I skimmed through Orlin's book, not giving it its due, but enjoying every part of it i read. ( )

Really interesting book which teaches ideas of calcus with stories and history. The book does not focus much on the application of calcus it's ubiquitous by now, but it gives historical reasoning behind some of the discoveries. It also talks about dichotomy of pure maths vs approximate maths of statistics. Overall nice read ( )

Lightweight, lighthearted book about calculus with two-color drawings that are really not that bad. Not a lot of truly technical matter, but some of it, like the bicycle discussion is downright puzzling. Maybe that _could_ have profited from a clearer diagram.

Chapter II is about gravitation and acceleration. It does some strong simplification, i.e., the apple on the earth's surface is compared to the apple at the moon's distance from the earth. This is Ok, because the inertia and the force should exactly balance, leaving the same acceleration for the apple as for the moon. So, in one second, at the moon's radius an apple/or the moon would fall toward the earth a distance that roughly corresponds to the thickness of a credit card, approximately 1 mm. Then the question is how far does it go in a direction orthogonal to this earthwards fall? The answer is given by a dubious trick with similar triangles. And the question is kind of wrong. The question is more like:
* the moon goes around the earth in approximately a circle
* the distance of the moon from the earth is pretty well estimated (since the time of the ancient Greeks)
Does the amount that the moon must move toward the earth in order to maintain this approximately circular orbit agree with what would be predicted if the moon's acceleration toward the earth varied according to the square of the distance?

We know the acceleration of an apple at the surface of the earth, it's g = Gm_E/(r^2) where m_E is the mass of the earth and r^2 is the radius of the earth squared. We get to ignore the mass of the apple because the force and the inertia cancel out. Note that G is not a dimensionless constant: its units must be force * (L^2 / m^2) to make the standard formula work out. We can drop apples and calculate their acceleration, although that is rather difficult with the then current timekeeping technology. Let's assume we know that it is 32 ft/s^2 or 10m/s^2. We can then calculate the acceleration at the distance of the moon, without knowing anything about the earth's mass, but only the distance to the moon from the center of the earth relative to the distance to the earth's surface. The moon is only 60 times further away. So, a_moon = Gm_E/((60r)^2) = Gm_E/(3600 * r^2) = 1/3600 * g or 3mm/s. So, in 1 second, by simple integration, the fall should be 3/2 mm, assuming the original velocity was 0.

If the moon always stays at about the same distance relative to the earth, then its acceleration must be exactly canceled. Otherwise, it would continue to accelerate toward the earth, approaching it more and more rapidly. So, every second, the moon's motion must make it stay at the same distance from the earth and ensure that it's velocity toward the earth is 0. So, the moon's velocity which is at any point tangent to the circle which is is traversing, by definition, must be enough to do the cancelling.

Chapter XIV: That's Professor Dog to You!
About a series of articles of a dog that minimizes the time to get the ball (which has been thrown and is floating in Lake Michigan). Here's one link: https://web.williams.edu/Mathematics/sjmiller/public_html/103/Pennings_DogsCalcu.... The purpose is to show how much interest there can be in a simple minimization problem. And it is interesting. If the dog is actually a seal, and its r is less than its s, then, for example, it must always minimize time by swimming immediately. In that case, setting the derivative to 0 yields an imaginary solution. Perhaps it is easy to graph the formula for varying values of r,s,x, and z. It might be nice to take a look.

Chapter XV: Calculemus!
Leibniz and his automatic language is discussed, but also the nice fact that derivation can really be made entirely automatic and really, mostly, syntactic.

ETERNITIES:
About integration rather than differentiation.

Chapter XVI: To Literary Circles
Using clever geometry and the method of exhaustion to find the area of a circle.

Chapter XVII: War and Peace and Integrals
About Tolstoy's theory of history. Should I read something by Tolstoy. I think I had to once, in high school.

Chapter XVIII: Riemann City Skyline
About Riemann and Lebesgue sums. Maybe someday I will study.

Chapter XIX: A Great Work of Synthesis
Introduces the fundamental theorem of calculus, loosely. I had thought that people had already gotten around to that, by then.

Chapter XX: What Happens under the Integral Stays under the Integral
Kind of as much about the gamma function, as about differentiating under the integral. Somewhat about the point that DeMorgan makes, that symbolic integration is not a turn-the-crank activity. It made me think to check out Wolfram Alpha, for the first time in years.

Chapter XXI: Discarding Existence with a Flick of His Pen
Einstein's relativity and the history of the cosmological constant until now.

Chapter XXII: 1994, the Year Calculus was Born
In a medical journal, someone explains that you can graph some biological value of someone after eating and then find the area under the curve by approximating with trapezoids. Valid, but over 2000 years old.

Chapter XXIII: If Pains Must Come
A Benthamite calculus. Not much to say on this one, but it has a John Stuart Mill quotation, so it can not be all bad. On the other hand, there's some assertions about how bad movies w/ good ending are more fondly remembered than overall good movings w/ bad endings; which is bogus and meaningless.

Chapter XXIV: Fighting with the Gods
Archimedes does clever geometry tricks with infinitely small slices. These are clever, but the solids of revolution trick covered in the next chapter really gives the same result. Archimedes is flashier, though.

Chapter XXV: From Spheres Unseen
Solids of revolution introduced via the novel "Flatland". This is kind of clever, and I can apparently still work out the volume of a sphere after all these years. Claims that this is what calculus is nice for, turning the crank and not having to think too much about the specific problem you're dealing with. But it turns out that even the expression arising from the integration is a bit cool: if you group the terms, you'll see that you're subtracting the volume of the stuff left out from the volume of a cylinder. Maybe this is mundane, but it is not completely boring.

Chapter XXVI: A Towering Baklava of Abstractions
An uncomplimentary review of David Foster Wallace's writing about math. Also, a remark that mathematics isn't just formalism, but many more exciting things, which I agree with.

Chapter XXVII: Gabriel, Blow your Horn
A discussion of Toricelli's Trumpet, which has finite volume, but infinite surface area.I have not so far forgotten my calculus that I couldn't find the volume. However, nobody on the internet seems to agree with me about the area; I calculated a simpler formula than everybody else. According to Julian Fleron, the standard method for calculating areas of revolution gives the standard formula, so it turns out I have just totally forgotten how to do it (http://www.matharticles.com/ma/ma044.pdf). I still reach the same conclusion: the volume is finite, but the area is infinite. The point of this chapter seems to be that paradox is cool. I was already familiar with Toricelli's trumpet, probably from the recent book "The Calculus Story: A Mathematical Adventure", although I'm not sure. I get the feeling that Toricelli's Trumpet is popular right now. Note, that there are other similar constructions: Toricelli's trumpet is also known as Gabriel's, there is also Gabriel's wedding cake, made of ever smaller cylinders, and probably more. In fact, Julian Fleron describes a Gabriel's wedding cake, which has an easy-to-calculate surface area (and a fairly easy-to-calculate volume). The Toricelli trumpet is contained within his wedding cake, thus the wedding cake has a greater, but still finite volume, which is: pi^3/6.

The book quotes Amir Alexander, who claims that Torricelli's new style of math was what really got him in
trouble with the church. But Gabriel's wedding cake can be constructed w/ good old geometry (and an infinite series), so I'm not sure that this is really true.

Chapter XXVIII: Scenes from an Impossibility
Introduces the Gaussian integral. The integral of the formula e^(-x^2) has no formal solution except when integrated over the interval -inf to +inf when its solution is square root of pi. What do we learn from this? That integration isn't as turn-the-crank as differentiation and not to be snotty about non-formal approximate solutions.