1640 in science

The year 1640 in science and technology involved some significant events.

Botany

John Parkinson publishes Theatrum Botanicum:The Theater of Plants, or, An Herbal of a Large Extent.

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 : an odd prime p is expressible as the sum of two squares.

The micrometer is developed.
A form of bayonet is invented; in later years it will gradually replace the pike.
The reticle telescope is developed and initiates the birth of sharpshooting.