\magnification=\magstep1
\def\start{\noindent}
\def\ssc{\bigskip\noindent{\it Some specific comments}\bigskip}
\def\finish{\bigskip\noindent{\it Grade:}\vfill\eject}
\start
Stephen Loewenthal -- Leibniz vs. Newton
\bigskip
This is a generally good summary of the contributions
of Newton and Leibniz to the development of calculus and
the ``priority dispute'' that erupted over the question
of who should get primary credit for the discovery of these
ideas.  Your writing is good although you need to be a bit
more careful about mechanics or proofreading at some points 
(plurals vs. possessives, etc.)  

One point I think you underestimate is the extent
to which Newton's work on the calculus actually came from his
studies of physics -- in particular his derivation of 
Kepler's laws for the motions of planets from the 
inverse square law for gravitation.  It is true that
he did not explain what he did using the language of 
calculus in his {\it Principia Mathematica}, but I think
most historians believe that he worked out his derivations
using fluxions and fluents, then translated them into 
a more traditional, purely geometric language for publication.
He may have done that to avoid the criticism that would 
probably have resulted from the unfamiliarity of the methods.  
And, it is probably also true that Newton's avoidance of
publishing these ideas in his own work until much later
probably was a major contributing factor in the priority
dispute -- the continental Europeans did not want to 
give Newton credit because they didn't know it, or
because they considered that Newton had
never subjected his work to the public scrutiny 
Leibniz had.


Newton also applied those methods
to things like the derivation of the binomial series
(which is the same as our Taylor series:
$$(1+x)^p= 1 + px + {p(p-1)\over 2}x^2 + {p(p-1)(p-2)\over 3!}x^3 + \cdots $$
by the way -- it would have been good to point that out!)  But
I don't think it is really accurate to focus on that as the 
main impetus for the development of Newton's ideas on calculus.

\ssc
\item{1)}  I'm not sure what you mean by saying that
as a result of Newton's and Leibniz's studies on series, 
they were no longer just ``approximating tools''.  
This is not very clear.  Do you mean to say that their
work focused peoples' attention on infinite series
``for their own sakes,'' as algebraic objects in their
own right?  I think that might be a rather anachronistic
point of view (the focus on series as a primary way
to represent functions really only came into its own
through work of later mathematics like Euler, Lagrange, 
Cauchy, and others.)
\item{2)}  Like many of the other people in this story,
Huygens was not only a mathematician.  He was also a 
physicist who studied optics, made lenses for telescopes
and microscopes, etc.  Interestingly, Leibniz himself
was probably atypical in that regard.  His ``outside''
interests were mainly philosophical rather than what
we would call scientific.
\item{3)}  Your description of the after-effects of
the Newton-Leibniz controversy on British mathematics
is good, but you should also be aware that the isolation
that British mathematicians imposed on themselves in 
time led to a very original approach to algebra, logic,
and other subjects by later British mathematicians in
the 19th century.  Isolation from prevailing fashions
is not always all bad!
\item{4)}  It would have been good to say more about
this to tie it into what we did in this course.  Mention
the ${dy\over dx}$ notation for derivatives, the 
$\int f(x)\ dx$ notation for integrals, etc. in more 
detail.  Ironically, the one place where Newton's 
fluxion notation survives is in some physics books
and papers where the $\dot{x}$ notation is still
used for the derivative of $x$ with respect to time.
\finish
\start
Micki Rupon -- The Newton-Leibniz Debate
\bigskip
This is a good summary of the Newton-Leibniz debate
and the contributions that both men made to the development
of calculus.  The main place for improvement, I think, 
would be to say a bit more explicitly about how the
each of their ideas influenced things we did in this
course.  For instance, you could have mentioned that
we used the ${dy\over dx}$ notation for derivatives, the 
$\int f(x)\ dx$ notation for integrals, etc. from Leibniz.  
Ironically, the one place where Newton's 
fluxion notation survives is in some physics books
and papers where the $\dot{x}$ notation is still
used for the derivative of $x$ with respect to time.

A technical comment: usually when you make use
of a direct quotation in a paper like this, you should
give not only the bibliography item you took the 
quotation from, but also a page number or some other
more specific reference.
\ssc
\item{1)} What you are referring to here is 
Newton's development of the general binomial series,
which is the same as our Taylor series:
$$(1+x)^p= 1 + px + {p(p-1)\over 2}x^2 + {p(p-1)(p-2)\over 3!}x^3 + \cdots $$
It would have been good to point out that connection.
In the case where $p$ is a positive integer, the series
terminates after a finite number of terms.  That form
of the binomial formula had been known long before Newton.
The new thing here was the inclusion of the cases where
$p$ was not a positive integer.
\item{2)}  I think you mean to say Leibniz's proof of
the indefinite integral formulas for 
$$\int \sqrt{a^2 \pm x^2}\ dx$$
(like the entries in our table of integrals!).  What you 
said seems like an incomplete thought.
\item{3)}  It was ``unfair'' primarily because Newton 
himself was an officer of the Royal Society and picked
the investigative committee!
\item{4)}   Your description of the after-effects of
the Newton-Leibniz controversy on British mathematics
is good, but you should also be aware that the isolation
that British mathematicians imposed on themselves in 
time led to a very original approach to algebra, logic,
and other subjects by later British mathematicians in
the 19th century.  Isolation from prevailing fashions
is not always all bad!
\item{5)}  This sort of priority dispute has happened
many other times as well in science (for instance there's
one in full swing right now over who should get credit
for the development of the idea of MRI medical imaging,
and who should have received the Nobel Prize given for 
that this very year!).  Maybe these
disputes do look ``trivial'' to outsiders, but when the
development of new ideas is the primary way that scientists'
careers are evaluated, giving credit where credit is due
becomes a very personal and emotional thing to the people
involved.
\finish
\start Lauren Stillwell -- Newton vs. Leibniz
\bigskip
This is a good summary of the Newton-Leibniz debate
and the contributions that both men made to the development
of calculus.  The main place for improvement, I think, 
would be to say a bit more explicitly about how the
each of their ideas influenced things we did in this
course.  For instance, you could have mentioned that
we used the ${dy\over dx}$ notation for derivatives, the 
$\int f(x)\ dx$ notation for integrals, etc. from Leibniz.  
Ironically, the one place where Newton's 
fluxion notation survives is in some physics books
and papers where the $\dot{x}$ notation is still
used for the derivative of $x$ with respect to time.
\ssc
\item{1)}  This is a good quotation!  But you should
identify the source.
\item{2)}  What you are referring to here is 
Newton's development of the general binomial series,
which is the same as our Taylor series:
$$(1+x)^p= 1 + px + {p(p-1)\over 2}x^2 + {p(p-1)(p-2)\over 3!}x^3 + \cdots $$
It would have been good to point out that connection.
In the case where $p$ is a positive integer, the series
terminates after a finite number of terms.  That form
of the binomial formula had been known long before Newton.
The new thing here was the inclusion of the cases where
$p$ was not a positive integer.
\item{3)}  Since we never discussed ``quadratures'' it
would have been good to explain this term in more 
detail.  In our terms, ``quadrature'' is essentially
an old-fashioned synonym for definite integration
to find an area -- the actual area between the graph
and the $x$-axis.  The connection with rectangles is
that you can think of the result of 
integration $\int_a^b f(x)\ dx$ to find an area
as equivalent to the determination of a rectangle
that has the same area as the region under the 
graph $y = f(x)$ -- this is the same as the idea
of the formula 
$$\overline{y} = {1\over b - a} \int_a^b f(x)\ dx$$
for the {\it average value} of $f(x)$ on the interval
$a \le x \le b$ that we talked about in class.
\item{4)}  It would have been good to say more about
this, since it is a nice development of things
we have used a lot in this course.  The ``previous
formula'' is just the product rule for derivatives:
$${d\over dx}(y_1y_2) = y_1{dy_2\over dx} + y_2{dy_1\over dx}$$
The general product rule comes from differentiating this
repeatedly.  For instance for the second derivatives:
$${d^2\over dx^2}(y_1y_2) = y_1{d^2y_2\over dx^2} + 2
{dy_1\over dx}{dy_2\over dx} +  y_2{d^2y_1\over dx^2}$$
and then similarly for all higher-order derivatives.  The
coefficients that appear here are the same as the binomial
coefficients appearing in the expansion of $(a+b)^n$ 
for positive integer $n$.
\item{5)}  Your description of the after-effects of
the Newton-Leibniz controversy on British mathematics
is good, but you should also be aware that the isolation
that British mathematicians imposed on themselves in 
time led to a very original approach to algebra, logic,
geometry, and other subjects by later British mathematicians in
the 19th century.  Isolation from prevailing fashions
is not always all bad!
\item{6)}  I'm not sure exactly what you mean by
``analytical'' here.  If you mean, ``more mechanical, 
hence easier to use'' then I think you're onto something
interesting.  Leibniz's notation for calculus was 
far more successful than Newton's ultimately because
it facilitated understanding, even for those who were
not Newtons or Leibnizes themselves(!)  
\finish
\start
Kristie Giannetto -- Newton vs. Leibniz
\bigskip
Your paper on the Newton-Leibniz controversy contains
a lot of good information and your conclusions are 
correct, I think.  But I think the organization of
the paper is not ideal -- at several points you tend to introduce a 
good idea briefly, then talk about something else,
then return to the original idea at greater length.
In a paper of this length, that's not a very good 
strategy because it tends to dilute the points you
are making.  I also think you could have done more
to talk about specific connections between the mathematics
that Newton and Leibniz did and topics we discussed
in this course.  For instance, you might have noted
Newton's work on infinite series and the general 
binomial series he developed:
$$(1+x)^p= 1 + px + {p(p-1)\over 2}x^2 + {p(p-1)(p-2)\over 3!}x^3 + \cdots $$
which is the same as one of our examples of Taylor series(!)
The discussion of Newton's fluxions and fluents could
also have made a connection with ideas about parametric
curves as we discussed them briefly.  (Indeed, I think
much of the ultimate explanation for the way Newton
approached calculus really came from the fact that 
he was thinking of applications to description of 
motion in physics.)
\ssc
\item{1)}   This analysis of the after-effects of
the Newton-Leibniz controversy on British mathematics
is good, but you should also be aware that the isolation
that British mathematicians imposed on themselves in 
time led in time to a very original approach to algebra, logic,
parts of geometry, and other subjects by British mathematicians 
later in the 19th century.  In time, this became a big part of 
20th century mathematics.  Isolation from prevailing fashions
is not always all bad!  (Organizational point:
You continue this discussion later
on page 5 of the paper -- it might have been better to 
consolidate those two sections; as it is, your paper
seems to be rather repetitive and not efficiently organized.) 
\item{2)}  I think another (maybe the most important?) reason
that Leibniz's approach eventually ``won out'' was that it
was just a lot easier to use.  You didn't have to be 
a Newton or Leibniz yourself to learn how to compute
derivatives and integrals using Leibniz's formulation
and notation!  You get to this point in the next paragraph,
but again, the organization of your paragraphs blunts
what could have been a very good discussion.
\item{3)}  Interestingly enough, Newton 
himself was an officer of the Royal Society and picked
the investigative committee!  Does that sound like a
fair investigation?
\finish
\start
Matthew Frigault -- Newton vs. Leibniz: Plagiarism or Coincidence?
\bigskip
This is a good summary of the Newton-Leibniz debate about
the origins of calculus.  I agree with your differentiation
between Newton's physical approach and Leibniz's more 
general mathematical approach.  The major place for 
improvement I see is that it would have been good to 
say more about specific relations between things we 
did in this course and the contributions of the two
men.  See comment 4 below for one suggestion.  Other
possibilities would have been to mention some of 
Newton's work on what we now call Taylor series (in particular
the binomial series we studied just recently), some
of Leibniz's integration formulas which closely
parallel entries in our table of integrals, etc.
\ssc
\item{1)}  I'm not sure why you say this is ``unfortunate''.
Part of the reason for publication is so that a review of the
ideas and methods being proposed in the new work can be made 
by other scientists.  Subjecting your work to that kind
of scrutiny is a way of checking its accuracy and establishing 
that it is of interest to others.  Newton's reluctance to publish can
be seen as selfish or even egotistical.  For instance, there is evidence
he thought his ideas would be misunderstood and he didn't
want to be bothered with trying to explain them to others
he considered his scientific inferiors.  I don't find that
very admirable personally!
\item{2)}  The main subject of {\it Principia Mathematica}
was actually Newton's derivation of Kepler's Laws of 
planetary motion from the inverse square law of gravitation.
Moreover, although most people think he developed all that
using calculus, he did not present it that way -- he 
relied mostly on traditional Euclidean geometry proofs.
\item{3)}  It's not surprising in a sense that neither
Newton nor Leibniz specialized exclusively in mathematics.
There was far less specialization in general at this time
and there were only a handful of university professorships
in existence, so the time when one could make a career
solely as a mathematician was still largely in the future.
\item{4)}  Note how much this sounds like the idea of 
parametric curves as we discussed them in this course(!)
It would have been good to point out that connection.
\item{5)}  This is not very clear.  I think what you 
want to say is that the process of integration for 
Newton is the ``inverse process'' of taking differences.
\item{6)}  Don't forget that ``our experience'' (I presume
you mean by this the way we discussed the derivative
as the limit
$$f'(x) = \lim_{h\to 0} {f(x+h) - f(x) \over h})$$
profits from the later developments of calculus by 
many generations of people after Newton and Leibniz!
It is typical in the history of mathematics for 
the intuition of how something should work (and even
methods for using it effectively) to come {\it before}
the precise understanding of how and why the 
idea works! 
\item{7)}  I think another (maybe the most important?) reason
that Leibniz's approach and notation 
eventually ``won out'' was that it
was just a lot {\it easier to use}.  You didn't have to be 
a Newton or Leibniz yourself to learn how to compute
derivatives and integrals using Leibniz's formulation
of calculus and the ${dy\over dx}$ and $\int y\ dx$
symbolism!   It's true that British mathematicians
tended to isolate themselves out of a sense of loyalty
to Newton, but you should also be aware that that
self-imposed isolation in 
time led in time to a very original approach to algebra, logic,
parts of geometry, and other subjects by British mathematicians 
later in the 19th century.  In time, this became a big part of 
20th century mathematics.  Isolation from prevailing fashions
is not always all bad!
\item{8)}  This is overstated.  I think most historians
now would agree with what you say elsewhere: Newton
had these ideas first but did not publish them.
In the meantime, Leibniz also developed many very
similar ideas and did publish them.  Leibniz's
name is not ``tarnished'' now.
\finish
\start
P.J. Brennan -- Bernhard Riemann
\bigskip
This is a reasonably good summary of Riemann's life and 
of some of his mathematical work most closely
related to things we have discussed in this course.
There is one major organizational problem though -- 
see comment 4 below.  
\ssc
\item{1)}  Be careful -- you are looking at 
this from a present-day point of view.  At Riemann's time,
this was probably within 5 years of the
average life expectancy in Europe.  Also, there were a 
number of other very famous and influential mathematicians
 in the first half
of the 19th century who died much earlier than Riemann.  
The French mathematician \'Evariste
Galois, and the Norwegian mathematician Niels Abel
are two examples.  
\item{2)}  It's good that you tried to include a discussion
of exactly what Riemann's ideas in geometry were, but
what's here doesn't really do them justice.  Here's
a more complete description of what's involved.  
For a parametric curve $(x(t),y(t))$ in the plane,
there is an arc-length formula that is similar to
what we did for graphs of functions:
$$s = \int_a^b \sqrt{\left({dx\over dt}\right)^2+\left({dy\over dt}\right)^2}\ dt$$
We can write
$$ds^2 = dx^2 + dy^2 = \left({dx\over dt}\right)^2+\left({dy\over dt}\right)^2$$
for the term under the square root.  Then the arc-length is
$s = \int_{t=a}^{t=b} ds$.  
From earlier work of Gauss, Riemann knew that there were similar
formulas for arclengths of curves on other surfaces, with
$$ds^2 = A dx^2 + B dx dy + C dy^2$$
where $A,B,C$ are functions of the coordinates $x,y$ on
the surface whose form depends on what the surface is.  Riemann's
idea was to base general geometry in any dimension on the form
of this so-called ``metric'' -- the $ds^2$ involved in the 
arclength integral.  The general form of a Riemannian metric
is 
$$ds^2 = \sum_{i,j=1}^n g_{ij} dx_i dx_j$$
Einstein then used this idea to describe the geometry
of the 4-dimensional space-time in his theory of general
relativity.
\item{3)}  It would have been good to mention exactly
what the Riemann hypothesis says -- namely that all the 
``non-trivial'' zeroes of the $\zeta(s)$ function occur at points
of the form $s = {1\over 2} + it$.  This is probably the 
most famous unsolved problem in mathematics at present.
It is one of the Clay Mathematics Institute's prize problems
(anyone coming up with a solution that is published and
not challenged for a period of 2 years will will a
million-dollar prize!)
\item{4)}  It would have been better from an organizational
point of view to combine these two portions of the 
discussion of Riemann's theory of integration.  Splitting
it this way makes the whole paper seem disjointed.  Also,
integrals of functions which have vertical asymptotes
are rather subtle.  Some of them do ``converge'' to a finite
value, and that can be seen by using Riemann integrals
but taking a limit:
$$\int_0^1 {1\over\sqrt{\sin(x)}}\ dx = 
\lim_{a\to 0} \int_a^1 {1\over\sqrt{\sin(x)}}\ dx\doteq 2.035$$
Others do not converge.  Also, much more severely discontinuous
functions are interesting as well, and Riemann's integral 
is not ``the last word'' on those.  Other mathematicians
have developed even more sophisticated integration theories
that can handle functions like 
$$f(x) = \cases{1 & if $x$ is a rational number\cr
                0 & if $x$ is an irrational number\cr}$$
that are not continuous anywhere.  According to Riemann's
theory this function is not integrable over any interval,
but according to later integration theories like Leb\`egue
integration, it is.
\finish 
\start
Robert Molt -- The Power of Newton's Mathematics (in) Science:
Justification of Kepler's Laws
\bigskip
This is a very good presentation of (a modern version of)
Newton's derivation of Kepler's Laws from the inverse
square law for the gravitational force.  It is interesting
to note that Newton also showed the converse -- namely 
that if we assume Kepler's Laws hold, then the only 
pure power gravitational formula that can be the correct
one is the inverse square law.  The only criticism I have
is that the long string of formulas at the top of page
2 of the text should still be broken up and more 
explanation of each step should be included.  For instance,
one standard method is to number previous equations
and refer back to them as you use those results to 
substitute in later, etc.   But it 
is clear that you have worked out each step and understand
what is going on.  Good work!
\ssc
\item{1)}  You don't need this line I think.  Namely,
from the line before, since $u_r$ and $u_\theta$
are perpendicular, it follows from vector algebra 
that the coefficients of $u_r$ and $u_\theta$ on
both sides must be the same, i.e.
$$F_\theta = m\left(2{dr \over dt}{d\theta\over dt}+r{d^2\theta\over dt^2}\right)$$
and 
$$F_r = m\left(2{d^2r \over dt^2}-r\left({d\theta\over dt}\right)^2\right),$$
which is what you want.
\item{2)}  Since presumably $m \ne 0$, it's not really
necessary to put in all three of these equations.  It would 
be OK to skip a step and just write the last one.
\item{3)}  It would be better to say:  ``Now we will choose
our coordinate system to make $\theta =0$ correspond
to the perihelion point (i.e. minimum of $r$, or maximum of $z = 1/r$).
\finish
\bye



