Chronology

Albert Girard publishes a treatise on trigonometry containing the first use of the abbreviations sin, cos, tan. He also gives formulas for the area of a spherical triangle.

Fermat works on maxima and minima. This work is an early contribution to the differential calculus.

Oughtred invents an early form of circular slide rule. It uses two Gunter rulers.
Mydorge works on optics and geometry. He gives an extremely accurate measurement of the latitude of Paris.

Harriot's contributions are published ten years after his death in Artis analyticae praxis (Practice of the Analytic Art). The book introduces the symbols > and < for "greater than" and "less than" but these symbols are due to the editors of the work and not Harriot himself. His work on algebra is very impressive but the editors of the book do not present it well.
Oughtred publishes Clavis Mathematicae which includes a description of Hindu-Arabic notation and decimal fractions. It has a considerable section on algebra.

Descartes discovers Euler's theorem for polyhedra, $V - E + F = 2$ .
Cavalieri presents his development of Archimedes' method of exhaustion in his Geometria indivisibilis continuorum nova. The method incorporates Kepler's theory of infinitesimally small geometric quantities.

Fermat discovers the pair of amicable numbers 17296, 18416 which were known to Thabit ibn Qurra 800 years earlier.

Descartes publishes La Géométrie which describes his application of algebra to geometry.

Desargues begins the study of projective geometry, which considers what happens to shapes when they are projected on to a non-parallel plane. He describes his ideas in Brouillon project d'une atteinte aux evenemens des rencontres du Cone avec un Plan (Rough draft for an essay on the results of taking plane sections of a cone).

Pascal builds a calculating machine to help his father with tax calculations. It performs only additions.

Torricelli publishes Opera geometrica which contains his results on projectiles. He investigates the point which minimises the sum of its distances from the vertices of a triangle.

Fermat claims to have proved a theorem, but leaves no details of his proof since the margin in which he writes it is too small. Later known as Fermat's last theorem, it states that the equation $x^{n} + y^{n} = z^{n}$ has no non-zero solutions for $x, y$ and $z$ when $n > 2$ . This theorem is finally proved to be true by Wiles in 1994. (See THIS LINK.)
Cavalieri publishes Exercitationes geometricae sex (Six Geometrical Exercises) which contains in print for the first time the integral from 0 to $a$ of $x^{n}$ .

Wilkins publishes Mathematical Magic giving an account of mechanical devices.
Abraham Bosse publishes a work containing Desargues' famous "perspective theorem" - that when two triangles are in perspective the meets of corresponding sides are collinear.

Van Schooten publishes the first Latin version of Descartes' La géométrie.
De Beaune writes Notes brièves which contains the many results on "Cartesian geometry", in particular giving the now familiar equations for hyperbolas, parabolas and ellipses.