by John Stillwell I. General Reaarb, Poincare's papers on Fuchsian and
Kleinian I1'OUps are of Il'eat interest from at least two points of
view: history, of course, but also as an inspiration for further
mathematical proll'ess. The papers are historic as the climax of the
ceometric theory of functions initiated by Riemann, and ideal
representatives of the unity between analysis, ceometry, topololY and
alcebra which prevailed during the 1880's. The rapid mathematical
prOll'ess of the 20th century has been made at the expense of unity and
historical perspective, and if mathematics is not to disintell'ate
altogether, an effort must sometime be made to find its, main threads
and weave them tocether 81ain. Poincare's work is an excellent example
of this process, and may yet prove to be at the core of a . new
synthesis. Certainly, we are now able to gather up, some of the loose
ends in Poincare, and a broader synthesis seems to be actually taking
place in the work of Thurston. The papers I have selected include the
three Il'eat memoirs in the first volumes of Acta Math. -tice, on-
Fuchsian groups, Fuchsian, functions, and Kleinian groups (Poincare
[1882 a, b,1883]). These are the papers which made his reputation and
they include many results and proofs which are now standard. They are
preceded by an, unedited memoir written by Poincare in May 1880 at the
height of his, creative ferment.
