  We apply Benacerraf's distinction between mathematical ontology and
mathematical practice (or the structures mathematicians use in practice) to
examine contrasting interpretations of infinitesimal mathematics of the 17th
and 18th century, in the work of Bos, Ferraro, Laugwitz, and others. We detect
Weierstrass's ghost behind some of the received historiography on Euler's
infinitesimal mathematics, as when Ferraro proposes to understand Euler in
terms of a Weierstrassian notion of limit and Fraser declares classical
analysis to be a "primary point of reference for understanding the
eighteenth-century theories." Meanwhile, scholars like Bos and Laugwitz seek to
explore Eulerian methodology, practice, and procedures in a way more faithful
to Euler's own.
  Euler's use of infinite integers and the associated infinite products is
analyzed in the context of his infinite product decomposition for the sine
function. Euler's principle of cancellation is compared to the Leibnizian
transcendental law of homogeneity. The Leibnizian law of continuity similarly
finds echoes in Euler.
  We argue that Ferraro's assumption that Euler worked with a classical notion
of quantity is symptomatic of a post-Weierstrassian placement of Euler in the
Archimedean track for the development of analysis, as well as a blurring of the
distinction between the dual tracks noted by Bos. Interpreting Euler in an
Archimedean conceptual framework obscures important aspects of Euler's work.
Such a framework is profitably replaced by a syntactically more versatile
modern infinitesimal framework that provides better proxies for his inferential
moves.
  Keywords: Archimedean axiom; infinite product; infinitesimal; law of
continuity; law of homogeneity; principle of cancellation; procedure; standard
part principle; ontology; mathematical practice; Euler; Leibniz
