Quick links to Front matter, Back matter, and:

**Part One**: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits

**Part Two**: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics

**Part Three**: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

**Page 29: Nothing could be simpler…**

**Pages 30-31: In the last chapter… // The big idea here…**

**Pages 32-33: Using this chopping technique… // The basic idea was nice expressed…**

*MB: I like Cavalieri! I had never heard of him, And the approximations to pi; very nice.*

**Pages 34-35: In fact, the same basic idea… // Their basic approach was to take a circle…**

*FC: P.34-40 – Q: Is the “ancient Chinese gent” wearing Japanese getas? – Traditional footwear cultural sensitivities.*

*DLJ: I’m 44 pages in, and one objection, one suggestion. (Overall it’s excellent — and I think your idea of getting the ideas first before the uses is absolutely first rate.)*

Here’s the objection. Historically the way they estimated PI was from both sides, i.e. calculating bigger than the inscribed polygons and smaller than the enclosing polygons of increasing numbers of tangents. You’ve elided this, and you shouldn’t.

One of the Great Lies Teachers Tell is that lying is simplifying and simplifying is making things easier to understand. The appropriate simplification is sometimes, but not always, pedagogically useful, but a lie is never a simplification. An untruth necessarily makes the whole story more complicated. (Remember Harry Truman, “I don’t have to take notes ’cause I don’t tell lies.”)

You’re going to get one kid in every classroom who knows you’ve cut corners and he’s going to put up his hand and cause a fuss with the teacher, and all the usual stuff from there on out…

I’m not suggesting a major revision. It shouldn’t need more than one or two panels of the comic, saying “And a bunch of other guys were doing it from the outside in…” Quite possibly you can use the outside and inside to make the point that although the two never meet, still they are approaching the same limit.

Here endeth my complaint. Now, my suggestion: Pythagoras appears in one panel as they’re calculating those smaller and smaller triangles, but you never say what he’s all about. It’s worth a note. There are thousands of proofs of a^2 + b^2 = c^2. (I don’t know whether you can fit one into the margin. Maybe squeeze one in by lubricating it with some Eul.) Einstein came up with one when he was a teenager and Ulysses. S. Grant did a different one, though I don’t know whether it was as General, President, or student.

[Later: I was wrong about Ulysses S Grant coming up with his own proof of Pythagoras. It was some other President, maybe Grover Cleveland. When I come across it again I'll get it right and to you. On Pythagoras, my idea was that he might be explained as a marginal note when you first us him to say a^2 + b^2 = c^2. I saw him as a head sticking itself out from the side of the page, bending the cartoon borders back to stick his head out -- so that you don't have to renumber all your pages among other things. Once you've had him stick his headout once, then you could make a running gag out of him, perhaps talking about math in the ancient world, or different proofs of the Theorem through the years, or...]

*Can you fit something about it in? One way might be to have a totally different character turning back the visual edge of the “paper of the page” and sticking its head out. It could maybe say “Hey, ya wanna hear about Pythagoras? I’ve hidden him on page…”)*

YB: (1) Note that on page 35 we hint at the idea that you can also bound pi from above: “We can also show that pi < 4!" (2) Note also that we're already making other simplifications, e.g., Archimedes (and I think Liu Hui as well) started with a hexagon, not with a square. (3) The idea about Pythagoras is interesting, but I'm not quite sure what you're suggesting. What would you like us to say about him?

**Pages 36-37: Their real genius… // …to a hexadecagon…**

**Pages 38-39: There are two obvious questions… // First, calculate the height.**

*WM: I like the details on Archimedes’ estimate of pi and how it relates to finding areas of nonlinear figures. ( I also like that you cued the reader that he/she could skip the details on how the areas of the triangles were calculated.)*

**Pages 40-41: The second obvious question is… // The way we use integral calculus…**

**Page 42: Next we’ll tie everything together…**

Quick links to Front matter, Back matter, and:

**Part One**: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits

**Part Two**: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics

**Part Three**: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion