Differential principle:

(aka differential calculus) 1 of 2 fields of calculus (the other being integral calculus), finds rates at which quantities change, aims to find the derivative of a function, the differential & its applications.  The derivative of a function at a chosen input value describes rates of change of the function at that input value.  The process of finding a derivative is differentiation; geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point (provided the derivative exists & is defined at that point); the derivative determines the best linear approximation to the function at that point; differentiation is the reverse process to integration.

“Euler… treated the differentials as zero”:

the “generality of algebra” was a method of argument used in the 18th century (Euler, Lagrange) in manipulating infinite series; assumed that algebraic rules true for a certain class of expressions were valid on a larger class of objects, even if the rules are no longer obviously valid.  Consequently they believed that they could derive meaningful results by applying the usual rules true for finite expansions when manipulating infinite expansions. Cauchy rejected the use of these methods & sought a more rigorous foundation.

Cauchy’s… limit idea:

The Cauchy property describes sequences that seem to converge without mentioning any limit.  It is a modification of the usual definition of convergence except that we cannot compare the values of the sequence to limit; instead we have to compare such values to each other.  In calculus, the epsilon–delta definition of limit is a formalization of the notion of limit. The concept is due to Cauchy, who occasionally used it in arguments of his proofs, although he never gave epsilon–delta the definition of limit in his writings.