The first section of medieval Europe

During the prosperous period of Babylonian civilization, Egyptian civilization, China civilization, Indian civilization, Greek civilization and Roman civilization, Europe (except Greece and Italy) was still in the primitive civilization period, and new cultures began to appear around 500 AD. The 5th ~11century was a dark period in European history. The Catholic Church became the absolute power of European society, and the feudal religious rule made ordinary people believe in the kingdom of heaven and pursue the afterlife, thus being indifferent to secular life and not interested in nature. The church preached the truth of the apocalypse and had absolute authority to explain this truth, which led to the repression of reason and the stagnation of European civilization throughout the Middle Ages.

Because the Romans emphasized practicality rather than developing abstract mathematics, it also had a certain influence on European mathematics after the collapse of the Roman Empire, and finally made Europe in the dark age accomplish nothing in the field of mathematics. However, due to the need of religious education, there are also some low-level arithmetic and geometry textbooks. Boyzi (about 480~524), a Roman, wrote textbooks on geometry and arithmetic in Latin based on Greek materials. The content of geometry only contains some propositions in Volume 1, Volume 3 and Volume 4 of geometry, and some simple measurements. Arithmetic is based on a simple book written by Nicomachus 400 years ago. Such a simple book has always been the standard teaching material of European missionary schools. In addition, English Bede (674 ~ 735) and French Gelber (about 950~ 1003, the first Christian to study in a Spanish Muslim school) also discussed mathematics during this period. The former studies the notation in arithmetic, and it is said that the latter may have brought Indian Arabic numerals to Europe.

It was not until the 12 century that European mathematics showed signs of recovery. The translation and dissemination of Arabic and Greek works began to stimulate this recovery. Around 1 100, Europeans contacted Arabs in the Mediterranean and the Near East and Byzantines in the Eastern Roman Empire through trade and tourism. The Crusades plundered the land and brought Europeans into the Arab world. From then on, Europeans learned Greek and oriental classics from Arabs and Byzantines, which stimulated their search, exploration and research on these academic works, and finally led to the upsurge of European mathematics in the Renaissance. As the outpost of the Renaissance, Italy is easy to connect with the external civilization because of its special geographical position, and Sicily has also become a melting pot of eastern and western cultures. The route of ancient academic exchanges in western Europe is shown in the following figure.

The translation of mathematical works mainly includes "Elements of Geometry" and "Chinese Astronomical Table" translated by Adelard of Bath (about 1 120). Italian Plato (65438+the first half of the 2nd century) translated Albategnius astronomy, Dior, Hughes, geometry of spheres and other works. Gladow (1114 ~187),1the greatest translator in the 2nd century, translated more than 90 Arabic works into Latin, including Ptolemy's Grand Collection. Therefore, it can be said that 12 century is the translation era of European mathematics.

After the dark ages in Europe, the first influential mathematician was Fibonacci (1 170 ~ 1250). In his early years, he studied arithmetic with Arabs in North Africa with his father, then traveled to Mediterranean countries and returned to Italy to write Abachi. 1202). This masterpiece is mainly the contents of mathematical works in ancient China, India and Greece, including Indian Arabic numerals, fractional algorithms, open methods, quadratic and cubic equations, indefinite equations, and most of the contents of geometry and Greek trigonometry (such as the "grandson problem" and "hundred chickens problem" in China's mathematics appear in this book). In particular, the book systematically introduces Indian numbers, which has influenced the face of European mathematics. The abacus book can be regarded as the clarion call for the revival of European mathematics after a long night.

The process of European mathematics recovery is very tortuous. From12nd century to15th century, scholasticism in the church used the negative elements in the reintroduced Greek works to resist the progress of science. In particular, they regard some academics of Aristotle and Ptolemy as absolutely correct dogmas, and try to continue to use this new authoritarianism to bind people's thoughts. The real recovery of European mathematics will be in 15 and 16 century. At the climax of the Renaissance, the development of mathematics was closely combined with scientific innovation, and the significance of mathematics in understanding nature and exploring truth was highly emphasized by Renaissance representatives. Da Vinci (1452~ 15 19) said: "If a person doubts the extreme reliability of mathematics, he will fall into chaos, and he will never be able to calm down the arguments in sophistry science, which will only lead to endless empty talk. ..... because people's discussions cannot be called science unless they are explained and demonstrated by mathematics. " Galileo simply thought that the universe "this book is written in mathematical language". The development of mathematization in science promotes the prosperity of mathematics itself. The following briefly introduces the important aspects of the development of mathematics in this period.

Section 2 Transition to Modern Mathematics

2. 1 algebra

The progress of Europeans in mathematics began with algebra, which was the most prominent and far-reaching field in the Renaissance and opened the prelude to modern mathematics. It mainly includes the solution of cubic and quartic equations and the introduction of symbolic algebra.

Translator Gladow (1114 ~1187) translated Hua Lazimi's Algebra into Latin and began to spread in Europe. However, until the15th century, people still thought that cubic and quartic equations were as difficult as the problem of turning a circle into a square. The first breakthrough was made by Scipionedel Ferro (1465 ~1526), a professor of mathematics at the University of Bologna, around 15 15. He discovered the shape of (algebraic solution of cubic equation of m, n). At that time, popular scholars did not disclose their research results, and Ferro secretly passed his solution to his student Antonio Maria Fio. At the same time, another Italian mathematician, Hotel Nigro fontana 1499? ~ 1557, nicknamed tartaglia, also claims to be decomposed into (m, n > 0) cubic equation. So Fio began to challenge Tattaglia to solve the thirteen cubic equations put forward by the other side. Results Tattaglia solved shape and shape (m, n > 0) quickly, while Feo could only solve the former type of equation. Tattaglia also did not publish his solution. At the repeated requests of the scholar G. cardano (150 1~ 1576), Tattaglia taught him the solution. Soon, Kadan broke his promise and published these solutions on ARSMANA, 1545). The solution of cubic equation x3+px= q contained in Dafa essentially considers the identity (a-b)3+3ab(a-b) = a3-b3.

If A and B are selected, 3ab= p, a3-b3 = q, (*)

It is not difficult to work out a and b from (*).

a = b=

Therefore, the obtained a-b is the expected X, which is later called the Caldan formula.

Shortly after the cubic equation was solved, in 1540, the Italian mathematician T. Dakoy put forward a problem of quartic equation to Caldan. For the solution, Caldan was solved by his student Lodovici Ferrari (1522 ~ 1565), and his solution was also written by Caldan in The Big Book. The solution is to simplify the general quartic equation by using transformation, and further

So, for any z, there is

Then choose a suitable z to make the right side of the above formula completely flat, which actually makes

Do it. This becomes the cubic equation of z.

The types of quartic equations discussed by Ferrari mainly include:

Of course, it is unfair to say that Caldan completely copied it, because he has indicated in his book that Taishan told him this solution, and Taishan has not given any proof. Caldan not only extended the Tadessie method to the general cubic equation, but also supplemented the geometric proof. The book confuses the so-called "irreducible" situation in solving cubic equations (irreducible situation is discriminant), which essentially involves the complex representation of real numbers. 1572, four years after Karl Marx's death, the Italian mathematician R.Bombelli (about 1526~ 1573) introduced imaginary numbers into his textbook Algebra to solve the irreducibility of cubic equations, and expressed it as DIMMRQ1/. -1 1. Kardan thinks that complex roots appear in pairs (this speculation was later proved by Newton (1642~ 1727) in his Arithmetic of Everything), and he realized that cubic equations have three roots and quartic equations have four roots. On this basis, Albert Girard (1593 ~1632), a Dutchman, made a further inference in The New Discovery of Algebra (1629): For polynomial equations of degree n, if the impossibility (complex roots) is taken into account, including multiple roots, there should be n roots. However, no evidence was given. Caldan also found that the sum of three roots of cubic equation is equal to the reciprocal of the coefficient of x2 term, the sum of every two products is equal to the coefficient of x term, and so on. The relationship between roots and coefficients was later solved by David (F. Vita, 1540 ~ 1603), Newton and Gregory (James Gregory, 1638).

In France, the mathematician Veda also wrote several books about the theory of equations, such as Introduction to Analytical Methods (159 1), Arrangement and Modification of Equations (16 15) and Effective Numerical Solution (15). The Vedas give an approximation of algebraic equations. 1637, Descartes (1596 ~ 1650) first decomposed the quartic equation into two quadratic equations by the method of undetermined coefficients. The factorization theorem mentioned today was first put forward by Descartes in Geometry. He said: f (x) is divisible by (x-a) if and only if A is the root of f (x) = 0. He also proved that if the cubic equation with rational coefficients has rational roots, the polynomial can be expressed as the product of rational coefficient factors, and cited the principle of undetermined coefficient method. Descartes didn't prove it with geometry. The conclusion that a polynomial equation of degree n should have n roots, and the so-called "Cartesian sign rule" today: the maximum number of positive roots of polynomial equation f (x) = 0 is equal to the number of sign changes of coefficients, and the maximum number of negative roots is equal to the number of consecutive occurrences of two positive signs and two negative signs. Throughout Descartes' work, it is not difficult to find that he has initially established a modern method for rational roots of polynomial equations.

The study of European equation theory and algebra in Renaissance is a wonderful page in the history of mathematics. The work of Italians in solving cubic and quartic equations is the starting point of a series of long and far-reaching explorations on the theory of higher algebraic equations in mathematics in the whole 17 and 18 centuries.

The progress in algebra also lies in the introduction of a better symbol system, which is very important for the development of algebra itself and analysis. It is because of the establishment of symbol system that algebra can become a science. One of the most obvious and prominent signs of modern mathematics is the extensive use of mathematical symbols, which embodies the high abstraction and simplicity of mathematics. Another great progress in Renaissance algebra was the systematic introduction of symbolic algebra.

Although Egyptians, Greeks and Indians used abbreviations and symbols sporadically, China Song and Yuan mathematicians introduced Tianyuan, Geo-element, Man-element and Matter-element to express unknowns, but they didn't realize the significance of doing so. Only Diophantine consciously uses symbols to make the thinking and writing of algebra more compact and effective. Perhaps it is the result of the introduction of printing into Europe. Although the writing forms of European mathematical works in the15th century and the early16th century were mainly articles, some special words and abbreviations of specific mathematical symbols were popular. In the summary of the nature of arithmetic by Italian monk L.Pacioli (about1445 ~1509) ~1567), geometry and proportion, comprehensive arithmetic (1544) and C. Rudolf (about/kloc).

The systematization of mathematical symbols is first attributed to the French mathematician Veda, whose introduction of symbol system led to the most significant change in algebraic properties. David was originally a lawyer and politician, and studied mathematics in his spare time. He worked in Brittany's parliament and later served as an adviser to Prince Henry's Council in Navarra. When he was frustrated politically, he devoted himself to mathematical research from 1584 to 1589, and studied Cardin, Tattaglia, Bombelli and Steven (Steven, 1549). From these works, especially the works of Diophantine, he got the idea of using letters. In Introduction to Analysis (159 1), he consciously used systematic algebraic letters and symbols for the first time, with consonants representing known quantities and vowels representing unknown quantities. He called symbolic algebra "arithmetic of classes". At the same time, the division between arithmetic and algebra is clarified, and it is considered that algebraic operation is applied to the class or form of things, and arithmetic operation is applied to specific numbers. This makes algebra a subject to study general types of forms and equations, and because of its abstraction, its application is more extensive.

David's method was appreciated by later generations, and was inherited by Gillard's New Algebra Discovery and Oughtred's Practical Analysis (Oughtred, 1575~ 1660), and was used flexibly, especially through the latter's works, the trend of using mathematical symbols became popular. The improvement of algebraic methods used by Vedas was completed by Descartes. He first used the first few Latin letters (A, B, C, D, …) to represent known quantities, and the last few letters (X, Y, Z, W, …) to represent unknown quantities, which became a habit today. He changed the Vedic convention and adopted the literal coefficient indiscriminately. David's symbolic algebra retains the principle of homogeneity, which requires all terms in the equation to be "homogeneous", that is, volume and volume are added, and area and area are added. This obstacle was also eliminated with the birth of Descartes' analytic geometry.

By the end of the seventeenth century, European mathematicians generally realized that the deliberate use of symbols in mathematics had a good effect. And generalizes mathematical problems. However, there were too many symbols introduced at random at that time, and the symbols we use today are actually left over after a long period of elimination.