Note: the following has been abstracted from the Grolier Encyclopedia.

History of Mathematics

Mathematics is as old as civilization itself. By the Neolithic Period, as life became settled and villages began to appear, writing and counting became increasingly useful, if not necessary. With counting, the history of mathematics began. To count the passage of time, to weave intricate patterns in baskets or fabrics, and to apportion goods, crops, and livestock required a basic sense of arithmetic. Similarly, even in the most rudimentary cultures, the ability to decorate pottery with intricate designs, to distinguish constellations among the stars, or to arrange stones, obelisks, and tombs in ritualistic formations indicates a sense of space and geometry.

Egyptian, Babylonian, and Greek Mathematics

The earliest knowledge of mathematics is preserved in Egyptian papyruses, Babylonian cuneiform tablets, and Greek manuscripts. They indicate that the first mathematical concerns involved ARITHMEtic, Algebra, Geometry, and Trigonometry.

Arithmetic and Algebra

Among the earliest surviving mathematical texts are the famous Rhind papyrus (c.1750 BC) and the Golonishev papyrus. They reveal that the Egyptians used a decimal system; the unit was represented by a single line, and tens, hundreds, and thousands by hieroglyphic symbols. Arithmetic for the Egyptians was essentially additive; repeated doubling was used for multiplication. Except for the fraction 2/3, for which there was a special hieroglyph, all fractions were expressed as unit fractions of the form 1/n; a relatively simple fraction like 2/59 was always handled in the more complex though equivalent form 1/36 + 1/236 + 1/531 = 2/59.

Unit fractions were extremely cumbersome and would not have facilitated computation or the advance of arithmetic. Even so, Egyptian mathematics was apparently suited for applications in commerce and agriculture. To deal with such problems as storing grain or apportioning loaves of bread, the Egyptians even applied a rudimentary algebra, although it did not advance beyond the simple linear equation in one unknown.

In contrast, Babylonian arithmetic, which made use of a place-valued sexagesimal system, made certain computations, such as multiplication and division, considerably easier than the Egyptian method. The Babylonian base 60 is still used in measuring time (1 hour = 60 minutes, 1 minute = 60 seconds) and in measuring the degrees of a circle. The Babylonians also surpassed the Egyptians in their use of algebra. Cuneiform tablets from the Hammurabic period (about 1950 BC) reveal an ability to solve even quadratic and simple cubic equations.

Cuneiform tablets from later periods (about 600 BC to AD 300) also reflect the algebraic-arithmetic strengths of the Babylonians and show the advances they made in applying their mathematics to astronomy. To facilitate their complicated computations, tables for multiplication, reciprocals, and square roots were prepared, as well as tables for solving certain basic forms of equations.

The first major discoveries in Greek mathematics are ascribed to Pythagoras of Samos and his followers. Pythagorean arithmetic regarded numbers as sums of units or points and consequently has often been interpreted as an abstract form of atomism. A group centered around Zeno of Elea (5th century BC) opposed this Pythagorean atomism and formulated Zeno's Paradoxes. The ultimate effect of Zeno's arguments was to stress the need to study the definitions and foundations of mathematics more closely. The Pythagoreans also provided the first general proof of the so-called Pythagorean Theorem and discovered the existence of the irrational number, then known as an incommensurable magnitude.

The discovery of incommensurable magnitudes was very troubling to Pythagorean philosophy, which asserted that all magnitudes could be expressed in terms of integers or ratios of integers. The discovery made it clear that Pythagorean arithmetic was insufficient to express such geometric quantities as the diagonal of a square. Some have called this the first great crisis in the history of mathematics. Although Eudoxus of Cnidus later solved the dilemma by working out a theory of proportion, after Pythagoras's time Greek mathematics became essentially geometric rather than algebraic. This trend was reinforced by Plato, the teacher of Eudoxus, who regarded geometry as the model of certain reasoning.

Geometry and Trigonometry

The best-known mathematician of antiquity is Euclid (fl. 3d century BC), whose Elements of Geometry provides a systematic treatment of geometry in the form of definitions, axioms, postulates, and theorems. The progression of Euclid's arguments was taken as a model of logical rigor in ancient times; since then axiomatization has represented the highest form of scientific argument.

Because Euclid was a compiler and editor of existing ideas, the greatest mathematician of ancient times judged by the quality of his own original work was Archimedes (287-212 BC), who applied the method of exhaustion to determine rigorously the areas and volumes of numerous geometric figures. His younger contemporary, Apollonius of Perga, introduced the terminology for the Ellipse, the Hyperbola, and the Parabola and determined the specific properties of each type of curve, all of which are Conic Sections. The circle remained the most important curve, because astronomy in ancient times was based upon the geometry of perfect circles and uniform circular motions. Eudoxus, Aristarchus of Samos, Hipparchus of Nicaea, and Claudius Ptolemy made fundamental contributions in developing geometric models for planetary motions.

The last of the noteworthy geometers of ancient times was Pappus (fl. 3d century AD), whose Collection is a final synthesis of the work of his predecessors since Euclid.

In the Almagest Ptolemy introduced a kind of trigonometry in terms of a table of chords that was equivalent to a sine table. Not only did the Almagest contain formulas for the sines and cosines of the sums and differences of angles, it also provided the rudiments of Spherical Trigonometry, the study of triangles projected onto the surface of a sphere. Spherical trigonometry was given considerable attention by Menelaus (about AD 100), who was responsible for showing the significance of the arcs of great circles in dealing with such problems as spherical triangles.

Although some work in algebra was carried out in this period, ancient mathematicians never managed to amalgamate geometry and algebra in the way that mathematicians in the 16th and 17th centuries did when they created Analytic Geometry. Instead, the Greeks confined most of their work to geometry, the only branch of mathematics in which they could deal successfully with continuous magnitudes and with quantities that were incommensurable.

Islamic and Medieval Mathematics

After the fall of the Roman Empire in the western Mediterranean, the Greco-Roman tradition was maintained and transmitted to the Latin West by the Byzantines in Constantinople and by scholars in intellectual centers such as Isfahan, Jundishapir, and Baghdad. Islamic scholars also helped to spread mathematical discoveries made in India and China. Among the earliest examples of such activity was Al-Farazi's translation of the Hindu Siddhantas. Al-Khwarizmi's book on arithmetic contained a thorough exposition of the Hindu system of numbers and brought the idea of place-valued decimal notation to the West.

Because Arabic science was greatly interested in astronomy, largely for reasons related to astrology and the casting of horoscopes, considerable attention was devoted to Ptolemy's Almagest and to the advance of trigonometry. The astronomer Al-Battani advanced the study of spherical trigonometry and produced a table of cotangents for use in astronomical computations. The Persian poet Omar Khayyam devoted considerable time to the study of algebra and geometry. Al-Haitham, also known as Alhazen, applied geometry to the study of light, and Abul-Wafa advanced spherical trigonometry.

Like their Islamic counterparts, scholars in the Latin West first brought new life to their mathematics by translating basic works from Greek or Arabic sources, especially after Toledo fell into Christian hands in 1085. Leonardo Pisano, better known as Fibonacci, wrote his Liber Abaci (c.1202) based on bits of arithmetic and algebra that he had accumulated during his travels. He brought both the Arabic place-valued decimal system and the use of Arabic numerals to the Latin West.

The most striking advance in medieval mathematics was its innovative application of mathematics to physics, particularly to the problems of uniform and accelerated motion. In this respect the French scholastic Nicole Oresme and a group of mathematicians, including Thomas Bradwardine, at Merton College, Oxford, are noteworthy.

With the fall of Constantinople (1453) many Eastern scholars left for western Europe and brought knowledge of Greek manuscripts (and often the manuscripts themselves) with them. The astronomer Georg von Peuerbach began a translation of Ptolemy's Almagest that one of his students, Regiomontanus, completed. In Italy artists studied Vitrivius, applied geometry to the construction of great buildings, and pioneered in the mathematical study of perspective. Leonardo da Vinci, Leon Battista Alberti, and Piero della Francesca wrote treatises on the mathematics of perspective.

Early in the 16th century great progress was made in algebra. In Italy Niccolo Tartaglia and Scipione Ferro discovered general solutions for cubic equations, which were eventually published by Gerolamo Cardano in his Ars magna (1545). Cases involving imaginary roots were treated by Rafaello Bombelli in his Algebra (1572). In the late 16th century Francois Viete demonstrated the value of symbols by using plus (+) and minus (-) signs for operations, and letters to represent unknowns. Viete's innovative notation helped to make possible the great mathematical advances of the 17th century.

Throughout the 16th century, in keeping with the importance of bookkeeping, calculation, and the preparation of trigonometric and astronomical tables in an increasingly commercial and mercantile economy, mathematicians sought better notation and quicker methods. With this incentive a Flemish mathematician, Simon Stevin of Bruges, introduced decimal fractions, and John Napier of Scotland invented the Logarithm.

Mathematics in the 17th and 18th Centuries

The greatest discoveries made in mathematics during the 17th century were stimulated by the revolution brought about in physics and astronomy by Copernicus, Kepler, and Galileo. By showing how mathematics could be applied to the analysis of uniform and accelerated motion, Galileo proved that the paths followed by projectiles were always parabolic. The 17th century also saw the birth of analytic geometry, calculus, number theory and probability theory.

Analytic Geometry

In his Discours de la methode (Discourse on Method, 1637) the French philosopher Rene Descartes championed the logic of mathematics as a paradigm for reasoning. The innovative La Geometrie, an appendix to the Discours, brought algebra and geometry together in the form of analytic, or coordinate, geometry. Analytic geometry made possible for the first time the graphic representation of functions and allowed the properties of a wide variety of curves to be determined systematically and with considerable precision. Among the most important contributions of analytic geometry before the invention of the calculus was its role in helping to solve the so-called problem of tangents, that is, the determination of a line that lies tangent to a given curve at a given point. Such mathematicians as Isaac Barrow, Bonaventura Cavalieri, Pierre de Fermat, Christian Huygens, Descartes, and William Wallace all worked on the subject.

Calculus

By the middle of the century the essential components for calculus--analytic geometry, infinitesimal methods, the study of areas, and the problem of tangents--were all present. Within a decade of each other, Sir Isaac Newton and Gottfried Wilhelm von Leibniz discovered independently the fundamental features of the calculus, as well as the important reciprocity that made integrations far easier to calculate in terms of their inverses, known as differentiations.

Newton discovered his fluxional calculus in 1665-66, having studied Barrow's work and the Arithmetic of John Wallis, which had established a link between the quadrature of areas and the drawing of tangents to curves. Newton, who approached his calculus with applications to physics clearly in mind, thought of curves as generated by the motion of points and viewed his derivatives as velocities. By contrast, the calculus of Leibniz, developed between 1673 and 1676, was influenced by the geometry of Descartes, Huygens, and Pascal. The first account of Leibniz's Differential Calculus was published in 1684, followed by his Integral Calculus in 1686. Leibniz invented symbols of such operational utility that they quickly became the standard notation for the new calculus. The symbols for differentials (dx and dy) and the sign for the integral appear in papers that Leibniz published on the calculus.

Eighteenth-century mathematics was characterized by a further elaboration of the differential and integral calculus. In general, mathematicians abandoned Newton's fluxional calculus in favor of the new methods that Leibniz had presented. Jakob Bernoulli and his son Johann studied Leibniz's papers and assiduously developed the techniques of the calculus and the integration of ordinary Differential Equations. At mid-century Leonhard Euler, who also studied infinite series, worked out many basic theorems of the calculus and developed a theory of differential equations.

Greatly influenced by Euler's prodigious work, the French mathematician Joseph Louis Comte de Lagrange sought to improve the rigor of mathematics by avoiding intuition in favor of purely analytic proofs. His great textbooks of 1797 and 1801 sought to place the calculus on a rigorous basis by developing the subject algebraically, with no references to geometry or to intuitions of any sort. Like the critics of Newton and Leibniz who distrusted the concepts of the limit and of the infinitesimal "becoming zero," Lagrange rejected the method of limits and instead approached the study of functions through their Taylor Series, thus making the study of functions possible in purely algebraic terms. But difficulties with the Convergence of such series and the discovery of functions that had no Taylor series limited the success of his approach. By the end of the century Pierre Simon de Laplace had established the Newtonian world system with the best mathematics of the day in his Traite de mecanique celeste (Treatise on Celestial Mechanics, 1799-1825).

Number Theory and Probability

There was also considerable interest in number theory and probability in the 17th and 18th centuries. Pierre de Fermat excelled in both. His study of probability arose in the context of games of chance and eventually found important applications in business, notably in insurance and in the calculation of mortality tables.

19th- and 20th-Century Mathematics

In the 19th century mathematicians were for the first time most commonly teachers in schools and universities rather than members of royal courts or academies. In addition, while mathematics continued to be applied to the standard problems of physics and astronomy, pure mathematics, divorced from physical problems, increasingly developed an impetus of its own. Calculus broadened into analysis (those areas making use of concepts of calculus such as the limit), and advances were made in geometry and number theory.

Analysis, Geometry, and Number Theory

In Germany the work of Carl Friedrich Gauss covered most major areas of pure and applied mathematics, bridging 18th-century mathematics to modern mathematics. His most impressive single work, published in 1801, provided a thorough and innovative treatment of number theory. Gauss also succeeded in giving a physical interpretation to complex numbers --those which contain both real and imaginary components--by representing them in terms of points on a two-dimensional plane, an achievement that greatly helped to establish the mathematical respectability of complex numbers. In applied mathematics Gauss studied geodosy and the motion of the planets and wrote a masterly treatise on the Least-Square Method. Gauss's surviving notebooks and papers show that he had also discovered Non-Euclidean Geometry.

The textbooks of Augustin Louis Cauchy, published in 1821 and 1823 and designed for students at France's famous Ecole Polytechnique (founded 1794), were concerned with developing the basic theorems of the calculus as rigorously as possible. Other French mathematicians who produced important books in connection with their teaching include A. M. Legendre, Gaspard Monge, and S. F. Lacroix. From Monge's work at the Ecole Polytechnique there proceeded a generation of geometers that included J. B. Biot, J. V. Poncelet, Charles Dupin, and Jean Hachette. The Ecole Polytechnique also produced work of equal importance in applied mathematics.

Mathematical physics made rapid progress in the hands of such theoreticians as Lagrange, Monge, Joseph Fourier, S. D. Poisson, and Cauchy, as well as through the work of A. M. Ampere, Gaspard de Coriolis, Louis Poinsot, and Jean Poncelet. In carrying out a mathematical treatment of heat, Fourier established that any arbitrary function could be represented by a trigonometric series of specific form, the so-called Fourier series central to Fourier Analysis.

The German mathematician P. G. L. Dirichlet was the first to develop rigorously the use of the Fourier series. Others making significant advances in analysis at this time were the Norwegian N. H. Abel and the German C. G. J. Jacobi. In addition to his work in analysis, Dirichlet also demonstrated the strength of applying techniques of analysis to the study of number theory.

The most immediate influence of Dirichlet was on his student Bernhard Riemann. In his study of complex functions, Riemann introduced the concept of the Riemann surface, thus relating Topology to analysis. Riemann also wrote papers on the foundations of geometry and the study of trigonometric series. In number theory he applied (1859) the theory of complex numbers to the distribution of prime numbers.

Riemann's discovery of a continuous, nondifferentiable function showed the inadequacy of geometric intuition as a guide in analysis; mathematicians had always assumed that any continuous function must possess derivatives. Among the most important mathematicians in the 19th century to stress the need for new methods of analysis was Karl Theodor Weierstrass, who emphasized the rigor of proceeding arithmetically--defining, for example, irrational numbers as limits of convergent series. Leopold Kronecker, a colleague of Weierstrass at Berlin, was greatly opposed to the sort of analysis developed by Weierstrass and advocated, instead of infinite processes, the reduction of all mathematics to arguments involving only the integers and a finite number of steps. Kronecker is well known for his remark: "God created the integers--all else is the result of man."

Kronecker's opposition to any use of the infinite in mathematics left him adamantly opposed to transfinite Set Theory, created by Georg Cantor in the 1880s. The great value of set theory, however, particularly in analysis and topology, ensured its eventual acceptance into mathematics despite initial opposition to the idea of the infinite. Cantor, as well as Weierstrass and Richard Dedekind, also developed a theory of irrational numbers.

As analysis was making rapid progress in the 19th century, so, too, were geometry and the new fields to which it gave rise. Projective Geometry was simultaneously discovered early in the century by Joseph Gergonne and Poncelet. In the 1820s several Germans, including Jacob Steiner and K. G. C. von Staudt, greatly extended synthetic geometry, while August Ferdinand Mobius and Julius Plucker in Germany, Michel Chasles in France, and Arthur Cayley in England emphasized algebraic geometry. Mobius is also known for his pioneering efforts in topology, and for his Mobius Strip, a theoretical surface having only one side.

Among the most controversial discoveries of the 19th century were non-Euclidean geometries, which were discovered as a result of futile attempts to prove Euclid's parallel postulate. Simultaneously, Nikolai Ivanovich Lobachevsky and Johann Bolyai realized (as had Gauss earlier) that alternative axioms could be introduced for which the resulting geometries were non-Euclidean but perfectly consistent. Later, Hermann Grassman further extended the scope of geometry; his work led to Vector Analysis of affine and metric spaces. Topology, known as analysis situs in the 19th century, also grew out of geometry, as did the 20th-century concept of fractional dimensions.

Late in the century, French mathematics came to rival that of Germany. Following the proof by Joseph Liouville of the existence of Transcendental Numbers, Charles Hermite proved in 1873 that e was such a number, and shortly thereafter Ferdinand Lindemann established in 1882 that pi was also transcendental. Hermite came to be the leading exponent of analysis in France at the end of the century. Among his contemporaries and followers were figures such as Rene Baire, Emile Borel, J. S. Hadamard, H. L. Lebesgue, and C. E. Picard. The greatest mathematician in France at the end of the century was Henri Poincare, whose interests covered nearly every field of creative mathematics. His observation that apparently deterministic systems could show chaotic behavior, for example, presaged the development of Chaos Theory in the later 20th century.

Whereas analysis and geometry received great attention on the Continent, British mathematicians tended to pursue algebra and its applications to geometry. Continental mathematics was eventually promoted in Britain, however, by such mathematicians as Charles Babbage, Sir John Herschel, and George Peacock, all leaders of the Analytical Society. Algebra in Britain also took the form of Boolean Algebra, promulgated in The Laws of Thought (1854) by George Boole. Boolean algebra was the first of a series of important contributions by Englishmen to symbolic logic; these contributions culminated in the work of Bertrand Russell and Alfred North Whitehead in the 20th century.

Other branches of mathematics that led to important areas of activity in the 20th century include the study of divergent series, Tensor analysis and Differential Geometry, and abstract algebra, including the study of fields, groups, and rings. Group theory, with which names such as Evariste Galois and Camille Jordan are associated, was one of the great discoveries and unifying principles of the late 19th century. Group theory made possible the unification of geometry and algebra; in this respect Hermann Helmholtz and Sophus Lie showed how Riemann's work was greatly enriched by the study of transformation groups. Lie's interests were subsequently advanced considerably by the French mathematician Elie Cartan.

Philosophy of Mathematics

The discovery of paradoxes in set theory has led to attempts to ensure that mathematics be kept free of paradoxes and contradictions. Gottlob Frege, Giuseppe Peano, Augustus De Morgan, and the English mathematicians Whitehead and Russell, working in the spirit of Boole, stressed logicism as the safe road to a certain mathematics. Logicism posits the priority of logic and assumes that mathematical objects can be defined within the framework of logic. Then the properties of these objects can be proven using normal logical methods. Russell and Whitehead's attempt at a logicist program in their Principia Mathematica, however, has proven inadequate and cumbersome.

A second approach followed the work of David Hilbert, who proposed to formalize relevant parts of mathematics with the aid of an artificial language of logic, and to prove by means of finite mathematics that no paradoxes can be derived in the formal system. Hilbert's work, known as formalism, was taken up by John von Neumann and Kurt Godel, among others, and was dealt a severe setback when Godel proved in 1931 with his incompleteness theorem that no axiomatic approach could be sufficient to determine the consistency of any branch of mathematics. This theorem did not affect the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than Hilbert's. Another approach followed the work of Kronecker, who believed that mathematics should deal only with finite numbers and with a finite number of operations. Kronecker was followed in this view (with certain variations) by Poincare and L. E. J. Brouwer, who placed particular emphasis upon intuition. This program, known as intuitionism, stresses that mathematics has priority over logic, that the objects of mathematics are constructed and operated upon in the mind by the mathematician, and that it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.

In a sense, the ultimate nature and foundations of mathematics remain open questions. As a result, mathematics will continue to grow and expand with a freedom limited only by consistent reasoning.

(c) 1996 vico65@aol.com