancient Chinese buildings, when viewed from the side, do not have "isosceles triangle" sides made of straight lines, but rather two segments of brachistochrones. Designed according to this principle, during heavy summer rainstorms, the rainwater falling on the roof can flow away at the fastest speed, thereby protecting the building.

Challenges in mathematical history have existed since ancient times, but this challenge of the Brachistochrone can be described as the most exciting one in mathematical history for several reasons:

First, the number of participants was large. Second, those who derived the correct results were all illustrious mathematicians. Newton and Leibniz independently founded calculus; the Bernoulli family, represented by the Bernoulli brothers, was a mathematical dynasty whose status was similar to the Bach family in music. L’Hôpital showed mathematical talent at a young age, solving Pascal’s cycloid problem at fifteen. The famous L’Hôpital’s rule is something everyone becomes familiar with when studying higher mathematics.

Third, the solutions of each person had their own merits. Johann Bernoulli’s solution was the most elegant; he used an analogy with Fermat’s Principle, merging physics and geometry, and reached the conclusion instantly using optical ideas (somewhat like a "stunt" in a Math Olympiad). Jakob’s method was the most generalized, embodying the ideas of the calculus of variations. Newton, Leibniz, and L’Hôpital all used calculus methods (relative to variations, calculus was the "traditional" method), but their steps were different.

Animation Demonstration:

Finally, this problem directly led to the debut of another once-in-a-century genius. The great mathematician Leonhard Euler (a student of Johann Bernoulli) also began publishing related works in 1726. In 1728, he began to re-examine problems such as the brachistochrone with his characteristic spirit of thoroughness, eventually establishing the general method for finding the extrema of integrals. Euler’s method was later developed by Lagrange, who first placed the calculus of variations on an analytical foundation. He also fully utilized the calculus of variations to construct his system of analytical mechanics, reducing all of mechanics to a single unified variational principle—the principle of virtual work.

These new branches, together with calculus itself, formed the vast field known as "Analysis," which stands alongside algebra and geometry as the three major disciplines of mathematics. In the 18th century, its prosperity far exceeded that of algebra and geometry.

Mathematicians of the 18th century not only greatly expanded the boundaries of analysis but also gave it a meaning distinct from geometry. They strove to use pure analytical techniques to escape dependence on geometric proofs. This tendency became another major characteristic of 18th-century mathematics and was most typically expressed in the work of Euler and Lagrange.

In the preface to Mécanique Analytique, Lagrange declared: "No diagrams will be found in this work. The methods that I explain require neither constructions nor geometrical or mechanical arguments, but only algebraic [analytical] operations, subject to a regular and uniform course."