Archimedes is one of the greatest mathematicians of all time. He became a popular figure as a result of his involvement in the defense of Syracuse against the Roman siege in the first and second punic wars. He is reputed to have held the Romans at bay single-handedly with war engines of his design; to have been able to move a full-size ship complete with crew and cargo by pulling a single rope; to have discovered the principle of buoyancy while taking a bath, taking to the streets naked calling "eureka" (I found it!); and to have invented the irrigation device known as Archimedes' screw.
In creativity and insight, he exceeds any other mathematician prior to the European renaissance. In a civilization with an awkward number system and a language in which "a myriad" (literally ten thousand) meant "infinity", he invented a positional numeral system and used it to write numbers up to 1064. He devised a heuristic method based on statics to do private calculation that we would classify today as integral calculus, but then presented rigorous geometric proofs for his results. To what extent he actually had a correct version of integral calculus is debatable. He proved that the ratio of a circle's perimeter to its diameter is the same as the ratio of the circle's area to the square of the radius. He did not call this ratio π but he gave a procedure to approximate it to arbitrary accuracy and gave an approximation of it as "exceeding 3 in less than 1/7 but more than 10/71". He was the first, and possibly the only, Greek mathematician to introduce mechanical curves (those traced by a moving point) as legitimate objects of study, and used Archimedes' spiral to square the circle. He proved that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. (This proposition must be understood as follows. The "base" may be taken to be a secant line of the parabola, not necessarily orthogonal to the axis of the parabola, but one must construe the word "base" in the formula to mean the component of its length in a direction orthogonal to the axis of the parabola, ignoring the component parallel to the axis; the "height" is the length of a segment parallel to the axis of the parabola, running from the midpoint of the base to the curve.) He proved that the area and volume of the sphere are in the same ratio to the area and volume of a circumscribed straight cylinder, a result he was so proud of that he made it his epitaph.
Archimedes is probably also the first mathematical physicist on record, and the best before Galileo and Newton. He invented the field of statics, enunciated the law of the lever, the law of equilibrium of fluids and the law of buoyancy. He gave the equilibrium positions of floating sections of paraboloids as a function of their height, base area and density using only Greek geometry, a feat that would be taxing to a modern physicist using calculus. He was the first to identify the concept of center of gravity. He found the centers of gravity of various geometric figures, assuming uniform density in their interiors, including triangles, paraboloids, and hemispheres.
Archimedes' works were not very influential, even in antiquity. He and his contemporaries probably constitute the peak of Greek mathematical rigour. Many of his works were lost when the library of Alexandria was destroyed and survived only in Latin or Arabic translations. During the middle ages the mathematicians who could understand Archimedes' work were few and far between. Also, his "mechanical method" was lost until around 1900, after the arithmetization of analysis had been carried out successfully. We can only speculate about the effect that the "method" would have had on the development of calculus had it been known in the 16th and 17th centuries.
In this work, which was thus unknown in the Middle Ages, but of which the importance was realised after its discovery, Archimedes pioneered the use of infinitesimals, showing how breaking up a figure in an infinite number of infinitely small parts could be used to determine its area or volume. Archimedes did probably consider these methods not mathematically precise, and it is assumed that he used these methods to find the laws of geometry, then used more traditional methods to prove them. This particular work is found in what is called the Archimedes Palimpsest. Some details can be found at how Archimedes used infinitesimals.