Today I thought it would be nice to investigate Newton's method of
fluxions, since all that most people are taught is the calculus of
Leibniz. Surprisingly, though, I didn't find much in the way of
explanation online. I did find Newton's first work on the method, and
his main summary.

http://books.google.com/books?id=1ZcYsNBptfYC&pg=PA400
http://www.archive.org/stream/methodoffluxions00newt

But these are difficult to read for the layperson and I expected
several tutorials on the method. Instead I didn't find any. The
closest sites that I found so far to giving any kind of explanation
are:

http://66.102.9.132/search?q=cache:http://school.maths.uwa.edu.au/~schultz/3M3/L20Newton.html
http://www.math.rutgers.edu/~cherlin/History/Papers1999/kijewski.html
http://cll.mcmaster.ca/multimedia_projects/sample/newton/fluxions.htm

There is also a more advanced description of the theory here:

http://www.math10.com/en/maths-history/history5/origins-differential-integral2.html

One of these articles mentions a book which is said to explain it quite well.

http://openlibrary.org/b/OL7780301M/The_Newton_Handbook
The Newton Handbook
by Gerek Gjertsen
Published in February 1987, Routledge & Kegan Paul Books Ltd

But it appears to be out of print.

One of the most interesting quotes in this connexion is as follows:

'The first edition claimed "Moments, as soon as they are of finite
magnitude, cease to be moments. To be given finite bounds is in some
measure contradictory to their continuous increase or decrease." On
the other hand, the second version read "Finite particles are not
moments, but the very quantities generated by moments." Apparently
Newton himself struggled with the meaning of it all, just as Leibniz
struggled with his ideas of differences and "differentials".'
(kijewski.html)

Any modern explanation of the method of fluxions would presumably have
to patch up such difficulties by using modern understanding and
techniques. For example, "there was a point of contention which
brought much criticism by other men of science, and this was the way
in which Newton divided by a small nonzero finite quantity o, and
then, a few lines later, set it equal to zero", (ibid.) which is what
would now be solved using a limit. Newton thought this part was
intuitive, but some might say he missed out an important step.

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