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Lagrange launched a challenge via the French Academy in the 1780s calling on authors to provide a clarification of the use of infinite and infinitesimal numbers in analysis. Carnot wrote an essay that …

If you are willing to extend "contests" to "disputations" as your latest answer seems to suggest, what certainly should be mentioned is the drawn-out disputation over the indivisibles that turned part …

Eratosthenes lived around -200 (more precisely -276 to -195/194) and was a correspondent of Archimedes. Archimedes communicated his Method to Eratosthenes in the form of a letter. Euclid lived around …

The "minimes" was a particular religious order that existed at the time, and is frequently mentioned in 17th century sources. It seems that he was a religious authority (prieur is somewhat less than …

The issue of Cauchy's understanding of continuity is a subject of lively historical debate. Grabiner represents only one view in this debate. Laugwitz has published a series of scholarly articles stud …

A colleague pointed me to a good source for historical documents at https://archive.org but you need to log in there.

We were able to obtain an audio file of Bishop's talk from the American Academy of Arts and Sciences. Our analysis will be published in Historia Mathematica and is available on the arxiv. We exami …

There is a nice recent article about Cauchy's contribution to integral geometry which is probably what you mean by Cauchy's Crofton formula: Hyksova, M.; Kalousova, A.; Saxl, I. “Early history of geo …

Desargues certainly pioneered original mathematics. The notion of a point at infinity in projective geometry is usually attributed to him. Kepler apparently did not work in projective geometry but r …

This is mainly a comment on the discussion following the original post. Leibniz did not use the $\Sigma^\infty$ notation. If you replace $\infty$ by an infinite hyperinteger, and interpret the sum a …

The idea that infinite sets, if they existed, would admit 1-1 correspondences with their proper parts was well understood already in the 17th century, and were the reason such sets were rejected as co …

"examples where someone proved some (important?) theorem, then (much?) later, someone else rediscovered the insights that led to the proof and built a new theory that eclipsed the original theorem." …

If the question concerns projective space specifically rather than projective geometry in a broader sense, then the answer would have to be Jean-Victor Poncelet (1788 – 1867). Desargues already introd …

If you want something simultaneously recent, readable, and controversial, you may want to take a look at Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M.; van Atten, M. "Two-track depictions of Leibn …

There is a relevant piece of information that does not seem to have been mentioned in the other answers so here it is. Fermat became acquainted with the book by Diophantus by studying in the library …