Practically speaking, you can accomplish a surprising amount using only geometry, basic mathematics and a few choice rules of thumb. Eventually, though, the need to calculate trickier matters — rates of change, curvatures, areas under curves — demands that you develop mathematics that can handle them. This is where calculus comes in.
Today, we credit its development to both Isaac Newton and Gottfried Wilhelm Leibniz, both of whom developed it independently around the turn of the 18th century, but in truth the world's intellects had been flirting with the idea for millennia. Let's start by returning briefly to the ancient Greeks. In their mathematical heyday, they produced Eudoxus (c. 408-355 BC), whose method of exhaustion edged right up to the notion of the limit, and Archimedes (c. 287-212 BC), who came up with practical methods of calculation that bore a family resemblance to integral calculus. Exhaustion would later spring from the mind of fifth-century Chinese mathematician Liu Hui, too, and similarly significant strides were taken among medieval Indian thinkers [sources: Boyer, SLU]
These forerunners to calculus, like so many advances from antiquity that were lost with the fall of Rome, hint at the knowledge that was lost to Europe when it entered the Dark Ages. But as ideas, they also went begging in their own time because their potential was never fully realized [sources: Boyer, SLU].