When I wrote earlier this week about Archimedes, I didn’t get into why I found the Nova piece so remarkable. I loved it in part because I remember the “aha” I had as a youngster when I got the notion of integration – of summing across a series of infinite slices.
Malcolm Tredinnick wrote to point out that this is in fact something that’s been invented multiple times – Egyptian and Chinese methematicians had at least part of the insight as well:
…. Some of the ancient Egyptian scrolls (two
famous ones are the Rhind — or Ames — papyrus and the Moscow papyrus) included various mathematical algorithms and puzzles. These included things like computing the volume of a pyramid. My recollection is that historians are not completely sure *how* the Egyptians discovered some of their algorithsm; just that they did write them down.
The Chinese, on the other hand, wrote down their working as well as the conclusions. They also knew how to compute things like the volume of a pyramid, which they did by repeated dissections and then taking the leap that the infinite series of dissections converged to something sensible (that was really Newton and Liebnizs’ breakthrough: they realised both that there was a question to be settled about the convergence of such series and they provided reasonably rigorous answers). The Chinese also appear to have come up with an early version of approximating a circle with succesively more sided polygons and computing the area that way. Archimedes did it better than they did, though.
Malcolm’s right when he points out that we should be careful not to “look through rose-colored historical glasses” and infer more about their early insight than is warranted. But it’s still cool.