mathematics

mathematics,

deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science.

Branches of Mathematics

Foundations

The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests (see logiclogic,
the systematic study of valid inference. A distinction is drawn between logical validity and truth. Validity merely refers to formal properties of the process of inference.
..... Click the link for more information. ; symbolic logicsymbolic logic
or mathematical logic,
formalized system of deductive logic, employing abstract symbols for the various aspects of natural language. Symbolic logic draws on the concepts and techniques of mathematics, notably set theory, and in turn has contributed to
..... Click the link for more information. ). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axiomsaxiom,
in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
..... Click the link for more information. effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of setsset,
in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g.
..... Click the link for more information. , originated by Georg Cantor, which now constitutes a universal mathematical language.

Algebra

Historically, algebraalgebra,
branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and
..... Click the link for more information. is the study of solutions of one or several algebraic equations, involving the polynomialpolynomial,
mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a₀xⁿ+a₁x
..... Click the link for more information. functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods.

Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groupsgroup,
in mathematics, system consisting of a set of elements and a binary operation a+b defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e.
..... Click the link for more information. (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), ringsring,
in mathematics, system consisting of a set R of elements and two binary operations, such that addition makes R a commutative group and multiplication is associative and distributes over addition (see commutative law; associative law; distributive law).
..... Click the link for more information. (of which the integers, or whole numbers, constitute a basic example), and fieldsfield,
in algebra, set of elements (usually numbers) that may be combined under the operations of addition and multiplication so that it constitutes an additive group, the nonzero elements form a multiplicative group, and multiplication distributes over addition.
..... Click the link for more information. (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics.

Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmeticarithmetic,
branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with numbers for practical purposes, e.g.
..... Click the link for more information. and number theorynumber theory,
branch of mathematics concerned with the properties of the integers (the numbers 0, 1, −1, 2, −2, 3, −3, …). An important area in number theory is the analysis of prime numbers.
..... Click the link for more information. , which are concerned with special properties of the integers—e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruences—are also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems.

Analysis

The essential ingredient of analysisanalysis,
branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit are subjected to rigorous examination; differential and integral
..... Click the link for more information. is the use of infinite processes, involving passage to a limitlimit,
in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1-2, 1-4, 1-8, 1-16, … are obviously getting smaller and smaller; since, if enough
..... Click the link for more information. . For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculuscalculus,
branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.
..... Click the link for more information. . The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vectorvector,
quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.
..... Click the link for more information. analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behavior of various physical systems. Calculus is one of the most powerful and supple tools of mathematics. Its applications, both in pure mathematics and in virtually every scientific domain, are manifold.

Geometry

The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometrygeometry
[Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
..... Click the link for more information. is concerned with the axiomatic study of polygons, conic sections, spheres, polyhedra, and related geometric objects in two and three dimensions—in particular, with the relations of congruence and of similarity between such objects. The unsuccessful attempt to prove the "parallel postulate" from the other axioms of Euclid led in the 19th cent. to the discovery of two different types of non-Euclidean geometrynon-Euclidean geometry,
branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
..... Click the link for more information. .

The 20th cent. has seen an enormous development of topologytopology,
branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size.
..... Click the link for more information. , which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry and differential geometrydifferential geometry,
branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian coordinates),
..... Click the link for more information. , in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development.

Applied Mathematics

The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computercomputer,
device capable of performing a series of arithmetic or logical operations. A computer is distinguished from a calculating machine, such as an electronic calculator, by being able to store a computer program (so that it can repeat its operations and make logical
..... Click the link for more information. science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems (e.g., differential equations, large systems of linear equations). It has a major use in technology for modeling and simulation. For example, the huge wind tunnelswind tunnel,
apparatus for studying the interaction between a solid body and an airstream. A wind tunnel simulates the conditions of an aircraft in flight by causing a high-speed stream of air to flow past a model of the aircraft (or part of an aircraft) being tested.
..... Click the link for more information. , formerly used to test expensive prototypes of airplanes, have all but disappeared. The entire design and testing process is now largely carried out by computer simulation, using mathematically tailored software. It also includes mathematical physics, which now strongly interacts with all of the central areas of mathematics. In addition, probabilityprobability,
in mathematics, assignment of a number as a measure of the "chance" that a given event will occur. There are certain important restrictions on such a probability measure.
..... Click the link for more information. theory and mathematical statisticsstatistics,
science of collecting and classifying a group of facts according to their relative number and determining certain values that represent characteristics of the group.
..... Click the link for more information. are often considered parts of applied mathematics. The distinction between pure and applied mathematics is now becoming less significant.

Development of Mathematics

The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia B.C., it was used for surveying and mensuration; estimates of the value of π (pipi,
in mathematics, the ratio of the circumference of a circle to its diameter. The symbol for pi is π. The ratio is the same for all circles and is approximately 3.1416.
..... Click the link for more information. ) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step.

Greek Contributions

A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by ThalesThales
, c.636–c.546 B.C., pre-Socratic Greek philosopher of Miletus and reputed founder of the Milesian school of philosophy. He is the first recorded Western philosopher. Thales taught that everything in nature is composed of one basic stuff, which he thought to be water.
..... Click the link for more information. (6th cent. B.C.), PythagorasPythagoras
, c.582–c.507 B.C., pre-Socratic Greek philosopher, founder of the Pythagorean school. He migrated from his native Samos to Crotona and established a secret religious society or order similar to, and possibly influenced by, the earlier Orphic cult.
..... Click the link for more information. , PlatoPlato
, 427?–347 B.C., Greek philosopher. Plato's teachings have been among the most influential in the history of Western civilization. Life

After pursuing the liberal studies of his day, he became in 407 B.C. a pupil and friend of Socrates. From about 388 B.
..... Click the link for more information. , and AristotleAristotle
, 384–322 B.C., Greek philosopher, b. Stagira. He is sometimes called the Stagirite. Life

Aristotle's father, Nicomachus, was a noted physician. Aristotle studied (367–347 B.C.
..... Click the link for more information. , and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period.

During the Golden Age (5th cent. B.C.), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of EleaZeno of Elea
, c.490–c.430 B.C., Greek philosopher of the Eleatic school. He undertook to support in his only known work, fragments of which are extant, the doctrine of Parmenides by demonstrating that motion and multiplicity are logically impossible.
..... Click the link for more information. proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as √2, also dates from this period. Eudoxus of CnidusEudoxus of Cnidus
, 408?–355? B.C., Greek astronomer, mathematician, and physician. From the accounts of various ancient writers, he appears to have studied with Plato in Athens, spent some time in Heliopolis, Egypt, founded a school in Cyzicus, and spent his later years
..... Click the link for more information. (4th cent. B.C.) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes.

The later (Hellenistic) period of Greek science is associated with the school of Alexandria. The greatest work of Greek mathematics, EuclidEuclid
, fl. 300 B.C., Greek mathematician. Little is known of his life other than the fact that he taught at Alexandria, being associated with the school that grew up there in the late 4th cent. B.C.
..... Click the link for more information. 's Elements (c.300 B.C.), appeared at the beginning of this period. Elementary geometry as taught in high school is still largely based on Euclid's presentation, which has served as a model for deductive systems in other parts of mathematics and in other sciences. In this method primitive terms, such as point and line, are first defined, then certain axioms and postulates relating to them and seeming to follow directly from them are stated without proof; a number of statements are then derived by deduction from the definitions, axioms, and postulates. Euclid also contributed to the development of arithmetic and presented a geometric theory of quadratic equations.

In the 3d cent. B.C., ArchimedesArchimedes
, 287–212 B.C., Greek mathematician, physicist, and inventor. He is famous for his work in geometry (on the circle, sphere, cylinder, and parabola), physics, mechanics, and hydrostatics.
..... Click the link for more information. , in addition to his work in mechanics, made an estimate of π and used the exhaustion theory of Eudoxus to obtain results that foreshadowed those much later of the integral calculus, and Apollonius of PergaApollonius of Perga,
fl. 247–205 B.C., Greek mathematician of the Alexandrian school. He produced a treatise on conic sections that included, as well as his own work, much of the work of his predecessors, among whom was Euclid.
..... Click the link for more information. named the conic sections and gave the first theory for them. A second Alexandrian school of the Roman period included contributions by Menelaus (c.A.D. 100, spherical triangles), Heron of AlexandriaHeron of Alexandria
or Hero,
mathematician and inventor. The dates of his birth and death are unknown; conjecture places them between the 2d cent. B.C. and the 3d cent. A.D. He is believed to have lived in Alexandria; although he wrote in Greek, his origin is uncertain.
..... Click the link for more information. (geometry), PtolemyPtolemy
(Claudius Ptolemaeus), fl. 2d cent. A.D., celebrated Greco-Egyptian mathematician, astronomer, and geographer. He made his observations in Alexandria and was the last great astronomer of ancient times.
..... Click the link for more information. (A.D. 150, astronomy, geometry, cartography), PappusPappus
, fl. c.300, Greek mathematician of Alexandria. He recorded and enlarged on the results of his predecessors, including Euclid and Apollonius of Perga, in his Mathematical Collection (8 books; date conjectural).
..... Click the link for more information. (3d cent., geometry), and DiophantusDiophantus
, fl. A.D. 250, Greek algebraist. He pioneered in solving a type of indeterminate algebraic equation where one seeks integer values for the unknowns; work in this field is known as Diophantine analysis. Only 6 of the 13 books with which he is credited are extant.
..... Click the link for more information. (3d cent., arithmetic).

Chinese and Middle Eastern Advances

Following the decline of learning in the West after the 3d cent., the development of mathematics continued in the East. In China, Tsu Ch'ung-Chih estimated π by inscribed and circumscribed polygons, as Archimedes had done, and in India the numerals now used throughout the civilized world were invented and contributions to geometry were made by AryabhataAryabhata
, c.476–550, Hindu mathematician and astronomer. He is one of the first known to have used algebra; his writings include rules of arithmetic and of plane and spherical trigonometry, and solutions of quadratic equations.
..... Click the link for more information. and BrahmaguptaBrahmagupta
, c.598–c.660, Indian mathematician and astronomer. He was among the first to meaningfully discuss the concepts of zero and of negative numbers. He wrote in verse the Brahma-sphuta-siddhanta
..... Click the link for more information. (5th and 6th cent. A.D.). The Arabs were responsible for preserving the work of the Greeks, which they translated, commented upon, and augmented. In Baghdad, Al-KhowarizmiAl-Khowarizmi
, fl. 820, Arab mathematician of the court of Mamun in Baghdad. His treatises on Hindu arithmetic and on algebra made him famous. He is said to have given algebra its name, and the word algorithm is said to have been derived from his name.
..... Click the link for more information. (9th cent.) wrote an important work on algebra and introduced the Hindu numerals for the first time to the West, and Al-BattaniAl-Battani
or Albatenius
, b. before 858, d. 929, Arab astronomer and mathematician. He is best known in astronomy for his improvements and corrections of the Ptolemaic tradition.
..... Click the link for more information. worked on trigonometry. In Egypt, Ibn al-HaythamIbn al-Haytham
or Alhazen
, 965–c.1040, Arab mathematician. Ibn al-Haytham was born in Basra, but made his career in Cairo, where he supported himself copying scientific manuscripts.
..... Click the link for more information. was concerned with the solids of revolution and geometrical optics. The Persian poet Omar Khayyam wrote on algebra.

Western Developments from the Twelfth to Eighteenth Centuries

Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo FibonacciFibonacci, Leonardo
, b. c.1170, d. after 1240, Italian mathematician, known also as Leonardo da Pisa. In Liber abaci (1202, 2d ed. 1228), for centuries a standard work on algebra and arithmetic, he advocated the adoption of Arabic notation.
..... Click the link for more information. ), who wrote on arithmetic and algebra (Liber abaci, 1202) and on geometry (Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccolò TartagliaTartaglia, Niccolò
, c.1500–1557, Italian engineer and mathematician. Largely self-educated, he taught mathematics at Verona, Brescia, and Venice. A pioneer in applying mathematics to artillery, he recorded his results in Della nova scientia (1537).
..... Click the link for more information. and Girolamo CardanoCardano, Girolamo or Geronimo
, 1501–76, Italian physician and mathematician. His works on arithmetic and algebra established his reputation.
..... Click the link for more information. , in trigonometry by François VièteViète or Vieta, François
, 1540–1603, French mathematician. As a founder of modern algebra, he introduced the use of letters as algebraic symbols and correlated algebra with geometry and trigonometry.
..... Click the link for more information. , and in such areas of applied mathematics as mapmaking by Mercator and others.

The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon StevinStevin, Simon
, 1548–1620, Dutch engineer and mathematician. His experiments in hydrostatics showed that the pressure exerted by a liquid is dependent only on its vertical height and not on the shape of the liquid's container, and demonstrated the principle of the
..... Click the link for more information. and logarithms by John NapierNapier, John
, 1550–1617, Scottish mathematician and theologian. He invented logarithms and wrote Mirifici logarithmorum canonis descriptio (1614), containing the first logarithmic table and the first use of the word logarithm.
..... Click the link for more information. and Henry BriggsBriggs, Henry,
1561–1630, English mathematician. He was the first professor of geometry at Gresham College, London (1596–1619), and Savilian professor of astronomy at Oxford (from 1619).
..... Click the link for more information. ; the beginnings of projective geometry were made by Gérard DesarguesDesargues, Gérard
, 1591–1661, French mathematician and engineer, a founder of modern geometry. He discovered the theorems on involutions and transversals known by his name and worked on conic sections. His writings, lost for a time, were republished in 1864.
..... Click the link for more information. and Blaise PascalPascal, Blaise
, 1623–62, French scientist and religious philosopher. Studying under the direction of his father, a civil servant, Pascal showed great precocity, especially in mathematics and science.
..... Click the link for more information. ; number theory was greatly extended by Pierre de FermatFermat, Pierre de
, 1601–65, French mathematician. A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory and probability theory.
..... Click the link for more information. ; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, GalileoGalileo
(Galileo Galilei) , 1564–1642, great Italian astronomer, mathematician, and physicist. By his persistent investigation of natural laws he laid foundations for modern experimental science, and by the construction of astronomical telescopes he greatly enlarged
..... Click the link for more information. and Johannes KeplerKepler, Johannes
, 1571–1630, German astronomer. From his student days at the Univ. of Tübingen, he was influenced by the Copernican teachings. From 1593 to 1598 he was professor of mathematics at Graz and while there wrote his Mysterium cosmographicum (1596).
..... Click the link for more information. made fundamental contributions.

The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by René DescartesDescartes, René
, Lat. Renatus Cartesius, 1596–1650, French philosopher, mathematician, and scientist, b. La Haye. Descartes' methodology was a major influence in the transition from medieval science and philosophy to the modern era.
..... Click the link for more information. and that of the calculus by Isaac NewtonNewton, Sir Isaac,
1642–1727, English mathematician and natural philosopher (physicist), who is considered by many the greatest scientist that ever lived. Early Life and Work
..... Click the link for more information. and, independently, by G. W. LeibnizLeibniz or Leibnitz, Gottfried Wilhelm, Baron von
, 1646–1716, German philosopher and mathematician, b. Leipzig.
..... Click the link for more information. . Descartes's invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. CavalieriCavalieri, Francesco Bonaventura
, 1598–1647, Italian mathematician, a Jesuit priest. Professor at Bologna from 1629, he invented the method of indivisibles (1635) that foreshadowed integral calculus.
..... Click the link for more information. , Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.

The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the BernoulliBernoulli
or Bernouilli
, name of a family distinguished in scientific and mathematical history. The family, after leaving Antwerp, finally settled in Basel, Switzerland, where it grew in fame.
..... Click the link for more information. family (especially Jakob, Johann, and Daniel), Leonhard EulerEuler, Leonhard
, 1707–83, Swiss mathematician. Born and educated at Basel, where he knew the Bernoullis, he went to St. Petersburg (1727) at the invitation of Catherine I, becoming professor of mathematics there on the departure of Daniel Bernoulli (1733).
..... Click the link for more information. , Guillaume de L'Hôpital, and J. L. LagrangeLagrange, Joseph Louis, Comte
, 1736–1813, French mathematician and astronomer, b. Turin, of French and Italian descent. Before the age of 20 he was professor of geometry at the royal artillery school at Turin.
..... Click the link for more information. . Important advances in geometry began toward the end of the century with the work of Gaspard MongeMonge, Gaspard, comte de Péluse
, 1746–1818, French mathematician, physicist, and public official. He was distinguished for his geometrical research, which laid the foundations of modern descriptive geometry, a field essential to mechanical drawing and architectural
..... Click the link for more information. in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. PonceletPoncelet, Jean Victor
, 1788–1867, French mathematician and army engineer. He taught at the school of mechanics at Metz and at the Faculté des Sciences and the École Polytechnique, both in Paris.
..... Click the link for more information. , who founded projective geometry (1822).

In the Nineteenth Century

The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. GaussGauss, Carl Friedrich
, born Johann Friederich Carl Gauss, 1777–1855, German mathematician, physicist, and astronomer. Gauss was educated at the Caroline College, Brunswick, and the Univ.
..... Click the link for more information. . In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. LobachevskyLobachevsky, Nikolai Ivanovich
, 1793–1856, Russian mathematician. A pioneer in non-Euclidean geometry, he challenged Euclid's fifth postulate that one and only one line parallel to a given line can be drawn through a fixed point external to the line; he developed,
..... Click the link for more information. (1826) and independently by János BolyaiBolyai
, family of Hungarian mathematicians. The father, Farkas, or Wolfgang, Bolyai, 1775–1856, b. Bolya, Transylvania, was educated in Nagyszeben from 1781 to 1796 and studied in Germany during the next three years at Jena
..... Click the link for more information. (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard RiemannRiemann, Bernhard
(Georg Friedrich Bernhard Riemann) , 1826–66, German mathematician. He studied at the universities of Göttingen and Berlin and was professor at Göttingen from 1859.
..... Click the link for more information. (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.

In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. HamiltonHamilton, Sir William Rowan,
1805–65, Irish mathematician and astronomer, b. Dublin. A child prodigy, he had mastered 13 languages by the age of 13 and was still an undergraduate when he became professor of astronomy at the Univ. of Dublin (1827).
..... Click the link for more information. , whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. GrassmannGrassmann, Hermann Günther
, 1809–77, German mathematician and Sanskrit scholar, educated in Berlin. He invented a new algebra of vectors (somewhat similar to quaternions), presented in his book Die Ausdehnungslehre (1844, 4th ed. 1969).
..... Click the link for more information. , who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.

These developments continued with the group theory of M. S. LieLie, Marius Sophus
, 1842–99, Norwegian mathematician. He is noted for his contributions to the theories of differential equations and continuous transformation groups.
..... Click the link for more information. in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg CantorCantor, Georg
, 1845–1918, German mathematician, b. St. Petersburg. He studied under Karl Weierstrass and taught (1869–1913) at the Univ. of Halle. He is known for his work on transfinite numbers and on the development of set theory, which is the basis of modern
..... Click the link for more information. , J. W. R. DedekindDedekind, Julius Wilhelm Richard
, 1831–1916, German mathematician. Dedekind studied at Göttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
..... Click the link for more information. , and K. W. WeierstrassWeierstrass, Karl Wilhelm Theodor
, 1815–97, German mathematician. From 1864 he was professor of mathematics at the Univ. of Berlin. His development of the modern theory of functions is described in his Abhandlungen aus der Funktionenlehre
..... Click the link for more information. . Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. CauchyCauchy, Augustin Louis, Baron
, 1789–1857, French mathematician. He was professor simultaneously (1816–30) at the École polytechnique, the Sorbonne, and the Collège de France in Paris. While a political exile (1830–38) he taught at the Univ.
..... Click the link for more information. 's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.

In the Twentieth Century

In the 20th cent. the trend has been toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach has been the use of formal axiomatics, in which the notion of axioms as "self-evident truths" has been discarded. Instead the emphasis is on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).

The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand RussellRussell, Bertrand Arthur William Russell, 3d Earl,
1872–1970, British philosopher, mathematician, and social reformer, b. Trelleck, Wales.
..... Click the link for more information. and A. N. WhiteheadWhitehead, Alfred North,
1861–1947, English mathematician and philosopher, grad. Trinity College, Cambridge, 1884. There he was a lecturer in mathematics until 1911. At the Univ.
..... Click the link for more information. , and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt GödelGödel, Kurt
, 1906–78, American mathematician and logician, b. Brünn (now Brno, Czech Republic), grad. Univ. of Vienna (Ph.D., 1930). He came to the United States in 1940 and was naturalized in 1948.
..... Click the link for more information. and A. Church.

Bibliography

See R. Courant and H. Robbins, What Is Mathematics? (1941); E. T. Bell, The Development of Mathematics (2d ed. 1945) and Men of Mathematics (1937, repr. 1961); J. R. Newman, ed., The World of Mathematics (4 vol., 1956); E. E. Kramer, The Nature and Growth of Mathematics (1970); M. Kline, Mathematical Thought from Ancient to Modern Times (1973); D. J. Albers and G. L. Alexanderson, ed., Mathematical People (1985).

Mathematics

Mathematics is the science of the quantitative relations and spatial forms of the real world.

“Pure mathematics studies the spatial forms and quantitative relations of the real world and thus its subject matter is rooted in reality. The highly abstract form of this subject matter hardly conceals its derivation from the outside world. However, in order to investigate these forms and relations as such, one must separate them entirely from their underlying content, which must be set aside as something irrelevant” (F. Engels in K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 37). The abstractness of mathematics does not, however, imply the separation of mathematics from material reality. The number of quantitative relations and spatial forms studied by mathematics is inseparably connected with the demands of technology and the natural sciences and is growing continually; therefore the definition of mathematics given above is endowed with ever richer content.

Mathematics and other sciences. The applications of mathematics are extremely varied. In principle, the domain of application of the mathematical method is unlimited: all types of motion of matter can be studied mathematically. However, the role and significance of the mathematical method vary with each case. No single mathematical scheme can cover all aspects of real phenomena, and therefore the process of cognition of the concrete always involves two conflicting tendencies: the isolation of the form of the phenomena under study and the logical analysis of this form, on the one hand, and the disclosure of aspects that do not fit into established forms and the transition to the consideration of new forms that are more flexible and more completely encompass the phenomena, on the other. If the difficulties in studying some range of phenomena bear largely on the second tendency in the sense that each new step in the investigation entails the consideration of qualitatively new aspects of the phenomena, then the mathematical method recedes into the background; in this case, dialectical analysis of all aspects of a phenomenon can only be obscured by mathematical schematization. On the other hand, if comparatively simple and stable basic forms of the phenomena under study encompass these phenomena with great accuracy and completeness and if within the limits of these forms there arise quite difficult and complex problems requiring special mathematical investigation, in particular, the creation of special symbolic notation and a special algorithm for their solution, then we enter the sphere of dominance of the mathematical method.

Celestial mechanics, particularly the study of planetary motion, is a typical example of the complete dominance of the mathematical method. The law of universal gravitation, which has a very simple mathematical expression, almost completely determines the range of phenomena studied under this heading. With the exception of the theory of lunar motion, we may, within the limits of observational accuracy, disregard the shape and dimensions of celestial bodies and think of them as “material points.” The solution of the resulting problem of the motion of n material points subject to gravitational forces presents great difficulties as early as the case of n = 3. However, every result obtained by mathematical analysis of the adopted model of the phenomenon describes a real event with remarkable accuracy: a logically simple model reflects well the selected range of phenomena, and all difficulties consist in extracting the mathematical consequences from the adopted model.

No appreciable reduction of the role of the mathematical method occurs upon transition from mechanics to physics, but the difficulties of application increase significantly. There is almost no field of physics that does not require the use of a sophisticated mathematical apparatus, but the main difficulty involved in the investigations often lies in the selection of the assumptions for mathematical treatment and in the interpretation of the results obtained by mathematical means rather than in the development of the mathematical theory.

The ability of the mathematical method to encompass the very process of transition from one level of insight into reality to the next higher, and qualitatively new, level, can be observed by using a number of physical theories as examples. A classical example is the relation between the macroscopic theory of diffusion, which presupposes a continuously distributed diffusing substance, and the statistical theory of diffusion, which proceeds from consideration of the motion of individual particles of the diffusing substance. In the first theory the density of the diffusing substance satisfies a specific partial differential equation and the study of various problems pertaining to diffusion reduces to finding solutions of this differential equation for appropriate boundary and initial conditions. The continuous theory of diffusion describes with very high precision the real course of events insofar as it deals with ordinary (macroscopic) spatial and time scales. However, for small regions of space (containing only a small number of particles of the diffusing substance), the very concept of density loses meaning. The statistical theory of diffusion proceeds from consideration of the microscopic random motions of the diffusing particles induced by the molecules of the solvent. We do not know precise quantitative laws governing these microscopic displacements. However, the mathematical theory of probability enables us (on the basis of general assumptions about the smallness of displacements over short time intervals and the independence of the displacements of a particle in two successive time intervals) to obtain definite quantitative consequences, that is, to determine (approximately) the laws governing the probability distribution of the displacements of particles over long (macroscopic) time intervals. Since the number of individual particles of the diffusing substance is very great, the laws governing the probability distribution of the displacement of individual particles lead, on the assumption that the displacements of each particle are independent of the others, to specific and nonrandom laws governing the motion of the diffusing substance as a whole. These laws turn out to coincide with the differential equations underlying the continuous theory. This example is quite typical in the sense that a statistic of random events frequently leads from one set of laws (in this case, the laws governing the motions of the individual particles of the diffusing substance) to a qualitatively different set of laws (in this case, the differential equations of the continuous theory of diffusion).

In the biological sciences the mathematical method plays a more subordinate role. In the social sciences and the humanities, to an even greater degree than in biology, the mathematical method gives way to the direct analysis of phenomena in all their specific complexity. The application of the mathematical method in the biological and the social sciences and in the humanities is accomplished chiefly through cybernetics. The importance of mathematics for the social sciences (as for the biological sciences) in the form of an auxiliary science—mathematical statistics—remains significant. But in the final analysis of social phenomena, aspects of the qualitative distinctness of each historical stage assume such a dominant position that the mathematical method often recedes into the background.

Mathematics and technology. The principles of arithmetic and elementary geometry arose, as will be seen from the history section below, in response to the needs of everyday life; the subsequent creation of new mathematical methods and ideas took place under the influence of the mathematical natural sciences, such as astronomy, mechanics, and physics, whose development responds to practical needs. But the direct ties between mathematics and technology have frequently taken the form of application of existing mathematical theories to technological problems. Let us point out, however, examples of the rise of new general mathematical theories in response to direct technological requirements: the development of the method of least squares was connected with geodetic work; the study of many new types of partial differential equations first began with the solution of technical problems; and operational methods of solving differential equations were developed in connection with electrical engineering. The requirements of communications led to the formation of a new branch of probability theory—information theory. The problems of synthesizing control systems led to the development of new branches of mathematical logic. Along with the needs of astronomy, technical problems also played a decisive role in the development of the methods of approximate solution of differential equations. Many methods of approximate solution of partial differential equations and of integral equations were developed entirely to meet technological needs. The problem of rapidly obtaining numerical solutions became quite acute with the growing complexity of technical problems. Numerical methods are acquiring ever greater importance in connection with the possibilities opened up by computers for the solution of practical problems. The high level of theoretical mathematics has made it possible to develop rapidly the methods of computer science, which has played a major role in the solution of a number of very important practical problems, including those in the use of atomic energy and in space exploration.

A clear understanding of the position of mathematics as an independent science with its own subject and method became possible only after considerable factual material was accumulated. Such an understanding came about in ancient Greece in the sixth and fifth centuries B.C. It is customary to place the development of mathematics before that time in the period of the rise of mathematics and to date the beginning of the period of elementary mathematics to the sixth and fifth centuries B.C. During these two early periods, mathematical investigations dealt almost exclusively with an extremely limited stock of basic concepts that had arisen in the early stages of historical development in connection with the simple requirements of domestic life, which reduced to the counting of objects, calculation of the number of goods and areas of plots of land, determination of the size of the individual parts of architectural structures, measurement of time, mercantile calculations, and navigational measurements. The early problems of mechanics and physics (with the exception of certain studies conducted by the Greek scholar Archimedes [third century B.C.] already requiring the rudiment of infinitesimal calculus) could still be satisfied with this stock of basic mathematical concepts. Astronomy, which was entirely responsible, for example, for the early development of trigonometry, was the only science that systematically placed its own particular and very great demands on mathematics long before the extensive development of the mathematical study of natural phenomena in the 17th and 18th centuries.

In the 17th century the new requirements of the natural sciences and technology forced mathematicians to focus their attention on creating methods that would make it possible to mathematically study motion, the processes of the change of quantities, and transformations of geometric figures (for example, in projection). The period of the mathematics of variable quantities begins with the use of variables in the analytic geometry of the French scientist R. Descartes and the creation of the differential and integral calculus.

The further expansion of the range of quantitative relations and spatial forms studied by mathematics led, at the start of the 19th century, to the necessity of consciously referring to the process of the expansion of the subject of mathematical investigations after setting the task of studying systematically, from a sufficiently general viewpoint, the possible types of quantitative relations and spatial forms. The creation of “imaginary geometry” by the Russian mathematician N. I. Lobachevskii, which subsequently found real applications, was the first significant step in this direction. The development of this type of investigation introduced such important new features in the structure of mathematics that it is natural to place the mathematics of the 19th and 20th centuries in the special period of modern mathematics.

Origin of mathematics. The counting of objects in the earliest stages of cultural development led to the creation of very simple concepts of the arithmetic of natural numbers. It was only on the basis of a developed system of oral reckoning that written numeration systems arose and the methods of performing the four arithmetic operations on natural numbers were gradually developed (only division was to present difficulties for a long time). The requirements of measurement (the amount of grain, the length of a road) led to the appearance of the names and designations for the simplest fractions and to the development of methods of performing arithmetic operations on fractions. In this way there accumulated material that gradually evolved into the oldest mathematical science—arithmetic. The measurement of areas and volumes and the requirements of construction and, somewhat later, of astronomy gave rise to the development of the rudiments of geometry. These processes progressed among many peoples largely independently and in parallel. The accumulation of arithmetic and geometric knowledge in Egypt and Babylonia was of particular importance to the subsequent development of science. In Babylonia, advanced techniques of arithmetic calculations were the basis of rudiments of algebra and, the requirements of astronomy, of rudiments of trigonometry.

Extant mathematical texts of ancient Egypt (the first half of the second millennium B.C.) consist primarily of examples dealing with the solution of individual problems and, at best, of prescriptions for their solution, which sometimes can be understood only by analyzing the numerical examples given in the texts. The term “prescription” for the solution of certain types of problems is precisely right, since there obviously was no mathematical theory in the sense of proofs of general theorems. The lack of any distinction between exact and approximate solutions attests to this. Nonetheless, the very stock of established mathematical knowledge was quite large, as can be inferred from well-developed construction techniques, the complexity of land relations, and the need for an accurate calendar.

There are more mathematical texts enabling us to assess mathematics in Babylonia than in Egypt. Babylonian cuneiform mathematical texts encompass the period from the second millennium B.C. to the rise and development of Greek mathematics. Babylonia of this period had acquired from the earlier Sumerian period a well-developed mixed decimal-sexagesimal numeration system that already incorporated positional notation (given symbols designated the same number of units of different sexagesimal orders). Tables of reciprocals reduced division to multiplication. In addition to tables of reciprocals, there were tables of products, squares, and square and cube roots. Of the achievements of Babylonian geometry, which exceeded the knowledge of the Egyptians, we note the well-developed measurement of angles and certain rudiments of trigonometry that apparently were connected with the development of astronomy. The Pythagorean theorem was known to the Babylonians.

Period of elementary mathematics. Mathematics arose as an independent science—with a clear understanding of the distinctness of its method and of the necessity of systematically developing its fundamental concepts and premises in a sufficiently general form—only after a large amount of specific material was accumulated in the form of separate methods of arithmetic calculations, methods of determining areas and volumes, and other procedures. With respect to arithmetic and algebra, this process may have started in Babylonia. However, this new trend, which consisted in the systematic and logically consistent construction of the foundations of the mathematical science, was fully defined in ancient Greece. The system of exposition of elementary geometry that was developed by the ancient Greeks served for two millennia as a model of the deductive construction of a mathematical theory. Number theory gradually grew out of arithmetic. The systematic study of quantities and measurement was created. The process of the formation (in connection with the problem of measuring quantities) of the concept of real numbers proved to be extremely protracted, owing to the fact that the concepts of irrational and negative numbers are among those more complex mathematical abstractions which, in contrast to the concepts of natural number, fraction, and geometric figure, are not an intrinsic part of prescientific human experience.

The creation of algebra as literal calculation concluded only at the end of the two-millennium period under consideration here. Special symbols for unknowns were developed by the Greek mathematician Diophantus (probably in the third century) and more systematically in India in the seventh century, but the representation of the terms of an equation by means of letters was introduced only in the 16th century by the French mathematician F. Vieta.

The development of geodesy and astronomy led early to the detailed elaboration of both plane and spherical trigonometry.

The period of elementary mathematics concluded (in Western Europe in the early 17th century) when mathematical interest shifted to the domain of the mathematics of variable quantities.

ANCIENT GREECE. The development of mathematics took an essentially different direction in ancient Greece than in the Orient. Although the level of Babylonian mathematics in the techniques of calculations, the art of solving algebraic problems, and the development of the mathematical tools of astronomy was reached and surpassed by the Greeks only in the Hellenistic era, mathematics in ancient Greece had entered a totally new stage of logical development much earlier. The need for clear mathematical proofs appeared, and the first attempts at a systematic construction of a mathematical theory were made. Mathematics, like all scientific and artistic endeavors, ceased to be impersonal as was the case in the ancient Orient. Now it was created by mathematicians who were known by name and who left mathematical works, which, unfortunately, have come down to us only in fragments preserved by later commentators.

In arithmetic, the Greeks considered themselves to be students of the Phoenicians, explaining the Phoenicians’ high development of arithmetic in terms of the requirements of their extensive commerce. Tradition links the beginnings of Greek geometry with the travels to Egypt in the seventh and sixth centuries B.C. of the first Greek geometers and philosophers Thales of Miletus and Pythagoras of Samos. In the Pythagorean school, arithmetic developed from the simple art of counting into number theory. Very simple arithmetic progressions (in particular, 1 + 3 + 5 + … + 2n — 1 =n²) were summed, the divisibility of numbers and different types of means (arithmetic, geometric, and harmonic) were studied, and the problems of number theory (such as the search for perfect numbers) were connected in the Pythagorean school to the mystical, magical nature ascribed to numerical relationships. In connection with the geometric Pythagorean theorem, a method of obtaining infinitely many triples of “Pythagorean numbers,” that is, sets of three integers satisfying the equation a² + b² = c², was found. In geometry, the problems that Greek geometers of the sixth and fifth centuries B.C. worked on after mastering the Egyptian heritage also arose naturally from the very simple requirements of construction, surveying, and navigation. Such, for example, were the problems of the relationship between the lengths of the legs and hypotenuse of a right triangle (which is expressed by the Pythagorean theorem), the relation between the areas of similar figures, squaring the circle, trisection of an angle, and duplication of the cube. However, the approach to these problems was necessarily new owing to the growing complexity of the object under study. Not confining themselves to approximate, empirically determined solutions, Greek geometers sought exact proofs and logically complete solutions of a problem. The proof of the incommensurability of the diagonal of a square with its side can serve as a clear example of this new trend. In the second half of the fifth century B.C. the philosophical and scholarly life of Greece was concentrated in Athens. Hippias of Elis and Hippocrates of Chios worked here. The first systematic textbook of geometry is ascribed to Hippocrates of Chios. A developed system of geometry that did not disregard such logical subtleties as proofs of the familiar congruence criteria for triangles undoubtedly had already been created by that time. The discovery of all five regular polyhedrons—perhaps the most noteworthy achievement of geometry in the fifth century B.C. and the result of the search for ideal simple bodies that could serve as the building blocks of the universe—was a reflection of the first, albeit purely speculative, attempts in mathematics to give a rational explanation of the structure of matter. At the turn of the fourth century B.C. Democritus created, on the basis of atomistic concepts, a method of determining volume that later served as the starting point for Archimedes in his development of the method of infinitesimals. In the fourth century B.C., the age of a certain subordination of mathematics to restrictions imposed by idealist philosophy commenced in an atmosphere of political reaction and the declining power of Athens. Here, the science of numbers is strictly separated from the “science of calculation,” and geometry from the “science of measurement.” Assuming the existence of incommensurable segments, areas, and volumes, Aristotle imposed a general ban on the application of arithmetic to geometry. Geometric constructions were limited to those that could be carried out with compass and straightedge. The investigations of Eudoxus of Cnidus, which were connected with the trend toward logical analysis of the foundations of geometry, may be considered the single most significant achievement of the mathematicians of the fourth century B.C.

HELLENISTIC AND ROMAN ERAS. For seven centuries after the third century B.C., Alexandria was the center of scientific and particularly mathematical investigations. Here, Greek mathematics reached its zenith amid an atmosphere of the fusion of various world cultures, major state requirements, extensive building, and governmental patronage of science that was unprecedented in its scope. Despite the spread of Greek education and scientific interests throughout the entire Hellenistic and Roman world, Alexandria, with its Museum, the first research institution in the modern sense of the word, and with its libraries, attracted nearly all the major scholars. Of the mathematicians mentioned below, only Archimedes remained faithful to his homeland of Syracuse. The first century of the Alexandrian era (third century B.C.) was distinguished by the highest intensity of mathematical creativity. Euclid, Archimedes, Eratosthenes, and Apollonius of Perga belonged to this century.

In his Elements, Euclid collected and subjected to definitive logical treatment the achievements of the previous period in geometry. At the same time he first laid the foundations of a systematic theory of numbers, proving the infinite nature of the sequence of primes and constructing a complete.theory of divisibility. Of the geometric studies of Euclid not included in the Elements and of the works of Apollonius of Perga, the creation of a complete theory of conic sections was of the greatest importance for the subsequent development of mathematics. The chief contribution of Archimedes in geometry lies in the determination of various areas and volumes (including the areas of a parabolic segment and the surface of a sphere and the volumes of a sphere, spherical segment, and segment of a paraboloid) and centers of gravity (of, for example, a spherical segment and a segment of a paraboloid). The spiral of Archimedes is just one example of the transcendental curves that were studied in the third century B.C. After Archimedes, Alexandrian science no longer reached its former completeness and depth, although the growth of the volume of scientific knowledge continued; the rudiments of infinitesimal analysis contained in the heuristic methods of Archimedes were not developed further.

It should be stated that the interest in the approximate measurement of quantities and in approximate calculations that arose from applied needs did not lead mathematicians of the third century B.C. to reject mathematical rigor. All the numerous approximate extractions of roots and even all astronomical computations were accomplished with a precise indication of the limits of error, as was the case with the celebrated Archimedean determination of the circumference of a circle in the form of the faultlessly proved inequalities

where p is the circumference of a circle with diameter d. This distinct understanding that approximate mathematics is not “unrigorous” mathematics was later forgotten.

The lack of a conclusively formulated concept of the irrational number was a significant shortcoming of all mathematics in antiquity. As has been indicated, this fact led philosophy in the fourth century B.C. to reject completely the legitimacy of applying arithmetic to the study of geometric quantities. In fact, in the theory of proportions and in the method of exhaustion, mathematicians of the fourth and third centuries B.C. did succeed in indirectly applying arithmetic to geometry. The centuries that followed did not bring a definitive resolution of the problem through the creation of a new fundamental concept (irrational number) but rather gradual oblivion, which became possible with the gradual loss of the concept of mathematical rigor. At this stage in the history of mathematics, however, the temporary rejection of mathematical rigor proved to be useful by opening up the possibility for the unhindered development of algebra, whose use, within the framework of the rigorous concepts of Euclid’s Elements, was restricted to the extremely confining “geometric algebra” of segments, areas, and volumes. Significant advances in this direction can be found in the Metrics of Hero of Alexandria (Heron). Independent and extensive development of real algebraic calculation, however, is found only in Diophantus’ Arithmetica, which is primarily devoted to the solution of equations. Classifying his investigations as pure arithmetic, Diophantus naturally restricted himself, in contrast to the utilitarian Hero, to rational solutions, thus excluding the possibility of geometric or mechanical applications of his algebra. In antiquity, trigonometry was perceived largely as a branch of astronomy and not as a part of mathematics. The requirements of complete rigor in formulations and proofs were not imposed on trigonometry just as they were not on Hero’s computational geometry. Hipparchus was the first to compile tables of chords that were the equivalent of our sine tables. The principles of spherical trigonometry were developed by Menelaus and Ptolemy.

In pure mathematics, the scholars of the last centuries of the ancient period (except for Diophantus) concentrated increasingly on providing commentary on those who preceded them. The works of the commentators of that period, such as Pappus (third century) and Proclus (fifth century), for all their universality could not lead, amid the decline of the ancient world, to the consolidation into a unified science of the algebra of Diophantus, the trigonometry incorporated in astronomy, and Hero’s frankly nonrigorous computational geometry, all of which developed separately.

CHINA. The Arithmetic in Nine Chapters, which was compiled from earlier sources by Chang Ts’ang and Ching Ch’ou-ch’ang in the second and first centuries B.C., reveals that Chinese mathematicians had highly developed computational techniques and an interest in general algebraic methods. This work describes, in particular, methods of computing square and cube roots of integers. A large number of problems are formulated in such a way that we must conclude that the authors knew the method of solving a system of linear equations by successive elimination of unknowns and thought of the problems as examples clarifying that method. Interest in problems of the following type arose in China in connection with calendar calculations: When a number is divided by 3 the remainder is 2, when divided by 5 the remainder is 3, and when divided by 7 the remainder is 2; what is the number? Sun-tzu (between the second and sixth centuries) and Ch’in Chiu-shao (13th century) by way of examples provided a description of a regular algorithm for solving such problems. The result obtained by Tsu Ch’ung-chih (second half of the fifth century) that the ratio of the circumference of a circle to its diameter lies in the range 3.1415926 < π < 3.1415927 can serve as an example of the high level of development of computational methods in geometry. The studies of the Chinese dealing with numerical solutions of equations are remarkable. Problems in geometry leading to cubic equations first appeared in the works of the astronomer and mathematician Wang Hsiao T’ung (first half of the seventh century). Methods of solving fourth- and higher degree equations were given by the 13th- and 14th-century mathematicians Ch’in Chiu-shao, Li Yeh, Yang Hui, and Chu Shih-chieh.

INDIA. The flowering of Hindu mathematics dates to the fifth to 12th centuries. The most important mathematicians of this period were Aryabhata, Bhaskara, and Brahmagupta. The Hindus made two important contributions. The first consisted in introducing into wide use the modern decimal system of notation including the systematic use of zero to designate the absence of units of a given order. The origin of the numbers used in India, now called Arabic, has not been fully elucidated. The second and more important contribution was the creation of an algebra that freely used not only fractions but irrational and negative numbers. However, negative solutions were usually considered impossible in the interpretation of solutions of problems. In general, it should be noted that while both fractions and irrational numbers were associated from the very time of their appearance with the measurement of continuous quantities, negative numbers arose chiefly out of the intrinsic requirements of algebra and only later (in the 17th century) acquired independent significance. In trigonometry the introduction of the sine, cosine, and inverse sine curves should be noted.

CENTRAL ASIA AND THE MIDDLE EAST. The Arab conquests and the brief consolidation of vast territories under the power of the Arab caliphs led to a situation in which the scholars of Central Asia, the Middle East, and the Iberian Peninsula used the Arabic language from the ninth to 15th centuries. Science developed in the commercial cities amid a cosmopolitan atmosphere and governmental support of major scientific undertakings. In the 15th century the work of Ulug Beg, who assembled more than 100 scholars at his court and observatory in Samarkand and who organized astronomical observations, the computation of mathematical tables, and other endeavors that long remained unsurpassed, were the brilliant culmination of this era.

In Western European science the opinion long prevailed that the role of the “Arab culture” in mathematics consisted chiefly in the preservation and transmission to Western European mathematicians of the ancient Greek and Hindu mathematical discoveries. For example, the works of Greek mathematicians first became known in Western Europe through Arabic translations. In reality, however, the contributions to the development of science of the mathematicians who wrote in Arabic, and in particular of the mathematicians who were of the peoples of what is now Soviet Middle Asia and the Caucasus (Khorezm, Uzbek, Tadzhik, Azerbaijan), were much greater.

In the first half of the ninth century, Muhammad ibn Musa al-Khwarizmi gave the first exposition of algebra as an independent science. The term “algebra” was taken from the title of Khwarizmi’s work Al-jabr w’almugabala, from which European mathematicians of the early Middle Ages learned of the solution of quadratic equations. Omar Khayyam systematically studied and classified cubic equations and determined the conditions for their solvability (in the sense of the existence of positive roots). In his algebraic treatise Khayyam states that he searched long for an exact solution of cubic equations. The search of Central Asian mathematicians for such a solution was not successful. However, these mathematicians were well acquainted both with geometric methods (using conic sections) and approximate numerical methods of solution. While they adopted the Hindu decimal system and the use of zero, the mathematicians of Central Asia and the Middle East primarily used the sexagesimal system in major scientific calculations (apparently because of the sexagesimal division of angles in astronomy).

Trigonometry was developed extensively in connection with astronomical and geodetic projects. Al-Battani introduced the use of the trigonometric functions of sine, tangent, and cotangent. Abu-al-Wafa used all six trigonometric functions and verbally expressed the algebraic relations between them, computed tables of sines for every ten minutes of angles with an accuracy of up to 1/60⁴ and tables of tangents, and established the law of sines for spherical triangles. Nasir al-Din al-Tusi (Nasir-Eddin) further developed spherical trigonometry and al-Kashi gave a systematic account of the arithmetic of decimal fractions, which he rightly considered to be more convenient than sexagesimal fractions. In connection with problems of the extraction of roots, al-Kashi verbally formulated Newton’s binomial formula and found the rule for forming the coefficients C^m_n = (C_n−1)^m + (C_n−1)^m−1 In the Treatise on the Circle (c. 1427), al-Kashi, found the value of π to 17 decimal places by computing the perimeters of inscribed and circumscribed polygons with 3 × 2²⁸ sides. In connection with the computation of extensive tables of sines, al-Kashi gave an extremely advanced iterative method for the numerical solution of equations.

WESTERN EUROPE BEFORE THE 16TH CENTURY. For Western European mathematics the 12th to 15th centuries were essentially a period of the mastery of the heritage of the ancient world and the Orient. In spite of the fact that no new significant results were established during this period, the general character of European mathematical culture at that time was distinguished by a number of substantial progressive features that made possible the rapid development of mathematics in subsequent centuries. The sophisticated requirements of the politically independent members of the Italian bourgeoisie, who were rapidly growing wealthy, led to the creation and extensive dissemination of textbooks that combined a practical general orientation with extreme detail and scientific orientation. In the 12th century, less than 100 years after the appearance of the first Latin translations of Greek and Arabic mathematical works, Leonardo of Pisa (Fibonacci) wrote Liber abaci (Book of the Abacus; 1202) and Practica geometriae (The Practice of Geometry; 1220), which expounded on arithmetic, mercantile arithmetic, algebra, and geometry. These books enjoyed great success.

By the end of this era, with the invention of the printing press, textbooks came into still wider use. During this period the universities became the centers of theoretical scientific thought. The progress of algebra as a theoretical discipline and not merely as a collection of practical rules for the solution of problems is evidenced by a clear understanding of the nature of irrational numbers as the ratios of incommensurable quantities (the English mathematician T. Bradwardine [first half of the 14th century] and N. Oresme [mid-14th century]) and especially in the introduction of fractional exponents (Oresme) and negative and zero exponents (the French mathematician N. Chuquet [end of the 15th century]). Here too arose the first ideas, which anticipated the following era, of infinitely large and infinitesimal quantities. The broad scope of scientific research during this era was reflected not only in numerous translations and editions of Greek and Arabic works but also in such undertakings as the compilation of extensive seven-place trigonometric tables by Regiomontanus (J. Müller). Mathematical symbols were greatly improved. Scientific criticism and polemics developed. The search for solutions of difficult problems, which was encouraged by the custom of public competitions, led to the first proofs of unsolvability. Thus, Leonardo of Pisa proved in the work Flos (Flower; 1225), a collection of problems that he proposed and brilliantly solved, that neither rational nor simple quadratic irrationalities (of the type , etc.) could be solutions of the equation x³ + 2x² + 10x = 20.

WESTERN EUROPE IN THE 16TH CENTURY. The 16th century was the first century of Western Europe’s superiority over ancient Greece and the Orient. This was true in astronomy (N. Copernicus’ discovery) and mechanics (the first investigations of Galileo appeared at the end of the century). On the whole, this was also the case in mathematics despite the fact that in some areas European science still lagged behind the achievements of 15th-century Central Asian mathematicians and that in actuality the major new ideas that determined the subsequent development of European mathematics were to appear only in the 17th century. However, in the 16th century it seemed that a new era was beginning in mathematics with the discovery of algebraic solutions of cubic equations (S. dal Ferro, c. 1515, and later and independently N. Tartaglia, c. 1530) and quartic equations (L. Ferrari, 1545), which for centuries had been considered impossible. G. Cardano investigated cubic equations and discovered the irreducible case in which the real roots of an equation are expressed in terms of complex numbers. This forced Cardano to acknowledge, albeit quite tentatively, the advantage of calculations involving complex numbers. Algebra was further developed by F. Vieta (Viete), the founder of true algebraic literal calculation (1591; before him letters were used only for unknowns). The concept of perspective, which was developing in geometry even before the 16th century, was set forth by the German artist A. Diirer (1525). In 1585, S. Stevin worked out the rules of arithmetic operations for decimals.

RUSSIA UNTIL THE 18TH CENTURY. Russian mathematical education in the ninth to 13th centuries was on the level of the most culturally advanced countries of the Orient and Western Europe. However, it was set back for a long time by the Mongolian invasion. In the 15th and 16th centuries, society’s requirements for mathematical knowledge increased significantly in connection with the strengthening and economic growth of the Russian state. Numerous manuscript handbooks of arithmetic and geometry containing rather extensive information necessary for practical activities (trade, taxation, construction, the use of artillery) appeared in the late 16th century and particularly in the 17th century.

A system of number symbols resembling the Greco-Byzantine system and based on the Slavic alphabet came into use in Ancient Rus’. Slavic numeration appears in Russian mathematical literature until the early 18th century, but beginning in the late 16th century this numeration was gradually replaced by the decimal system now in use.

The oldest known mathematical work dates from 1136 and was written by the Novgorod monk Kirik. It is devoted to arithmetic-chronological calculations, which show that at this time it was already known in Rus’ how to solve the complex problem of determining the date of Easter each year, a problem that reduces to solving indeterminate linear equations over the integers. The arithmetic manuscripts of the late 16th century and of the 17th century contain, in addition to a description of Slavic and Arabic numeration, arithmetic operations involving positive integers and a detailed exposition of the rules of operation on fractions, the rule of three, and a solution of linear equations in one unknown using the rule of false position. For easier application of the general rules, many practical examples were discussed in the manuscripts and a counting board—the prototype of the Russian abacus—was described. The first, arithmetic, part of the celebrated Arithmetic of L. F. Magnitskii (1703) was organized in the same manner. Manuscripts on geometry, which for the most part pursued practical goals, contained descriptions of rules, often approximate, for determining areas and volumes and employed the properties of similar triangles and the Pythagorean theorem.

Period of the development of the mathematics of variable quantities. An essentially new period of mathematical development begins with the 17th century. “The Cartesian variable quantity was a turning point in mathematics. Because of this, motion and thus dialectics entered mathematics and for the same reason the differential and integral calculus became immediately necessary” (F. Engels in K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 573). The range of quantitative relations and spatial forms now studied by mathematics is no longer exhausted by numbers, quantities, and geometric figures. This was due chiefly to the explicit introduction into mathematics of the ideas of motion and change. The idea of the relation between quantities (for example, the value of a sum depends on the values of the addends) was already implicit in algebra. However, in order to encompass quantitative relations in the process of their change, the very relations between quantities had to be made an independent subject of study. Therefore, the concept of function, which subsequently played the same role of a primary and independent subject of study as the concepts of quantity or number had previously, moved into the foreground. The study of variables and of functional relations led to the fundamental concepts of mathematical analysis, which in explicit form introduce the idea of the infinite into mathematics, and to the concepts of limit, derivative, differential, and integral. Infinitesimal analysis was created, primarily in the form of the differential calculus and the integral calculus, which makes it possible to relate finite changes of variables to their behavior in the immediate vicinity of the individual values that they assume. The fundamental laws of mechanics and physics were written in the form of differential equations, and the problem of integrating these equations was advanced as one of the most important tasks of mathematics. The search for unknown functions defined by different types of conditions is the subject of the calculus of variations. Thus, in addition to equations in which the unknowns are numbers, there appeared equations in which the unknowns to be determined are functions.

The subject studied by geometry likewise expanded considerably with the penetration into geometry of the ideas of motion and transformation of figures. Geometry began to study motion and transformations as such. For example, in projective geometry projective transformations of a plane or space are themselves one of the main objects of study. The conscious development of these ideas dates only to the late 18th century and the early 19th. Much earlier, with the creation of analytic geometry in the 17th century, the relationship of geometry to all other mathematics changed fundamentally: a universal method of translating the problems of geometry into the language of algebra and of analyzing and solving them by purely algebraic and analytic methods was found, and the possibility emerged of representing (illustrating) algebraic and analytic facts geometrically, for example, by graphic representation of functional relations.

In the 17th and 18th centuries algebra was largely devoted to consequences stemming from the possibility of studying the left-hand side of the equation P(x) = 0 as a function of the variable x. This approach made it possible to study the problem of the number of real roots and to develop methods of separating and approximately calculating these roots. In the complex domain this approach led the French mathematician J. D’Alembert to a proof of the fundamental theorem of algebra, which states that every algebraic equation has at least one root. Although not strictly rigorous, this proof was nevertheless sufficiently convincing for 18th-century mathematicians. Also significant in the 17th and 18th centuries were the achievements of “pure” algebra, which does not require the concepts of continuous change of quantities that were borrowed from analysis. (Suffice it to mention here the solution of arbitrary systems of linear equations by means of determinants, and the elaboration of the theory of the divisibility of polynomials and the elimination of unknowns.) However, the conscious separation of purely algebraic facts and methods from the facts and methods of mathematical analysis was typical only of the second half of the 19th century and of the 20th century. In the 17th and 18th centuries, algebra was construed largely as the first chapter of analysis, in which one is limited to algebraic relations and equations, in distinction to analysis, where one investigates arbitrary relations between quantities and the solution of arbitrary equations.

The scientists of the advanced Western European countries, above all I. Newton and G. von Leibniz, endeavored in the 17th century to create the new mathematics of variable quantities. In the 18th century the St. Petersburg Academy of Sciences, where many prominent foreign mathematicians (L. Euler, D. Bernoulli) worked, became one of the main centers of mathematical investigations, and the Russian mathematical school, which produced brilliant investigations in the early 19th century, gradually took shape.

17TH CENTURY. The new stage of mathematical development described above was closely linked with the creation in the 17th century of mathematical natural science, whose goal is the elucidation of the course of individual natural phenomena in terms of general, mathematically formulated natural laws. Throughout the 17th century truly profound and extensive mathematical investigations were conducted only in two fields of the natural sciences, namely mechanics and optics. In mechanics, Galileo discovered the laws of falling bodies (1632-38); J. Kepler, the laws of planetary motion (1609-19); and Newton, the law of universal gravitation (1687). In optics, Galileo (1609) and Kepler (1611) constructed telescopes; Newton developed optics on the basis of the corpuscular theory, and C. Huygens and R. Hooke on the basis of wave theory. Nevertheless, 17th-century rationalist philosophy advanced the idea of the universality of the mathematical method (Descartes, B. Spinoza, Leibniz), which imparts a certain brilliance to this essentially philosophical era in the development of mathematics.

Serious new mathematical problems were posed in the 17th century by navigation (the necessity of improving clock-making and of designing accurate chronometers) and by cartography, ballistics, and hydraulics. Seventeenth-century authors appreciated and insisted on emphasizing the great practical importance of mathematics. By relying on its close connection with natural science, 17th-century mathematics was able to rise to a new stage of development. New concepts, which did not fit into the old formal-logical categories of mathematics, were justified since they corresponded to the relationships of the real world. For example, the “reality” of the concept of the derivative stemmed from the “reality” of the concept of velocity in mechanics; therefore, the problem was not whether this concept could be justified logically but only how this was to be done.

The mathematical achievements of the 17th century began with the discovery of logarithms by J. Napier, who published his tables in 1614. In 1637, Descartes published Geometry, which contains the foundations of coordinate geometry and a classification of curves in which curves were subdivided into algebraic and transcendental. The real roots of an equation of any degree were investigated in algebra in close relation to the possibility of representing the roots of an equation P(x) = 0 by the points of intersection of the curve y = P(x) with the axis of the abscissas (Descartes, Newton, M. Rolle). P. Fermat’s investigations of maxima and minima and his search for tangents to curves in essence contained the methods of the differential calculus, but the methods themselves had not yet been isolated or developed. Another source of infinitesimal analysis was the development of the method of “indivisibles” by Kepler (1615) and B. Cavalieri (1635), both of whom applied this method to the determination of the volumes of solids of revolution and other problems. Thus, the foundations of the differential calculus were essentially created in geometric form.

A parallel development was the study of infinite series. The properties of very simple series, beginning with the geometric progression, were studied by J. Wallis (1685). N. Mercator (1668) expanded In (1 + x) in a power series. Newton obtained (1665-69) the binomial formula for any exponent and the power series expansions of the functions e^x, sin x, and arc sin x. Nearly all 17th-century mathematicians, for example, Wallis, Huygens, Leibniz and Jakob Bernoulli, contributed to the subsequent development of the concept of infinite series.

With the creation of the coordinate method and the dissemination of the concept of directed mechanical quantities (velocity, acceleration), the concept of negative number became clear and acquired intuitive content. On the other hand, complex numbers remained a by-product of the algebraic apparatus, and for the most part continued to be merely an object of unproductive disputes.

The discovery of the differential and integral calculus in the proper sense of the terms dates to the last third of the 17th century. Leibniz, who gave an extensive exposition of the fundamental ideas of the new calculus in articles published between 1682 and 1686, holds priority with respect to this discovery insofar as publication is concerned. But insofar as the time of obtaining the actual results is concerned, there is every reason to believe that priority rests with Newton, who arrived at the fundamental ideas of the differential and integral calculus in 1665-66. In 1669, the manuscript of Newton’s On Analysis by Means of Equations was submitted to the British mathematicians I. Barrow and J. Collins and was acclaimed by British mathematicians. The Method of Fluxions, in which Newton gave a definitive systematic exposition of his theory, was written in 1670-71 but was published only in 1736. Leibniz, however, began his investigations of infinitesimal analysis only in 1673.

Newton and Leibniz were the first to consider in general form the operations of the differentiation and integration of functions, which were basic to the new calculus. They established the relation between these operations (the Newton-Leibniz formula) and worked out a general uniform algorithm for them. However, Newton and Leibniz differed in their approach. The concepts of “fluent” (a variable quantity) and its “fluxion” (rate of change) are the basic concepts for Newton. To the direct problem of finding fluxions and the relations between fluxions for given fluents (differentiation and construction of differential equations), Newton contrasted the inverse task of finding fluents from given relations between fluxions, that is, the general problem of integrating differential equations. The problem of finding the antiderivative appears here as a particular case of integration of the differential equation dy/dx = f(x). This point of view was quite natural for Newton as the creator of mathematical physics: his calculus of fluxions was simply a reflection of the idea that the primary laws of nature are expressed by differential equations, and prediction of the course of the processes described by these equations requires their integration.

Leibniz focused on the problem of the transition from the algebra of the finite to the algebra of infinitesimals. He perceived the integral as the sum of an infinitely large number of infinitesimals, and regarded differentials—infinitesimal increments of variable quantities—as the fundamental concept of the differential calculus. In contrast, Newton, after introducing the corresponding concept of “moment,” strove to move away from it in his later works.

The publication of Leibniz’ works inaugurated a period of intensive collective development of the differential and integral calculus, the integration of differential equations, and the geometric applications of analysis by Jakob Bernoulli, Johann Bernoulli, G. L’Hôpital, and other mathematicians of Continental Europe. The modern approach to the study of mathematics whereby the results obtained are published immediately in journals and are used by other scientists in their own research soon after publication developed at this time.

In addition to analytic geometry, differential geometry developed in close connection with algebra and analysis. The foundations for the subsequent development of pure geometry, chiefly in the sense of the creation of the basic concepts of projective geometry, were laid in the 17th century. Also of particular importance were the investigations in number theory (B. Pascal, Fermat) and the development of the fundamental concepts of combinatorial analysis (Fermat, B. Pascal, Leibniz). The 17th century also saw works on probability theory (Fermat, B. Pascal), which culminated at the end of the century in a result of fundamental importance—the discovery of the simplest form of the law of large numbers (Jakob Bernoulli, published 1713). We should also note the construction by B. Pascal (1641) and Leibniz (1673-74) of the first calculating machines, which, incidentally, long had no practical consequences.

18TH CENTURY. In the early 18th century the general mode of mathematical research gradually changed. The successes of the 17th century, brought about primarily by methodological innovation, stemmed from the boldness and depth of the general ideas. This brought mathematics closer to philosophy. By the beginning of the 18th century, the new fields of mathematics created in the 17th century reached a level at which further advancement required skill in the use of the mathematical apparatus and inventiveness in the finding of surprising sophisticated ways of solving difficult problems. Of the two greatest mathematicians of the 18th century, L. Euler and J. Lagrange, the former is the more brilliant representative of this virtuosic trend. However, Lagrange, while perhaps yielding to Euler in the number and diversity of problems solved, combined masterful techniques with the broad generalizing concepts that were typical of the French mathematical school of the second half of the 18th century, which was closely connected with the extensive philosophical movement of French enlighteners and materialists. Fascination with the extraordinary power of the apparatus of mathematical analysis naturally led to the belief in the possibility of its purely automatic development and in the infallibility of mathematical calculations even when the symbols involved were meaningless. During the period of creation of infinitesimal analysis, mathematicians relied on ideas that carried strong intuitive conviction even though they were unable to provide logical justification for these ideas. Now they openly advocated the right to manipulate according to conventional rules mathematical expressions lacking direct meaning without considering the element of intuitive appeal and without providing any logical justification for the legitimacy of such manipulations. Among the older generation, Leibniz, who in reference to the integration of rational fractions by expanding them into imaginary expressions spoke in 1702 of the “marvelous intervention of the ideal world,” became increasingly inclined in this direction. The more realistically minded Euler did not speak of miracles but interpreted the legitimacy of operations involving imaginary numbers and divergent series as an empirical fact that was confirmed by the correctness of results obtained by similar transformations. Although work was initiated to provide rational clarification of the foundations of infinitesimal analysis, systematic logical substantiation of analysis was accomplished only in the 19th century.

The most prominent mathematicians of the 17th century often were also philosophers or experimental physicists; in the 18th century, however, mathematics became an independent profession. Eighteenth-century mathematicians came from diverse segments of society. They early revealed their mathematical abilities and moved rapidly along in their academic careers. For example, Euler, the son of a preacher in Basel, at the age of 20 was invited to the St. Petersburg Academy of Sciences as an adjunct, at 23 became a professor there, and at 39 became chairman of the physicomathematical division of the Berlin Academy of Sciences. Lagrange, the son of a French official, was a professor in Turin at 19, and at 30 chairman of the physicomathematical division of the Berlin Academy of Sciences. P. Laplace, the son of a French peasant, was a professor at a military school in Paris at 22 and a member of the Paris Academy of Sciences at 36. Mathematical natural science (mechanics, mathematical physics) and the technical applications of mathematics remained in the sphere of activity of mathematicians. For example, Euler worked on problems in shipbuilding and optics, Lagrange created the foundations of analytic mechanics, and Laplace, who considered himself primarily a mathematician, was also a major astronomer and physicist.

In the 18th century many outstanding results were obtained in mathematics. Through the works of Euler, Lagrange, and A. Legendre, number theory became a systematic science. Lagrange gave (1769, published 1771) a general solution of indeterminate quadratic equations. Euler established (1772, published 1783) the law of quadratic reciprocity. He also used the zeta function in the study of prime numbers, thus giving rise to analytic number theory.

By means of expansions into continued fractions, Euler proved (1737, published 1744) that e and e² are irrational and J. Lambert demonstrated (1776, published 1768) that π is irrational. In algebra, G. Cramer introduced (1750) determinants for the solution of systems of linear equations. Euler considered it an empirically established fact that every algebraic equation has a root of the form . The conviction grew that imaginary expressions, not only in algebra but also in analysis, in general are always reducible to the form D’Alembert proved (1748) that the absolute value of a polynomial cannot have a nonzero minimum (the D’Alembert lemma), considering this as proof that every algebraic equation has a root. The formulas of A. De Moivre and of Euler relating the exponential and trigonometric functions of a complex variable led to further expansion of the applications of complex numbers in analysis. Newton, J. Stirling, Euler, and Laplace laid the foundations of the calculus of finite differences. B. Taylor discovered (1715) a formula for expanding an arbitrary function into a power series.

For 18th-century mathematicians, especially Euler, series became one of the most powerful and flexible tools of analysis. The serious study of the condition for the convergence of series was undertaken by D’Alembert. Euler, Lagrange, and particularly Legendre laid the foundations for the study of elliptic integrals—the first type of nonelementary function to be subjected to an intensive study.

Considerable attention was devoted to differential equations. In particular, Euler devised (1739, published 1743) the first method of solving a linear differential equation of any order with constant coefficients, D’Alembert examined systems of differential equations, and Lagrange and Laplace developed the general theory of linear differential equations of any order. Euler, G. Monge, and Lagrange laid the foundations of the general theory of first-order partial differential equations; the same was done for second-order equations by Euler, Monge, and Laplace. Of particular interest is the introduction into analysis of the expansion of functions into trigonometric series since it led to a controversy between Euler, D. Bernoulli, D’Alembert, Monge and Lagrange which centered on the concept of function. This controversy prepared the groundwork for the fundamental results of the 19th century concerning the relationship between functions defined by means of analytic expressions and functions defined in an arbitrary manner. Finally, a new branch of analysis—the calculus of variations—was created by Euler and Lagrange in the 18th century. On the basis of various achievements of the 17th and 18th centuries, De Moivre, Jakob Bernoulli, and Laplace formulated the basic principles of probability theory.

In geometry Euler brought the system of elementary analytic geometry to its culmination. Euler, A. Clairaut, Monge, and J. Meusnier laid the groundwork for the differential geometry of space curves and surfaces. Lambert developed the theory of perspective, and Monge gave descriptive geometry its final form.

The survey presented above shows that 18th-century mathematics, while proceeding from the ideas of the 17th century, far surpassed previous centuries in the scale of work. This flowering of mathematics was primarily connected with the activities of various academies, and universities played a lesser role. The remoteness of the major mathematicians from university teaching was compensated for by the energy with which, beginning with Euler and Lagrange, they all wrote textbooks and extensive treatises on individual topics.

All the branches of mathematical analysis that were created in the 17th and 18th centuries continued to develop intensively in the 19th and 20th centuries. The range of their applications to the problems advanced by natural science and technology also expanded greatly. However, in addition to this quantitative growth, a number of essentially new features can be observed in the development of mathematics in the late 18th century and in the early 19th century.

Expansion of the subject of mathematics. The enormous amount of factual material that was accumulated in the 17th and 18th centuries necessitated a deeper logical analysis and consolidation of the material from new viewpoints. Among the new developments at the turn of the 19th century were discovery and introduction into use of the geometric interpretation of complex numbers by the Danish surveyor C. Wessel (1799), and the French mathematician J. Argand(1806), N. Abel’s proof (1824) of the unsolvability of a general fifth-degree algebraic equation by radicals, development by A. Cauchy of the foundations of the theory of functions of a complex variable and Cauchy’s works on the rigorous substantiation of infinitesimal analysis, the creation of non-Euclidean geometry by N. I. Lobachevskii (1826, published 1829-30) and J. Bolyai (1832), and the works of K. Gauss (1827) on the intrinsic geometry of surfaces.

The close relation of mathematics to natural science now acquired more complex forms. Major new theories arose not only as a result of the direct requirements of natural science and technology but also from the intrinsic requirements of mathematics itself. Such was the development of the theory of functions of a complex variable, which occupied the central position in all mathematical analysis in the first half of the 19th century. Lobachevskii’s “imaginary geometry” is another noteworthy example of a theory that arose as a result of the development within mathematics itself.

Still another example can be cited of how the reevaluation begun at the end of the 18th century and the first half of the 19th of previously obtained concrete mathematical facts from more general points of view found strong support in the second half of the 19th century and in the 20th in the new requirements of natural science. Group theory originated with Lagrange’s examination (1771) of groups of substitution in connection with the problem of the solvability of algebraic equations of higher degrees by radicals. E. Galois (1830-32, published 1832, 1846), using the theory of substitution groups, gave a conclusive answer to the question of the conditions for the solvability by radicals of algebraic equations of any degree. In the mid-19th century A. Cayley gave a general “abstract” definition of group. S. Lie worked out the theory of continuous groups on the basis of general problems of geometry. Only after this did E. S. Fedorov (1890) and the German scientist A. Schonflies (1891) establish that the structure of crystals conforms to theoretical group principles; still later, group theory became a powerful research tool in quantum physics.

The development of vector calculus and tensor calculus was more directly related to the requirements of mechanics and physics. The extension of vector and tensor concepts to infinite-dimensional quantities took place within the framework of functional analysis and was closely connected to the requirements of modern physics.

Thus, the range of quantitative relations and spatial forms studied by mathematics expanded greatly as a result both of the requirements of mathematics itself and of the new needs of natural science. It included such problems as the relations among the elements of an arbitrary group, vectors, and operators in function spaces and the various forms of spaces of arbitrary dimension. With this broad understanding of the terms “quantitative relations” and “spatial forms,” the definition of mathematics given at the beginning of the article is also applicable to the new, modern stage of the development of mathematics.

An essential innovation in 19th-century mathematical development was that problems related to the necessary expansion of the range of quantitative relations and spatial forms to be studied became the objects of conscious and active interest on the part of mathematicians. Whereas previously the introduction, for example, of negative and complex numbers and the precise formulation of the rules governing operations involving such numbers required much work and effort, mathematics now required the development of techniques for the planned creation of new geometric systems and new “algebras” with noncommutative or even nonassociative multiplication needed in one or another connection. For example, the creation of a new “algebra” with new rules of operation in order to analyze and synthesize some new type of relay-contact circuits is now a matter of everyday scientific technical practice and no longer generates particular surprise. It would be difficult to overrate the importance of the reorganization of the entire frame of mathematical thinking that had to take place during the 19th century for this change in attitude to set in. From this conceptual viewpoint, the discovery of Lobachevskii’s non-Euclidean geometry was the most significant discovery of the early 19th century. It was precisely on the basis of this geometry that belief in the inviolability of the axioms that had been elucidated by a millennium of mathematical development was overcome and the possibility of creating essentially new mathematical theories by selectively setting aside previously imposed restrictions that had no intrinsic logical necessity was understood; it was also perceived that with time such an abstract theory may find increasingly broad and quite specific applications.

The extraordinary expansion of the subject of mathematics in the 19th century resulted in increased attention to the problems of the “substantiation” of mathematics, that is, to a critical reevaluation of initial assumptions (axioms), to the construction of a rigorous system of definitions and proofs, and to a critical examination of the logical means used in these proofs. Works on the rigorous substantiation of various branches of mathematics rightly occupied a significant place in 19th- and 20th-century mathematics. The results obtained in the substantiation of the foundations of analysis (the theory of real numbers, the theory of limits, and the rigorous substantiation of all methods of the differential and integral calculus) are now given with greater or lesser completeness in most textbooks (even purely practical textbooks). However, there are still instances where the rigorous substantiation of a mathematical theory arising from practical requirements is delayed. As late as the turn of the 20th century this was the case with operational calculus, which found extremely broad applications in mechanics and electrical engineering. A logically irreproachable exposition of mathematical probability theory was given only after a long delay. And even now there is no rigorous substantiation of many mathematical methods that are used extensively in modern theoretical physics, where many valuable results are obtained by “illegal” mathematical procedures.

The standards of mathematical rigor currently observed by practicing mathematicians took shape only toward the end of the 19th century. These standards are based on the set-theoretic conception of the structure of any mathematical theory. From this point of view any mathematical theory deals with one or several sets of objects that are interconnected by certain relations. All formal properties of these objects and relations necessary for the development of the theory are stated in the form of axioms that do not touch upon the specific nature of the objects and relations themselves. A theory is applicable to any system of objects connected by relations that satisfy its fundamental system of axioms. Accordingly, a theory is considered to be logically rigorous if it employs only those properties of the objects or of the relations that are mentioned in the axioms and if all additional objects or relations that are introduced in the course of the development of the theory are defined formally in terms of the axioms.

Mathematical logic clarifies another aspect of the structure of any mathematical theory. A system of axioms in the (set-theoretic) sense outlined above now merely imposes “outer” limits on the domain of applications of a given mathematical theory. It indicates the properties of the system of objects and relations under study but provides no clues concerning the logical means by which this mathematical theory may be developed. For example, the properties of the system of natural numbers can be determined up to the accuracy of isomorphism by using a very simple system of axioms. Nevertheless, it often proves to be difficult to solve problems the answer to which in principle is uniquely predetermined by the adoption of this system of axioms: number theory abounds in problems that were posed and very simply formulated long ago but that have not yet been solved. The question naturally arises whether this is so because the solution of a few simply formulated problems of number theory requires a very long chain of arguments composed of elementary links that are known and already have come into use, or because essentially new, not previously used, methods of logical derivation are required to solve certain problems of number theory.

Modern mathematical logic has provided a specific answer to this question: no single deductive theory can exhaust the diversity of the problems of number theory. More accurately, even within the theory of natural numbers it is possible to formulate a sequence of problems p₁,p₂, …, p_n, … such that for any deductive theory there will be found among these problems one that is unsolvable within the limits of the given theory (K. Gödel). Here, “deductive theory” implies a theory that is developed from a finite number of axioms by constructing chains of arguments of arbitrary length consisting of links belonging to a finite number of elementary methods of logical deduction that are fixed for the given theory.

Thus it was found that the concept of a mathematical theory in the sense of a theory encompassed by a single system of set-theoretic axioms is substantially broader than the logical concept of a deductive theory: unlimited use of essentially new methods of logical arguments lying outside any finite set of standardized methods is inevitable even in the development of the arithmetic of natural numbers.

All results that can be obtained within the framework of a single deductive theory can also be obtained by calculations based on a fixed set of rules. If a rigorously defined prescription is given for the solution of some class of problems, then we speak of a mathematical algorithm. From the very creation of an adequately developed system of mathematical symbols, the problem of constructing sufficiently general and at the same time brief algorithms has occupied a major place in the history of mathematics. But only in the last decades have the general theory of algorithms and the theory of “algorithmic solvability” of mathematical problems been developed. The practical prospects for these theories are apparently great, particularly in connection with the current development of computing techniques that make it possible to replace complex mathematical algorithms with computer programs.

History of mathematics in the 19th century and early 20th century. FIRST HALF OF THE 19TH CENTURY. A considerable expansion of the domain of applications of mathematical analysis took place in the early 19th century. In physics before this, mainly mechanics and optics required a large mathematical apparatus; but now such an apparatus also became a requirement of electrodynamics, the theory of magnetism, and thermodynamics. Very important branches of the mechanics of continuous media were developed extensively; only the hydrodynamics of an incompressible ideal fluid had been created as early as the 18th century by D. Bernoulli, Euler, D’Alembert, and Lagrange. The mathematical requirements of technology also grew rapidly. At the start of the 19th century these included problems of thermodynamics of steam engines, technical mechanics, and ballistics. The theory of partial differential equations and especially the potential theory were developed more intensively as the fundamental apparatus for the new branches of mechanics and mathematical physics. Most of the leading analysts of the first half of the century—Gauss, J. Fourier, S. Poisson, Cauchy, P. Dirichlet, G. Green, and M. V. Ostrogradskii—worked in this direction. Ostrogradskii laid the foundations of the calculus of variations for functions of several variables. Vector analysis arose as a result of investigations of the equations of mathematical physics by G. Stokes and other British mathematicians.

Despite the mechanistic conviction of the possibility of describing all natural phenomena with differential equations that prevailed in natural science at the beginning of the 19th century, probability theory was further developed under the pressure of practical requirements. Laplace and Poisson created a powerful new analytic apparatus for this purpose. P. L. Chebyshev gave a rigorous justification of the elements of probability theory and proved his celebrated theorem (1867) combining all previously known forms of the law of large numbers in one general formulation.

As was indicated above, in addition to working on the new problems posed by natural science and technology, mathematicians focused attention on problems of the rigorous substantiation of analysis (Cauchy, 1821, 1823) from the very start of the 19th century. Lobachevskii (1834) and, later, Dirichlet (1837) clearly formulated the definition of function as a wholly arbitrary correspondence. In 1799, Gauss published the first proof of the fundamental theorem of algebra, cautiously formulating the theorem, however, in purely real terms (the decomposability of a real polynomial into real factors of the first and second degrees). Only much later (1831) did Gauss clearly set forth the theory of complex numbers.

The theory of functions of a complex variable arose on the basis of a clear understanding of the nature of complex numbers. Gauss possessed great knowledge in this field but published almost nothing. The general foundations of the theory were laid by Cauchy, and the theory of elliptic functions was developed by Abel and G. Jacobi. Already at this early stage, in contrast to the purely algorithmic approach of the 18th century, attention was focused on clarifying the distinctiveness of the behavior of functions in the complex domain and the fundamental geometric principles that predominate here (beginning with the dependence discovered by Cauchy of the radius of convergence of a Taylor’s series on the location of singular points). This somewhat “qualitative” and geometric character of the theory of functions of a complex variable was intensified still further in the mid-19th century by B. Riemann. Here it turns out that the natural geometric carrier of a many-valued analytic function is not the complex plane but the Riemann surface that corresponds to the given function. K. Weierstrass achieved the same generality as Riemann but on the basis of analysis alone. However, the geometric ideas of Riemann subsequently dominated the entire approach to the theory of functions of a complex variable.

During the period of preoccupation with the theory of functions of a complex variable, Chebyshev was the most prominent investigator of the specific problems of the theory of functions in the real domain. The most explicit expression of this trend was the theory of best approximations, which was formulated (beginning in 1854) by Chebyshev on the basis of the requirements of the theory of mechanisms.

In algebra, after the aforementioned proof of the unsolvability by radicals of a fifth-degree general equation (P. Ruffini, Abel), Galois demonstrated that the Galois group of an equation is the key to its solvability by radicals. The general abstract study of groups was proposed by Cayley. It should be noted that even in algebra the importance of group theory was universally recognized only after the works of C. Jordan appeared in the 1870’s. The concept of the field of algebraic numbers, which led to the creation of a new science—the algebraic theory of numbers— also originated in the works of Galois and Abel. Work on the old problems of number theory, which were connected with the simplest properties of ordinary integers, also rose to an essentially new level in the 19th century. In 1801, Gauss worked out the theory of representation of numbers by quadratic forms, and in 1848 and 1850, Chebyshev obtained fundamental results on the density distribution of prime numbers in the sequence of natural numbers. In 1837, Dirichlet proved the theorem of the existence of an infinite number of primes in arithmetic progressions.

The differential geometry of surfaces was created by Gauss (1827) and K. M. Peterson (1853). As indicated above, the creation of non-Euclidean geometry by Lobachevskii was of fundamental importance to the development of new views on the subject of geometry. Projective geometry, which is also associated with significant changes in the old views of space, developed in parallel with and for a long time independently of non-Euclidean geometry (J. Poncelet, J. Steiner, K. von Staudt). J. Pliicker constructed a geometry by considering lines as the fundamental elements, and H. Grassmann created the affine and metric geometry of n-dimensional vector space.

Already in the Gaussian intrinsic geometry of surfaces, differential geometry was in essence freed from its supposedly unbreakable bond with Euclidean geometry: it is incidental to this theory that a surface lies in three-dimensional Euclidean space. On this basis, Riemann created (1854 published 1856) the concept of an n -dimensional manifold with a metric geometry defined by a differential quadratic form. This gave rise to the general differential geometry of n-dimensional manifolds. Riemann was also responsible for the first ideas on the topology of n -dimensional manifolds.

Late 19th century and early 20th century. It was only in the early 1870’s that F. Klein obtained a model of Lobachevskii’s non-Euclidean geometry that finally eliminated all doubts regarding its consistency. In 1872, Klein subordinated the study of the various “geometries” of spaces of different dimensions that had been constructed by that time to the study of the invariants of a given group of transformations. At the same time (1872), works on the substantiation of analysis acquired the necessary foundation in the form of the rigorous theory of irrational numbers (R. Dedekind, G. Cantor, and Weierstrass). The fundamental works of Cantor on the general theory of infinite sets were published between 1879 and 1884. Only then could the modern general concepts of the subject of mathematics, the structure of mathematical theory, the role of axiomatics, and so on be formulated. The extensive dissemination of these concepts required several more decades. The general recognition of the modern concept of the structure of geometry is usually associated with the publication in 1899 of D. Hibert’s Foundations of Geometry.

The further investigations into the foundations of mathematics concentrated on surmounting the logical difficulties that arose in general set theory and studying the structure of mathematical theory and the methods of constructive solution of mathematical problems by the procedures of mathematical logic. These investigations grew into a large independent branch of mathematics—mathematical logic. The foundations of mathematical logic were created in the 19th century by G. Boole, P. S. Poretskii, F. Schroder, G. Frege, and G. Peano. The beginning of the 20th century saw major achievements in this field (Hilbert’s theory of proofs, the intuitionist logic created by L. Brouwer and his adherents).

At the end of the 19th century and the beginning of the 20th, all branches of mathematics, beginning with the oldest—number theory—underwent extraordinary development that surpassed previous periods not only in the number of works but also in the perfection and power of the methods and the conclusiveness of the results. E. Kummer, L. Kronecker, Dedekind, E. I. Zolotarev, and Hilbert laid the foundations of the modern algebraic theory of numbers. In 1873, C. Hermite proved that e is a transcendental number and in 1882, the German mathematician F. Lindeman proved that π is transcendental. J. Hadamard (1896) and C. J. de la Vallée Poussin (1896) completed Chebyshev’s investigation of the law of diminishing density distribution of the primes in the sequence of natural numbers. H. Minkowski introduced geometric methods into number-theoretic investigations. In Russia number theory after Chebyshev was developed brilliantly by A. N. Korkin, G. K. Voronoi, and A. A. Markov, as well as by E. I. Zolotarev.

The focus of algebraic investigations shifted to new domains, such as the theory of groups, fields, and rings. Many of these branches of algebra found wide application in natural science: in particular, group theory found application in crystallography and later in the problems of quantum physics.

At the boundary between algebra and geometry, S. Lie created (beginning in 1873) the theory of continuous groups, whose methods later penetrated into newer domains of mathematics and natural science.

Elementary and projective geometry attracted the attention of mathematicians chiefly from the standpoint of the study of their logical and axiomatic foundations. However, the greatest scientific efforts were concentrated on differential and algebraic geometry. The differential geometry of Euclidean three-dimensional space was systematically developed by E. Beltrami and J. G. Darboux, among others. Later, the differential geometry of various groups of transformations broader than Euclidean motion developed rapidly, especially the differential geometry of h-dimensional spaces. Stimulated by the emergence of the general theory of relativity, geometric investigations in this direction were initiated by T. Levi-Civita, E. Cartan, and H. Weyl.

Because of the development of more general viewpoints of set theory and the theory of functions of a real variable, the theory of analytic functions at the end of the 19th century lost its previous position as the core of all mathematical analysis. However, it continued to be developed intensively both in accordance with its intrinsic requirements and because of its newly discovered connections with other branches of analysis and natural science. Especially significant in this last direction was the elucidation of the role of conformal mappings in the solution of boundary value problems for partial differential equations (for example, Dirichlet’s problems for the Laplace equation), in the study of planar flows of an ideal fluid, and in problems of the theory of elasticity.

Klein and H. Poincare created the theory of automorphic functions in which Lobachevskii’s geometry found remarkable applications. C. E. Picard, Poincare, Hadamard, and E. Borel worked out in depth the theory of entire functions, which in particular made it possible to derive the theorem of the density distribution of primes that was mentioned above. Poincare, Hilbert, and others developed the geometric theory of functions and the theory of Riemann surfaces. Conformal mappings found application in aeromechanics (N. E. Zhukovskii, S. A. Chaplygin).

As a result of systematic development of mathematical analysis, a new branch of mathematics—the theory of functions of a real variable—arose on the basis of the rigorous arithmetic theory of irrational numbers and set theory. Whereas previously only functions arising “naturally” from various special problems were studied systematically, interest in the complete elucidation of the true extent of the general concepts of analysis is typical of the theory of functions of a real variable. (At the very start of the theory’s development B. Bolzano and later Weierstrass found, for example, that a continuous function may not have a derivative at any point.) Investigations in the theory of functions of a real variable led to general definitions of the concepts of the measure of a set, measurable functions, and integral, which play an important role in modern mathematics. The foundations of the modern theory of functions of a real variable were laid by mathematicians of the French school (Jordan, Borel, H. Lebesgue, R. Baire), and later the leading role was assumed by the Russian and Soviet school.

In addition to its immediate interest, the theory of functions of a real variable greatly influenced the development of many other branches of mathematics. Its methods proved to be essential in the construction of the foundations of functional analysis. Although the methods of functional analysis developed under the influence of the theory of functions of a real variable and set theory, the content and nature of the problems of functional analysis border directly on classical analysis and mathematical physics. Functional analysis is virtually indispensable (chiefly in the form of the theory of operators) in quantum physics. The first conscious isolation of functional analysis as a special branch of mathematics was made by V. Volterra in the late 19th century. The calculus of variations and the theory of integral equations, whose systematic development was also begun by Volterra and continued by E. Fredholm, were now perceived as parts of functional analysis. The most important special case of operators in Hilbert space, whose primary role was brought out in the works of Hilbert on integral equations, was analyzed with particular intensity.

Most of the mathematical problems posed by natural science and technology reduce to the solution of differential equations, both ordinary (in the study of systems having a finite number of degrees of freedom) and partial differential equations (in the study of continuous media and in quantum physics). Therefore, all directions of investigations of differential equations were intensively developed during this period. The methods of operational calculus were created to solve complex linear systems. The method of parametric expansion was used widely in the study of nonlinear systems having small nonlinearity. The analytic theory of ordinary differential equations continued to be developed (Poincare and others). However, in the theory of ordinary differential equations, problems of the qualitative investigation of their solutions now attracted the greatest attention: the classification of singular points (Poincare and others) and problems of stability, which were studied in great depth by A. M. Liapunov.

The qualitative theory of differential equations served for Poincaré as a starting point for a broadly conceived continuation of the investigations, barely outlined by Riemann, into the topology of manifolds, particularly the study of the fixed points of continuous mappings of manifolds onto themselves. The “combinatorial,” “homologous,” and “homotopic” methods of modern topology originated here. Another direction arose in topology based on set theory and functional analysis and led to the systematic development of the theory of general topological spaces.

The theory of partial differential equations assumed an essentially new form at the end of the 19th century owing to intense interest in boundary value problems and the elimination of the restriction to analytic boundary conditions. The analytic theory, which originated with Cauchy, Weierstrass, and S. V. Kovalevskaia, did not lose its importance in the process. However, it recedes somewhat into the background, since it was found that it did not guarantee correctness in the solution of boundary value problems, that is, it did not guarantee the possibility of finding an approximate solution if the boundary conditions were known only approximately, although without this possibility a theoretical solution has no practical value. The picture is more complex than it appears from the standpoint of analytic theory: the boundary value problems that can be posed correctly for different types of differential equations vary. Direct recourse to the corresponding physical concepts (such as wave propagation, heat flow, and diffusion) proves to be the most reliable guide in selecting appropriate boundary value problems for each type of equation. The related transformation of the theory of partial differential equations primarily into the theory of equations of mathematical physics has great positive significance. Works on individual types of equations of mathematical physics rightly constitute a major portion of all mathematical output. Poincare, Hadamard, Lord Rayleigh, (J. W. Strutt), Lord Kelvin (W. Thompson), Hilbert, Liapunov, V. A. Steklov, and others worked on the equations of mathematical physics after Dirichlet and Riemann.

The methods of probability theory significantly supplement the methods of differential equations in the study of nature and the solution of technical problems. In the early 19th century, probabilistic methods were mostly used in the theory of artillery fire and the theory of errors. However, at the end of the 19th century and the start of the 20th, probability theory acquired many new applications as a result of the development of statistical physics and mechanics and the elaboration of the apparatus of mathematical statistics. The most profound theoretical investigations of the general problems of probability theory in the late 19th century and early 20th were accomplished by the Russian school (Chebyshev, Markov, Liapunov).

The practical use of the results of theoretical mathematical investigation necessitates that answers to problems be given in numerical form. However, even after exhaustive theoretical analysis this often proves to be a difficult task. At the end of the 19th century and the beginning of the 20th, the numerical methods of analysis grew into an independent branch of mathematics. Here, particular attention was paid to the methods of numerical integration of differential equations (the methods of Adams, Stormer, Runge, and others) and to quadrature formulas (Chebyshev, Markov, Steklov). The extensive development of works requiring numerical calculations led to the necessary compilation and publication of an ever-growing number of mathematical tables.

Intensive work on the problems of the history of mathematics was begun in the second half of the 19th century.

Based on A. N. KOLMOGOROV’S article in the 2nd edition of the Bol’shaia Sovetskaia Entsiklopediia

CONCLUSION. The foregoing discussed the main features of modern mathematics and listed the main directions of mathematical research according to branches as they had formed by the early 20th century. This division into branches largely still exists despite the rapid development of mathematics in the 20th century, especially since World War II. (The current state of mathematics and the achievements of various schools and individuals are reflected in the corresponding articles on number theory, algebra, logic, geometry, topology, function theory, functional analysis, differential equations, equations of mathematical physics, probability theory, mathematical statistics, and computer science.)

The requirements of the development of mathematics itself, the “mathematization” of various fields of science, the penetration of mathematical methods into many spheres of practical endeavor, and the rapid progress of computer technology have led mathematicians to redirect their efforts within the various branches of mathematics and to develop new mathematical disciplines, such as the theory of algorithms, information theory, game theory, operations research, and cybernetics.

Discrete, or finite, mathematics has arisen on the basis of the problems of the theory of control systems, combinatorial analysis, the theory of graphs, and coding theory.

The problems of the optimal (in various senses of the word) control of physical or mechanical systems, which are described by differential equations, have led to the creation of the mathematical theory of optimal control. The related problems of the control of objects in conflict situations have resulted in the creation and development of the theory of differential games.

Studies in control theory and in related fields, combined with the progress in computing technology, have provided the foundations for automating new spheres of human activity.

Soviet mathematics holds a leading place in world mathematical science. The works of Soviet scientists play a decisive role in many fields. The advances of prerevolutionary Russian mathematics were connected with the work of outstanding individuals and encompassed a narrow range of topics. There were scientific mathematical centers in only a few cities (St. Petersburg, Moscow, Kazan, Kharkov, Kiev). The main achievements were associated with the work of the St. Petersburg school. After the Great October Socialist Revolution a number of important new directions were followed in the Moscow mathematical school. Such universities as St. Petersburg, Moscow, and Kazan were the main centers of mathematical research in prerevolutionary Russia. Since 1917 the development of mathematical research and applied mathematics has been closely linked with the development and growth of the Academy of Sciences of the USSR. Studies in mathematics are largely concentrated in the institutes of mathematics of the Academy of Sciences of the USSR, the academies of sciences of the Union republics, and leading universities. During the Soviet era the rise of many scientific schools in cities where previously no significant work was done in mathematics has been an important feature of the mathematical development in the Soviet Union. Such are the mathematical schools in Tbilisi, Yerevan, Baku, Vilnius, Tashkent, Minsk, Sverdlovsk, and other cities and the scientific school that was created in the 1960’s at Akademgorodok, near Novosibirsk.

Mathematical investigations are conducted abroad both in institutes of mathematics and at universities (especially in capitalist countries).

The first mathematical societies, which now exist in many countries, appeared as early as the turn of the 18th century. Extensive reports on world achievements in mathematics and its applications, as well as communications on the most interesting works of individual scientists, are read and discussed at the international congresses of mathematicians, held every four years (since 1898). The organization and encouragement of international cooperation in mathematics, the preparation of scientific programs for the international congresses of mathematicians, and other functions are tasks of the International Mathematical Union. Articles about current mathematical investigations, as well as news about mathematical events in various countries, are published in mathematical journals, which now total more than 250 (as of the early 1970’s).

REFERENCES

Philosophy and history of mathematics
Kolmogorov, A. N. “Matematika.” In Bol’shaia Sovetskaia entsiklopediia, 2nd ed., vol. 26. Moscow, 1954.
Matematika, ee soderzhanie, melody i znachenie, vols. 1-3. Moscow, 1956.
Zeuthen, H. Istoriia matematiki v drevnosti i v srednie veka, 2nd ed.
Moscow-Leningrad, 1938. (Translated from French.)
Zeuthen, H. G. Istoriia matematiki v XVI i XVII vekakh, 2nd ed. Moscow-Leningrad, 1938. (Translated from German.)
Van der Waerden, B. L. Probuzhdaiushchaiasia nauka. Matematika Drevnego Egipta, Vavilona, Gretsii. Moscow, 1959. (Translated from Dutch.)
Kol’man, E. Istoriia matematiki v drevnosti. Moscow, 1961.
Iushkevich, A. P. Istoriia matematiki v srednie veka. Moscow, 1961.
Wileitner, H. Istoriia matematiki ot Dekarta do serediny XIX stoletiia, 2nd ed. Moscow, 1966. (Translated from German.)
Wileitner, H. Khrestomatiia po istorii matematiki, sostavlennaia po pervoistochnikam …, 2nd ed. Moscow-Leningrad, 1935. (Translated from German.)
Klein, F. Lektsii o razvitii matematiki v XIX stoletii, part 1. Moscow-Leningrad, 1937. (Translated from German.)
Rybnikov, K. A. Istoriia matematiki, vols. 1-2. Moscow, 1960-63.
Bourbaki, N. Ocherki po istorii matematiki. Moscow, 1963. (Translated from French.)
Struik, D. J. Kratkii ocherk istorii matematiki, 2nd ed. Moscow, 1969. (Translated from German.)
Istoriia matematiki s drevneishikh vremen do nachala XIX stoletiia, vols. 1-3. Moscow, 1970-72.
Cantor, M. Vorlesungen über Geschichte der Mathematik, 3rd ed., vols. 1-4. Leipzig, 1907-13.
Surveys and encyclopedias
Vinogradov, I. M. “Matematika i nauchnyi progress.” In Lenin i sovremennaia nauka, book 2. Moscow, 1970.
Matematika. [Collection of articles.] Moscow-Leningrad, 1932. (Nauka v SSSR za 15 let, 1917-1932.)
Matematika v SSSR za tridtsat’ let, 1917-1947. (Collection of articles.) Moscow-Leningrad, 1948.
Matematika v SSSR za sorok let, 1917-1957, vol. 1. (Collection of articles.) Moscow, 1959.
Weyl, H. “A Half-century of mathematics.” American Mathematical Monthly, 1951, vol. 58, no. 8, pp. 523-53.
Entsiklopediia elementarnoi matematiki, books 1-5. Moscow-Leningrad, 1951-66.
Weber, G., and I. Welstein. Entsiklopediia elementarnoi matematiki, 2nd ed., vols. 1-3. Odessa, 1911-14. (Translated from German.)
Enzyklopädie der mathematischen Wissenschaften, mil Einschluss ihrer Anwendunger, vols. 1-6. Leipzig, 1898-1934.
Ibid., 2nd., vol. 1. Leipzig, 1950—.
Encyclopedic des sciences mathematiques pures et appliquees, vols. 1-7. Paris-Leipzig, 1904-14.
Mathematik, 6th ed. Leipzig, 1971. (Kleine Enzyklopddie.)
Mathematisches Wörterbuch, 2nd ed. vols. 1-2. Berlin-Leipzig, 1962.

mathematics

[¦math·ə¦mad·iks] (science and technology) The deductive study of shape, quantity, and dependence; the two main areas are applied mathematics and pure mathematics, the former arising from the study of physical phenomena, the latter the intrinsic study of mathematical structures.

mathematics

1. a group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation 2. mathematical operations and processes involved in the solution of a problem or study of some scientific field
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noun a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

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