WEIERSTRASS,K.T.W.(1815-1897) and RIEMANN,G.F.B.(1826-1866)

    A misdirected youth spent in studying the law and finane gave Weierstrass a late start in mathematics, and it was not until he was forty that he fonally emancipated himself from secondary teaching by obtaining an instructorship at the University of Berlin, and another eight years passed before, in 1864, he was awarded a full professorship at the university and could finally devote all his time to advanced mathematics. Weierstrass never regretted the years he spent in elementary teaching,

and he later carried over his remakable pedagogical abilities into his university work, becoming probably the greatest teacher of advanced mathematics that the world has yet known.
    Weierstrass wrote a number of early papers on hyperelliptic integrals, Abelian functions, and algebraic differntial equations, but his widest known contribution to mathematics is his construcition of complex functions by means of power series.  This, in a sense, was an extension to the complex plane of the idea earlier attempted by Lagrange, but Weierstrass carried it through with absolrte rugor.
    In algebra, Weierstrass was perhaps the first to give a so-called posrulational definition of a determinsant. He defuned the determinant of a square matrix Aas a polynomial in the elements of A, which is homogeneous and linear in the elements of each row of A, which merely changes sign when two rows of Aare permuted, and which reduces to I when A is the correspending identity matrix.
  Weiestrass was a very influential teacher, and his meticulously prepared lectures established an ideal for many future mathmaticians;"Weierstrassean rigor" became synonymous with "extremely careful reasoning." weiestrass was "the mathemarical conscience par excellence," and he became known as "the father of modern analysis."  He died in Berlin in 1897, just one hrndred years after the furst publication, in 1797 by Lagrange, of an attempt to rigorize the calculus.

    Along with this rigorization of mathematics, there appeared a tendency toward abstract generalization, a process that has becomevery pronounced in present-day mathenatics.  Perhaps the German mathematician Georg Friedrich Bernhard Riemann influenced this feature of midern mathrmatics mire than any other nineteedth-century mathematician.  He certainly wielded a profound influence on a number of branches of mathematics, particularly geometry and function theory, and few mathematicians have bequeathed to their successors a richer legacy of ideas for further develipment.

    Riemann was born in 1826 in a small cillage in Hanover, the son of a Lutheran pastor.  In manner, he was always shy; in health, he was always frail.   In spote of the very modest cirsumstances of his father.  Riemann managed to decure a good education, first aty the University of Berlin and then at the University of Gottingen.  He took his dectoral degree at the latter institution with a brilliant thesis in the field of complex-function theory.  In this thesis, one finds the so-called Carahy-Riemann differential equations(Known, however, before Riemann's time) that guarantee the analyticity of a complex varible, and the highly fruitful cincept of a Riemann surface,ch introduced topological cinsiderations into analysis.  Riemann clarified the cincept of integrability by the definition of what we now knowas the Rirmann integral, which led, in the twentieth century, to the more general Lebesgue integral,and thence to futher generalizatiens of the integral.
    Later, Albert Einstein and othera found Riemann's broad cincept of space and geometry the mathematical milieu needed for general relativity theory. Riemann himself contribrted in a number of directions to theoretical physics; he was the first, for example, to give a mathematical treatmint of shock waves.
    Famius in mathematical literature are the so-called Riemann zeta function and associated Riemann hypotheses.  The latter is a celebrated unproved conjecture that is to classical analysis what Fermat'slast "theorem" is to number theory.

    In 1857, riemann was appointed assistant prefessor at Gottingen, and then, in 1859, full professor, succeeding Dirichlet in the chair once occupied by Gauss.  Riemann died of tuberculosis in 1866, when only years of age, in nothern Italy, where he had gone to seek an improvement in his health.