  We examine prevailing philosophical and historical views about the origin of
infinitesimal mathematics in light of modern infinitesimal theories, and show
the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal
mathematics in a new light. We also detail several procedures of the historical
infinitesimal calculus that were only clarified and formalized with the advent
of modern infinitesimals. These procedures include Fermat's adequality;
Leibniz's law of continuity and the transcendental law of homogeneity; Euler's
principle of cancellation and infinite integers with the associated infinite
products; Cauchy's infinitesimal-based definition of continuity and "Dirac"
delta function. Such procedures were interpreted and formalized in Robinson's
framework in terms of concepts like microcontinuity (S-continuity), the
standard part principle, the transfer principle, and hyperfinite products. We
evaluate the critiques of historical and modern infinitesimals by their foes
from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the
consistency, as distinct from the issue of the rigor, of historical
infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in
this connection.
