This is the first of the two volumes (for a review of the second volume see the following review) of a course of mathematical analysis teached by Roger Godement during thirty-five years at the University of Paris. The contents is quite classical: sets and functions, convergence of sequences and series, continuous and differentiable functions, elementary functions. The treatment is less classical: precise although unpedantic (rather far from the style definition-theorem-corollary), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. The ones on set theory, real numbers, harmonic series, uniform convergence, Cauchy criterion, differentiable functions, logarithmic function, strange identities can be recommended. The author also gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction contains also comments which are very unusual in a book on mathematical analysis, going from pedagogy to critics of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such away that the reader is less surprised than he should. (ZDM/Mathdi)