From today's point of view it's fair to call the 1859 note minimal (with non-minimal contributions to other things than the inequality). The correct historical assessment is a different question that I do not have enough information to answer. For the Cauchy-Schwarz inequality Buniakovsky could be seen as packaging a fact about collections of $n$ numbers as a statement about familiar individual objects, the integrals. I don't know what else he might have done on the same subject in other papers prior to Schwarz. For the translations, his French was bad, or he got the math wrong? @katz

The transition from a uniformly-spaced Riemann sum to an integral was not something in need of proof by the standards of the time, and I don't think Buniakovsky can be faulted for treating it superficially. What needs proof is his later assertion in the 1859 paper, that the equality condition is the same for the integral form of the inequality. He could have done this by taking the limit of some steps in Cauchy's proof, ending up with an integral form of the sum-of-squares identity. If that too is taken as obvious then it's even less clear what was the contribution to the inequality. @katz

Buniakovsky's 1859 note says nothing about the possibility of applying inequalities to probability (statistics is not mentioned at all). He opens the article by saying that the arithmetic mean is a useful thing, that probability is an example of a field where the arithmetic mean is useful, and that his book on probability is a reference for those remarks. Nowhere in the article is there any connection of inequalities to probability theory. @MargaretFriedland

Actually, I would agree, if there were evidence that Buniakovsky-1859 catalyzed changes in thinking for reasons similar to what you wrote. I have no proof, but your comment strikes me as probably a reverse history, where earlier events are interpreted in terms of modern understanding of the subject. The idea that the Cauchy inequality was fundamental, and its integral form therefore an important fact for analysis, was unavailable at the time. The long time lapses from C to B to S are an indication of that; if people saw the importance they would have worked on it. @5xum

In hindsight. It's not clear that Buniakovsky or his readers would have noticed anything new and different in his result other than an interesting fact about integrals, and it's possible that 100 percent of the shift in thinking started later (such as after Schwarz' paper). Calculus as a subject in need of inequalities was a point of view that came later, with real analysis. As I suggested in the question, Buniakovsky's paper may be significant in raising continuous inequalities to an object of study, but that is a different thing from the ideas around the CS inequality. @5xum

@5xum: are you saying Cauchy and his contemporaries might not have considered it obvious that the average of continuous $f(x)$ at $n$ equally spaced points on an interval converges to $\int f$ for large $n$? It is assumed in Bunyakovsky's article prior to the discussion of the inequality, and used in an informal or intuitive way. I believe it was understood at that level since Newton or earlier, maybe much earlier, and it may have been dealt with formally by Cauchy's time. Nothing else is used in Bunyakovsky's paper from1859.

@grand_chat, the Wikipedia page on Buniakovsky has the exaggerated statement that he "is credited with an early discovery of the Cauchy–Schwarz inequality, proving it for the infinite dimensional case in 1859, many years prior to Hermann Schwarz's works on the subject." Wikipedia's article on the Cauchy-Schwarz inequality is more precise: it says that B was C's student, who "noted that by taking limits one can obtain an integral form of Cauchy's inequality."

@Jonas, the 1859 paper may well deserve more appreciation as a founding article for the field of integral inequalities, and for further developing the notion of a functional, along with the capital-letter notation for functionals and its use in a nonlinear case. Those contributions seem to be overshadowed by the relatively simple move of taking a limit of the Cauchy inequality. re: integrals, Buniakovsky uses only the unformalized, unproved statement that by assigning "consecutive values" of $f(x)$ in the inequality, leaving to the reader to understand it as an average, one gets $\int f(x)$.

Suggestions for canonical Latinization of Buniakovsky vs Buniakowski vs ... are welcome.