the whole argument is about whether a sentence like "Hypatia was a first-rank mathematician" is accurate. Obviously that hinges on the definition of "first-rank" since no one is disputing her actual accomplishments or claiming that they really compare to all-time greats like Gauss or Archimedes.

It was the person I'm replying to who used "first-rate" actually. I don't think there's a big difference in meaning between the two.

Usage examples show nuance in the meaning of a word that are not captured in the definition.

>Obviously that hinges on the definition of "first-rank"

When figuring out the definition for a term used, the best source is the context it is used. In this case, Hypatia is the benchmark given for what qualifies as "first rank."

You can then argue the author is misusing that term, but that's a different argument that the article doesn't make.

My understanding is that they're synonyms, more or less:

“By the early years of King Charles I's reign, these four groups had been renamed to a numerical sequence. The royal ships were now graded as first rank, the great ships as second rank, the middling ships as third rank, and the small ships as fourth rank. Soon afterwards, the structure was again modified, with the term rank now being replaced by rate, and the former small ships now being sub-divided into fourth, fifth and sixth rates” --wikipedia