The Bernoulli brothers left a deep impression on the mathematics of their time, including harmonic series and many other contributions. However, there is another story about these two competing and difficult brothers, which must be the most fascinating story in the whole history of mathematics.

The story begins in June 1696, when johann bernoulli published a challenge in Leibniz's Teacher's Magazine. Obviously, the tradition of open challenge began in the era of Fiore and tartaglia. Although the current debate is a quiet pen battle in academic magazines, it still has the ability to make or destroy a person's reputation, as John himself said:

"... to be sure, it is the difficulties before us that are also useful problems, which inspire outstanding people to strive for enriching human knowledge, and they will become famous in one fell swoop and be immortal. "

John's challenge is wonderful. He imagines two points A and B at different heights on the ground, and don't let one of them be directly above the other. Connecting these two points, of course, we can make infinitely many different curves, from straight lines and circular arcs to countless other curves and wavy lines. Now imagine a ball rolling along a curve from point A to the lower point B. Of course, the time required for the ball to roll depends entirely on the shape of the curve. Bernoulli's challenge to mathematics is to find a curve AMB, so that the time required for the ball to roll along this curve is the shortest. He called this curve "steepest descent line", which was synthesized by the Greek words "shortest" and "time".

Obviously, the first guess is to connect two points A and B into a straight line AMB. However, John warned against trying to adopt this simplistic approach:

"... don't make a hasty judgment. Although straight line AB is indeed the shortest route connecting point A and point B, it is not the shortest route. AMB curve is well known to geometricians. If no one else can find this curve before the end of the year, I will announce the name of this curve. "

John is scheduled to announce the answer to the mathematics community at 1697 65438+ 10/month 1. However, by the time of writing, he only received the answer from the famous Leibniz, and Leibniz

"humbly ask me to extend the deadline to Easter so that when the answer is announced ... no one will complain that the time given is too short. I not only agreed to his request, but also decided to personally announce the extension period to see who can finally solve this wonderful problem after such a long time. "

Then, in order to ensure that people don't misunderstand this difficult problem, John repeated it again:

"In an infinite number of curves connecting two known points ... choose a curve. If this curve is replaced by a thin tube or slot, and a ball is placed in the thin tube or slot and allowed to roll freely, then the ball will roll from one point to another in the shortest time. "

At this time, John began to enthusiastically advocate rewarding those who solved his fastest descent line problem. Don't forget, he knows the answer himself, so his words about mathematics honor are hard to avoid boasting:

"I hope someone can win the championship quickly. Of course, the prize is neither gold nor silver, because these things can only arouse the interest of humble people ... On the contrary, because virtue itself is the best reward and reputation is the strongest stimulus, our reward for noble winners is honor, praise and recognition ... "

In this passage, John seems to think that he has won another victory in the face of his poor brother Jacob. However, he has another goal in mind. John wrote:

"... few people can solve our unique problems, even those who claim not only to explore the secrets of geometry through special methods, but also to expand the territory of geometry in extraordinary ways. These people think that their great theorem is unknown, but it has already been published. "

Who else can doubt that the "theorem" he said refers to the flow number method, and the object he despises is isaac newton? Newton claimed to have discovered this theory before Leibniz published his calculus paper in 1684. There is no doubt that John's challenge goal is very clear. He copied a copy of his fastest descent line problem, put it in an envelope and sent it to England.

Of course, in 1697, Newton was busy with the affairs of the mint, and, as he himself admitted, his mind was not as alert as it was in its heyday. At that time, Newton and his niece Catherine condit lived in London. Catherine described the story:

"/kloc-One day in 0/697, when he received a question from Bernoulli, Sir isaac newton was busy minting new coins. He didn't get home exhausted until four o'clock, but he didn't go to bed until he solved the problem. It is already four o'clock in the morning. "

Even in his later years, exhausted after a hard day's work, isaac newton succeeded in solving many problems that Europeans failed to solve! This shows the strength of this great British genius. He obviously felt that his reputation and honor had been challenged; Moreover, Bernoulli and Leibniz are still eagerly waiting to announce their answers. Therefore, Newton did his part and solved the problem in only a few hours. However, Newton was a little angry. It is said that he once said, "I don't like ... teasing foreigners on math problems."

Let's go back to Europe. As Easter approached, johann bernoulli received several letters. The curve that each of them seeks is an inverse trochoid, which is indeed a "curve well known to geometricians". We noticed that Pascal and Huygens studied this important curve, but they didn't realize that cycloid was still the fastest declining curve. John wrote in an exaggerated tone: "... you will be shocked if I clearly state that Huygens' ... cycloidal line is the steepest descent line we are looking for." "

By Easter, the deadline for the challenge has arrived. John received five answers. Including his own answer and Leibniz's answer. His brother Jacob gave a third answer (which might upset John) and the Marquis of Lopita gave a fourth answer. The final answer stamped the envelope with a British postmark. When John opened it, he found that although the answer was anonymous, it was completely correct. He obviously met his opponent, isaac newton. Although the answer is unsigned, it is obviously written by a brilliant genius.

It is said (perhaps not completely reliable, but interesting) that John put down his anonymous answer in embarrassment and awe and said knowingly, "I recognized the lion from his claws."