[math-fun] yet another calculus question
Another question that came up today is, why introduce Riemann sums at all? "It looks like Newton got by just fine without them, if he died before Riemann was born!" Of course, Newton could have used specific Riemann sums without the full general picture Riemann developed, but it looks as if the student may have been right: I just poked around a little bit on the web to see how Newton defined integration, and was a chagrined to learn that he *defined* the definite integral of f(x) from x=a to x=b as F(b)-F(a), where F is a primitive (also known as antiderivative) of f! But surely this can't be the whole story. After all, Newton used the calculus to show that the attraction between two balls is the same if you replace either or both by a point-mass, and my impression is that he did it by dividing the balls into spherical shells, dividing the shells into bands, etc. That sounds like a (multi-dimensional) Riemann sum to me (albeit perhaps a non-rigorous one, if Newton used infinitesimal elements)! So, what's the deal with these sums? Who first used them, and who first tried to make them rigorous? If I'm hogging too much bandwidth with calculus stuff, or if this latest more-historical-than-theoretical question is inappropriate for this forum, let me know. Jim Propp
On Feb 8, 2008 9:02 AM, James Propp <jpropp@cs.uml.edu> wrote:
But surely this can't be the whole story. After all, Newton used the calculus to show that the attraction between two balls is the same if you replace either or both by a point-mass, and my impression is that he did it by dividing the balls into spherical shells, dividing the shells into bands, etc. That sounds like a (multi-dimensional) Riemann sum to me (albeit perhaps a non-rigorous one, if Newton used infinitesimal elements)!
I think Newton's argument, if I recall correctly (I don't have my Principia handy), was more like "Each shell acts like a point at the center, therefore the whole ball does" and for each shell, it was something to do with finding corresponding areas that together acted like a central point -- yes, probably with infinitesimals, and with the two rings of the shell that are cut by two concentric cones, if that's a clear enough picture of the diagram. He certainly had the idea that an antiderivative is an integral, even if he didn't explicitly show an integral as a limit of a sum but reasoned infinitesimally instead. By the way, speaking of calculus history, I highly recommend _The Archimedes Codex_. A fun read, coauthored by a historian of mathematics and a conservator of medieval manuscripts. (http://www.stanfordalumni.org/news/magazine/2007/sepoct/features/archimedes.... has some more info) --Joshua Zucker
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