Editing 1640 in science (section)

==Mathematics==
* The 16-year-old [[Blaise Pascal]] demonstrates the properties of the ''[[hexagrammum mysticum]]'' in his ''Essai pour les coniques'' which he sends to [[Marin Mersenne|Mersenne]].
* October 18 – [[Fermat]] states his "[[Fermat's little theorem|little theorem]]" in a letter to [[Bernard Frénicle de Bessy|Frénicle de Bessy]]: if ''p'' is a [[prime number]], then for any [[integer]] ''a'', ''a''<sup>&nbsp;''p''</sup>&nbsp;−&nbsp;''a'' will be divisible by ''p''.
* December 25 – Fermat claims a proof of the [[Fermat's theorem on sums of two squares|theorem on sums of two squares]] in a letter to Mersenne ("Fermat's Christmas Theorem"): an [[Even and odd numbers|odd]] [[prime number|prime]] ''p'' is expressible as the sum of two squares.