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The real start of algebra in Europe was only after the famous debate between Caldano and Tartalya. In order to understand the context of this debate that shook the mathematical community, we have to talk about it separately.

In the early years of the 16th century, at the oldest University of Bologna in Italy, there was a professor of mathematics named Ferlo,1465-1526 who devoted himself to studying the formula solution of the unary cubic equation, which was a difficult problem in the world at that time.

As we all know, although the solutions of some unary quadratic equations have been mastered in ancient Babylon and ancient China, the formula solutions of unary quadratic equations were given by the Central Asian mathematician Al-Khowarizmi in 825. Hua Lazimi rewrites the equation x2+ax+b=0 to

So that the two roots of the equation are obtained.

After Hua Lazimi, many mathematicians have made unremitting efforts to explore the mystery of the solution of cubic equations. However, in the course of 700 years, no one has made substantial progress except for the special solutions of individual equations. In the face of the grim reality, some people are deterred, they suspect that such a formula solution does not exist. However, Filo disagreed and still pursued it persistently. Heaven lived up to the kind-hearted man, and he finally made a major breakthrough at the age of forty.

In 1505, Filo announced that he himself had found a solution to a special case of the cubic equation in the form of x3+px=q. At that time, mathematicians tried to keep the secrets they had discovered in order to shine in the popular math competition at that time, so it was not surprising that Filo did not publish his results publicly at that time. But Filo finally failed to find an opportunity to show his talent and resentment passed away, so that people are still unable to fully solve the mystery of Filo's solution. However, people seem to know for sure that Ferro taught his method to a proud pupil, Floridus of Venice.

Now the conversation turns to the other side. In Bricia in northern Italy, there is a young man of little fame named Nicolo Tartalya. His original name was Fontana, he lost his father when he was young, his family was poor, and he suffered a near death. Pain, fear and shock left him with a problem of inarticulation. Later, he simply changed his name to "Tartalya", which means "stuttering" in Italian.

Nicolo Tartaria Jr. Tartalya is talented and studious. He studied physics and mathematics, and soon showed his superhuman talent. In particular, some of the papers he published were famous for their strange ideas, high ground and high ground, showing their quite deep mathematical attainments.

Tartalya's self-taught talents were despised and envied by some people with professional background at that time.

In 1530, Cora, a math teacher in Brissia, asked Tartalya two challenging questions to stump each other. These two questions are:

(1) find a number, whose cube plus 3 times the square equals 5.

(2) find three numbers, in which the second number is 2 greater than the first number, and the third number is 2 larger than the second number, and their product is 1000.

These are actually two problems for finding the real root of the cubic equation. If the first number in the question is x, then the equation of the first question is x3+3x2-50.0and the equation of the second question is x3+6x2+8x-1000=0. Tartalya won the challenge by finding the real roots of these two equations, and became famous for it.

The news spread to the University of Poliya. Ferro's student, Floridus, felt a little uncomfortable when he heard that someone in Bricia could solve the cubic equation. He originally thought that he had a single biography of a famous teacher, and this life should be unique, but he killed a "Cheng Yaojin" along the way, and he was also a small potato who did not enter the hall of elegance. How could he be convincing? So after several negotiations, they finally decided to hold an open math competition in Milan, the second largest city in Italy, on February 22, 1535. The two sides each asked 30 questions and decided the outcome within 2 hours.

As the competition approached, Tartalya felt a little nervous because he came out of the mountain by self-study. He thought, "Floridus is a disciple of Ferro, and it is impossible to say that he will embarrass himself by solving the cubic equation, so how will he deal with it?" He thought, "what is the difference between the kind of solution he has mastered and that of Philo?" He thought hard, and the ideas in his mind were constantly carrying on various new combinations, and these new combinations finally hit the spark of inspiration.

Eight days before the game, Tartalya finally found a way to further solve the cubic equation. He was ecstatic for this and made full use of the remaining eight days to familiarize himself with his new method while carefully constructing 30 problems that could only be solved by the new method.

On February 22, 1535, the Gothic Marble Church in Milan was crowded and bustling, and everyone waited for the competition to come. The game began, and the 30 questions raised by both sides were dazzling cubic equation problems. But see Tartalya leisurely, carry pen such as flying, in less than 2 hours, solved all the problems of Floridus. At the same time, Floridus picked up his pen and picked up the paper. He sighed and lost by 0 ∶ 30.

The mathematical circle was shocked by the news. There was a man in Milan who couldn't sit still. He was Caldano, a well-known doctor in Europe at that time. Caldano is not only highly skilled in medicine, but also good at mathematics. He has published many mathematical papers and carefully studied the problem of cubic equations, but found nothing. So when he heard that Tartalya had mastered the solution of the cubic equation, he was full of hope to share the result.

However, Tartalya was already famous in Europe at that time, so he did not intend to publish his results immediately, but was obsessed with completing the giant translation of the original Geometry. He turned away many people who asked for advice. Caldano, a former doctor, familiar with the essentials of psychology, moved Tartalya with diligence, hard work and sincerity, so that he seemed to see the shadow of his childhood, thus becoming the only exception. In 1539, at Caldano's repeated pleas, Tartalya finally agreed to teach him his secret, but on the condition that he should keep it a secret.

However, Caldano did not keep this promise. In 1545, he published under his own name ArsMagna, which means great skill, in which he introduced the solution of incomplete cubic equations and wrote:

"about 30 years ago, Ferro of Bologna discovered this law and taught it to Floridus in Venice, who competed with Tartalya in mathematics. Tartalya also discovered this method. Tartala told me the method at my request, but did not prove it. With the help of this, I have found a number of proofs, which are described below because they are very difficult."

The above is the reason why the root formula of the cubic equation is called Caldano formula.

Caldano pointed out that for incomplete cubic equations

Its solution is given.

By the way, from the complete cubic equation ax3+bx2+cx+d=0 to the incomplete cubic equation, only one transformation y=x+b / 3a is needed. It's actually only one step away.

In the second year of the publication of Dafa, Tartalya published "questions and inventions", denouncing Caldano's treachery and demanding an open competition with Caldano in Milan.

However, on the day of the competition, it was not Caldano himself, but his gifted student, Ferrari (Ferrari,1522-1565), a young man who had been a servant since childhood and was chosen by Caldano for his outstanding talent. At this time, Ferrari was in his prime, quick in mind and eloquent. He not only mastered the essentials of understanding the cubic equation, but also found a very ingenious solution of the quartic equation. Stammering Tartalya, which is Ferrari's opponent, is naturally vulnerable to defeat!

Since then, although Tartalya devoted himself to the great work of algebra, he was so hurt by this setback that he died in 1557 at the age of 58.

Source: "Mathematics Story Book for Children" author: Zhang Yuannan Zhang Chang part of the picture comes from the network copyright belongs to the original author Editor: Zhang Runxin this article comes from Wechat official account: Origin Reading (ID:tupydread), author: Zhang Yuannan Zhang Chang, Editor: Zhang Runxin

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