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{\it Abigail Luhn -- Geometry, the Foundation of Renaissance Art}
\bigskip
Good work on an interesting topic! This is an excellent paper on the way 
that geometry formed the basis
of several developments in Renaissance art including the use of linear 
perspective in drawing and painting.  You write very well and your
use of footnotes and citation is exactly correct.  I hope you found 
that looking into this subject was a rewarding complement to things
we discussed in the class.
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\noindent
{\it Specific Comments}
\bigskip
\item{1.}  You're right that Alberti's title recalls (and was probably
inspired by) the title of Euclid's most famous work.  It is, even now, still 
a tradition, of course, to name books intended as systematic introductions
to subject with titles of the form {\it Elements of ... }.  
\item{2.}  It might be better to say ``... all lines orthogonal to the picture plane'' 
here.  {\it Orthogonal} just means ``meeting at a right angle,'' so if you 
don't specify what the lines are meeting, it's not entirely clear what is meant.
\item{3.}  Leonardo also used a great deal of geometry in some of the 
architectural drawings he made, and is even sometimes credited as the 
originator of some results about mathematical symmetry for that reason.
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\noindent
{\it Grade:}  
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\noindent
{\it Hannah Smith -- The Mathematics of M.C. Escher's Artwork}
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\noindent
Very good paper on Escher and his mathematically-inspired artworks.  The 
quotation from Escher's writing about his periodic tilings of the plane
that we discussed in class:  
``Mathematicians have opened the gate leading to an extensive domain, 
but they have not entered this domain themselves. By their very nature 
they are more interested in the way in which the gate is opened than in 
the garden lying behind it''  is one of the clearest statements of how
he saw his work in relation to the mathematical ideas it embodies.  But 
sometimes people who create things don't understand all of the influences
that go into their work or the ways that what they create ``works'' on 
deeper levels.  See comment 6 below.

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\noindent
{\it Specific Comments}
\bigskip
\item{1.}  This is an interesting and subtle question.  Although Escher had
essentially no formal mathematical training beyond the high school level, 
he did come from a family of scientists and engineers.  Moreover, as you discuss later, 
he did have a number of contacts with professional mathematicians (including Poly\'a
and Coxeter) and tried
to incorporate what he learned from them (or what he intuited from examples
they showed him) into his works.  And I would say that many 
of his ``habits of mind,'' including most spectacularly the way he approached
his classification of periodic tilings of the plane, show similarities with
the ways mathematicians think.  So I don't think there is much argument
that he was trying to portray ideas that had a lot in common with mathematics.
\item{2.} I think it's also possible that he realized the transformations 
needed to continue the pattern into the center would produce elements of the 
image on such small distance scales that they would not be intelligible for
the viewer of the print.  So he might have ``punted'' for that reason too.
\item{3.}  This is a good observation, and many of Escher's other works
address this idea in other ways, too.  
\item{4.}  There is also a mathematical theory of knots that Escher may have
heard about from some of his mathematical contacts.  The knot that he
pictured is one of the simplest ones, called the {\it trefoil knot}.
Mathematicians have developed many other examples and classified the simpler
possible configurations of knots.
\item{5.}  As I said in class, I think many of Escher's most interesting 
works are the ones where regular subdivisions are used as elements of the 
overall design (as in {\it Reptiles} or {\it Night and Day}).  
\item{6.}  Even though Escher never claimed to be thinking mathematically, 
I think his art shows that he probably did not even understand himself
how saturated his art was with ideas coming from mathematics and with
mathematical ways of thinking.  Sometimes artists themselves do not understand
their work completely!
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\noindent
{\it Grade:}
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\noindent
{\it Luke Melody -- M.C. Escher and Mathematics}
\bigskip
Mostly good paper on an interesting and subtle question.  Although Escher had
essentially no formal mathematical training beyond the high school level,
he did come from a family of scientists and engineers.  Moreover, as you discuss,
he did have a number of contacts with professional mathematicians (including George 
Poly\'a -- whom you did not mention -- and H.S.M. Coxeter -- whom you did). 
He tried to incorporate what he learned from them (or what he intuited from examples
they showed him) into his works, but what he did was not mathematics in 
a conventional sense.  And yet, I would say that many
of his ``habits of mind,'' including most spectacularly the way he approached
his classification of periodic tilings of the plane, show similarities with
the ways mathematicians think.  So I agree that he was trying to portray ideas 
that had a lot in common with mathematics and that by nature he had much in 
common with mathematicians.

I think it might have been better to cast your net a bit wider with your sources, 
though.  The information you have found is good but it only scratches the surface.
Some of the articles in the books I put on reserve for the class's use would have
given you some additional points of view and examples to draw on.

Another important technical point: the way you make use of a number of 
direct quotations and other paraphrases of your sources without providing full
{\it in-text or footnote citations} to identify the where in the sources
the quotations or the information came from is not really sufficient as
documentation of your sources.  You need to provide
a full reference {\it for each quote you use} so that a reader could go
to the source and check to see that you are quoting accurately
and interpreting what the other author was trying to say 
correctly too.
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  I don't see much {\it arithmetical} content in Escher's work -- not much
about numbers or their properties.  His work is much more geometric and visual than
it is arithmetic.
\item{2.}  Actually, I think Escher has a rather low reputation among mainstream
artists.  His work is too mathematical to be attractive to people who value emotional
expressiveness or artistic flair more than systematic geometric thinking or incorporation
of other mathematical concepts or constructions.  His drawing skills often seem limited
and ``stiff'' to art critics not attuned to the mathematical concepts he is trying 
to illustrate.
\item{3.}  Good observation, although I think all artists would say there is 
some element of ``problem-solving'' in what they do too, at least some of the time.  
\item{4.}  ``Colleagues'' might be a bit too strong here.  Escher did conduct exchanges
of correspondence with several mathematicians.  But the information they wanted to 
share was often not anything Escher could understand on a technical level because
he essentially had only a high school-level training in mathematics.   
\bigskip\noindent
{\it Grade:}
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\noindent
{\it Lucius White -- M.C. Escher: Artist, Mathematician, or Both?}
\bigskip
Mostly good paper on an interesting and subtle question.  As you have found, 
although Escher had
essentially no formal mathematical training beyond the high school level,
he did come from a family of scientists and engineers.  Also, although you 
did not mention this, he did have a number of contacts with professional mathematicians 
(including George Poly\'a and H.S.M. Coxeter) who admired Escher's work and 
who shared information with him about ideas he inquired about or that they 
saw expressed in his work.
He also tried to incorporate what he learned from them (or what he intuited from examples
they showed him) into his works.   Ultimately, what he did was not mathematics in
a conventional sense.  And yet, I would say (as you report) that many
of his ``habits of mind,'' including most spectacularly the way he approached
his classification of periodic tilings of the plane, show similarities with
the ways mathematicians think.  So I agree that he was trying to portray ideas
that had a lot in common with mathematics and that by nature he had much in
common with mathematicians.

I think it might have been better to cast your net a bit wider with your sources,
though.  The information you have found is good but it only scratches the surface.
Some of the articles in the books I put on reserve for the class's use would have
given you some additional points of view and examples to draw on.
\bigskip
\noindent
{\it Specific Comments:}
\bigskip
\item{1.}  I think the most you can really say here is that his family background 
might have meant that he was exposed to some mathematical ideas early in his life.
Escher did not study mathematics past the high school level in a formal way, though.
\item{2.}   Actually, I think Escher has a rather low reputation among mainstream
artists.  His work is too mathematical to be attractive to people who value emotional
expressiveness or artistic flair more than systematic geometric thinking or incorporation
of other mathematical concepts or constructions.  His drawing skills often seem limited
and ``stiff'' to art critics not attuned to the mathematical concepts he is trying
to illustrate.  It is true that his art has been popular at some times, but 
``influential artist'' is too strong here.  
\bigskip
\noindent
{\it Grade:}  
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\noindent
{\it Michelle McGahan -- Evolution of Symbolic Algebra: The Quadratic Formula}
\bigskip
This is a very good ``case study'' of the way algebraic thinking evolved over
time, from the {\it rhetorical} stage of the Egyptians and Babylonians, to the 
fully {\it symbolic} stage of today's quadratic formula
$$x = {-b \pm \sqrt{b^2 - 4ac} \over 2a}$$
for the solutions of $ax^2 + bx + c = 0$.  I especially like the way 
you developed your own topic here and tied it both to things we discussed
in class, and to your own interest in mathematics education.  Good work!
  
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  I think you could say that ``rhetorical algebra'' is the 
standard way of presenting mathematics in both the Egyptian and 
Babylonian traditions (as well as in ancient Indian and Chinese
mathematics).  It is sort of the first stage in the development 
of algebra almost everywhere that we know about.  In your research, I'm guessing you
might have run across the idea that there is a sort of intermediate {\it ''syncopated''}
stage that some historians identify as well, in which some of the quantities
involved in solving equations are represented by symbols or words, while the
solution process is not fully expressed yet as a single formula.  We saw
some examples of this sort of thing in class when we discussed the evolution
of algebra in Europe after the Renaissance, and you discuss some aspects
of this later too.
\item{2.}  It seems that you are missing a ``not'' in this sentence.
From the surviving evidence, the Babylonian approach to problems of this
type was mostly rhetorical (i.e. they knew processes for solving the 
equations, expressed as algorithms), rather than symbolic.  Although
the process is, in a way, equivalent to what our quadratic formula
says, that's not the way the Babylonians would have thought about
what they were doing.  
\item{3.}  Something got garbled here.
\item{4.}  The question you raise in this last section is a good one. 
It seems to me that a good education in mathematics should allow students
to develop skill in applying formulas, but also give them the tools 
to allow them to understand where the formulas come from and why 
they ``work.''  If no justification is given (if students just 
memorize, ``plug and chug,'' then I think something important is
lost in the process and it becomes harder to motivate why learning
those skills is important.  
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\noindent
{\it Grade:}
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\noindent
{\it Melody Wauke -- Christopher Clavius' Influence on Mathematical Education}
\bigskip
Very good paper on Christopher Clavius and his contributions to mathematics and
to Jesuit education.  Your writing is good and the paper flows well, although
there are a few places where you might have expanded things or explained points
in more detail.
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.} It might be good to say a few words about who Gottfried Leibniz was
here -- he was a philosopher and mathematician who is generally credited as
being a co-inventor of calculus, together with Isaac Newton.
\item{2.} ``Saving the phenomena'' (that is, developing mathematical theories
of the solar system, and the cosmos more generally, that were consistent with 
astronomical observations) was a major concern of theoretical astronomy going
all the way back to Greek science.  For instance, Claudius Ptolemy's system
of epicycles was developed to explain how the retrograde motions of some of 
the planets might work, as observed from the Earth.
\item{3.}  Strictly speaking, I am not sure that requiring an examination
in mathematics ``proves'' that mathematics is necessary for a well-rounded
education.  I think proposing the idea of requiring such an exam is evidence that he
thought it was a necessary component.
\item{4.}  It might have been interesting to say more about why this was 
the case (and why it is actually even still the case that some philosophers
see mathematics as somewhat trivial and not especially interesting).  
\item{5.}  This seems like a bit of a stretch, I think, unless you explain
your idea more fully.  It is certainly true
that ``finding God in all things'' is a hallmark of Jesuit thinking in 
general and a goal for their educational efforts.  But what is the
connection between that goal and the study of mathematics?  Does mathematics
help us understand God or God's handiwork in some special way?  If so, how?
\bigskip
\noindent
{\it Grade:}
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\noindent
{\it Emily Liguori -- Christopher Clavius and His Contribution to the Mathematics World}
\bigskip
Good paper on Christopher Clavius and his contributions to mathematics and
to Jesuit education.  Your writing is good and the paper flows well, although
there are a few places where you might have expanded things or explained points
in more detail.
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.} This is an interesting and curious point.  As I guess you read, ``Clavius'' 
is thought to be a Latinized version of his original family name.  But that could be
any one of several different German words, so it's not known exactly what that name would 
have been.
\item{2.}  Modern mathematicians would dispute this point.  Clavius is relying too much
on his physical intuition here, though he was not out of line at his time in history.
Later developments showed, in fact, that it is possible to have {\it any number} of 
mutually orthogonal coordinate axes in a mathematical space and to do geometry, in 
some form, in those spaces.  This is part of the idea behind the geometry of 
Bernhard Riemann that we mentioned briefly in class.  Realizing that this is 
possible is part of the idea that geometry does not necessarily describe the 
physical space we live in. 
\item{3.}  The formal title of the document was the {\it Ratio Studiorum} (``Plan of 
Studies'').
\item{4.}  Tom Cecil in the Holy Cross Mathematics and CS Department has been an 
active member of the Clavius Group.  He helps to organize and attends the meetings 
almost every summer.  The group has met at Holy Cross a number of times.
\item{5.}  Avoid using the word ``genius'' as an adjective in formal writing.  
Also, the Gregorian calendar, while logical, was not accepted immediately everywhere.
Some Protestant countries waited a long time to adopt it.  For instance, England
used the Julian calendar until 1752 (i.e. for more than 150 years after the time
of Clavius).  When they did adopt it, the fact that uneducated workers thought they 
``lost'' almost two weeks of pay because of the adjustment in dates led to some unrest.

\bigskip
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{\it Grade:}
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\noindent
{\it Brooke Tranten -- Father Matteo Ricci, Ming China, and Euclid}
\bigskip
Very good paper on Matteo Ricci and his experiences in China, including the translation
of Euclid's {\it Elements} into Chinese.  Your writing is generally good, and you 
do an especially good job of documenting your sources with footnotes.  There are quite
a few small typos and other minor slips, though, so in the future be more careful about
proofreading your work before you submit it.
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  It's better to say that the postulates are to be accepted without proof.
In other words, the question of whether they could be explained or not does not 
really even come up because (for the Greeks at least) the truth of a postulate should 
be self-evident or obvious.  So they are taken as given to start the reasoning.  This 
is one reason for the way Postulate 5 was singled out for special attention almost
from the start of the history of the {\it Elements}.  It didn't ``feel like'' something
that should be used as a postulate.  
\item{2.}  I think almost any translation project involves more linguistic work than 
anything else.  After all, the goal was to take the existing text of Euclid's
{\it Elements} and to find Chinese equivalents or substitutes for everything there.  
When there is not an existing tradition of that sort of mathematical proof to draw on, finding
the language to make it accessible becomes a difficult job.
\item{3.}  There's a lot going on in this sentence and it ends up being a bit unclear on
its own because it is only the rest of the paragraph that explains everything in more detail.  
The first sentence of a paragraph does not always need to summarize everything that 
appears later in the paragraph.  And it is interesting that the Jesuits ran into 
some controversy within the wider Catholic Church because of the ways that they 
learned and used Chinese culture to gain entry into the higher levels
of that society.
\item{4.} OK, I looked it up and ``disbandment'' is actually a word, but it still 
looks awkward to me.  My understanding of the history (I'm not an expert by any means) 
is that the Jesuits were expelled from countries in many parts of the world at this time for a whole
set of overlapping reasons that had more to do with European politics than anything 
about what they had done in their missionary work in China.  Having said that, there
was also a lot of criticism of Ricci (and the Jesuits more generally) within the Church
because it was felt that they had gone to far in adapting and modifying elements of Christian 
doctrine to make them seem more compatible with existing elements of Chinese culture and 
understandable to the Chinese.   
\bigskip
\noindent
{\it Grade:}
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\noindent
{\it John Diepenbrock -- Confucius and Euclid; A Bridge Between}
\bigskip
This is a good account of the way Matteo Ricci brought the {\it Elements} of 
Euclid to China and used that mathematical knowledge as part of his approach
to introducing Catholicism in that part of the world.  I think the only slightly
weak point is that you do not really discuss what the Chinese mathematical tradition
looked like (apart from the passage on page 6 where you talk about the way Ricci's
``student'' Qu Taisu would have appreciated the contents of  Book I of the {\it Elements}). 
So it is not so clear from what you were saying why Euclid would
make such a strong impression on the Chinese.  Also, the discussion on pages 6 and 7 does not
really make it clear that Qu Taisu's translation effort was probably intended
only for his own study.  Moreover, whatever he produced has not survived. 
The later translation of Books I - VI with Xu Guangqi was published and that was
the work that had the widest influence
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  Astrolabes are not especially technologically advanced, even for Ricci's time.
Versions of these instruments had been used since the time of the Greeks.  The first telescopes
were produced in Europe during Ricci's lifetime, though it is not certain that he would 
have heard about them in China.  
\item{2.}  This degree of engagement with the local culture was very unusual
for Europeans at the time, but it was a standard technique used by Jesuit missionaries.  
It was one of the reasons why Ricci's efforts (and Jesuit efforts
more generally) were so successful in many parts of the world.  
\item{3.}  You are referring to Clavius's translation of Euclid's {\it Elements}
but something got left out here, I think!
\item{4.}  The published Chinese translation that Ricci coauthored with
Xu Guangqi included only the text of Books I - VI, that is,
only the first {\it six} books and their treatment of topics from plane geometry.
(There are only 13 books altogether in modern standard versions of the {\it Elements} of 
Euclid.  Two other books that were included in some older versions are now thought to be 
additions by later authors and not part of the original work.)
\bigskip
\noindent
{\it Grade:}
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\noindent
{\it Amado Fores -- The Chinese Remainder Theorem}
\bigskip
This is a good discussion of one of the distinctive contributions from the 
Chinese mathematical tradition.  You have included a number of good worked-out
mathematical examples and highlighted some of the modern developments and 
applications of the theory.  The only slightly weak point is the one
addressed in comment 1 below.  
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  It would have been good to try to explain exactly how this sort of congruence 
problem relates to questions about calendrical cycles, since as you say that was apparently 
the motivation for the Chinese to study this sort of problem.  It's not exactly clear from 
what you say what the connection is because the concrete problem is not phrased
in calendar language, but more abstractly.  To do that, you would have to restate the problem something
like this:  In overlapping calendar cycles of length 3, 5, and 7 starting on the 
same day, a certain day is the 2nd day in the cycle of length 3, the 3rd day
in the cycle of length 5, and the 2nd day in the cycle of 7.  How many days after
the start of all three cycles did this day occur?    
\item{2.}  This is an interesting point and it's good that you included the brief
description on this page and the next.  ``Coding'' and ``Cryptography'' are two
connected but different subjects, though.  (Error-control) coding theory is the 
study of methods for reliable transmission of information over noisy channels, while
cryptography is the study of methods for secret transmission of information 
over channels where adversaries may be intercepting messages.  
\bigskip
\noindent
{\it Grade:}
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\noindent
{\it Laurie Ulysse -- The Roots of Ancient Chinese Mathematics: Linear Equations}
\bigskip
Very good job on this paper about the way the Chinese tradition included systematic
methods for solving systems of linear equations that are strikingly similar to methods
that were only developed in Europe much later.  It's interesting that they were able
to do this even without a well-developed form of symbolic algebra to express the
equations the way we would (before passing to the coefficient matrices).  
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  I'm not sure exactly what you mean by ``knowledge based on the current dynasty''
here.  I have read that the knowledge tested in the exams that people had to take to 
obtain jobs in the Imperial government changed very little over time and became quite old-fashioned
after a time.  It was a very traditional and conservative society in general so things 
might not change very much even when a new dynasty of Emperors would come into power.
\item{2.}  This is not a complete sentence, and I'm not sure what you are trying to 
say here.  Why is it important whether or not the ``Arithmetic Classic of the Gnomon'' can 
be dated to a pre-Euclid era?
\item{3.}  Do you mean that even if the ideas of solving systems of linear equations 
arose independently in different cultures at different times, that is only to be expected
because there is only one way to go about these problems -- the ``all roads lead to Rome''
idea?  I'm not sure I would agree with that since even in our mathematics classes, there
are alternate methods that are used to solve equations like this.  Methods having
connections with the idea of Gaussian elimination are only one possibility.
\item{4.}  See the previous comment -- it's not really true that we ``must'' use
Gaussian elimination on these sorts of problems.  There are other methods that do 
not involve the elimination of variables or creation of zeroes in the coefficient matrix
by doing row operations.  However, it does turn out that the Gaussian elimination method
is the most efficient one for large systems of equations (that is, lots of equations and
lots of variables).  So such methods are often used today even when the computations
are done on computers.  
\item{5.}  This should be more like 2000 years before, not 3000.
\item{6.}  Almost any method for these calculations is prone to error if done by 
hand by a human being(!)  Large systems are almost always solved by computer today
for this reason! 
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{\it Grade:}
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\noindent
{\it Abby Henry -- Learning and Teaching Mathematics}
\bigskip
You tie some of the historical topics we discussed with aspects of the Jesuit 
{\it Ratio Studiorum} concerning the teaching of mathematics together in an interesting way. 
But I'm not sure the paper as a whole hangs together very well.  One issue for me is that while I agree that 
teaching only rote procedures without understanding does not give students as 
deep an understanding of mathematics as they could get otherwise, saying (or implying) that the 
Egyptians and Babylonians were somehow wrong for doing what they did is not taking
into account that they just didn't have much of an alternative(!)  That's what their
mathematics was, so to speak, so that's how they had to teach it.  Our understanding
of mathematics has been heavily influenced by the Greek contributions, and by 
Euclid in particular.  The ideas expressed in the {\it Ratio Studiorum}, for instance
can hardly be discussed without mentioning that Euclid formed a big part of 
the subject matter that was taught in Jesuit schools at that time.    
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  But making up the examples included in the data on Plimpton 322 must have taken
a lot of thought and effort because of the sizes of some of the numbers involved(!)  I think
it would be more accurate to say that this list of data from reciprocal pairs would give a
teacher a large stock of examples to draw on to give many different problems of the same
type and check the answers.  But there would have been a major effort involved in creating
the list of problems at some point.  
\item{2.}  Easier from {\it our} point of view, yes.  But not easier if the whole 
infrastrucure of equations, variables, symbolic formulas, etc. was not there to begin
with.  Thinking all of that up would be harder than just learning to solve a group of set problems
by seeing the same procedures repeated several times. 
So I think it's probably a form of conceptual anachronism to say this.  
\bigskip
\noindent
{\it Grade:}
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\noindent
{\it Matt McCormick -- Islamic Influences on Geometry During the European Dark Ages}
\bigskip
This is a good discussion of Islamic influences on geometry and the way Islamic
mathematicians did more than just ``transmit'' Greek knowledge back to Europe
toward the end of the middle ages.  For the future, though, be aware that some 
of what you were doing to document your sources would be seen as insufficient 
by some professors.  When you use a direct quotation, for instance, you need
to include enough information that the reader could (if desired) look up 
the location of the original text and check that you have used it appropriately.
This means providing page numbers, or other ways to specify locations within
the source text.  
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  The way you are thinking that Western dominance in world politics over the
past few centuries has put Western viewpoints at the forefront of scientific research
is somewhat unclear.  I think there are several components to this.  First, of course,
Western countries have had the resources to train scientists and to help sponsor the 
conditions that have led to much research (either directly through government funding,
or indirectly through maintaining the conditions allowing for private companies to 
invest in research).  Political power has helped those countries to obtain and keep 
those resources.  But it is also true that other parts of the world (the Islamic
world in particular) have been less inclined to try to do anything like this in recent
times.  The fundamentalist brands of Islam that have been powerful in places like 
Iran and Saudi Arabia have actively discouraged a lot of possible research agendas
and suppressed their societies' ability to participate in science.  The conditions
during the Abbasid caliphate that you describe later in the paper were much different.
\item{2.}  I think you mean Eudoxus here(!)
\item{3.}  For papers like this, it is better to identify the authors you quote
more fully.  Don't assume that the reader has been in on our class discussions and
knows who Kline is the first time you mention him.
\item{4.}  I'm not necessarily saying that this is incorrect, but it would be good
to provide evidence for statements like this.  How do we know that Euclid
made a big impression?  Also, who is ``the public?'' Don't forget that the component
of society who would have access to texts like this would be much smaller than
at present.  
\item{5.}  I don't think it is correct to say that the Arabs were the first people
to apply geometry to architecture.  In fact, the Greeks themselves, and the Romans
following them, certainly did so too.  In fact, this was almost the only way 
the Romans made use of the Greek mathematical tradition they inherited -- in 
engineering, architecture, and the like.
\item{6.}  Technical point:  When you give a citation to one of your sources, the
citation should identify the source clearly in your list of references, and it should
also give something like a page number showing where the location from which 
you took the quotation.  Be more careful about this, especially in papers for
courses in humanities departments!!!
\item{7.}  I think you could argue that the Islamic contributions had a big 
effect on Europe somewhat later (for instance, after 1200 CE or so).  But at the time,
there just was not that much contact or sharing of information between 
Western Europe and the Islamic world.
\bigskip
\noindent
{\it Grade:}
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{\it Meg Norton -- Euclid's {\sl Elements} and Modern Methods of Mathematics Education}
\bigskip
I have the feeling that a lot of the topics you address here are going to be
ones that you will want to continue to address as you move into teaching
at the secondary level:  To what extent should mathematics instruction
focus on developing skills?  To what extent should mathematics instruction
focus on having students complete projects with specified products or goals
as the end-result?  How should deductive, theoretical approaches (as in 
traditional geometry classes) by balanced by practically-oriented topics
with connections to science, finance, etc.?  I can tell you have been thinking
about all this and have some definite opinions.  My advice would be to be always
open to revising those opinions as you learn more and gain more experience.
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  I think it is anachronistic to refer to the Alexandrian Museum as a
``university.''  While there were some similarities -- scholars were in residence 
and teaching took place there -- 
many of the structures of modern universities do not appear to have been present.
\item{2.}  I am not familiar with this term.  What do you mean by a ``situational''
method?  Does this mean a method that is tailored to the specific problem, possibly 
without any justification given through general theory?  
\item{3.}  It strikes me that there are two things going on in what you are saying
here that are not necessarily connected the way you seem to be saying that they 
are.  It is true that most teaching in this period consisted of having students
memorize parts of texts (including the {\it Elements}).  But I think that the teachers
would have justified this by saying something like ``you don't really know a subject
unless you have it by heart.''  So the first step would be the memorization, and 
then (perhaps only especially bright) students could go farther by proving new
results based on what they had learned.  Some (very) ``old school'' teachers even today  
approach things that way (although it's admittedly much rarer now than before).  In other words, there 
was not necessarily any connection between that teaching method and the fact that 
the mathematics presented in Euclid was not geared toward solving practical problems with 
actual numbers, realistic measurements, etc.  There's a difference between
the teaching method and the content that was being taught.   
\item{4.}  This is a subtle question, and it goes far beyond mathematics, I think.
In my own experience I do know that some of the things I know the 
best were topics that I had to learn, often on my own, to solve particular problems
I was interested in.  I can understand why some educators want to try to harness
that sort of curiosity by setting up classroom experiences where students will derive motivation
for learning things by having concrete questions that they really {\it want to figure 
out} and answer.
To me, though, there are problems with using that approach {\it exclusively}. First, 
there are times when it is just not possible to make much progress in a reasonable 
amount of time without a substantial base of information and technique to 
build on and draw from.  I think this is what you were thinking when you pointed
out the ``waste of time'' aspect.  But there is a second issue too: not everyone
will be motivated by {\it the same} questions.  Getting all, or even enough, students
in a class to be interested about any particular project to make it worthwhile 
for them and for the whole class may not be easy either.  If we think that there
are certain basic skills and ideas everyone needs to master, then this might not
be the right approach there either. 
\bigskip
\noindent
{\it Grade:}
\noindent
\vfill\eject
\noindent
{\it Lauren Clair -- Mathematics in Education}
\bigskip
Good paper on Christopher Clavius and his contributions to mathematics and
to Jesuit education.  Your writing is mostly quite good, although
there are a few places where you might have been clearer, expanded things or explained points
in more detail.
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.} I think it would be more usual to say that this went through 
16 editions between 1570 and 1618.  The {\it Sphaera} of Johannes de Sacrobosco
 itself was written in the early 13th century 
and was used as a textbook of astronomy in Europe for centuries after 
that time, right down to the time of Clavius.  
\item{2.}  Clavius's views about the importance of including mathematics
in the standard curriculum in the Jesuit schools was somewhat unusual
for his time.  But that was mostly because mathematics was not considered
to be that important by many people coming out of the religious and 
philosophical traditions of his time.  There had been earlier periods 
(think the high point of the Greek tradition, for instance) where mathematics
was held in much higher regard and where the subject was a standard part
of education.  So Clavius was, in effect, urging a return to an earlier
tradition rather than proposing something that was completely new. 
\item{3.}  Clavius also wrote several other documents/books urging the
inclusion of mathematics, so you could say they were also probably 
important in the development of the final version of the {\it Ratio
Studiorum}.  This was true even though he was not a formal member of 
the group that prepared the final draft of that document.
\item{4.} This seems to be circling back to the ideas you discussed
earlier around the place I marked comment 2 above.  The organization 
could be a bit tighter here.  
\item{5.}  Be careful about using words like ``monumental,'' ``uncountable
number of people,'' ``incredible,'' and so forth.  This actually takes
away from the strength of your conclusions in a way, since it seems
as though you are overstating your conclusions.  Formal writing is usually
somewhat more measured and sober than this.  
\bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Dillon Jordan -- Rome's Logical Revolution}
\bigskip
I think your main point here -- that the Romans did use mathematics
in some important ways, even if they did not contribute very much in 
the way of abstract pure mathematical research in the way the Greeks
did -- is a mostly valid one.  The way you focus on military matters is 
interesting.  In a way, though, I think there might even have been stronger
points to be made along these lines if you had brought in some of the big civil 
engineering and architectural projects the Romans constructed throughout
their empire.  Those required practical mathematics in even stronger
and more direct ways than the organization of their fighting forces.  
Either way, though, I agree that it is not correct to say that the 
Romans did {\it no mathematics}.  It was just mostly a different style
of mathematics and for different purposes.

Your writing is good and I can see that your History courses have prepared
you well for the technical details of citing sources and using footnotes.  
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.} I think the Greek reliance on the ``relatively unimaginative'' 
phalanx warfare style might be explained by 
the fact that the Greek city states did not generally keep standing armies.  
Their land fighting forces were composed of groups of their able-bodied male 
citizens organized for particular campaigns.  So I expect they had to make do with
formations and tactics that could be mastered in a relatively short period of time
by men who were not professional soldiers.  The Roman legions were very 
different organizations in several ways.  So it is not surprising that they 
evolved different and more flexible tactics that could be explained
using some mathematical ideas.    
\item{2.}  This is the same person as the Heron we discussed briefly 
in class.  (``Heron'' is a Greek form of the name which is often 
Latinized to ``Hero.'')  Heron was famous as both a mathematician 
and an engineer and inventor.
\item{3.}  Technical point:  "mathematics" is not a plural noun. 
So this last sentence should say ``... perfected its practical use.''
But the point is well-taken.
 \bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Dillon Jordan -- Rome's Logical Revolution}
\bigskip
I think your main point here -- that the Romans did use mathematics
in some important ways, even if they did not contribute very much in 
the way of abstract pure mathematical research in the way the Greeks
did -- is a mostly valid one.  The way you focus on military matters is 
interesting.  In a way, though, I think there might even have been stronger
points to be made along these lines if you had brought in some of the big civil 
engineering and architectural projects the Romans constructed throughout
their empire.  Those required practical mathematics in even stronger
and more direct ways than the organization of their fighting forces.  
Either way, though, I agree that it is not correct to say that the 
Romans did {\it no mathematics}.  It was just mostly a different style
of mathematics and for different purposes.

Your writing is good and I can see that your History courses have prepared
you well for the technical details of citing sources and using footnotes!  
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.} I think the Greek reliance on the ``relatively unimaginative'' 
phalanx warfare style might be explained by 
the fact that the Greek city states did not generally keep standing armies.  
Their land fighting forces were composed of groups of their able-bodied male 
citizens called up specifically for particular campaigns.  So I suspect 
they had to make do with formations and tactics that could be mastered in a 
relatively short period of time by men who were not full-time professional soldiers.  
The Roman legions were very different organizations in several ways.  So it 
is not surprising that they evolved different and more flexible tactics that 
could be explained using some mathematical ideas.    
\item{2.}  This is the same person as the Heron we discussed briefly 
in class.  (``Heron'' is a Greek form of the name which is often 
Latinized to ``Hero.'')  Heron was famous as both a mathematician 
and an engineer and inventor.
\item{3.}  Technical point:  "mathematics" is not a plural noun. 
So this last sentence should say ``... perfected its practical use.''
But the point is well-taken.
 \bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it David Odell -- No title}
\bigskip
This is a mostly good summary of the arguments about Plimpton 322 
presented by Eleanor Robson in several of her articles.  I agree
they are pretty persuasive.  There are a few places where what you 
say is not quite correct, though.  See in particular point 3 below. 
Neugebauer was not ``making up'' the theory of generating functions
for Pythagorean triples to explain Plimpton 322.  Instead, he
was proposing that that method might have been used to generate
the examples on Plimpton 322 because it was a technique that was
known in antiquity and he (Neugebauer) thought it was conceivable 
that the Babylonians might have known about it too. 
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.}  Not sure exactly what you mean by ``encapsulating'' here.  I was using this
quote to drive home the point that in looking at mathematics of the past, we have
to remember that people from ancient civilizations were not necessarily approaching
things from the same points of view that we would today.  This is true even though 
many people tend to think of mathematics as something like a ``universal language''
that doesn't change over time or differ between cultures.  Eleanor Robson makes 
very similar points in her work, of course. 
\item{2.}  ``Document'' sounds a bit strange here.  But of course that word
is technically perfectly correct because the clay tablets used by the 
Old Babylonians served much the same purposes that our paper or electronic
documents would today(!) 
\item{3.}  This is not actually correct.  The idea that Pythagorean triples
can be generated by the formulas 
$$x = p^2 - q^2, y = 2pq, z = p^2 + q^2$$
is actually much, much older.  It appears, for instance, in Proposition 29 of Book X
of Euclid's {\it Elements} (in the part of that work devoted to questions in 
number theory)!  It may even go back much farther than that.  The idea was
that Otto Neugebauer proposed that this may have been used to generate the 
triples found on Plimpton 322 {\it because he knew that this method had been 
known for a long time}.
\item{4.}  Maybe.  To me, it is even more persuasive to note that the 
Old Babylonians do not seem to have had or used any idea of {\it measure for angles}.
So the idea of constructing trigonometric tables (at least the way we
think of the trig functions as functions of an angle measured in terms
of degrees or radians or some other system) was just not part of 
their mindset or mathematical toolkit.
\item{5.}  This is a slightly strange choice of words.  I'm not sure
it makes sense to say that anything is being ``espoused'' in the data
on Plimpton 322.  It's essentially the record of some numerical calculations
and not a statement of something we can recognize as a theory.  
\item{6.}  Maybe not, but I think we can say that whatever the purpose
of the tablet was, it contains records of some rather impressive
computations.  Look back at some of the numbers involved and think 
about the process of computing squares of those numbers in base 60(!)
If there's no mathematical ``genius'' there, there is at least a 
tremendous amount of persistence and skill!
\bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Keshia Bynoe -- Eleanor Robson: Plimpton 322}
\bigskip
What you have to say here is mostly a good summary of the 
arguments about Plimpton 322 presented by Eleanor Robson in 
several of her articles.  

The main issue I have, though, is 
the way you make use of a number of direct quotations and other 
paraphrases of your sources without providing proper {\it in-text
or footnote citations} to identify the where in the sources
the quotations or the information came from.  You need to provide
a reference {\it for each quote you use} so that a reader could go
to the source and check to see that you are quoting accurately
and interpreting what the other author was trying to say 
correctly.
 
I know you were not trying to pass off the information you were
presenting as your own work, so I'm not going to invoke 
the academic integrity policy in this case.  But you should 
be aware that doing similar things in papers for other courses 
{\it might have that consequence}.  For papers in college-level 
courses, to be safe and avoid questions
of plagiarism, you need to familiarize yourself
with the proper methods for citing sources and use them(!)
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.}  I think it might be more accurate to say that the data
from Plimpton 322 could have been used to generate other problems
similar to the one from YBC 6967.  We would say this involved
solving quadratic equations; the Babylonians probably thought 
of what they were doing more along the lines of the ``cut and paste
geometry'' ideas we discussed. 
\item{2.}  This is not too clear.  The idea that Pythagorean triples
can be generated by the formulas 
$$x = p^2 - q^2, y = 2pq, z = p^2 + q^2$$
is actually very old.  It appears, for instance, in Proposition 29 of Book X
of Euclid's {\it Elements} (in the part of that work devoted to questions in 
number theory)!  It may even go back much farther than that.  The idea was
that Otto Neugebauer proposed that this may have been used to generate the 
triples found on Plimpton 322 {\it because he knew that this method had been 
known for a long time}.
\bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Jonathan Paulino -- Eleanor Robson's Reinterpretation of Plimpton 322}
\bigskip
This is a mostly good summary of the arguments about Plimpton 322 
presented by Eleanor Robson in several of her articles.  I agree
they are pretty persuasive.  There are a few places where what you 
say is not quite correct, though.  See in particular point 2 below. 
Neugebauer was not ``making up'' the theory of generating functions
for Pythagorean triples to explain Plimpton 322.  Instead, he
was proposing that that method might have been used to generate
the examples on Plimpton 322 because it was a technique that was
known in antiquity and he (Neugebauer) thought it was conceivable 
that the Babylonians might have known about it too. 
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.}  We know that the Babylonian scribe who created Plimpton 322 
was using the base 60 positional system because that was the standard
system used by everyone in that tradition.  It's not necessary to rely
just on the numbers appearing in the ``digits'' on the tablet to draw
that conclusion.  It is an accepted fact that this was the system
the Babylonians used for all of their numerical calculations.  
\item{2.}  This is not too clearly stated.  The fact that Pythagorean triples
are {\it always} generated by the formulas 
$$x = p^2 - q^2, y = 2pq, z = p^2 + q^2$$
is actually very old.  It appears, for instance, in Proposition 29 of Book X
of Euclid's {\it Elements} (in the part of that work devoted to questions in 
number theory)!  It may even go back much farther than that.  The idea 
Otto Neugebauer proposed was that this method may have been used to generate the 
triples found on Plimpton 322.  While it's possibly an anachronism, it
was at least a superficially reasonable thing to suggest {\it because 
Neugebauer knew that this method had been 
known for a long time} and it was at least conceivable that the creator
of Plimpton 322 knew and used that method too.  
Note also that it is not really
the algebraic form of the equations given above that is the issue, either.  
For instance, Euclid did not have
that form, yet he was able to prove that an equivalent form always generated
Pythagorean triples.  (By the way, the statement of his Proposition needs some reinterpretation
before it can be seen to be equivalent to this way of expressing the values of 
$x,y,z$ above, but it is the same from our point of view.)
\bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Shane Taylor -- Plimpton 322: Eleanor Robson's Missing Artifacts}
\bigskip
This is a mostly good summary of the arguments about Plimpton 322 
presented by Eleanor Robson in several of her articles, although I'm not sure
exactly what ``missing artifacts'' you mean to point out in your title.  I have
a feeling that is connected to your idea that she has not quite
proved her case because you don't see that she has identified the particular
problems that the data on Plimpton 322 could be used for. It's good that
you tried to make a critical appraisal of her work, but there is one
aspect of what you are saying that does not quite hold true because there
is a connection you did not quite appreciate.  
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.}  I think ``mistakes'' is probably too strong a word here.  In studying history,
you would not normally say that someone who proposes a reasonable explanation for 
something in the past has made a ``mistake'' if that proposal 
later turns out to be incorrect because new evidence comes to light.  
In history, things are not as clear-cut as things tend to be in mathematics 
where the typical questions often have just one correct answer.  To give 
just one example of what I mean, consider the way Otto Neugebauer sought to 
explain the way the numbers on Plimpton 322 may have been generated.  
The fact the formulas 
$$x = p^2 - q^2, y = 2pq, z = p^2 + q^2$$
always generate Pythagorean triples (i.e. for all choices of integers $p$ and $q$)
is actually very old.  A form of it appears, for instance, in Proposition 29 of Book X
of Euclid's {\it Elements} (in the part of that work devoted to questions in 
number theory)!  It may even go back much farther than that.  The idea 
Neugebauer proposed was that this method may have been used to generate the 
triples found on Plimpton 322.  While Robson would argue that that is an anachronism, 
it was at least a superficially reasonable thing to suggest {\it because 
Neugebauer knew that this method had been 
known for a long time} and it was at least conceivable that the creator
of Plimpton 322 knew and used that method too.  
\item{2.}  Your main point seems to be that Robson does not really provide
any concrete suggestion about {\it what problems} might have been based
on the data contained in Plimpton 322.  But in fact she didn't really need
to do that because there is one very well-known and common type of problem that 
was exactly the type you wanted to see.  I'm thinking of problems 
like the one from YBC 6967 that we discussed in class:  If $60/x$ exceeds
$x$ by 7, find $x$.  Note that the $60/x$ and $x$ form exactly the sort
of reciprocal pair that Robson discusses and every row of Plimpton
322 gives a way to find one of those pairs.  If the reciprocal pair is
known, the difference is also known, so that data can be used to set up 
a problem like the one from YBC 6967, and all of those problems can be solved by the 
same sort of procedure as the one given on that tablet.  This is the ``missing
link'' I think you were looking for, and I think Robson doesn't discuss it in
a lot of detail because it is so obvious to her (as an expert in the 
sorts of problems that appear in various Babylonian texts).  The one you 
suggest from the much later Seleucid period is somewhat similar, but of
course, the date is roughly 1300 years after the presumed date of 
Plimpton 322, so it is hard to argue that there is any direct connection.
 \bigskip
\noindent
{\it Grade:}
\vfill\eject
{\it Declan Downey -- Euclid and Plato: Perfect Polyhedra}
\bigskip
This is a mostly good summary of the way Euclid presented facts about the 
five regular polyhedra (``Platonic'' solids) in Book XIII of the {\it Elements}. 
It is good that you chose a topic different from everyone else in the class.  The 
only thing I think you might have pursued in more detail was the reasoning by which 
Euclid sought to demonstrate that there are {\it only five different types of such
polyhedra}.  This is interesting because it shows that some aspects of the 
{\it Elements} have a sort of unfinished feel and don't have quite the same
level of logical completeness that most of it does.  
\bigskip
\noindent
{\it Specific Comments}
\bigskip 
\item{1.}  I think it is more accurate to say that Book XIII gives constructions
of the five solids.  It is not really necessary to prove that they exist since
that is clear by making models or by considering objects in the real world
with these forms.
\item{2.}  Note that six equilateral triangles do fit exactly around a point
in the plane and form a regular hexagon.  
So the whole plane can be tiled with equilateral triangles.  Similarly, the 
whole plane can be tiled with squares.  Those arrangements are sometimes considered
as partsof the same family of geometric configurations as the regular
polyhedra.
\item{3.}  This is a bit akward because you only describe what is meant by 
the ``dual'' of a polyhedron on the next page.  In writing about mathematics, it
is better to introduce the necessary definitions of terms as soon as possible
so that you are not using words whose precise meanings may not be clear. 
\item{4.}  Wrong word (it's not the fish or the bottom of a shoe!)
\item{5.}  Nice observation.  In fact the word ``elements'' was used
by the Greeks for both the constituent substances making up the whole physical
universe and for the basic building-blocks of knowledge in some area.  It is, even now, still 
a tradition, of course, to name books intended as systematic introductions
to subject with titles of the form {\it Elements of ... } because of this
sort of analogy.
 \bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Yingying Hao -- An introduction to ancient Chinese mathematics, ... }
\bigskip
Good job on this paper about some aspects of the Chinese mathematical tradition and the ways
the Chinese came into contact with developments in Western mathematics through their
contacts with Matteo Ricci, S.J. and other Europeans in the 17th century.  Your writing 
is quite good although I know English was not your first language.  The way that shows
up here is that there are a few places where your choice of words seems a bit awkward
and there are other places where it is not entirely clear what you meant to say.  But getting
more comfortable writing is mostly a matter of practicing enough that it comes to 
seem natural.   
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  Using the approximate value $\pi \doteq 3$ would be a case of ``rounding
down'' rather than ``rounding up.''
\item{2.}  The standard notation for congruences modulo an integer $m$ is something 
like $N \equiv 2 \pmod{3}$ (note it's just ``mod,'' not ``mode'').  Also,  
it would have been good to try to explain exactly how this sort of congruence 
problem relates to questions about calendrical cycles, since that was apparently 
the motivation for the Chinese to study this sort of problem.  
To do that, you would have to restate the problem something
like this:  In overlapping calendar cycles of length 3, 5, and 7 starting on the 
same day, a certain day is the 2nd day in the cycle of length 3, the 3rd day
in the cycle of length 5, and the 2nd day in the cycle of 7.  How many days after
the start of all three cycles did this day occur?  It is thought that this was the 
reason questions like this were first studied, and that they were later generalized
and ``abstracted'' into the type of problem you discuss.
\item{3.}  The way that Ricci immersed himself in traditional Chinese culture and
made contacts within the imperial administration is an interesting one.  The fact
that he was so willing to meet the Chinese on their own terms was a big factor in the 
success of his efforts.
\item{4.}  I'm not quite sure what you mean by saying the proof is ``strictly 
rational.''  It's different from a Euclidean style proof in several ways and
leads to something we would naturally express with algebraic equations leading
to the relation $c^2 = a^2 + b^2$ for the sides $a,b,c$ of the right triangle.
But it does give a general argument showing that the ``Pythagorean Theorem''
holds for all right triangles.  
\bigskip
\noindent
{\it Grade:}
\vfill\eject
\noindent
{\it Shaniquea Ramos -- Jesuits and the Importance of Math}
\bigskip
You discuss a number of the important points regarding the 
role of Christopher Clavius in the introduction of instruction
in mathematics in the Jesuit schools in the early history of the 
Society of Jesus quite well.  

There are a number of consistent grammatical and other technical errors in your 
writing that you should try to ``clean up'' as soon as possible, though. I'm thinking
of the way you use ``seen'' instead of the past tense ``saw'' 
and mix tenses in the same sentence or in successive sentences.
This detracts from what you have to say.  
  
Another important technical point: the way you make use of a number of 
direct quotations and other paraphrases of your sources without providing full
{\it in-text or footnote citations} to identify the where in the sources
the quotations or the information came from is not really sufficient as
documentation of your sources.  You need to provide
a full reference {\it for each quote you use} so that a reader could go
to the source and check to see that you are quoting accurately
and interpreting what the other author was trying to say 
correctly too.

\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  ``Real men'' has a lot of other side meanings or connotations that are not
what you meant here.  Something like ``well-balanced men'' would be closer
to what I think you were trying to say.
\item{2.}  The equation here shows a quite accurate approximation to the actual length in 
days of the solar year (the time needed for the Earth to travel one complete circuit
around the Sun).  The fact that this is not exactly 365.25 is the reason that the 
Julian calendar was slowly getting more and more out of step with the seasons and
the reason that the Gregorian calendar reform was instituted.  Clavius was responsible
for the calculations show that this more precise way of adding leap years would solve
the problem.  He also suggested the one-time correction of skipping 11 or 12 days
to bring the calendar back into sync.  
\item{3.}  Careful, Clavius did not actually personally contribute to the writing 
of the {\it Ratio Studiorum}, although it is thought that his writings on the importance
of mathematics and his personal influence on the Jesuits who did write the successive
drafts of the document did have a major effect on what the final version said
about the role of mathematics in the Jesuit schools.  
\item{4.}  And conversely, mathematics was viewed as less important because it was 
more difficult to see how it related to the ``big questions'' that the humanities
dealt with.
\bigskip
\noindent
{\it Grade:}
\bye
