  In this paper we use theoretical frameworks from mathematics education and
cognitive psychology to analyse Cauchy's ideas of function, continuity, limit
and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the
development of mathematical thinking from human perception and action into more
sophisticated forms of reasoning and proof, offering different insights from
those afforded by historical or mathematical analyses. It highlights the
conceptual power of Cauchy's vision and the fundamental change involved in
passing from the dynamic variability of the calculus to the modern
set-theoretic formulation of mathematical analysis. This offers a re-evaluation
of the relationship between the natural geometry and algebra of elementary
calculus that continues to be used in applied mathematics, and the formal set
theory of mathematical analysis that develops in pure mathematics and evolves
into the logical development of non-standard analysis using infinitesimal
concepts. It suggests that educational theories developed to evaluate student
learning are themselves based on the conceptions of the experts who formulate
them. It encourages us to reflect on the principles that we use to analyse the
developing mathematical thinking of students, and to make an effort to
understand the rationale of differing theoretical viewpoints.
