A fascinating article just appeared in the New York Times Science section and concurrently in Science magazine. There is suddenly compelling evidence that ancient Babylonians living in modern day Iraq used concepts of Calculus to compute the orbit of the planet Jupiter on the sky.
If this result holds, then it would mean the concept of computing areas under curves, or integration, was discovered and in use 1400 years earlier than previously thought. Interestingly, the application for this was not terrestrial, but astronomical, so astronomy may have formed the motivation for developing the mathematical concepts central to Calculus!
According to the information derived from fragments of cuneiform tables, these people had the concept of how how to graph the time dependence of various quantities. For example, they could plot speed on the y-axis and time on the x-axis. Further, they knew that a curve of speed as a function of time had a particular use. If they would sum up the area under the curve, then the result would be a measure of distance!
You can try this yourself by, for example, timing how long it takes to jog 1 mile. Let us say for simplicity, and so you do not actually have to do this today, that you can jog 1 mile in 10 minutes, or 0.1 miles/minute. Let us further say that you can jog 2 miles in 20 minutes, or 0.1 miles/minute, and 3 miles in 30 minutes or 0.1 miles/minute. Now if you make a graph (go ahead!) of velocity
on the y-axis and time on the x-axis, you can put on the graph the following coordinates (x,y) = (10, 0.1), (20, 0.1), and (30,0.1). If you draw a line through the points, which shape does it make (Answer: it should be a straight horizontal line). Finally, if you sum up the area under the curve from 0 to say 20 minutes (which is just a rectangle) you get: area = length x width = 20 min x 0.1 miles/minute = 2 miles. And indeed recall that we had assumed that in 20 minutes we could run 2 miles.
These ancient people had this concept well understand. Further, they used this concept to track the orbit of Jupiter across the sky in a way that was not known ever to the ancient Greeks or other ancient peoples. So did the mathematicians of the middle ages independently discover this concept, or did they borrow it? This is not known, although time will tell and comments/suggestions from any experts out there are very welcome.