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¥¶. The Nineteenth - Century Mathematics of Germany :
Modern Mathematics

¢º Characteristic of the 19th Century Mathematics
¢º The Early Nineteenth Century
¢º Non-Euclidean Geometry
¢º The Abstraction of Algebra
¢º The Arithmetization of Analysis
¢º Mathematicians ranges from the 19th to the 20th century
¢º The Three Greatest Mathematicians of Womens

¡ß Characteristic of the Nineteenth - Century mathematics

    If we call that 1600's was the times of the establishement of mathematics and that 1700's was the times of development of 1600's.
    We can say that 1800's was the times of continuation of founding again. In 1800's, the Germans advanced in their studies about mathematics, but on the other hand, the French progressed in 1700's Gauss, Weierstrass, Riemann, Dedekind, Cantor, Klein and Hilbert established the basic of modern mathematics.     The mathematical points in 1800's are the number theory of Gauss, the research of a French Cauchy about analysis, the work of a Hungarian Bolyai about Non-Euclidean, the contribution of the theory of the equation and the theory of the group of a French Galois the analysis of Weierstrass and Riemann and the origination of Riemann's geometry.
    The 19th century was the greatest, because of the release of geometry, the abstraction of algebra and the changing in to arithmetic of analysis.   In this century, mathematics was given chase to strictness, abstraction and universality.

¡ßThe Early Nineteenth Century

¡Ý Gauss :   He is along with Archimedes and Isaac Newton, as one of the three greatest mathematicians of all time, The diversion of the 18th century mathematics to the 19th century depended largely on Gauss.
    Famous is Gauss' assertion that "mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics."
    He understands that complex number is a real object of mathematics and thinks it a mathematical reality.
    And so on, Gauss gave the first wholly satisfactory proof of the <fundamental theorem of algebra> (that a polynomial equation with complex coefficients and of degree n>0 has at least one complex root).

¡Û Cauchy : With Lagrange and Gauss, the nineteenth-centuryrigorization of analysis got under way.  This work was considerably furthered and strengthened by the great French mathematician Augustin-Louis Cauchy, the most outstanding an aalyst of the first half of the nineteenth century.
    Father of theory of functions, Cauchy defined function as relationship between variables.
    His name is met in calculus, 'Cauchy root test' and 'Cauchy ration test', in complox function theory, 'Cauchy inequality' and 'Cauchy's integral fomula.
    Cauchy defined the derivative, with respect to x, of y = f(x) as the limit, when ¡âx ¡æ 0, of the difference quotient.


¡Ý Abel and Galois: These two men, like a streaking meteor in the mathematical heavens, flashed to an early brilliance and then was suddenly and pathetically extinguished by premature death, leaving remarkable material for future mathematicians to work upon.
    He adopted concept called 'group'as basic one of theory of eguation.
    He contributed to the mathematics today through the study of group theory.
¡Ý Jacobi : He used the terms, determinant, and contributed to the determinant theory.

¡ßNon-Euclidean Geometry

Parallel postulate, Euclidean the 5th postulate, (¡¸Through a given point not on a given line can be drawn 'just one' line parallel to the given line.¡¹).  This situation is equivalent, respectively, to the fact that the sum of three internal angles of a triangle is 180 degrees.
    This parallel postu late is so complex that it seems like theorem.
    Many mathematicians, therefore, made on effort to prove the postulate but satisfactory results were not made.
    So mathematlcians used another way, indirect proof, to deny the postulate and find out contradictions.
    But they only failed to prove the postulate further more they got new theorems through the way
    Finally, they found these new theorems called non-Euclidean geomerry.
¡Ý Saccheri, Girolamo (1667~1733) : These three possibilities are refered to by Saccheri as the 'hypothesis of the acute angle,' the 'hypothesis of the right angle', and the 'hypothesis of the obtuse angle'.  He tried to prove these three possibilities were contradictions ,but rather had admitted his inability to find one, Saccheri would today unquestionably be credited with the discovery of non-Euclidean geometry.

¡Ý Lobachevsky : Lobachevsky and Bolyai asserts that parallel postulate is axiom not tneorem.
    He said "Through a given point not on a given line can be drawn 'more than one' line parallel to the given line".
    He discovered new geometry.

¡Ý Riemann : He asserted that "Through a given point not on a given line can be drawn 'no' line parallel to the given line." He discovered consistent geometry.
    Ridigity and generalization of abstraction are the characteristics of Riemann's mathematics.
    He widened the geometry to the variety of space.
¡Ý Klein He classified Euclidean geometry and non-Euclidean geometry in 1871.




    Non-Euclidean geometry means the possibility of another geometry's existence except Euclidean geometry.
    It, therefore, is clear that mathematicians don't have to cling to practical physical space.



¡ßThe Abstraction of Algebra

¡Ý Hamilton : As Euclidean geometry was regarded as an only geometry, so arithmetic algebra was regarded as an only algebra until the early 17th century.
    But new algebraic system in which commutative law for multiplication is broken is observed.
 Hamilton's quaternions(ordered real number guadruples)_the commutative and assocrative law for addition and the assocrative and distributive for multiplication is broken.

¡Ý Cayley : One more noncommutative algebra the matric algebra devised by the English mathematician Arthur Cayley (1821-1895) in 1857.

¡Ý Boole : Boole maintained that the essential character of mathematics lies in its form rather than in its content; mathematics is not (as some dictionaries today still assert) merely "the science of measurement and number," but, more broadly, any study consisting of symbols along with precise rules of operation upon those symbols, the rules being subject only to the requirement of inner consistency.
    He established both formal logic and a new algebra_the algebra of ests known today as Boolean algebra.  In more recent times, Boolean algebra has found a number of applica tions, such as to the theory of electric switching circuits.


¡ßThe Arithmetization of Analysis

    The demand for an even deeper understanding of the foundations of analysis was strikingly brought out in 1874 with publicizing of an example, due to the German mathematician Karl Weierstrass, of a continuous function having no derivative, or, what is the same thing, a continuous curve possessing no tangent at any of its points.
    It became clear that the theory of limits, continuity, and differentiability depend upon more recondite properties of the real number system than had been supposed.  Accordingly, Weierstrass advocated a program wherein the real number system itself should first be rigorzed, then all the basic concepts of analysis should be derived from this number system.  This remarkable program, known as the arithmetization of analysis, proved to be difficult and intricate, but was ultimately realized by Weierstrass and his followers, so that today all of analysis can be logically derived from a postulate set characterizing the real number system.



¡ßMathematicians ranges from the 19th to the 20th century

    This section will be devoted to a brief consideration of Georg Cantor and Henri Poincare, two mathematicians with life spans astride the nineteenth and twentieth centuries, and who exerted a considerable influence on much of the mathematics of present times.  It is also natural to insert a few words about Leopold Kronecker, the harsh and relentless critic of Cantor's of Cantor's mathematics of the infinite.
¡Ý Cantor : He commenced in 1874 his revolutionary work on set theory and the theory of the infinite.  With this latter work,  Cantor created a whole new field of mathemaitcal research.
    He classified infinite set according to power.
    He tought if one - to - one correspondence is possible between two sets, the two sets have the same numbers of element countable set: N~Z~Q, non-conutable set:(0,1)~R
    Today,   Cantor's set theory has penetrated into almost every branch of mathematics, and it has proved to be of particular importance in topology and in the foundations of real function theory.

¡Ý Kronecker : As a finitist, he condemned the work of Cantor, regarding it as theology and not as mathematics.  Believing that all of mathemat ics must be based by finite methods upon the whole numbers, he was a nine teenth-century Pythagorean.  "God made the whole numbers, all the rest is the work of man"
¡Ý Poincare : Poincare has been described as the last of the universalists in the field of mathematics.  It is certainly true that he commanded and enriched and astodnishing range of subjects.
    He was also one of the ablest popularizer's of mathematics and scirence.


¡ßThe Three Greatest Mathematicians of Womens

¡Ý Hypatia : Theon's (Theon lived in the turbulent colsing period of the fourth century A.D.)daughter.  Hypatia, was distinguished in mathematics, medicine, and philosophy, and wrote commentaries on Diophantus Arithmetica and Apollonius, Conic Sections.  She is the first woman mathematician to be men tioned in the history of mathematics.  Her life and barbarous murder by a mob of fanatical Christians in March, 415, are reconstructed in Charles Kingsley's novel.

¡Ý Kovalevsky : She had earlier studied under Karl Weierstrass, contributed to the field of partial differential equations.

¡Ý Noether : Amalie Emmy Noether, one of the most outstanding mathematicians.
    Although Noether was a poor lecturer and lacked pedagogical skill, she managed to inspire a surprising number of students who also left marks in the field of abstract algebra.