it is occupied by Aristotle, a disciple of Plato.
Aristotle laid the foundations of the science of logic and expressed a number of ideas regarding definitions, axioms, infinity and the possibility of geometric constructions.
The greatest of the Greek mathematicians of the classical period, second in importance to the results obtained only to Archimedes, was Eudoxus (c. 408-355 BC).
It was he who introduced the concept of magnitude for such objects as straight lines and angles.
Having the concept of magnitude, Eudoxus logically strictly justified the Pythagorean method of dealing with irrational numbers.
The works of Evdox made it possible to establish the deductive structure of mathematics on the basis of explicitly formulated axioms.
He also took the first step in creating mathematical analysis, since it was he who invented the method of calculating areas and volumes, called the "exhaustion method".
This method consists in constructing inscribed and described flat figures or spatial bodies that fill ("exhaust") the area or volume of the figure or body that is the subject of the study.
Eudoxus also belongs to the first astronomical theory explaining the observed motion of the planets.
The theory proposed by Eudoxus was purely mathematical; it showed how combinations of rotating spheres with different radii and axes of rotation could explain the seemingly irregular movements of the Sun, Moon and planets.
Around 300 BC, the results of many Greek mathematicians were brought together into a single whole by Euclid, who wrote the mathematical masterpiece of the Beginning.
From a few shrewdly selected axioms, Euclid derived about 500 theorems that covered all the most important results of the classical period.
Euclid began his essay with the definition of such terms as a straight line, an angle and a circle.
Then he formulated ten self evident truths, such as "the whole is greater than any of the parts".
And from these ten axioms, Euclid was able to deduce all the theorems.
For mathematicians, the text of Euclid's Principles served as a model of rigor for a long time, until in the 19th century it was discovered that there were serious shortcomings in it, such as the unconscious use of explicitly unformulated assumptions.
Apollonius (c. 262-200 BC) lived in the Alexandrian period, but his main work is sustained in the spirit of classical traditions.
His proposed analysis of conic sections - a circle, an ellipse, a parabola and a hyperbola was the culmination of the development of Greek geometry.
Apollonius also became the founder of quantitative mathematical astronomy.
The Alexandrian period.
During this period, which began around 300 BC, the character of Greek mathematics changed.
Alexandrian mathematics arose as a result of the merger of classical Greek mathematics with the mathematics of Babylonia and Egypt.
In general, the mathematicians of the Alexandrian period were more inclined to solve purely technical problems than to philosophy.
The great Alexandrian mathematicians Eratosthenes, Archimedes, Hipparchus, Ptolemy, Diophantus and Papp demonstrated the power of the Greek genius in theoretical abstraction, but they were just as willing to apply their talent to solving practical problems and purely quantitative problems.
Eratosthenes (c. 275-194 BC) found a simple method for accurately calculating the circumference of the Earth, he also owns a calendar in which every fourth year has one day more than the others.
The astronomer Aristarchus (c. 310-230 BC) wrote an essay On the sizes and distances of the Sun and Moon, which contained one of the first attempts to determine these sizes and distances; by its nature, Aristarchus ' work was geometric.
The greatest mathematician of antiquity was Archimedes (c. 287-212 BC).
He owns the formulations of many theorems about the areas and volumes of complex figures and bodies, which he proved quite strictly by the method of exhaustion.
Archimedes always tried to get exact solutions and found upper and lower bounds for irrational numbers.
For example, working with the correct 96 square, he perfectly proved that the exact value of the number ?
it is located between 31/7 and 310/71.
Archimedes also proved several theorems containing new results of geometric algebra.
He is responsible for the formulation of the problem of dissecting the ball with a plane so that the volumes of the segments are located among themselves in a given ratio.
Archimedes solved this problem by finding the intersection of a parabola and an isosceles hyperbola.
Archimedes was the greatest mathematical physicist of antiquity.
To prove the theorems of mechanics, he used geometric considerations.
His essay On floating bodies laid the foundations of hydrostatics.
According to legend, Archimedes discovered a law bearing his name, according to which a repulsive force equal to the weight of the liquid displaced by him acts on a body immersed in water, while bathing, while in the bathroom, and unable to cope with the joy of discovery that seized him, he ran naked into the street shouting: "Eureka!"("Opened!")
In the time of Archimedes, they were no longer limited to geometric constructions that could be carried out only with the help of compasses and a ruler.
Archimedes used a spiral in his constructions, and Diocles (the end of the 2nd century BC) solved the problem of doubling the cube with the help of a curve introduced by him, called a cissoid.
In the Alexandrian period, arithmetic and algebra were considered independently of geometry.
The Greeks of the classical period had a logically sound theory of integers, but the Alexandrian Greeks, having adopted Babylonian and Egyptian arithmetic and algebra, largely lost their already developed ideas about mathematical rigor.
Heron of Alexandria, who lived between 100 BC and 100 AD, transformed a significant part of the geometric algebra of the Greeks into frankly lax computational procedures.
However, when proving new theorems of Euclidean geometry, he was still guided by the standards of logical rigor of the classical period.
The first rather voluminous book in which arithmetic was presented independently of geometry was an Introduction to arithmetic by Nicomachus (c. 100 AD).
In the history of arithmetic, its role is comparable to the role of Euclid's Principles in the history of geometry.
For more than 1000 years, it served as a standard textbook, since it clearly, clearly and comprehensively expounded the doctrine of integers (simple, composite, mutually simple, as well as proportions).
Repeating many Pythagorean statements, the Introduction of Nicomachus at the same time went further, since Nicomachus saw more general relations, although he cited them without proof.
A significant milestone in the algebra of the Alexandrian Greeks was the work of Diophantus (ca. 250).
One of his main achievements is connected with the introduction of the principles of symbolism into algebra.
In his works, Diophantus did not offer general methods, he dealt with specific positive rational numbers, and not with their letter designations.
He laid the foundations of the so called Diophantine analysis the study of indefinite equations.
The highest achievement of the Alexandrian mathematicians was the creation of quantitative astronomy.
To Hipparchus (c. 161-126 BC) we owe the invention of trigonometry.
His method was based on a theorem stating that in such triangles, the ratio of the lengths of any two sides of one of them is equal to the ratio of the lengths of the two corresponding sides of the other.
In particular, the ratio of the length of the cathet lying against the acute angle A in a right triangle to the length of the hypotenuse should be the same for all right triangles having the same acute angle A.
This ratio is known as the sine of the angle A.
The ratios of the lengths of the other sides of a right triangle are called the cosine and tangent of the angle A. Hipparchus invented a method for calculating such ratios and compiled their tables.
With these tables and easily measurable distances on the Earth's surface, he was able to calculate the length of its great circle and the distance to the Moon.
According to his calculations, the radius of the Moon was one third of the Earth's radius; according to modern data, the ratio of the radii of the Moon and the Earth is 27/1000.
Hipparchus determined the duration of the solar year with an error of only 61/2 minutes; it is believed that it was he who introduced the latitudes and longitudes.
Greek trigonometry and its applications in astronomy reached their peak in the Almagest of the Egyptian Claudius Ptolemy (died 168 AD).
The theory of the motion of celestial bodies was presented in the Almagest, which prevailed until the 16th century, when it was replaced by the theory of Copernicus.
Ptolemy sought to build the simplest mathematical model, realizing that his theory was just a convenient mathematical description of astronomical phenomena, consistent with observations.
The Copernican theory prevailed precisely because it turned out to be simpler as a model.
The decline of Greece.
After the conquest of Egypt by the Romans in 31 BC, the great Greek Alexandrian civilization fell into decline.
Cicero proudly claimed that, unlike the Greeks, the Romans are not dreamers, and therefore apply their mathematical knowledge in practice, extracting real benefits from them.
However, the contribution of the Romans to the development of mathematics itself was insignificant.
The Roman numeral system was based on cumbersome notation of numbers.
Its main feature was the additive principle.
Even the subtractive principle, for example, writing the number 9 in the form of IX, came into wide use only after the invention of typesetting letters in the 15th century.
The Roman notation of numbers was used in some European schools until about 1600, and in accounting a century later.
* 2. THE MIDDLE AGES AND THE RENAISSANCE* Medieval Europe.
The Roman civilization did not leave a noticeable trace in mathematics, because it was too concerned with solving practical problems.
Civilization, with the early Medieval Europe (c. 400-1100) was not productive for the exact opposite reason: intellectual life focused almost exclusively on theology and the afterlife.
The level of mathematical knowledge did not rise above arithmetic and simple sections from the Principles of Euclid.
The most important branch of mathematics in the Middle Ages was considered astrology; astrologers were called mathematicians.
And since medical practice was based mainly on astrological indications or contraindications, doctors had no choice but to become mathematicians.
Around 1100, the almost three century period of mastering the heritage of the Ancient World and the East preserved by the Arabs and Byzantine Greeks began in Western European mathematics.
Since the Arabs owned almost all the works of the ancient Greeks, Europe received an extensive mathematical literature.
The translation of these works into Latin contributed to the rise of mathematical research.
All the great scientists of that time admitted that they drew inspiration from the writings of the Greeks.
The first European mathematician worth mentioning was Leonardo of Pisa (Fibonacci).
In his work The Book of Abacus (1202), he introduced Europeans to Indo Arabic numerals and calculation methods, as well as Arabic algebra.
Over the next few centuries, mathematical activity in Europe waned.
The code of mathematical knowledge of that era, compiled by Luca Pacioli in 1494, did not contain any algebraic innovations that Leonardo did not have.
Rebirth.
Among the best geometers of the Renaissance were artists who developed the idea of perspective, which required geometry with converging parallel lines.
The artist Leon Battista Alberti (1404-1472) introduced the concepts of projection and section.
Rectilinear rays of light from the observer's eye to various points of the depicted scene form a projection; the cross section is obtained when the plane passes through the projection.
In order for the painted picture to look realistic, it had to be such a cross section.
The concepts of projection and section gave rise to purely mathematical questions.
For example, what are the general geometric properties of the section and the original scene, what are the properties of two different sections of the same projection formed by two different planes intersecting the projection at different angles?
From such questions, projective geometry arose.
Its founder is Zh.
Desargues (1593-1662), using proofs based on projection and section, unified the approach to various types of conic sections, which the great Greek geometer Apollonius considered separately.
* 3. THE BEGINNING OF MODERN MATHEMATICS* The beginning of the 16th century in Western Europe was marked by important achievements in algebra and arithmetic.
Decimals and the rules of arithmetic operations with them were introduced into circulation.
The real triumph was the invention of logarithms in 1614 by J. R. R. Tolkien.
Neperom.
By the end of the 17th century, the understanding of logarithms as exponents of a power with any positive number other than one as the base was finally formed.
Since the beginning of the 16th century, irrational numbers have become more widely used.
B. Pascal (1623-1662) and I. Barrow (1630-1677), a teacher of I. Newton at the University of Cambridge, argued that such a number as can be interpreted only as a geometric quantity.
However, in the same years, R. Descartes (1596-1650) and J. Wallis (1616-1703) believed that irrational numbers are permissible by themselves, without reference to geometry.
In the 16th century, disputes continued over the legality of the introduction of negative numbers.
Even less acceptable were the complex numbers that arose when solving quadratic equations, such as those called "imaginary"by Descartes.
These numbers were under suspicion even in the 18th century, although L. Euler (1707-1783) successfully used them.
Complex numbers were finally recognized only at the beginning of the 19th century, when mathematicians got used to their geometric representation.
Achievements in algebra.
In the 16th century, Italian mathematicians N. Tartaglia (1499-1577), S. Dal Ferro (1465-1526), L. Ferrari (1522-1565) and D. Cardano (1501-1576) found general solutions of equations of the third and fourth degrees.
To make algebraic reasoning and its recording more accurate, many symbols were introduced, including +, -, ?, , =, > and <.
The most significant innovation was the systematic use of the French mathematician F.Vietom (1540-1603) letters for the designation of unknown and constant quantities.
This innovation allowed him to find a single method for solving equations of the second, third and fourth degrees.
Then the mathematicians turned to equations whose degrees are higher than the fourth.
Working on this problem, Cardano, Descartes and I. Newton (1643-1727) published (without proof) a number of results concerning the number and type of roots of the equation.
Newton discovered the relation between the roots and the discriminant [b^2-4ac] of the quadratic equation, namely, that the equation ax^2 + bx + c = 0 has equal real, different real or complex conjugate roots, depending on whether the discriminant b^2-4ac is zero, greater than or less than zero.
In 1799, K. Friedrich Gauss (1777-1855) proved the so called basic theorem of algebra: every polynomial of the nth degree has exactly n roots.
The main task of algebra - the search for a general solution of algebraic equations - continued to occupy mathematicians in the early 19th century.
When talking about the General solution of the second degree equation ax^2 + bx + c = 0, mean that each of the two roots can be expressed using a finite number of operations of addition, subtraction, multiplication, division and extraction of roots produced on the coefficients a, b and C. Young Norwegian mathematician N. Abel (1802-1829) proved that it is impossible to obtain the General solution of equations of degree higher than 4 with a finite number of algebraic operations.
However, there are many equations of a special form of degree higher than 4 that allow such a solution.
On the eve of his death in a duel, the young French mathematician E. Galois (1811-1832) gave a decisive answer to the question of which equations are solvable in radicals, i.e. the roots of which equations can be expressed in terms of their coefficients using a finite number of algebraic operations.
In Galois theory, substitutions or permutations of roots were used and the concept of a group was introduced, which has found wide application in many areas of mathematics.
The development of group theory is a good example of the continuity of creative work in mathematics.
Galois built his theory based on the work of Abel, Abel relied on the work of J.Lagrange (1736-1813).
In turn, many outstanding mathematicians, including Gauss and A. Legendre (1752-1833), implicitly used the concept of a group in their works.
Newton was not overly modest when he stated: "If I saw further than others, it was because I stood on the shoulders of giants."
Analytical geometry.
Analytical, or coordinate, geometry was created independently by P. Fermat (1601-1665) and R. Descartes in order to expand the possibilities of Euclidean geometry in construction problems.
However, Fermat considered his works only as a reformulation of the work of Apollonius.
The real discovery - the realization of the full power of algebraic methods - belongs to Descartes.
Euclidean geometric algebra for each construction required the invention of its original method and could not offer the quantitative information necessary for science.
Descartes solved this problem: he formulated geometric problems algebraically, solved an algebraic equation, and only then built the desired solution - a segment that had the appropriate length.
Analytic geometry itself arose when Descartes began to consider indefinite construction problems, the solutions of which are not one, but many possible lengths.
Analytical geometry uses algebraic equations to represent and study curves and surfaces.
Descartes considered acceptable a curve that can be written using a single algebraic equation with respect to x and y.
This approach was an important step forward, because it not only included such curves as the conchoid and the cissoid among the permissible ones, but also significantly expanded the area of curves.
As a result, in the 17th and 18th centuries.
many new important curves, such as the cycloid and the chain line, have entered scientific use.
It seems that the first mathematician who used equations to prove the properties of conic sections was J. R. R. Tolkien.
Wallis.
By 1865, he had algebraically obtained all the results presented in the V Book of Euclid's Principles.
Analytic geometry has completely changed the roles of geometry and algebra.
As the great French mathematician Lagrange noted, " while algebra and geometry were moving each in their own way, their progress was slow, and their applications were limited.
But when these sciences combined their efforts, they borrowed new vital forces from each other and since then they have been moving towards perfection with rapid steps."
Mathematical analysis.
The founders of modern science - Copernicus, Kepler, Galileo and Newton approached the study of nature as mathematicians.
Studying motion, mathematicians have developed such a fundamental concept as a function, or a relation between variables, for example , d = kt^2, where d is the distance traveled by a freely falling body, and t is the number of seconds that the body is in free fall.
The concept of a function immediately became central to determining the speed at a given time and the acceleration of a moving body.
The mathematical difficulty of this problem was that at any moment a body passes a zero distance in a zero period of time.
therefore, determining the value of the speed at a time by dividing the path by time, we will come to the mathematically meaningless expression 0/0.
The problem of determining and calculating the instantaneous rates of change of various quantities attracted the attention of almost all mathematicians of the 17th century, including Barrow, Fermat, Descartes and Wallis.
The disparate ideas and methods proposed by them were combined into a systematic, universally applicable formal method by Newton and G. Leibniz (1646-1716), the creators of differential calculus.
On the issue of priority in the development of this calculus, there were heated debates between them, with Newton accusing Leibniz of plagiarism.
However, as the research of historians of science has shown, Leibniz created mathematical analysis independently of Newton.
As a result of the conflict, the exchange of ideas between mathematicians of continental Europe and England was interrupted for many years to the detriment of the English side.
English mathematicians continued to develop the ideas of analysis in the geometric direction, while the mathematicians of continental Europe, including I. Bernoulli (1667-1748), Euler and Lagrange achieved incomparably greater success following the algebraic, or analytical, approach.
The basis of all mathematical analysis is the concept of a limit.
The speed at a time is defined as the limit to which the average speed d/t tends, when the value of t is getting closer to zero.
Differential calculus provides a convenient general method for calculating the rate of change of the function f (x) for any value of x.
This speed is called the derivative.
From the generality of the notation f (x), it can be seen that the concept of a derivative is applicable not only in problems related to the need to find speed or acceleration, but also in relation to any functional dependence, for example, to some relation from economic theory.
One of the main applications of differential calculus is the so called maximum and minimum problems; another important range of problems is finding the tangent to a given curve.
It turned out that using a derivative specially invented for working with motion problems, it is also possible to find areas and volumes bounded by curves and surfaces, respectively.
The methods of Euclidean geometry did not have the proper generality and did not allow obtaining the required quantitative results.
The efforts of mathematicians of the 17th century created numerous special methods that allowed finding the areas of figures bounded by curves of one kind or another, and in some cases the connection of these problems with the problems of finding the rate of change of functions was noted.
But, as in the case of differential calculus, it was Newton and Leibniz who realized the generality of the method and thereby laid the foundations of integral calculus.
The Newton Leibniz method begins by replacing the curve bounding the area to be determined with a sequence of polylines approaching it, similar to how it was done in the exhaustion method invented by the Greeks.
The exact area is equal to the limit of the sum of the areas of n rectangles when n turns to infinity.
Newton showed that this limit can be found by reversing the process of finding the rate of change of the function.
The operation that is the reverse of differentiation is called integration.
The statement that summation can be carried out by reversing differentiation is called the main theorem of mathematical analysis.
Just as differentiation is applicable to a much broader class of problems than the search for velocities and accelerations, integration is applicable to any problem related to summation, for example, to physical problems for adding forces.
* 4. MODERN MATHEMATICS* The creation of differential and integral calculus marked the beginning of"higher mathematics".
The methods of mathematical analysis, in contrast to the concept of the limit underlying it, looked clear and understandable.
For many years, mathematicians, including Newton and Leibniz, tried in vain to give an exact definition of the concept of limit.
And yet, despite numerous doubts about the validity of mathematical analysis, it was increasingly used.
Differential and integral calculus became the cornerstones of mathematical analysis, which eventually included such subjects as the theory of differential equations, ordinary and partial derivatives, infinite series, calculus of variations, differential geometry and much more.
A strict definition of the limit was obtained only in the 19th century.
Non Euclidean geometry.
By 1800, mathematics rested on two "whales" - on the numerical system and Euclidean geometry.
Since many properties of the numerical system were proved geometrically, Euclidean geometry was the most reliable part of the building of mathematics.
Nevertheless, the axiom of parallel ones contained a statement about straight lines extending to infinity, which could not be confirmed by experience.
Even the version of this axiom belonging to Euclid himself does not at all assert that some straight lines will not intersect.
Rather, it formulates a condition under which they will intersect at some endpoint.
For centuries, mathematicians have been trying to find a suitable replacement for the axiom of parallel systems.
But there was always a gap in each option.
The honor of creating non Euclidean geometry fell to N. I. Lobachevsky (1792-1856) and Ya.
Boyai (1802-1860), each of whom independently published their own original presentation of non Euclidean geometry.
In their geometries, infinitely many parallel lines could be drawn through a given point.
In the geometry of B. Riemann (1826-1866), no parallel line can be drawn through a point outside a straight line.
No one seriously thought about the physical applications of non Euclidean geometry.
The creation of the general theory of relativity by A. Einstein (1879-1955) in 1915 awakened the scientific world to the realization of the reality of non Euclidean geometry.
Non Euclidean geometry became the most impressive intellectual achievement of the 19th century.
It clearly demonstrated that mathematics can no longer be considered as a set of indisputable truths.
At best, mathematics can guarantee the validity of a proof based on unreliable axioms.
But mathematicians henceforth gained the freedom to explore any ideas that might seem attractive to them.
Each mathematician individually was now free to introduce his own new concepts and establish axioms at his discretion, taking care only that the theorems arising from the axioms did not contradict each other.
The grandiose expansion of the circle of mathematical research at the end of the last century was essentially a consequence of this new freedom.
Until about 1870, mathematicians were convinced that they were acting according to the predestinations of the ancient Greeks, applying deductive reasoning to mathematical axioms, thereby providing their conclusions with no less reliability than that possessed by the axioms.
Non Euclidean geometry and quaternions (algebra in which the property of commutativity does not hold) made mathematicians realize that what they took for abstract and logically consistent statements is actually based on an empirical and pragmatic basis.
The creation of non Euclidean geometry was also accompanied by the realization of the existence of logical gaps in Euclidean geometry.
One of the disadvantages of Euclidean Principles was the use of assumptions that were not explicitly formulated.
Apparently, Euclid did not question the properties that his geometric figures possessed, but these properties were not included in his axioms.
In addition, proving the similarity of two triangles, Euclid used the superposition of one triangle on another, implicitly assuming that the properties of the figures do not change when moving.
But in addition to such logical gaps, there were several erroneous proofs in the Beginnings.
The creation of new algebras, which began with quarternions, gave rise to similar doubts about the logical validity of arithmetic and algebra of an ordinary numerical system.
All the numbers previously known to mathematicians had the property of commutativity, i.e. ab = ba.
Quaternions, which revolutionized the traditional concepts of numbers, were discovered in 1843.Hamilton (1805-1865).
They turned out to be useful for solving a number of physical and geometric problems, although the commutativity property was not fulfilled for quaternions.
Quarternions forced mathematicians to realize that, except for the part devoted to integers and far from perfect Euclidean Principles, arithmetic and algebra do not have their own axiomatic basis.
Mathematicians freely handled negative and complex numbers and performed algebraic operations, guided only by the fact that they work successfully.
Logical rigor gave way to a demonstration of practical usefulness.
