In 1945, the Italian physicist and mathematician Girolamo Cardano published a book called "The Rules of Great Derivation or Algebra", which caused an uproar in the mathematical community. At the beginning of the book, the solution of standard linear and quadratic equations is introduced, followed by the complete process of solving the three- and four-degree algebraic equations in the unknown domain of the time. Since its inception, the book influenced the development of mathematics throughout Europe in the 16th century. It was not until the advent of Veda and Descartes that the book's place was replaced. The Great Derivative Had just been published, and an Italian mathematician, Tartaglia, began attacking Cardano. Although Cardano states on several occasions in the book, some of the basic solutions to the cubic equations in the book should be attributed to Tartaglia. Tartaglia insisted that when he showed Cardano its solution, Cardano, as a Christian and gentleman, promised to publish his own book after Tartaglia's book.
The European Renaissance began in the 14th century as the awakening and rebirth of European wisdom that had been dormant for 1,000 years. Towards the end of the 15th century, the first exciting moment in the history of mathematics was seen. As early as ancient Greece, Arab and Indian mathematicians were able to solve linear and quadratic equations.
As early as the 9th century, Arab mathematicians gave geometric solutions and even conjectures to certain cubic equations. It was not until between 1510 and 1515 that Pione del Ferro, a professor of mathematics at the University of Bologna, proposed an algebraic solution to the first cubic equation, a real breakthrough. He invented an algebraic formula to solve the "cubic equation of compression".
The early 16th century was not an era of enthusiastic publications, with no peer-reviewed journals and no Internet. If anyone can make a new discovery, they will try to keep it strictly confidential and use it publicly only when it is really advantageous. Isn't that understanding!
Moreover, at that time, the mathematical open challenge was popular, and the result of the challenge was directly related to the status and reputation of mathematics. Del Ferro, who died in 1526, passed on the solution thesis of the "compressed cubic equation" to his son-in-law, Annibele della Naf, and another student, Antonio Maria Fell.
Phil returned to his hometown of Venice with the "cheats" and became a math teacher, and spoke high-profile: he had a special talent for solving cubic equations. I overheard that another Venetian math teacher, Tartaglia, also had this ability. Or say that your peers are wrongdoers! So Fair made an open challenge to Tartaglia.
Tartaglia was born in Brescia, northern Italy, in 1499, to a poor family, so mathematics and science were self-taught. After settling in Venice, Tartaglia made a living teaching mathematics. Math teachers at that time, by participating in public polemics and debates, did everything in their power to preserve their reputation in public.
Tartaglia
Tartaglia had hinted at a colleague that he had solved the form
The equation, which poses a direct challenge to Fer. In early 1535, Fair openly challenged Tartaglia. According to their agreement: each person gives 30 questions to each other, and after 30 days, whoever solves the most problems is the winner. Don't worry, no one can get help and trick the old man.
No one knows how much Thalia really knew in the polemics, but on the night of February 13, 1535, he solved two forms of constant cubic equations, which was a great success.
That means he can solve all of Fair's problems. Know thyself and know the other, and never lose a battle. Tartaglia, on the other hand, was fully aware that Fair could only solve one cubic equation. As a result, Tartaglia won a great victory, and Fer disappeared from the public eye.
The winning Tartaglia became famous and the number of students grew dramatically. Tartaglia chose to keep his new approach strictly secret to ensure he remained victorious in the algebra controversy. As we all know, mathematics is a practical tool. At a time when the Western powers were vying for Italy, the application of mathematics to ballistics was a hot topic, and Tartaglia was also involved. In 1537, Tartaglia stopped focusing on cubic equations and went on to study ballistics and published a book, New Science. Just when the future was bright, another dangerous opponent, Girolamo Cardano, came in.
Cardano was born in Pavia in 1501, where his father taught him initial mathematics and, after the age of 12, learned Euclid geometry. Cardano's father was a well-educated lawyer and lecturer in geometry. Cardano's father wanted him to study law and, despite his talent in mathematics, wanted to make a career in medicine. He entered the University of Pavia at the age of 19 and at the age of 21 began participating in public debates and lectures on Euclidean geometry. He later transferred to the University of Padua, where he received his medical degree at the age of 25.
Cardano
Cardano, impatient, vengeful, prone to anger, is often involved in serious quarrels. From 1524 to 1547, he was embroiled in lawsuits, and more importantly, he won all of them. It is not difficult to imagine that if he could study law as his father expected, he would surely become a good judge or lawyer.
In the medical community, Cardano's reputation is growing. By the 1530s, he probably had become the most popular physician in northern Italy. In 1537, he was invited to teach medicine at the University of Pavia, but he refused. What a character, wayward!
Cardano is called a Renaissance man because of his extraordinary achievements in several very different fields, in addition to medicine. For example, he was also a veteran gambler and published a very popular and practical gambler's manual. This includes some advanced knowledge of probability, and of course a thousand techniques.
Cardano was born into a superstitious family and inherited the family's traditions. He also gave horoscopes to dignitaries. People who are proficient in mathematics and astronomy often do this. While Cardano was proud of his abilities, these trades got him into all sorts of trouble. Outsmart oneself.
Cardano's Great Derivation. Making algebraic methods of cubic and quadratic equations known is a huge contribution.
After hearing of Tartaglia's victory over Fair, Cardano had asked Tartaglia for permission to disclose Tartaglia's three-dimensional equation solution in his book, promising that it would be entirely attributable to Thalia.
Tartaglia then replied that she was planning to write a book of her own that would clearly explain the solution of cubic equations. But since there is still a lot of work to be done, it is not said when the book will be published. Cardano was not satisfied with this reply, so he pleaded indomitably.
In the surviving letters, it can be seen that the relationship between the two people is sometimes tense and sometimes relaxed. Cardano claims that Tartaglia is greedy and stingy in helping. For example, if he violently criticizes some of the achievements in Tartaglia's ballistic writings, Tartaglia will naturally take the sword and make a fierce reply. Neither of them is a good stubble, it is really a chess opponent.
Cardano had a plan and a premeditated plan to achieve his goals. As part of the plan, Cardano kindly invited Tartaglia to his home, losing no time in pointing out that he would have more to say. Cardano claimed that he was very interested in settling purely academic disputes, and that his ruse succeeded, writing to Tartaglia in the name of a great man, Alfonso de Avalos: "It must first be stated that I have great respect for you, and as soon as the book on ballistics came out, I bought one and gave one to Monsieur De Avalos). De Avalos was the Spanish viceroy, the head of the Imperial army, and one of the most influential figures in Italy at the time.
In another letter of 13 March 1539, Cardano wrote: "De Avalos urgently ordered me to write this letter to you immediately in his name, and advised you to come to Milan when you saw the letter, for he would like to speak with you very much." Tartaglia was very aware that his friendship with de Avalos would be of great help to him, and replied: Although I don't want to go over there, I will go.
Unsurprisingly, when Tartaglia arrived in Milan, de Avalos was not there. It's hard to say whether this is deliberate deception or a hasty schedule for dignitaries? If it is deception, it will be a complicated and dangerous scheme, because Tartaglia, who holds the invitation, if he writes directly to de Avaros, Cardano is moving a stone and dropping it on his own feet. It seems that Cardano can also do this kind of clever thing that is mistaken by cleverness.
Cardano managed to get the "cubic equation solution" out of Tartaglia, and Tartaglia wouldn't be stupid enough to hand it over. What he gave Cardano was a solution to the compressed equation rules, and it was given in the form of encrypted verses. Although he later made it clear to Cardano.
In September 1539, Cardano's Practice of Universal Arithmetic was published, without Tartaglia's solution. Tartaglia gives some explanations for the errors in the book. In fact, he was making fun of Cardano's book. In the following period, Cardano was preparing another book, The Great Derivation, which contained many basic results. He was well aware that the solution of the cubic equation was significant to the success of the book. So together with his able assistant Ludovico Ferrari, he spent years figuring out and expanding the meaning of Tartaglia's verses.
The content of cubic equations appears in chapter 11, which is named "About cubes plus one power equals constants". Tartaglia's law for Cardano covers the three basic forms of the cubic equation.
Mathematicians at that time did not use negative coefficients, did not have the algebraic notation we use now, so did not take a simple, general form:
Cardano's The Great Derivation uses a great deal of knowledge of geometry. As William Dunham put it in his Journey to Genius: "His proofs are purely geometric, including cubes written in words and their volume." In fact, we are not surprised when we recall the backwardness of the number symbols of the times and the lofty status of ancient Greek geometry in the minds of Renaissance mathematicians. ”
In each chapter of the Great Derivation, Cardano first gives a geometric explanation of a special numerical cubic equation, then writes out the general method of solving the equation, then gives one or more examples and applies the law to solve it. Because coefficients of 0 and negative numbers have not yet appeared, Cardano can only elaborate on 13 different cubic equations, each of which uses only positive coefficients and is independent of chapters. However, the geometric solution is inflected and lengthy, making it difficult for everyone to understand.
There is no doubt that Cardano has made a great contribution in this field, but it seems a little perfidious to Tartaglia, if there is a sacred promise of secrecy between them! Based on Tartaglia's statement, there is such a commitment. In the second year after the publication of the Great Yan, Tartaglia published Problems and Inventions, which preserved a detailed account of their meetings and the promise made to him by Cardano: "I swear by the Holy Gospels, in the name of a gentleman, that it will never be published, not only after you have told me of your discovery; and I promise in the name of a true Christian that they will be hidden in the heart as if they were codes, even after my death no one will be able to understand them." ”
It was pointed out that Cardano's secretary and assistant Ludovico Ferrari was also present when Cardano and Tartaglia met, and Ferrari later said in a resounding curse that Cardano had never sworn like that. Indeed, no one can argue that Tartaglia would make distorted changes to the records of the meeting, let alone tartaglia in anger.
Six years after the publication of the Great Derivative, Cardano and Frari heard that someone else had a solution to the cubic equation, so they came to Bologna in 1543 to visit their colleague Anne Belle della Naf. There, they were allowed to view the manuscripts of Cipione del Ferro, from which they learned that Del Ferro was the first to solve such equations using algebraic methods, not Tartaglia. Based on this, they argue that Tartaglia was not the original creator of this method, and that even if Cardano had taken the oath, it would have been an invalid oath.
Still, Cardano was cautious about announcing that he had discovered a law for solving cubic equations. In three places, he cites the work of previous generations in solving cubic equations. Shortly after the beginning of chapter 1, he writes: "In our time, Cipione del Ferro of Bologna has solved the situation in which cubic plus one power equals constant, which is a remarkable and admirable achievement. ”
At the beginning of chapter 11, it is written that Tartaglia, at my request, gave me the solution. With the help of this law, I found empirical evidence, which is indeed a difficult thing.
With the help of His secretary and assistant Ludovico Ferrari, he found solutions to three other cubic equations by using the methods he began to use as a stepping stone, by applying the appropriate substitution method and turning them into the forms we know. With Ferrari's help again, Cardano also shows how to find a solution to the 4th magnitude based on these cubic solutions. And Cardano points out that cubic equations should have three roots.
Despite Cardano's recognition, Tartaglia was furious. In the second half of his book Problems and Inventions, Tartaglia devoted himself to criticizing Cardano and his Great Derivation. Tartaglia ridiculed Cardano's mathematical abilities, denied ever giving Cardano a promise, and accused him of plagiarism.
The grumpy Ferrari responded to this situation on 10 February 1547 with a letter of challenge, declaring that Tartaglia could challenge him to debate him on almost any scientific subject. Ferrari passed on the Challenge to many scholars and dignitaries throughout Italy at the same time, so much so that Tartaglia could hardly refuse. Ferrari's offensive was ferocious, believing that Tartaglia had used attacks on others to erect a biography of himself; publishing an unacknowledged proof in his book that his work had been plagiarized; ironically, his book was full of errors.
Tartaglia responded that he wanted to challenge Cardano, not his student Ferrari. Nevertheless, Tartaglia reluctantly agreed and accepted the challenge. Ferrari insisted that Cardano's solution was due to Del Ferro and Fair, since both had found solutions to the cubic equations before Tartaglia, and we did not need to swear to keep it secret. In the next 8 months of fighting, Tartaglia was defeated, Ferrari declared the winner, and Cardano did not participate at all.
Tartaglia lost his position in his hometown, and Ferrari received many well-paid invitations. Tartaglia lost the public respect and reputation for him, and a bitter Tartaglia attempted to vent his pain on Carcano. He retreated to a corner of the circle, waiting angrily, watching, plotting.
Tartaglia manages to seduce Cardano with a sophisticated scheme and exploit Cardano's friendship with Gonazzaga and the Pope's indulgence in astrology. Gonazzaga was the governor of Milan and a rival of the Pope. So Tartaglia had an idea: through ingenious arrangement, to ask Cardano to serve the Pope as an astrologer and doctor. This can provoke some kind of accusation of political conspiracy. But the triumphant Cardano, who did not need any protector, declined the invitation for personal reasons. Tartaglia then indoctrinated that Cardano wanted to offend His Holiness the Pope by refusing the invitation. Tartaglia also copied an astrological analysis of Caldano's early years of Jesus, intended to show that it was blasphemous. The situation was not good for Cardano, but not enough to knock Cardano down. Tartaglia, who would not give up, then planned another battle.
Cardano's career was in full swing, controlling the medical lecturer at the University of Pavia, which earned him material wealth, writing time and prestige, and various invitations followed. Seeing the rivals of the scenery, Tartaglia tried to restrain himself, just like Cardano had wrapped around Tartaglia, how could the stubborn Tartaglia let go of Caldano, who had ruined himself. Just as the so-called Heavenly Dao cycle, sooner or later it will be repaid when it comes out of the mix. Finally, the goddess fell in love with him.
Cardano was lucky in many ways, but his two sons and a daughter were paving the way for him to perish. Thanks to Cardano's prominence, his daughter Ziara married an aristocratic family. Unfortunately, Ziara is a woman living a life of debauchery. As a result, not only was the marriage dissolved, but cardano also required to make compensation. In 1557 and 1558, Cardano was embroiled in a dispute between law and the church, and his once-glorious reputation began to be destroyed. Another trouble came, and in December 1557 his favorite eldest son, Jane Batista, got married. Cardano had hoped that the boss would marry a good woman, but chose a daughter of a poor and depressed family, and the family expected Cardano to support them, and Cardano did spend a huge amount of money on them. Two years later. Jane Batista's wife died of arsenic poisoning, and Cardano's son was arrested for murder. Cardano did everything he could to prove his son's innocence, but Jane Battis confessed himself and was executed. Cardano was shrouded in this shadow, which at the same time greatly affected his sanity, and began to become suspicious and suspicious of someone plotting against him. His other alcoholic and gambler son, Aldo, has brought him new problems, and he is in debt and unable to get along.
Now there was an attempt to oust Cardano from the medical faculties at the University of Pavia, followed by the deprivation of his seat in the more prestigious university of Milan, and in 1653 he was disqualified as a lecturer and charged with several crimes.
On 13 October 1570, Tartaglia dealt Cardano a double whammy. Using Cardano's own son Aldo as a whistleblower to reveal Cardano's whereabouts, Tartaglia handed him over to the Inquisition. And collected evidence against Cardano for many years, one of the "incriminating evidence" is that Cardano refused the Pope's invitation not to be his astrologer and doctor. Carcano's analysis of Jesus' life plate is disastrous, and some out-of-context statements are interpreted as blasphemous. Cardano was put on trial, and fortunately Cardano was not executed, but thrown into prison. In desperation, he sought help and was learned by Archbishop Hamilton of the Church, and was eventually rescued. At this point, Tartaglia's revenge was finally a success. Carnot died in 1576, and less than a year later, Tartaglia went with him.
Tartaglia's great and creative solution to cubic equations is epoch-making for the development of mathematics! And Cardano, a "versatile" Euclid talent with various shortcomings, also made a huge contribution to mathematics. The grumpy geniusEs Ferrari and Cardano and Tartaglia failed to cooperate and communicate fully in that era, and consumed each other in the struggle, which made the development of mathematics and may also limit the development of mathematics.