.. arly 19th century. The 16th century also saw the beginnings of modern algebraic symbolism (Mathematical Symbols), as well as the remarkable work on the solution of equations by the French mathematician Franois Vite. His writings influenced many mathematicians of the following century, including Pierre de Fermat in France and Isaac Newton in England. Mathematics Since the 16th Century Europeans dominated in the development of mathematics after the Renaissance. 17th Century During the 17th century, the greatest advances were made in mathematics since the time of Archimedes and Apollonius.

The century opened with the discovery of logarithms by the Scottish mathematician John Napier, whose continued utility prompted the French astronomer Pierre Simon Laplace to remark, almost two centuries later, that Napier, by halving the labors of astronomers, had doubled their lifetimes. (Although the logarithmic function is still important in mathematics and the sciences, logarithmic tables and their instrumental form-slide rules-are of much less practical use today because of electronic calculators.) The science of number theory, which had lain dormant since the medieval period, illustrates the 17th-century advances built on ancient learning. It was Arithmetica by Diophantus that stimulated Fermat to advance the theory of numbers greatly. His most important conjecture in the field, written in the margin of his copy of the Arithmetica, was that no solutions exist to an + bn = cn for positive integers a, b, and c when n is greater than 2. This conjecture stimulated much important work in algebra and number theory but is still unproved.

Two important developments in pure geometry occurred during the century. The first was the publication, in Discourse on Method (1637) by Descartes, of his discovery of analytic geometry, which showed how to use the algebra that had developed since the Renaissance to investigate the geometry of curves. (Fermat made the same discovery but did not publish it.) This book, together with short treatises that had been published with it, stimulated and provided the basis for Isaac Newton’s mathematical work in the 1660s. The second development in geometry was the publication by the French engineer Grard Desargues in 1639 of his discovery of projective geometry. Although the work was much appreciated by Descartes and the French philosopher and scientist Blaise Pascal, its eccentric terminology and the excitement of the earlier publication of analytic geometry delayed the development of its ideas until the early 19th century and the works of the French mathematician Jean Victor Poncelet. Another major step in mathematics in the 17th century was the beginning of probability theory in the correspondence of Pascal and Fermat on a problem in gambling, called the problem of points.

This unpublished work stimulated the Dutch scientist Christiaan Huygens to publish a small tract on probabilities in dice games, which was reprinted by the Swiss mathematician Jakob Bernoulli in his Art of Conjecturing. Both Bernoulli and the French mathematician Abraham De Moivre, in his Doctrine of Chances in 1718, applied the newly discovered calculus to make rapid advances in the theory, which by then had important applications in the rapidly developing insurance industry. Without question, however, the crowning mathematical event of the 17th century was the discovery by Sir Isaac Newton, between 1664 and 1666, of differential and integral calculus (Calculus). In making this discovery, Newton built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as Descartes, Francesco Bonaventura Cavalieri, Johann van Waveren Hudde, and Gilles Personne de Roberval. About eight years later than Newton, who had not yet published his discovery, the German Gottfried Wilhelm Leibniz rediscovered calculus and published first, in 1684 and 1686.

Leibniz’s notation systems, such as dx, are used today in calculus. 18th Century The remainder of the 17th century and a good part of the 18th were taken up by the work of disciples of Newton and Leibniz, who applied their ideas to solving a variety of problems in physics, astronomy, and engineering. In the course of doing so they also created new areas of mathematics. For example, Johann and Jakob Bernoulli invented the calculus of variations, and French mathematician Gaspard Monge invented differential geometry. Also in France, Joseph Louis Lagrange gave a purely analytic treatment of mechanics in his great Analytical Mechanics (1788), in which he stated the famous Lagrange equations for a dynamical system. He contributed to differential equations and number theory as well, and he originated the theory of groups.

His contemporary, Laplace, wrote the classic Celestial Mechanics (1799-1825), which earned him the title the French Newton, and The Analytic Theory of Probabilities (1812). The greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in these areas. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas. That is, Newton’s own accounts were based on kinematics and velocities, Leibniz’s explanation was based on infinitesimals, and Lagrange’s treatment was purely algebraic and founded on the idea of infinite series.

All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not resolved until the following century. 19th Century In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus. He based his approach only on finite quantities and the idea of a limit. This solution posed another problem, however; that of a logical definition of real number. Although Cauchy’s explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W.

R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. This definition is still taught, but other definitions were given at the same time by the German mathematicians Georg Cantor and Karl T. W. Weierstrass. A further important problem, which arose out of the problem-first stated in the 18th century-of describing the motion of a vibrating string, was that of defining what is meant by function. Euler, Lagrange, and the French mathematician Jean Baptiste Fourier all contributed to the solution, but it was the German mathematician Peter G.

L. Dirichlet who proposed the definition in terms of a correspondence between elements of the domain and the range. This is the definition that is found in texts today. In addition to firming the foundations of analysis, as the techniques of the calculus were by then called, mathematicians of the 19th century made great advances in the subject. Early in the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a whole new field for analysis, one that was developed in the work of Cauchy, Weierstrass, and the German mathematician Georg F. B.

Riemann. Another important advance in analysis was Fourier’s study of infinite sums in which the terms are trigonometric functions. Known today as Fourier series, they are still powerful tools in pure and applied mathematics. In addition, the investigation of which functions could be equal to Fourier series led Cantor to the study of infinite sets and to an arithmetic of infinite numbers. Cantor’s theory, which was considered quite abstract and even attacked as a disease from which mathematics will soon recover, now forms part of the foundations of mathematics and has more recently found applications in the study of turbulent flow in fluids. A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry.

In non-Eculidean geometry, more than one parallel can be drawn to a given line through a given point not on the line. Evidently this was discovered first by Gauss, but Gauss was fearful of the controversy that might result from publication. The same results were rediscovered independently and published by the Russian mathematician Nikolay Ivanovich Lobachevsky and the Hungarian Jnos Bolyai. Non-Euclidean geometries were studied in a very general setting by Riemann with his invention of manifolds and, since the work of Einstein in the 20th century, they have also found applications in physics. Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era.

While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1. In his doctoral dissertation he gave the first satisfactory proof of the fundamental theorem of algebra. Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying. Of more importance for algebra itself than Gauss’s proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems.

A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many, but not all, of the properties of the real numbers. Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann Gnther Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange.

Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula. Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to a geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. In the 20th century, algebra has also been applied to a general form of geometry known as topology. Another subject that was transformed in the 19th century, notably by Laws of Thought (1854), by the English mathematician George Boole and by Cantor’s theory of sets, was the foundations of mathematics (Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantor’s theory.

One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set ( Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising. They left open the question, however, of whether other paradoxes might arise in these restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, proved in 1931 by the American logician Kurt Gdel, that in any axiom system complicated enough to be interesting to most mathematicians, it is possible to frame propositions whose truth cannot be decided within the system.

Current Mathematics At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at Gttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilbert’s address at Gttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century.

When news breaks that another of the Hilbert problems has been solved, mathematicians all over the world await the details of the story with impatience. Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (Computer). Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbage’s imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms.

It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois. Mathematical knowledge in the modern world is advancing at a faster rate than ever before.

Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications. Mathematics.