Andrew Marzo

Your response to the essay question on the midterm is very good. You have definitely “hit all of the high points” and provided just the right level of detail about the way the different historians we discussed would interpret “quadratic algebra” problems such as the one on the YBC 6967 tablet. The only substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Catherine Caracciolo

You have a very good response to the essay question on the midterm. You have definitely “hit all of the high points” and provided the right level of detail about the way the different historians we discussed would interpret “quadratic algebra” problems such as the one on the YBC 6967 tablet.

The substantive comments I have are first that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources like this. The problem is one of solving an equation however you slice it. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Next, it would be better to qualify some of the statements such as: “Neugebauer is the only one” who would agree with the quote. He is the only historian we discussed who would do so. But his viewpoint was actually very influential because he and his coworkers were the first to translate and analyze these tablets (also the first to recognize that they were, in fact, records of mathematical calculations, and not just record-keeping). His interpretation was the first one to appear and it set the tone for scholarship about Old Babylonian mathematics for many years.

Finally, it is not really correct to say that Neugebauer’s and Hoyrup’s interpretations are not supported elsewhere. There are many examples of problems similar to the one on YBC 6967 that are known, and Neugebauer and Hoyrup have published interpretations of them along the same lines. Plimpton 322 can be taken as evidence that the Babylonians knew “something close to” the Pythagorean theorem, but there are other interpretations of that tablet too.

Dan Finn

You have a very good response to the essay question on the midterm. You have definitely “hit all of the high points” and provided just the right level of detail about the way the different historians we discussed would interpret “quadratic algebra” problems such as the one on the YBC 6967 tablet. The only substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources for problems like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Sarah Cahn

You have a very good response to the essay question on the midterm. You have definitely “hit all of the high points” and provided just the right level of detail about the way the different historians we discussed would interpret “quadratic algebra” problems such as the one on the YBC 6967 tablet. The only substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources for problems like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Also, instead of saying “Neugebauer would solve the problem this way” or “Hoyrup would solve the problem that way” you really mean to say that they are claiming the ancient Babylonians would have thought about the solution of the problem in those terms. Don’t forget that all of these methods are correct and essentially equivalent from the point of view of modern mathematics.

Kyle Kadlick

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the connection between the identity

(a + b)^{2} = (a − b)^{2} + 4ab

and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. What is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

Also, there is a bit more behind Rudman’s reasons for rejecting Hoyrup’s “cut and paste” visualization than just that his proposed diagram “looked better.” I think if you look back it his reasoning you will see that he thought Hoyrup’s way of seeing this was essentially “too clever” to be the basis of a general method for solving these problems. That’s questionable, of course. But it goes a little deeper than you are indicating.

Tom Santa Maria

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the actual mathematics. First, the key identity that comes into play in these problems can be stated as

(a + b)^{2} = (a − b)^{2} + 4ab.

I was also looking for the connection between this and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. That is, what is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

The other substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources for problems like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Astrid Ludwig

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the connection between the identity

(a + b)^{2} = (a − b)^{2} + 4ab

and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. What is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

Peter Zona

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the actual mathematics. First, the key identity that comes into play in these problems can be stated as

(a + b)^{2} = (a − b)^{2} + 4ab.

I was also looking for the connection between this and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. That is, what is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

Xuan Ha Nghiem

Your response to the essay question on the midterm is very good. You have definitely “hit all of the high points” and provided just the right level of detail about the way the different historians we discussed would interpret “quadratic algebra” problems such as the one on the YBC 6967 tablet. The only substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Kylee Sullivan

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the connection between the identity

(a + b)^{2} = (a − b)^{2} + 4ab

and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. What is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

Lucia Bostrom

This is a good response to the essay question on the midterm. You have hit most of the important points well, except for the “cut and paste geometry” interpretation of the YBC 6967 tablet offered by Jens Hoyrup. This similar to Rudman’s ideas in that it gives a proposed geometric visualization of the steps in the Babylonian solution of the YBC 6967 problem. But it is also different because the geometric connection is a different one. That is the main additional thing I was looking for here.

It is good, though, that you explained about the exact connection between the identity

(a + b)^{2} = (a − b)^{2} + 4ab

and the solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet by giving the connection between the x here and the a, b in the identity (or in the geometric diagrams).

Matt Kasuba

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the actual mathematics. First, the key identity that comes into play in these problems can be stated as

(a + b)^{2} = (a − b)^{2} + 4ab.

I was also looking for the connection between this and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. That is, what is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

The other substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources for problems like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Muhammad Javaid

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the actual mathematics. I was also looking for the connection between this and the actual solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. That is, what is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

The “with geometry in the background” is a good way of putting the ideas that Hoyrup and Rudman have about these examples of Babylonian mathematics. They certainly accept that there is some algebra involved, since the problem is one of solving an algebraic equation. But they would say that the method of solution shows that the Babylonians were thinking in geometric terms in one way or another.

John Macomber

Your response to the essay question on the midterm is very good. You have definitely “hit all of the high points” and provided just the right level of detail about the way the different historians we discussed would interpret “quadratic algebra” problems such as the one on the YBC 6967 tablet. The only substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources for problems like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.

Kevin Sutman

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the actual mathematics. First, the key identity that comes into play in these problems can be stated as

(a + b)^{2} = (a − b)^{2} + 4ab.

I was also looking for the connection between this and the solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. That is, what is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

Max Pettinelli

This is a good response to the essay question on the midterm. You have hit most of the important points well. The only thing I would have liked to see in addition was a bit more about the actual mathematics. First, the key identity that comes into play in these problems can be stated as

(a + b)^{2} = (a − b)^{2} + 4ab.

I was also looking for the connection between this and the solution of the equation x^{2} − 7x − 60 = 0 from the YBC 6967 tablet. That is, what is the connection between the x here and the a, b in the identity (or in the geometric diagrams)?

The other substantive comment I have is that even people in the “geometry camp” (like Hoyrup and Rudman) would acknowledge that there is some element of algebra content in Babylonian sources for problems like this. So the picture is a bit less “black and white” than you are making it. Neugebauer might say there is only algebra going on here. But Hoyrup and Rudman would say that there is a geometric motivation behind the method of solution of the algebra question.