Abigail Luhn – Geometry, the Foundation of Renaissance Art

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.

Specific Comments

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 Elements of ... .

2. It might be better to say “... all lines orthogonal to the picture plane” here. 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.

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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Hannah Smith – The Mathematics of M.C. Escher’s Artwork

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.

Specific Comments

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 Polyá 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.

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.

3. This is a good observation, and many of Escher’s other works address this idea in other ways, too.

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 trefoil knot. Mathematicians have developed many other examples and classified the simpler possible configurations of knots.

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 Reptiles or Night and Day).

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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Luke Melody – M.C. Escher and Mathematics

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 Polyá – 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 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 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.

Specific Comments

1. I don’t see much 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.

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.

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.

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.

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Lucius White – M.C. Escher: Artist, Mathematician, or Both?

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 Polyá 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.

Specific Comments:

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.

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.

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Michelle McGahan – Evolution of Symbolic Algebra: The Quadratic Formula

This is a very good “case study” of the way algebraic thinking evolved over time, from the rhetorical stage of the Egyptians and Babylonians, to the fully symbolic stage of today’s quadratic formula

x = \frac{−b ± \sqrt{b^{2} − 4ac}}{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!

Specific Comments

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 ”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.

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.

3. Something got garbled here.

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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Melody Wauke – Christopher Clavius’ Influence on Mathematical Education

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.

Specific Comments

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.

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.

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.

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).

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?

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Emily Liguori – Christopher Clavius and His Contribution to the Mathematics World

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.

Specific Comments

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.

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 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.

3. The formal title of the document was the Ratio Studiorum (“Plan of Studies”).

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.

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.

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Brooke Tranten – Father Matteo Ricci, Ming China, and Euclid

Very good paper on Matteo Ricci and his experiences in China, including the translation of Euclid’s 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.

Specific Comments

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 Elements. It didn’t “feel like” something that should be used as a postulate.

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 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.

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.

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.

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John Diepenbrock – Confucius and Euclid; A Bridge Between

This is a good account of the way Matteo Ricci brought the 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 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

Specific Comments

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.

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.

3. You are referring to Clavius’s translation of Euclid’s Elements but something got left out here, I think!

4. The published Chinese translation that Ricci coauthored with Xu Guangqi included only the text of Books I - VI, that is, only the first six books and their treatment of topics from plane geometry. (There are only 13 books altogether in modern standard versions of the 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.)

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Amado Fores – The Chinese Remainder Theorem

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.

Specific Comments

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?

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.

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Laurie Ulysse – The Roots of Ancient Chinese Mathematics: Linear Equations

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).

Specific Comments

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.

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?

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.

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.

5. This should be more like 2000 years before, not 3000.

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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Abby Henry – Learning and Teaching Mathematics

You tie some of the historical topics we discussed with aspects of the Jesuit 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 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.

Specific Comments

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.

2. Easier from 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.

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Matt McCormick – Islamic Influences on Geometry During the European Dark Ages

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.

Specific Comments

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.

2. I think you mean Eudoxus here(!)

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.

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.

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.

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!!!

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.

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Meg Norton – Euclid’s Elements and Modern Methods of Mathematics Education

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.

Specific Comments

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.

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?

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 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.

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 want to figure out and answer. To me, though, there are problems with using that approach 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 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.

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Lauren Clair – Mathematics in Education

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.

Specific Comments

1. I think it would be more usual to say that this went through 16 editions between 1570 and 1618. The 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.

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.

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 Ratio Studiorum. This was true even though he was not a formal member of the group that prepared the final draft of that document.

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.

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.

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Dillon Jordan – Rome’s Logical Revolution

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 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.

Specific Comments

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 ablebodied 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.

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.

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.

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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 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!

Specific Comments

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 ablebodied 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.

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.

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.

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David Odell – No title

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.

Specific Comments

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.

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(!)

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 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 because he knew that this method had been known for a long time.

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 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.

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.

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!

Grade: 21 Keshia Bynoe – Eleanor Robson: Plimpton 322

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 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 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 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(!)

Specific Comments

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.

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 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 because he knew that this method had been known for a long time.

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Jonathan Paulino – Eleanor Robson’s Reinterpretation of Plimpton 322

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.

Specific Comments

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.

2. This is not too clearly stated. The fact that Pythagorean triples are 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 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 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.)

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Shane Taylor – Plimpton 322: Eleanor Robson’s Missing Artifacts

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.

Specific Comments

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 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 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.

2. Your main point seems to be that Robson does not really provide any concrete suggestion about 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 wellknown 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.

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Declan Downey – Euclid and Plato: Perfect Polyhedra

This is a mostly good summary of the way Euclid presented facts about the five regular polyhedra (“Platonic” solids) in Book XIII of the 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 only five different types of such polyhedra. This is interesting because it shows that some aspects of the Elements have a sort of unfinished feel and don’t have quite the same level of logical completeness that most of it does.

Specific Comments

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.

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.

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.

4. Wrong word (it’s not the fish or the bottom of a shoe!)

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 Elements of ... because of this sort of analogy.

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Yingying Hao – An introduction to ancient Chinese mathematics, ...

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.

Specific Comments

1. Using the approximate value π ≐ 3 would be a case of “rounding down” rather than “rounding up.”

2. The standard notation for congruences modulo an integer m is something like N ≡ 2 (mod 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.

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.

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.

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Shaniquea Ramos – Jesuits and the Importance of Math

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 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 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.

Specific Comments

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.

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.

3. Careful, Clavius did not actually personally contribute to the writing of the 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.

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.

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