Now I just found out that Newton wrote more about limits than we're usually led to believe. In 1687, Newton wrote:

"Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...."

This quote comes from Bruce Porciau's paper, Newton and the Notion of Limit, in Historia Mathematica. He gives much more evidence that Newton understood the limit concept pretty well.

I guess I can still say that it took the best minds in all the world 150 years to come up with a precise definition of limit. But Bishop Berkeley's complaint ...

"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"... now seems to me more the product of a small mind and less the careful quest for precision of a mathematician. Now I lean more toward thinking Newton (and Leibniz?) got it, but it took 150 years for a mathematician to create a precise definition that would convince all the other mathematicians.

Regarding Leibniz, see this article from the Notices: http://www.ams.org/notices/201307/rnoti-p886.pdf where they contend Leibniz "got it" too. However, one should point out that apparently Newton didn't do a very good job of explaining any of this to his contemporaries, even if he "got it"!

ReplyDeleteThank you! That looks good! (I've read the first page or two so far...)

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