Now, about that infinitesimals stuff. I made some scribbles on
paper when I prepared the lecture, but never cared to type them into the
computer since it's sort of a classical material anyway. There are a few
books that you can consult. The simplest  treatment is in
Lectures on the Hyperreals : An Introduction to Nonstandard Analysis
(Graduate Texts in Mathematics, Vol 188) 1998, Springer. by Robert Goldblatt.
The theories of reals and hyperreals are discussed in the  first 3 very
short chapters of this book. The exposition is very elementary and 
accessible, close in spirit to the explanations I tried to give in my
presentation. There is also a recent translation of a book originally
published in Russian by my former high school classmate Oleg Ivanov,
the Russian title evades me, but the English title is "Easy as Pi"
(Springer 1999). There is also the first chapter of Stroyan and Luxemburg,
Introduction to Infinitesimals (Academic Press, 1978) which is very
readable. As for the original treatment by Abraham Robinson in his
Nonstandard Analysis, 2-nd edition, 1974, reprinted by Princeton Landmarks
in Mathematics in 1996, it's a tough book, the whole treatment relies on
a rather technical and lengthy ( > 40 pages long) first chapter "Tools from
Logic", it's difficult to read unless you are familiar with the technical
aspects of formal logic, but you can find some details about who
Mr. Robinson was. Also this book is written by the guy who invented the
whole thing and it presents the idea the way it was discovered (which is
almost never the easiest way to present it). There is also a recent
biography by Joseph Warren Dauben, entitled Abraham Robinson, The creation
of nonstandard analysis, a personal and mathematical odyssey (Princeton
University Press, 1995). I guess you can find some info about him on the
net, but  at the time he invented the nonstandard analysis he was working
in formal set theory and formal logic.
	As for my presentation, it evolved from an e-mail discussion with
one of my former professors at Brandeis. He asked me how my algebraic
treatment of calculus would relate to the infinitesimals, but it turned
out that both of us had a rather superficial idea of what infinitesimals
are. So I looked up a definition in "A Panorama of Pure Mathematics" by
Jean Dieudonne and tried to make it more tangible. It turned out that
the algebraic approach is indeed related to the infinitesimals, but is
simpler.
	Here is  the simplest approach to infinitesimals that I could
think of. You may remember that the sequences converging to zero are
sometimes called infinitely small. These are almost the infinitesimals
in a sense that the true infinitesimals are the classes of sequences
that contain a sequence converging to zero. Let me explain where these
classes come from. Let us consider first the sequences of real numbers.
They form a ring, i.e. you can multiply, add and subtract them, but not
always divide by non-zero. The real numbers are considered as the constant 
sequences, the sequence is zero if it's identically zero, i.e. a_n=0 for
all n. Now we want to make a field out of this ring. How do we do that?
By declaring some of the sequences to be zero. It is necessary because
in our ring we can get ab=0 with neither a nor b  =0, and we want to get
rid of this trash, so if ab=0, one of them should be declared zero.
Surprisingly enough, there is a consistent way to do it, and after we do
it, the classes of sequences that differ by some sequence declared zero
form a field which is called a hyperreal field and it contains the
infinitesimals in addition to the ordinary real numbers, as well as
infinitely large numbers (the reciprocals of infinitesimals, they are
represented by the sequences converging to infinity), also every
hyperreal which is not infinitely large is infinitely close to an
ordinary real, i.e. it is an ordinary real + an infinitesimal.
	This construction is parallel to the construction or the reals
from the rationals given by Cantor. He started with the ring of the Cauchy
sequences of rationals and declared all the sequences that converge to zero
to be zero. The result is the reals.
	To continue the construction of hyperreals, let us consider the
zero sets of our sequences, i.e. the z(a)={i: a_i=0}, i.e. z(a) is the set
of indexes i for which a_i=0. It's clear that if ab=0 the union of z(a)
and z(b) is N (the set of all natural numbers), so: 

	(i) one of the sequences that vanish on 2 complementary sets
	    should be declared zero
also 
	(ii) if a is declared zero, ab should be declared zero too, no
	     matter what b is. 
and 
	(iii) if both a and b are declared zero, then a*a+b*b should
	      also be declared zero.

	Now the idea is to single out a bunch U of subsets X of N and to
declare that a=0 if and only if z(a) belongs to U. From our conditions
(i), (ii) and (iii) we can see that (exercise):

	(i) From 2 complementary sets one belongs to U
	(ii) Any set containing any set that belong to U, belongs to U.
	(iii) An intersection of any 2 sets belonging to U belongs to U.
Also
	(iv) we don't want an empty set to belong to U
because then everything becomes zero because every set contains
an empty set.

	Any family of sets that satisfies (ii)-(iv) is called a filter
(an example: the complements to the finite sets, it's called the Frechet
filter and it is used in the usual limit theory). If (i) holds, U is
called an ultrafilter (because you can add no more sets to it without
breaking it). The only explicitly known example of an ultrafilter is
the family of sets containing a given element (in our case, say, the
number 10) such ultrafilters are called trivial, and if we use it in
our construction, we come back to the ordinary real numbers (exercise). 
Any ultrafilter containing a finite set is trivial (exercise).
It is known that any filter can be extended to an ultrafilter,
but the proof uses the axiom of choice. The existence of a nontrivial
ultrafilter can be added as an extra axiom, it's weaker than the axiom
of choice (that says that for any bunch of nonempty sets there is a
function that picks an element from any of them, f(X) is an element
of X).
	Now if we take a nontrivial ultrafilter (which is an extension of
the Frechet filter, exercise) and do our construction, we get the hyperreal
numbers as a result. The infinitesimals can be represented by the
non-vanishing sequences converging to zero in the ordinary sense, i.e.
with respect to the Frechet filter (exercise).
	If f is a real function of a real variable x then f naturally
extends to a hyperreal function of a hyperreal variable by composition:

f({x_n})={f(x_n)} where {...} means "the class of ... relative to our
ultrafilter".

	All the arithmetical expressions and formulas make sense for
hyperreals and hold true if they are true for the ordinary reals
(exercise, use (i)). You can prove that any finite (i.e such that
|x| < a for some ordinary real a) hyperreal x will be of the form y+d
where y is an ordinary (called standard) real and d is an infinitesimal.
It's parallel to the proof of the Bolzano-Weierstrass lemma that says
that you can pick a convergent subsequence form any bounded sequence,
done by bisection, the property (i) of the ultrafilters is again crucial.
	I guess now you can see that f is continuous means that f(a)-f(x)
is infinitely small whenever x-a is and f is differentiable means that
(f(x)-f(a))/(x-a)-f'(a) is infinitely small whenever x-a is. The funny
thing is that if we allow a to be hyperreal then the derivative will be
automatically continuous (because f'(x)-(f(x)-f(a))/(x-a) is infinitely
small when x-a is, therefore f'(x)-f'(a) is also infinitely small).

	All we did, really -- we took the ring of real sequences and
stamped out the zero divisors. It turned out to be a neat way to sweep
the limits under the rug.
	You can also see that if we can divide f(x)-f(a) by x-a as the
algebraic expressions or as continuous functions, then we don't have to
worry about all these fine points involving the infinitesimals. This is
the algebraic approach.
	Anyhow, I don't expect the algebraic approach to become popular
any time soon. The most I hope for is that it will become just another
way to explain the subject, and it looks like it will be easier for some
people to understand it this way because it is more flexible and doesn't
require outright the sophisticated tools from analysis.