Viète is much praised by historians of mathematics for his contributions to algebra, generally in a fairly obscure manner that doesn't cast much light on his actual accomplishments; and his introduction to the analytic art, more of a manifesto than a t ext, is not much help either. To complicate matters, the insightful commentary by Jacob Klein is couched for certain awkward historical reasons in the least helpful terms, though you did find a few immediately lucid passages.
In any case, a careful perusal of the rather limited secondary literature and some inspection of the texts themselves clarified matters considerably (and brought us to the point where Jacob Klein could be appreciated). I think your essay gives a clear indication of what Viète was in fact up to. It does not give any indication what his primary mathematical accomplishments were, as applications of his methodology, but that is just as well; he was a brilliant mathematician but it would needlessly complic ate matters to try to bring in examples of that brilliance in any detail. He is remembered for the usefulness of his system, not for his specific ideas; and perhaps he is really remembered for the system that Descartes built on his foundation.
The story of how this man, who published little in his lifetime, came to be so highly appreciated by the leading lights of the next generation, promises to be an interesting one as well, worth further investigation. References
The Jacob Klein is very good, though obscure at first reading. We were fortunate to locate the translation by Witmer of most of his work. Witmer has a habit of convincing himself rather easily that words like "latus" can be well represented by "x" (and t ook far greater liberties in his Cardan translation, severely diminishing its value) rather than leaving that question open for the reader, but we had the opportunity to check his translation against the original and avoided the corresponding errors. Beca use of the importance of that use of the Latin original, it would be better to include that edition as well in the list of references.
The Calinger is not much help in this instance, and the Burton is risky. He seems to be rather careless, based on the extracts I've seen to date.
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