Muhammad Javaid – Looking Further into Plimpton 322

What you say about Robson’s reinterpretation of Plimpton 322 and her ideas about the correct approach to the study of ancient mathematics is generally excellent. You have understood her main points well and summarized them clearly. This paper also seems more carefully written than the second paper for our course was – it reads much more smoothly, so good work!

However, I would say that the paper does not go quite far enough in some cases. Since we discussed some of this in quite a bit of detail in class, it would have been good to say more. First, you need to be careful with the “reciprocal pair” terminology since the way the actual numbers on Plimpton 322 are derived is not just the listing of x, 1/x pairs as on the MLC 1670 tablet that Robson reproduces in her Figure 9. It would be more accurate to say that the numbers on Plimpton 322 are derived from reciprocal pairs by the same sort of formulas we saw in the analysis of YBC 6967: from the pair x, 1/x, the numbers

\begin{aligned}c & = (x + 1/x)/2 \\ a & = 1 \\ b & = (x − 1/x)/2 \end{aligned}

are formed and then c^{2} = a^{2} + b^{2}. It is essentially the numbers c and b (possibly scaled) that occur on Plimpton 322. The goal of a problem text like YBC 6967 is then to derive the x and 1/x from knowledge of c and b. It would have been good to say something about Rudman’s alternative explanation for the genesis of Plimpton 322 – see page 97 in his book. Second, since you mention the p, q in the generating formulas for Pythagorean triples in the course of discussing Robson’s thoughts about Neugebauer’s interpretation, it would have been good to say more there too.

Specific Comments

1. See above – it would be more accurate to say that these scholars see Plimpton 322 as a list of information derived from reciprocal pairs – the numbers on the tablet are not the numbers x, 1/x in the reciprocal pair themselves.

2. Technical comment – In a paper like this, when you draw on a statement in a source for one of the things you are claiming, even if you are only giving a paraphrase (a rewording in your own words), you need to provide a reference to the place in the source (for instance a page number) you are referring to. Although you identified the source by name, you did not provide that. You should get into the habit of doing this as a matter of course for research papers in all of your classes. I am only lowering you score in the “cites quotations properly” category a bit for this. But be careful about this point in the future.

3. This is an accurate and well-written summary of Robson’s argument for rejecting the trigonometry table idea. However, I think it is important to realize that there are possible objections to what she is saying too. For instance, when she says that the the Babylonians did not have a notion of measuring angles and did not visualize circles as being traced out by a radius making a varying angle with the positive x-axis, she is certainly correct. But after that she seems to be almost doing the same sort of anachronistic reasoning from modern-day mathematical ideas for which she criticizes the others when she is claiming that there could not be any connection with what we would call trigonometry. It is certainly within the Babylonian scheme of things to think of describing the shape of triangles or rectangles by using ratios of the lengths of their sides (or the square of such a ratio). This effectively leads to the same sorts of quantities we call trigonometric functions of the angle. It also gives another possible explanation for the left-hand column of the remaining part of Plimpton 322 that would make sense in the Babylonian context. Whether you call the tablet a trigonometric table or not based on this is a matter of terminology, I would say. But I think that it easily could have been a table of quantities related to differently shaped rectangles or triangles to the Babylonian scribe who wrote it.

4. And by the way, this would also give a way to reconcile the idea that Plimpton 322 deals with rudimentary forms of things that later became formalized in trigonometry with Robson’s ideas about the table format. Suppose the relevant number to the Babylonians was the ratio c^{2}/b^{2} – a sort of “shape parameter” – (not the p or q from Neugebauer’s generating formulas for Pythagorean triples). Then the remaining part of the table would fit the “formatting rules” Robson infers.

5. See the general comments above about reciprocal pairs and their relation to the numbers on Plimpton 322.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately and completely presents facts and viewpoints of sources – 15

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas

–10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 9

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15 4. (10) Lacks incomplete sentences – 9

5. (10) Lacks run-on or overly complicated sentences – 9

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 14

8. (10) Cites quotations and/or references correctly – 7

Totals: Content and Structure – 95; Style and English Usage – 91.

Peter Zona – The Final Word on Plimpton 322

What you say about Robson’s reinterpretation of Plimpton 322 and her ideas about the correct approach to the study of ancient mathematics is generally excellent and well written. You have understood her main points well and summarized them clearly.

However, I would say that the paper does not go quite far enough in some cases. Since we discussed some of this in quite a bit of detail in class, it would have been good to say more. First, you need to be careful with the “reciprocal pair” terminology since the way the actual numbers on Plimpton 322 are derived is not just the listing of x, 1/x pairs as on the MLC 1670 tablet that Robson reproduces in her Figure 9. The numbers on Plimpton 322 are really derived from reciprocal pairs by the same sort of formulas we saw in the analysis of YBC 6967: from the pair x, 1/x, the numbers

\begin{aligned}c & = (x + 1/x)/2 \\ a & = 1 \\ b & = (x − 1/x)/2 \end{aligned}

are formed and then c^{2} = a^{2} + b^{2}. It is essentially the numbers c and b that occur on Plimpton 322. The goal of a problem text like YBC 6967 is then to derive the x and 1/x from knowledge of c and b. It would have been good to say something about Rudman’s alternative explanation for the genesis of Plimpton 322 – see page 97 in his book.

More importantly, in a paper like this, when you draw on a statement in a source for one of the things you are claiming, even if you are only giving a paraphrase (a rewording in your own words), you need to provide a reference to the place in the source you are referring to. I did not see any of this in your paper. You should get into the habit of doing this as a matter of course for research papers in all of your classes. Not doing this, in fact, can be considered a form of plagiarism and as such it might be dealt with using the Academic Honesty policy. I know you are not trying to conceal your sources and I did not find any direct copying. So I am only lowering you score in the “cites quotations properly” category for this. But be careful about this point in the future.

Specific Comments

1. You have clearly understood and appreciated Robson’s ideas about the importance of taking the historical context and what is known about ancient languages into account in order to avoid anachronistic interpretations of ancient mathematics based on modern mathematical ideas. But it is important, I think, to realize that the people Robson is criticizing here are primarily people who were trained as mathematicians and then have done some exploration of ancient mathematics. Some of them have been “amateur” historians, as you say, but that is not true of Neugebauer, who devoted his whole professional career to this kind of study. In general, mathematicians are somewhat notorious for this sort of ahistorical approach to the history of mathematics, so this sort of criticism is probably something we should hear more often! On the other hand, if it were not for the mathematicians who have looked at this material, we might not know anything about the mathematical ideas involved. Many of the “straight” historians who would be in the position to make more historically accurate appraisals do not know any mathematics! Robson is one of a group of pioneers in this area.

2. This is an accurate and well-written summary of Robson’s argument for rejecting the trigonometry table idea. However, I think it is important to realize that there are possible objections to what she is saying too. For instance, when she says that the the Babylonians did not have a notion of measuring angles and did not visualize circles as being traced out by a radius making a varying angle with the positive x-axis, she is certainly correct. But to me that seems to be almost the same sort of anachronistic reasoning from modern-day mathematical ideas for which she criticizes the others. It is certainly within the Babylonian scheme of things to think of describing the shape of triangles or rectangles by using ratios of the lengths of their sides. This effectively leads to the same sorts of quantities we call trigonometric functions of the angle. It also gives another possible explanation for the left-hand column of the remaining part of Plimpton 322 that would make sense in the Babylonian context. Whether you call the tablet a trigonometric table or not based on this is a matter of terminology, I would say. But I think that it easily could have been a table of quantities related to differently shaped rectangles or triangles to the Babylonian scribe who wrote it. And by the way, that would also give a way to reconcile this interpretation with Robson’s ideas about the table format. Suppose the relevant number to the Babylonians was the ratio c^{2}/b^{2} – a sort of “shape parameter” – (not the p or q from Neugebauer’s generating formulas for Pythagorean triples). Then the remaining part of the table would fit the “formatting rules” Robson infers.

3. See general comments above about the reciprocal pairs.

4. That is actually understandable since the article you read was published in 2002, and Rudman’s book appeared in 2009.

5. A technical point – this is fine as a citation for the MAA web site where you can download the article. However, since the article appeared first in a print journal, I think it would be more appropriate to give the reference for the original print form:

Author, “Title of Article.” Name of Journal, volume number (year), pages.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately and completely presents facts and viewpoints of sources – 15

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas

d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

–10 c. (10) Transitions between paragraphs are logical – 10 Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 15

8. (10) Cites quotations and/or references correctly – 3

Totals: Content and Structure – 95; Style and English Usage – 93.

Matt Kasuba – “No Luck for Sherlock”

What you say about Robson’s reinterpretation of Plimpton 322 and her ideas about the correct approach to the study of ancient mathematics is generally excellent and well written. You have understood her main points well and summarized them clearly.

However, the paper does not go quite far enough in some cases. (This would also have “filled out” the paper somewhat; it is significantly under the target length.) Since we discussed some of this in quite a bit of detail in class, it would have been good to say more. First, you need to be careful with the “reciprocal pair” terminology since the way the actual numbers on Plimpton 322 are derived is not just the listing of x, 1/x pairs as on the MLC 1670 tablet that Robson reproduces in her Figure 9. The numbers on Plimpton 322 are really derived from reciprocal pairs by the same sort of formulas we saw in the analysis of YBC 6967: from the pair x, 1/x, the numbers

\begin{aligned}c & = (x + 1/x)/2 \\ a & = 1 \\ b & = (x − 1/x)/2 \end{aligned}

are formed and then c^{2} = a^{2} + b^{2}. It is essentially the numbers c and b that occur on Plimpton 322. The goal of a problem text like YBC 6967 is then to derive the x and 1/x from knowledge of c and b. It would have been good to say something about Rudman’s alternative explanation for the genesis of Plimpton 322 – see page 97 in his book.

More importantly, in a paper like this, when you draw on a statement in a source for one of the things you are claiming, even if you are only giving a paraphrase (a rewording in your own words), you need to provide a reference to the place in the source you are referring to. I did not see any of this in your paper. You should get into the habit of doing this as a matter of course for research papers in all of your classes. Not doing this, in fact, can be considered a form of plagiarism and as such it might be dealt with using the Academic Honesty policy. I know you are not trying to conceal your sources and I did not find any direct copying. So I am only lowering you score in the “cites quotations properly” category for this. But be careful about this point in the future.

A final technical point: The entries in the bibliography are not complete. A standard format for citing journal articles is:

Author, “Title of Article.” Name of Journal, volume number (year), pages.

Specific Comments

1. You have caught Robson’s ideas about the importance of taking the historical context and what is known about ancient languages into account in order to avoid anachronistic interpretations of ancient mathematics based on modern mathematical ideas. But it is important, I think, to realize that the people Robson is criticizing here are primarily people who were trained as mathematicians and then have done some exploration of ancient mathematics. Neugebauer and Buck certainly fit into that category. (And in general, mathematicians are somewhat notorious for this sort of ahistorical approach to ancient mathematics, so this sort of criticism is something we need to hear!) On the other hand, if it were not for the mathematicians who have looked at this material, we might not know anything about the mathematical ideas involved. Many of the historians who would be in the position to make more historically accurate appraisals do not know any mathematics! Robson is one of a group of pioneers in this area.

2. This is an accurate summary of Robson’s argument for rejecting the trigonometry table idea. However, I think it is important to realize that there are possible objections to what she is saying too. For instance, when she says that the the Babylonians did not have a notion of measuring angles and did not visualize circles as being traced out by a radius making a varying angle with the positive x-axis, she is certainly correct. But to me that is almost the same sort of anachronistic reasoning from modern-day mathematical ideas that she accuses Neugebauer of doing. It is certainly within the Babylonian scheme of things to think of describing the shape of triangles or rectangles by using ratios of the lengths of their sides. This effectively leads to the same sorts of quantities we call trigonometric functions of the angle. But it also gives another possible explanation for the left-hand column of the remaining part of Plimpton 322 that would make sense in the Babylonian context. Whether you call that a trigonometric table or not is a matter of terminology, I would say. But it easily could have been a table of differently shaped rectangles or triangles to the Babylonians!

3. (Continuing from the previous comment): And by the way, that would also give a way to reconcile this interpretation with Robson’s ideas about the table format. Suppose the relevant number to the Babylonians was the ratio c^{2}/b^{2} – a sort of “shape parameter” – (not the p or q from the generating formulas for Pythagorean triples). Then the remaining part of the table would fit the “formatting rules” Robson infers.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately and completely presents facts and viewpoints of sources – 15

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas

–10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper 1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 8

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 14

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 9

7. (15) Is free of jargon, cliches, and misused words – 15

8. (10) Cites quotations and/or references correctly – 3

Totals: Content and Structure – 95; Style and English Usage – 89.

Dan Finn – “Last Word on Plimpton 322”

This is a very good paper on Eleanor Robson’s ideas about Plimpton 322’s place in Babylonian mathematics. You have caught her ideas about the importance of taking the historical context and what is known about ancient languages into account in order to avoid anachronistic interpretations of ancient mathematics based on modern mathematical ideas. (Mathematicians are somewhat notorious for doing that, by the way, so it is something we need to hear!)

The major comments I have are the following. First, it would been better to acknowledge that, even though you may be convinced by Robson’s reasoning, this is still an area of active debate, and her ideas have by no means been completely accepted. For instance, see point 3 below for a possible objection to her rejection of the trigonometry table interpretation. Second, you need to be a bit more careful with the “reciprocal pair” terminology since the way the actual numbers on Plimpton 322 are derived is not just the listing of x, 1/x pairs as on the MLC 1670 tablet that Robson reproduces in her Figure 9. The numbers on Plimpton 322 are really derived from reciprocal pairs by the same sort of formulas we saw in the analysis of YBC 6967: from the pair x, 1/x, the numbers

\begin{aligned}c & = (x + 1/x)/2 \\ a & = 1 \\ b & = (x − 1/x)/2 \end{aligned}

are formed and then c^{2} = a^{2} + b^{2}. It is essentially the numbers c and b that occur on Plimpton 322. The goal of a problem text like YBC 6967 is then to derive the x and 1/x from knowledge of c and b. Third and finally, it would have been good to say something about Rudman’s alternative explanation for the genesis of Plimpton 322 – see page 97 in his book.

Specific Comments

1) I think Robson lists Joyce’s web site as a source for this idea because it is so readily accessible. However, there are other historians who have proposed the trigonometry table interpretation as well.

2) When you say the “first column” here, I think you are referring to the conjectural missing part of the tablet on the left side. (I saw the actual Plimpton 322 last weekend at the exhibit in New York that I mentioned briefly in class, and it is very clear from the shape of the back that there must have originally been an additional section on the left, perhaps as wide as one third of the remaining tablet – certainly enough space for two additional columns of numbers). When and why that portion got separated and what it contained may never be known (unless we get really lucky and someone finds the missing piece and recognizes that it fits with the other portion!).

3) This is an accurate summary of Robson’s argument for rejecting the trigonometry table idea. However, I think it is important to realize that there are possible objections to what she is saying too. For instance, when she says that the the Babylonians did not have a notion of measuring angles and did not visualize circles as being traced out by a radius making a varying angle with the positive x-axis, she is certainly correct. But to me that is almost the same sort of anachronistic reasoning from modern-day mathematical ideas that she accuses Neugebauer of doing. It is certainly within the Babylonian scheme of things to think of describing the shape of triangles or rectangles by using ratios of the lengths of their sides. This effectively leads to the same sorts of quantities we call trigonometric functions of the angle. But it also gives another possible explanation for the left-hand column of the remaining part of Plimpton 322 that would make sense in the Babylonian context. Whether you call that a trigonometric table or not is a matter of terminology, I would say. But it easily could have been a table of differently shaped rectangles or triangles to the Babylonians! And by the way, that would also give a way to reconcile this interpretation with Robson’s ideas about the table format. Suppose the relevant number to the Babylonians was the ratio c^{2}/b^{2} – a sort of “shape parameter” – (not the p or q from the generating formulas for Pythagorean triples). Then the remaining part of the table would fit the “formatting rules” Robson infers.

4) First modern owner would be more accurate(!)

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 17

3. (10) Provides sufficient textual evidence about the main topic – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 7

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 8

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 13 8. (10) Cites quotations and/or references correctly – 10 Totals: Content and Structure – 92; Style and English Usage – 96.

Astrid Ludwig – “Plimpton 322: Evaluating Theories

This is a good paper on Eleanor Robson’s ideas about Plimpton 322’s place in Babylonian mathematics. You have caught her ideas about the importance of taking the historical context and what is known about ancient languages into account in order to avoid anachronistic interpretations of ancient mathematics based on modern mathematical ideas. (Mathematicians are somewhat notorious for doing that, by the way, so it is something we need to hear!) I like the way you tried to think independently about Robson’s conclusions and evaluate her reasoning. That is just what a good research paper should do.

The major criticism I have, though, is that for completeness it would have been good to go into somewhat more detail about the technical points. First, it would have been better to explain the p, q in Neugebauer’s interpretation and how they generate Pythagorean triples via the formulas a = p^{2} − q^{2}, b = 2pq and c = p^{2} + q^{2}. Second, see point 1 below. Third and finally, it would have been good to consult another source that has a lot to say on these questions – the Rudman book we used as a text for the course. For instance, it would have been very good to say something about Rudman’s alternative explanation for the genesis of Plimpton 322 – see page 97 in his book.

On the technical side, you need to be more careful in using direct quotations! When you use one, you should always identify the source. As described in the assignment, an acceptable way would be to say (Robson, 108) to identify a quotation from Robson’s article taken from page 108. Do this even if you have only one source. By the way, the name of the journal Robson’s article appeared in is the American Mathematical Monthly. The Mathematical Association of America is the professional organization that publishes it and holds the copyright to materials published there.

Specific Comments

1. You need to be a bit more careful with the “reciprocal pair” terminology since the way the actual numbers on Plimpton 322 are derived is not just the listing of x, 1/x pairs as on the MLC 1670 tablet that Robson reproduces in her Figure 9. The numbers on Plimpton 322 are really derived from reciprocal pairs by the same sort of formulas we saw in the analysis of YBC 6967: from the pair x, 1/x, the numbers

\begin{aligned}c & = (x + 1/x)/2 \\ a & = 1 \\ b & = (x − 1/x)/2 \end{aligned}

are formed and then c^{2} = a^{2} + b^{2}. It is essentially the numbers c and b that occur on Plimpton 322. The goal of a problem text like YBC 6967 is then to derive the x and 1/x from knowledge of c and b.

2. This seems like a relatively minor point to me.

3. This is an accurate summary of Robson’s argument for rejecting the trigonometry table idea. However, I think it is important to realize that there are possible objections to what she is saying too. For instance, when she says that the the Babylonians did not have a notion of measuring angles and did not visualize circles as being traced out by a radius making a varying angle with the positive x-axis, she is certainly correct. But to me that is almost the same sort of anachronistic reasoning from modern-day mathematical ideas that she accuses Neugebauer of doing. It is certainly within the Babylonian scheme of things to think of describing the shape of triangles or rectangles by using ratios of the lengths of their sides. This effectively leads to the same sorts of quantities we call trigonometric functions of the angle. But it also gives another possible explanation for the left-hand column of (the remaining part of) Plimpton 322 that would make sense in the Babylonian context. Whether you call that a trigonometric table or not is a matter of terminology, I would say. But it easily could have been a table of differently shaped rectangles or triangles to the Babylonians! And by the way, that would also give a way to reconcile this interpretation with Robson’s ideas about the table format. Suppose the relevant number to the Babylonians was the ratio c^{2}/b^{2} – a sort of “shape parameter” – (not the p or q from the generating formulas for Pythagorean triples). Then the remaining part of the table would fit the “formatting rules” Robson infers.

4. See comment immediately above!

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 7

2. (20) Accurately presents facts and viewpoints of sources – 17

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10.

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 14

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 8

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 13 8. (10) Cites quotations and/or references correctly – 5 Totals: Content and Structure – 94; Style and English Usage – 88.

Sarah Cahn – “Investigating Different Interpretations of Plimpton 322”

This is a very good paper on Eleanor Robson’s ideas about Plimpton 322’s place in Babylonian mathematics. You have caught her ideas about the importance of taking the historical context and what is known about ancient languages into account in order to avoid anachronistic interpretations of ancient mathematics based on modern mathematical ideas. (Mathematicians are somewhat notorious for doing that, by the way, so it is something we need to hear!) I like the way you tried to think independently about Robson’s conclusions and evaluate her reasoning (following my suggestions on your first draft). That is just what a good research paper should do. However, I think it would also have been good to bring some of Rudman’s ideas into the mix here. See comment 3 below. And Robson’s interpretation of Plimpton 322 is not really that it is a set of solutions to problems as you say in the conclusion. It’s really more like the raw material or data for creating statements of problems of a particular type. Specific Comments

1. In addition to the uses the Babylonian scribe might have made of this tablet, there are also the question of just how the information on it was generated, of course!

2. Robson’s reasoning is somewhat more careful than this, of course. She is referring specifically to tablets generated at that particular time in the temple at Larsa.

3. This section is not quite correct. There is only one problem on YBC 6967. It is of a particular type and the numbers on Plimpton 322 could have been used to generate similar problems with different numerical values. (This is something like the way WebAssign can generate different individually-tailored calculus problems of the same type for all the students using it!) It might have been good here to consider Rudman’s proposed reconstruction of how Plimpton 322 came to be as well – see page 97 in his book The Babylonian Theorem.

4. The sexagesimal system the Babylonians was used consistently throughout this period. The key point here is not that the sexagesimal computations were used in generating the tables of reciprocal pairs. It’s really that the tables of reciprocal pairs were a key component of the way they approached numerical computations using their sexagesimal system.

5. This is an accurate summary of Robson’s argument for rejecting the trigonometry table idea and a possible objections to what she is saying too. For instance, when she says that the the Babylonians did not have a notion of measuring angles and did not visualize circles as being traced out by a radius making a varying angle with the positive x-axis, she is certainly correct. But to me that is almost the same sort of anachronistic reasoning from modern-day mathematical ideas that she accuses Neugebauer of doing. It is certainly within the Babylonian scheme of things to think of describing the shape of triangles or rectangles by using ratios of the lengths of their sides. This effectively leads to the same sorts of quantities we call trigonometric functions of the angle. But it also gives another possible explanation for the left-hand column of (the remaining part of) Plimpton 322 that would make sense in the Babylonian context. Whether you call that a trigonometric table or not is a matter of terminology, I would say. But it easily could have been a table of differently shaped rectangles or triangles to the Babylonians! And by the way, that would also give a way to reconcile this interpretation with Robson’s ideas about the table format. Suppose the relevant number to the Babylonians was the ratio c^{2}/b^{2} – a sort of “shape parameter” – (not the p or q from the generating formulas for Pythagorean triples). Then the remaining part of the table would fit the “formatting rules” Robson infers.

6. This citation is not complete. A standard format for citing journal articles is:

Author, “Title of Article.” Name of Journal, volume number (year), pages.

It would also have been good cite Rudman as a reference and take what he has to say about this into account.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 17

2. (20) Accurately presents facts and viewpoints of sources – 16

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 9

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 9

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 14

8. (10) Cites quotations and/or references correctly – 7

Totals: Content and Structure – 92; Style and English Usage – 93.

Tom Santa Maria – “Hypatia”

This is a very good and balanced paper on Hypatia. I’m glad that you were able to incorporate some comments on the film Agora, and I agree with you (for the most part) about the liberties the authors of the screenplay took with the invented characters and the depiction of Hypatia’s possible discovery of the elliptical orbits of planets (the “wanderers”). There is no historical evidence for Hypatia anticipating Kepler by about 1200 years. But I would say it is an interesting speculation since we do know:

1. about Hypatia’s interest in astronomy,

2. the fact that the previous ideas of Aristarchus on a heliocentric solar system would have been known at her time and in her intellectual circles, and

3. about Hypatia’s own expertise in the theory of conic sections (based on her commentary on Apollonius).

The role of the slave Davus in the film is entirely fictional, as you point out, but inserted to further the plot and to personalize some of the conflicts that people around Hypatia would have felt.

The only critical comments I have are all technical writing issues. First, although you are a strong and fluent writer, I think there are times when you tend to say things in an overly flowery, slightly self-consciously “learned” way that actually comes across differently than you might intend. I’m thinking of things like the sentence “Alexandria was Christianizing fanatically and with celerity.” There are times when simpler and more direct ways of saying things are preferable, and this is one, at least to me.

Second, I think you should review what a good English style manual has to say about commas and semicolons. I’m looking at an old copy of the Harbrace College Handbook that says, essentially:

1. Use a semicolon between main clauses (i.e. complete sentences in themselves) not joined by and, but, or, nor, for, so or yet, and

2. Do not use a semicolon between two parts of unequal grammatical rank. For instance: “Then came the attack; this frightened the unprepared” (two equal main clauses). On the other hand: “Then came the attack, frightening the unprepared” (second clause subsidiary, modifying the noun “attack”).

3. Semicolons may also be used for clarity when the clauses to be separated themselves contain commas. “However” only needs a semicolon when it introduces a second complete clause.

There are times when you have semicolons that should be commas, and vice versa.

Specific Comments

1. Technical point – a “see” is the office or position of bishop (essentially the same as saying “bishopric”). The person holding the office is a bishop. That is what you mean here.

2.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 20

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 7

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 8 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 16

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 9

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 6

7. (15) Is free of jargon, cliches, and misused words – 14

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 98; Style and English Usage – 90.

Catherine Caracciolo – The Infamous Life of Hypatia of Alexandria

This is a good summary of Hypatia’s life and contributions, but it does not show much independent thought or evaluation of what you found in the different sources. On the other hand, it is definitely good that you looked beyond the easy Internet sources about her – I think you got a more balanced and complete picture that way. Still, it might have been good to try to say more about what Hypatia’s mathematics consisted of and how it connected with topics we have discussed in this course.

I don’t think the word “infamous” means quite what you think it does: “having an extremely bad reputation, deserving of or causing an evil reputation, detestable.” If you do mean to say this about Hypatia, for instance if you are using the word “infamous” ironically, then it can only be from the point of view of someone like Cyril or one of the Christians responsible for her death. It is questionable, then, whether those opinons of her are justified(!)

On the technical side, as you get into the paper, the paragraphs start to get very long. The division into paragraphs does not respect the logical structure of what you are saying either. This is something to work on.

Specific Comments

1. Synesius is probably best known (apart from his relation with Hypatia) as bishop of Ptolemais in Egypt. His role in possible inventions such as the astrolabe and hydroscope as (as far as I know) mostly the product of his contact with Hypatia.

2. As noted above, most people think that Synesius’s role in these inventions came mostly or entirely through his relation with Hypatia, and that Hypatia might in fact be responsible for the main ideas. On the other hand, some people doubt that Hypatia originated either of these ideas. Astrolabes, for instance, might have been constructed quite a long time before Hypatia’s time.

3. I’m not entirely sure what this phrase “person intellectual stimulation” means. Are you thinking that she advocated that women should have the freedom to pursue their intellectual development?

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 5

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

8 c. (10) Transitions between paragraphs are logical – 8 Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 8

7. (15) Is free of jargon, cliches, and misused words – 13

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 88; Style and English Usage – 94.

Kyle Kadlick – Hypatia of Alexandria

This is a reasonably good paper about the life and contributions of Hypatia, but it does not show much independent thought or evaluation of what you found in the different sources. I do like that you tried to include some of the actual mathematics that has been attributed to her personally with the discussion of problems on conics. However there is more you could have said there (see comment 2 below).

There is also a technical point about citations that you need to think about. The Deakin article you looked at is not really cited correctly. A standard format for citing journal articles is:

Author, “Title of Article.” Name of Journal, volume number (year), pages.

Specific Comments

1. “Family of mathematicians” might be an overstatement, since we do not know anything about others in her family beyond Theon.

2. This is good, but you might have said more. For instance, if you think in geometric terms (in Cartesian coordinates), the solutions of these equations are essentially finding the intersections of a line and a hyperbola. With modern algebra, of course, the solution is not difficult. You can solve the equation of the line for y, substitute into the other equation, and then factor or use the quadratic formula to solve for x.

3. This really has to be treated as a legend. Saying she was physically beautiful is one way that some people have tried to make her seem more like a tragic heroine.

4. There is also a major question whether Cyril was directly responsible for Hypatia’s death or not. Some people would say that he just provoked others to do it, or even just helped to create the conflict that led to it, without giving a direct order for her death.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 9

2. (20) Accurately presents facts and viewpoints of sources – 16

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 5

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 8 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 8 d. (10) Is concise, clear, and not too repetitive – 7 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

Style and English Usage – The paper

sented – 8 1. (20) Is clear and direct in its use of language – 16

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 8

7. (15) Is free of jargon, cliches, and misused words – 13

8. (10) Cites quotations and/or references correctly – 8

Totals: Content and Structure – 81; Style and English Usage – 88.

Kylee Sullivan – “Reserve your Right to Think”

This is a good paper about the life and contributions of Hypatia, but it does not show much independent thought or evaluation of what you found in the different sources. It is true that her mathematical attainments were very unusual for her time and place, and that aspect of her story is an appealing one for many people today who want to encourage women to pursue careers in mathematics and the sciences. It would have been good, though if you ahd tried to include some of the actual mathematics that has been attributed to her personally with a discussion of problems on conics from her commentary on Apollonius. I think that the sources you consulted were probably not sufficient for getting a complete picture of what is known, though.

Specific Comments

1. But this is somewhat questionable. For instance, astrolabes might have been known quite a bit earlier. And the discussion of the hydrometer comes from one of the letters Synesius wrote to her, asking her to construct one of the devices. So it is not clear whether she was responsible for the idea behind the device, or whether she was just in a better position to have one made or provide advice on how they should be made.

2. “Neo-Platonist School” sounds like a formal title, which would not have been used at the time. Be more careful about things like this.

3. Cyril was bishop of Alexandria. I think that even at this time, the term Pope was probably only used for the bishop of Rome.

4. There is also a major question whether Cyril was directly responsible for Hypatia’s death or not. Some people would say that he just provoked others to do it, or even just helped to create the conflict that led to it, without giving a direct order for her death. So this is another point that is open to debate.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 7

2. (20) Accurately presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 8

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 5

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10 1. (20) Is clear and direct in its use of language – 10

2. (10) Is free of misspellings – 7

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 13

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 88; Style and English Usage – 93.

Kevin Sutman – The Life and Times of Hypatia of Alexandria

This is a decent, but rather perfunctory paper about the life and contributions of Hypatia. It does not show much independent thought or evaluation of what you found in the different sources you consulted. The whole paper reads, in fact, like an almost undigested collection of paraphrases from those sources.

I also think it would have been much better to consult more authoritative sources than some of the ones you used. For example, your first source is essentially a student paper written by an undergraduate at Agnes Scott College. It is superficial at best. Only the third one is really suitable, and that only for a start. There were several more scholarly sources for this topic on reserve in the science library that you could have consulted for more complete and more balanced presentations.

Specific Comments

1. This is an overstated cliché. Avoid trite phrases like this in formal writing.

2. I think this is a matter of interpretation. Whether there was a formal post of leader is not clear. She may very well have been an informal leader by reason of the respect she commanded.

3. “Efficiency as a teacher” sounds strange here, and I do not think it is what you mean to say. (What does “efficiency as a teacher” mean? Good time management in class? Returning graded assignments quickly?) I think it would be more accurate to say it was a testament to the strength of her character, or to the respect that she was able to inspire in her students.

4. If I understand you, “legendary” is another cliché.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 3

2. (20) Accurately presents facts and viewpoints of sources – 15

3. (10) Provides sufficient textual evidence to support the argument – 8

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 5

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper 1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 9

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 9

6. (10) Uses correct punctuation – 8

7. (15) Is free of jargon, cliches, and misused words – 11

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 81; Style and English Usage – 89.

Lucia Bostrom – Hypatia of Alexandria

This is a good, but rather perfunctory paper about the life and contributions of Hypatia. It does not show much independent thought or evaluation of what you found in the different sources you consulted. The whole paper reads, in fact, like an almost undigested collection of paraphrases from those sources.

I also think it would have been much better to consult more authoritative sources than some of the ones you used. For example, several of your sources are essentially student paper written by undergraduates. They are is superficial at best. Only a few are really suitable, and that only for a start. There were several more scholarly sources for this topic on reserve in the science library that you could have consulted for more complete and more balanced presentations.

More importantly, in a paper like this, when you draw on a statement in a source for one of the things you are claiming, even if you are only giving a paraphrase (a rewording in your own words), you need to provide a reference to the place in the source you are referring to. I did not see any of this in your paper. You should get into the habit of doing this as a matter of course for research papers in all of your classes. Not doing this, in fact, can be considered a form of plagiarism and as such it might be dealt with using the Academic Honesty policy. I know you are not trying to conceal your sources and I did not find any direct copying. So I am only lowering you score in the “cites quotations properly” category for this. But be careful about this point in the future.

Specific Comments

1. This should be Ptolemais (the name of a city, not the name of the astronomer, or the general).

2. This is an overstated cliché. Avoid trite phrases like this in formal writing.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 7

2. (20) Accurately presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 5

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 5

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented –10 1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 8

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 13

8. (10) Cites quotations and/or references correctly – 3

Totals: Content and Structure – 85; Style and English Usage – 89.

Andrew Marzo – Consequences of Non-Euclidean Geometry

What you have to say about the topic is often good. See the comments below for some exceptions though, where I disagree with what what I think you are trying to say. It may be that I am misunderstanding what you meant to say in some of those cases. However, then the comment I would have is that you need to be really careful about how you discuss some of these ideas because they are pretty deep and complicated.

The structure of the paper itself is so repetitive, though, that it seems as though you may have been concerned that you needed to “pad” it to get to some required length. There was not any need to do that here.

Specific Comments

1. This is slightly incorrect historically. Plato lived about two generations before the time of Euclid. So “Euclidean” has to be qualified here as referring to the mathematical structure itself, not the specific presentation of geometry in the work of Euclid. It did not exist yet.

2. This is the main disagreement I have with your claims. It is not correct to say that the discovery of non-Euclidean geometry “disproved Euclidean geometry.” What it disproved was the idea that there could only be one kind of geometry that made sense as mathematics. Even though non-Euclidean geometry is now around, both it and Euclidean geometry are equally consistent. Neither disproves the other!

3. There seems to be a missing citation here.

4. I don’t agree with the first part of this either. The existence of several different forms of geometry as mathematical theories only says that which geometry describes physical space is a separate question! It is probably correct to say that this destroyed one of the bases for believing in rationalism, though.

5. This is a real “which came first – the chicken or the egg?” situation. Is Euclidean geometry based on our perception of space, or is it the other way around? Is our understanding of the properties of the counting numbers based on our perception of time, or is it the other way around? In order to believe as Plato or Kant did, you have to believe that geometry and arithmetic have some kind of an existence independent of human beings. I think that is somewhat hard to imagine!

6. The discovery of non-Euclidean geometries came after Kant’s death, so this sounds strange from the historical point of view. He never had to grapple with the implications of this discovery, and he may very well have changed his mind if he knew about it.

7. This is not really correct either. It is not the non-Euclidean geometry itself that is only valid in the extreme circumstances. Non-Euclidean properties of actual, physical space are only really observable when you get outside the normal range of human experience (over most of human history, anyway).

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 13 3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 4 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 8

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 8

7. (15) Is free of jargon, cliches, and misused words – 15

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 85; Style and English Usage – 93.

John Macomber – Mathematics and Its Philosophical Consequences

What you have to say about the topic is often very good. See the comments below for some small exceptions though, where I disagree with what what I think you are trying to say. It may be that I am misunderstanding what you meant to say in some of those cases. However, then the comment I would have is that you need to be really careful about how you discuss some of these ideas because they are pretty deep and complicated.

The major criticism I have about the paper is that the concluding paragraph seems to go off in a slightly new direction that addresses your personal reactions to some of the ideas. That doesn’t seem appropriate here, since the rest of the paper has not addressed that sort of question.

Specific Comments

1. I think “circumstances” is the wrong word here. (Either that, or I don’t understand what you mean by it.) Don’t you mean the original square of side 1 can be cut into two triangles? Then four copies of that triangle can be reassembled to make a square of side \sqrt{2}. This is the heart of the mathematical reasoning behind the dialog between Socrates and the slave boy in the Meno.

2. This is a real “which came first – the chicken or the egg?” situation. Is our understanding of the properties of the counting numbers based on our perception of time, or is it the other way around? Is Euclidean geometry based on our perception of space, or is it the other way around? In order to believe as Plato or Kant did, you have to believe that arithmetic and geometry have some kind of an existence independent of human beings. I think that is somewhat hard to imagine!

3. I would agree with this if you deleted the word “mathematical” here. Mathematicians have usually tended to ignore (even scoff at) some of Kant’s ideas, especially after the properties of non-Euclidean geometry got to be better known and it got to be hard to defend Kant’s idea that Euclidean geometry was the only conceivable one.

4. It is true that logicism and formalism did have some successes. But since you brought this up, they ultimately failed. As you may have read in the Davis and Hersh book, Kurt Gödel eventually proved a result called the Incompleteness Theorem which says, in effect, that in any axiomatic mathematical system powerful enough to describe all of the arithmetic of the integers, there are true statements that cannot be proved to be true within the system and false statements that cannot be proved to be false within the system. Moreover, he also showed slightly later that one of these “undecidable” statements is that that the system itself is consistent (that is, that it contains no contradictions). Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure: a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 5

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 9

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 9

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 13

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 93; Style and English Usage – 92.

Max Pettinelli – Euclid’s Number Theory

Your paper on some of the highlights of Euclid’s treatment of number theory is very good and well-written for the most part. I have a technical comment about citing your sources (see comment 1 below). What you said about the Riemann zeta-function is not quite correct either (see comment 4 below). But I am glad you chose this topic (you were the only one who did!) and I hope you found it interesting.

Specific Comments

1. This is mostly a technical point, but it is important. In a paper like this, when you draw on a statement in a source for one of the things you are claiming, even if you are only giving a paraphrase (a rewording in your own words), you need to provide a reference to the place in the source you are referring to. You do not do this consistently in this paper. For instance, the location where I marked comment 1 has a direct quotation for which you only identify the name of the author of the web page in the text of your paper. David Joyce’s web site should also be listed in the Works Cited, of course. The footnote you give on page 4 is closer to what should be done. The point is that you should get into the habit of including complete references as a matter of course for every quotation or paraphrase you use for research papers in all of your classes. Not doing this, in fact, can be considered a form of plagiarism. As such it might be dealt with using the Academic Honesty policy. I know you are not trying to conceal your sources. So I am only lowering you score in the “cites quotations properly” category for this. But be careful about this point in the future.

2. I think it would be better to say it like this: While Euclid gives the argument for the three primes A, B, C, the argument can be generalized to show that for any finite list of primes with a n entries, a number can be constructed that must be divisible by some prime number not on the list. Hence the number of primes cannot be finite.

3. This sounds rather harsh. It would be better to say something more neutral here – perhaps “Proving this statement has turned out to be extremely difficult since all of the infinitely many even numbers must be accounted for.”

4. The only place where you have run into some mathematical difficulties is here in the description of the Riemann zeta-function. What you say is not correct, but I have not penalized you too much for this because this was certainly far beyond anything we did in the course. The idea is that ζ(s) is defined as the sum of an infinite series:

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}}

for real s > 1. (You may have seen series like this if you studied this topic in a calculus course.)

Riemann (by the way, this is the same Riemann we mentioned briefly toward the end of the course in connection with non-Euclidean geometry) showed that it is possible to make sense of this as a function of a variable s taking complex number values s = u+it.

The statement of the Riemann Hypothesis is then that the only “non-trivial” zeroes of the function ζ(s) occur at numbers of the form s = \frac{1}{2} +it. This turns out to be related to properties of prime numbers because there is another formula for ζ(s) coming from work of an earlier mathematician named Leonhard Euler:

ζ(s) = ∏_{p prime} (1 − \frac{1}{p^{s}} )^{−1} .

(The ∏ means to take the product of these factors.) This is certainly the most famous currently unsolved problem in mathematics and you might be interested to know that there is a $1 million prize offered for a solution by the Clay Institute of Mathematics (whose web site you consulted).

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 18

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 9

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 15

8. (10) Cites quotations and/or references correctly – 5

Totals: Content and Structure – 98; Style and English Usage – 92.

Xuan Ha Nghiem – Platonic Solids

Your paper on the Platonic solids and some of the ways people have attempted to use these shapes to understand properties of the physical world (some valid and some invalid) is very good. I especially like that you chose a topic that no one else did and that you tied the topic of the paper to our discussion of Raphael’s The School of Athens in class. Your writing is also very strong here!

The only thing I would have liked to see more of was more of the discussion of the Euler relation and the resulting proof that only 5 platonic solids can exist. See comment 3 below for a sketch of the details that you asked about in your email message.

Specific Comments

1. It would be better to say “...with their hands in gestures symbolizing different aspects of their philosophies.” (It is not really possible to “gesture at an aspect of a philosophy” in the literal sense!)

2. I think you mean that if you had four polygons meeting at a vertex with angles of 90^{∘}, then the surface would have to be a part of a plane – that is, it could not curve in three-dimensional space to enclose a volume since the four faces meeting at that vertex would all be in the same plane.

3. This is the point you asked about in your email. This is one way to see it. First, it can be seen that the numbers

V = number of vertices
E = number of edges
F = number of faces

of a convex polyhedron must always satisfy

V − E + F = 2.

This is called the Euler relation, after the mathematician Leonhard Euler. For a Platonic solid, where the sides, angles, and edges are all congruent, each vertex must have the same number m ≥ 3 of edges meeting there. Similarly, each face must have the same number n ≥ 3 of edges. Each edge has two different endpoints, so

2E = mV,

since the total number of edges meeting at all V vertices is mV, but that counts each edge twice. Similarly, each edge is a part of two different faces, so

2E = nF.

Solve these equations for V and F and substitute into the Euler relation:

\frac{2E}{m} − E + \frac{2E}{n} = 2.

Then divide by 2E and rearrange to get

\frac{1}{m} + \frac{1}{n} = \frac{1}{E} + \frac{1}{2}.

Then the idea is that there are only a few different ways to get solutions of this equation with whole numbers m, n, E. Note first that the right side is always strictly bigger than 1/2. This says that m, n cannot both be large. The only solutions, in fact, are:

m n E polyhedron
3 3 6 tetrahedron
3 4 12  cube
4 4 12 tetrahedron
3 5 30 dodecahedron
5 3 30 icosahedron

4. You should also give a source for this quotation. Where did you find it?

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 10

2. (20) Accurately presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 8

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 10

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas

–10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 10

7. (15) Is free of jargon, cliches, and misused words – 15

8. (10) Cites quotations and/or references correctly – 9

Totals: Content and Structure – 96; Style and English Usage – 97.

Kaitlin O’Hagan –

This is a good summary of Hypatia’s life and contributions. It does not really show a lot of independent thought or evaluation of what you found in the different sources. On the other hand, it is definitely good that you looked beyond the easier Internet sources about her – I think you got a more balanced and complete picture that way. As a technical point, I think it was probably not necessary to list some of the sources you put in your Works Cited list. Since you do have more authoritative sources, for example, the student paper written by an undergraduate at Agnes Scott College could have been omitted. It is superficial at best and the information about the commentary on Apollonius is much better and more complete in several of the other sources you list.

It might have been good to try to say more about what is known about what Hypatia’s mathematics consisted of (in particular the commentary on Apollonius’s Conics) and how it connected with topics we have discussed in this course.

Your paper is very well-written.

Specific Comments

1. Some people doubt that Hypatia originated the astrolabe. Similar devices might have been constructed quite a long time before Hypatia’s time. What we do know is that Synesius credited Hypatia with the idea, as you say.

2. I would say that this really has to be treated as a legend. Saying she was physically beautiful is one way that some people have tried to make her seem more like a tragic heroine.

3. This is really an interpretation and a statement of opinion, of course. I do not believe there is any direct evidence we have about Cyril’s emotional state at the time. So this probably deserves a reference to the source where you found the interpretation, and a qualification that there is really no way that we can know this for sure.

4. The fact that Cyril was canonized (much) later is an interesting sidelight to the story. It probably deserves more comment than the quick footnote that you give it. People inclined to blame Cyril for Hypatia’s death have used this to criticize the role of Christianity in this story, for instance. Because Cyril did not become a saint until long after his death, it would probably be better either to always refer to him by this title or not to do it at all. Throwing in the title at this point seems a bit odd.

Content and Structure – The paper

1. (10) Draws on suitable and sufficient sources – 8

2. (20) Accurately and completely presents facts and viewpoints of sources – 18

3. (10) Provides sufficient textual evidence to support the argument – 10

4. (10) Gives evidence of independent evaluation of the claims in the sources and original thought – 7

5. Has a clearly articulated and logical structure:

a. (10) First paragraph lays out the topic or question – 10 b. (10) Later paragraphs have topic sentences and clearly identifiable main ideas –

10 c. (10) Transitions between paragraphs are logical – 10 d. (10) Is concise, clear, and not too repetitive – 10 e. (10) Has a clearly identifiable conclusion that draws together the evidence pre-

sented – 10

Style and English Usage – The paper

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (10) Lacks run-on or overly complicated sentences – 10

6. (10) Uses correct punctuation – 9

7. (15) Is free of jargon, cliches, and misused words – 13

8. (10) Cites quotations and/or references correctly – 10

Totals: Content and Structure – 93; Style and English Usage – 97.