Johann Carl Friedrich (Gauss):
Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. Gauss was the premier number theoretician of all time, Other contributions of Gauss include hypergeometric series, foundations of statistics, and differential geometry. He also did important work in geometry, providing an improved solution to Apollonius' famous problem of tangent circles, stating and proving the Fundamental Theorem of Normal Axonometry, and solving astronomical problems related to comet orbits and navigation by the stars. (The first asteroid was discovered when Gauss was a young man; he famously constructed an 8th-degree polynomial equation to predict its orbit.) Gauss also did important work in several areas of physics, and invented the heliotrope.
Archimedes:
Archimedes' methods anticipated both the integral and differential calculus. He was similar to Newton in that he used his (non-rigorous) calculus to discover results, but then devised rigorous geometric proofs for publication. His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it). Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb.
Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day, though Apollonius soon surpassed it. That Archimedes shared the attitude of later mathematicians like Hardy and Brouwer is suggested by Plutarch's comment that Archimedes regarded applied mathematics "as ignoble and sordid ... and did not deign to [write about his mechanical inventions; instead] he placed his whole ambition in those speculations the beauty and subtlety of which are untainted by any admixture of the common needs of life."