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

• John Parkinson publishes Theatrum Botanicum:The Theater of Plants, or, An Herbal of a Large Extent.^[1]

• 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.

• 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.

• April 1 – Georg Mohr, Danish mathematician (died 1697)
• December 13 (bapt.) – Robert Plot, English naturalist and chemist and illustrator of the first dinosaur fossil (died 1696)
• Elias Tillandz, Swedish physician and botanist in Finland (died 1693)

• December 22 – Jean de Beaugrand, French mathematician (born c. 1584)