I found this paper by John Cleave, <A HREF="https://www.jstor.org/stable/pdf/686641.pdf">Cauchy, Convergence, and Continuity</A> (1971) quite illuminating.

> According to our present-day (Weierstrassian) conception of the
> continuum, Cauchy's 1821 theorem is false – one must impose the
> condition of uniform convergence to get a correct statement. Lakatos
> (1966) first pointed out that the theorem is a perfectly correct
> statement about a Leibnizian continuum – an extension of the
> Weierstrassian continuum in which there are infinitely large and
> infinitely small numbers. He shows that if "the neighbourhood of a
> particular point" is understood as the set of points infinitely close
> to that value, and if the usual definition of convergence is assumed
> for sequences of numbers in the extended continuum, then Cauchy's
> proof is correct. 
> 
> The aim of this paper is to examine Lakatos' claim more closely. We
> show that Cauchy's notions can be comfortably interpreted in terms of
> non-standard analysis and, in particular, that convergence of a series
> of functions in the infinitesimal neighbourhood of a point in Cauchy's
> sense is equivalent to the notion of "point of uniform convergence" in
> the Weierstrassian sense. If the correctness of the interpretation of
> Cauchy by non-standard analysis is granted one must therefore concede
> that the notion of uniform convergence was implicit in Cauchy's work
> of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms
> by Seidel (1847) or Weierstrass.

See also a subsequent analysis in the same direction by Cutland et al. <A HREF="https://www.jstor.org/stable/pdf/687214.pdf">
On Cauchy's notion of infinitesimal</A> (1988).