The Calculus describes a set of powerful analytical techniques, including differentiation and integration, that utilize the concept of a limit in the mathematical description of the properties of functions, especially curves. The formal development of the calculus in the later half of the 17th century, primarily through the independent work of English physicist and mathematician Sir Isaac Newton (1642-1727) and German mathematician Gottfried Wilhelm Leibniz (1646-1716), was the crowning mathematical achievement of the Scientific Revolution. The subsequent advancement of the calculus profoundly influenced the course and scope of mathematical and scientific inquiry.
Important mathematical developments that laid the foundation for the calculus of Newton and Leibniz can be traced back to mathematical techniques first advanced in Ancient Greece and Rome. In addition to existing methods to determine the tangent to a circle, the Greek mathematician and inventor Archimedes (c.290-c.211B.C.), developed a technique to determine the tangent to a spiral, an important component of his water screw.
The majority of other ancient fundamental advances ultimately related to the calculus were concerned with techniques that allowed the determination of areas under curves (principally the area and volume of curved shapes). In addition to their mathematical utility, these advancements both reflected and challenged prevailing philosophical notions regarding the concept of infinitely divisible time and space. Two centuries before the work of Archimedes, Greek philosopher and mathematician Zeno of Elea (c.495-c.430 B.C.) constructed a set of paradoxes that were fundamentally important in the development of mathematics, logic and scientific thought. Zeno's paradoxes reflected the idea that space and time could be infinitely subdivided into smaller and smaller portions and these paradoxes remained mathematically unsolvable in practical terms until the concepts of continuity and limits were introduced. more