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,
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.
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.
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.