The theory of series in the 17th and 18th centuries poses several
interesting problems to historians. Indeed, mathematicians of the time
derived num- ous results that range from the binomial theorem to the
Taylor formula, from the power series expansions of elementary functions
to trigonometric series, from Stirling's series to series solution of
di?erential equations, from
theEuler-MaclaurinsummationformulatotheLagrangeinversiontheorem, from
Laplace's theory of generating functions to the calculus of operations,
etc. Most of these results were, however, derived using methods that
would be found unacceptable today, thus, if we look back to the theory
of series
priortoCauchywithoutreconstructinginternalmotivationsandtheconc- tual
background, it appears as a corpus of manipulative techniques lacking in
rigor whose results seem to be the puzzling fruit of the mind of a -
gician or diviner rather than the penetrating and complex work of great
mathematicians. For this reason, in this monograph, not only do I
describe the entire complex of 17th- and 18th-century procedures and
results concerning series, but also I reconstruct the implicit and
explicit principles upon which they are based, draw attention to the
underlying philosophy, highlight competing approaches, and investigate
the mathematical context where the series t- ory originated. My aim is
to improve the understanding of the framework of 17th- and 18th-century
mathematics and avoid trivializing the complexity of historical
development by bringing it into line with modern concepts and views and
by tacitly assuming that certain results belong, in some unpr- lematic
sense, to a uni?ed theory that has come down to us today.
