1640 in science
- The 16-year-old Blaise Pascal demonstrates the properties of the hexagrammum mysticum in his Essai pour les coniques which he sends to Mersenne.
- October 18 – Fermat states his "little theorem" in a letter to Frénicle de Bessy: if p is a prime number, then for any integer a, a p − a will be divisible by p.
- December 25 – Fermat claims a proof of the theorem on sums of two squares in a letter to Mersenne ("Fermat's Christmas Theorem"): an odd prime p is expressible as the sum of two squares.
- Tran, Linh. "Theatrum Botanicum: The Theater of Plants, or, An Herbal of a Large Extent". Texas A&M University Bioinformatics Working Group. Archived from the original on November 5, 2008. Retrieved 2011-04-01.