Tall, Dark, and Mysterious


Something for you to do while I vacation yet some more

File under: Queen of Sciences, What I Did On My Summer Vacation. Posted by Moebius Stripper at 9:57 pm.

In a few days I’ll be setting out to visit some more of the Gulf Islands. One of them doesn’t allow cars, and is powered entirely by generators. At least two lack bank machines. One is rumoured to have “honesty stands” where local artists leave their work unattended and trust visitors to pay for what they take. Only one has more than a thousand full-time residents. I may at some point pop into a cafe for a few minutes on a rainy day to avail myself of the dial-up connection on the island’s IBM 486 , but don’t hold your breath. I will be making notes on postcards, so holler if you want one.

In the meantime, some of you may be able to help me with a project I’m thinking about working on one of these days book I have no excuse not to work on now that I don’t have a pesky job to worry about. A bit of background: for the last five years, I have worked at an academic summer camp for mathematically gifted high school students. One of my favourite classes to teach - and one of the most popular among the campers - has been one that I call “Calculus Without Calculus”. Those of my readers who know me in real life are well aware that I…well, I don’t hate calculus, so much as I think that calculus doesn’t need me to love it. Calculus gets more than enough attention in the thousands of high schools and universities that inflict it upon every other student that passes through their doors, the overwhelming majority of whom don’t learn it properly and wouldn’t ever use it again even if they did. Nevertheless, calculus is a natural choice for students who have not learned to think mathematically: for all the terror it strikes in students’ hearts, it’s one of the easier branches of mathematics to reduce to mindless algorithms in a low-level course. Need to maximize some quantity? Set a derivative to zero, and solve. What were we trying to do again?

Calculus Without Calculus is a collection of methods of solving typical calculus problems without taking a single derivative. The two main methods involve inequalities, or exploiting geometric properties of figures, particularly symmetry. The old “maximize area with given cost of rectangular fence” problem? You can complete the square, take a derivative - or you can apply the AM-GM inequality. The question about getting the best view of the statue on a pedestal, that appears in the chapter on inverse trig functions in every single calculus book? Solvable using elementary circle geometry that could be found in every grade ten math text before it was decreed that geometry should no longer be taught. There are tons of these. I know of three calculus-free methods of finding tangents to ellipses: one using transformations of circles, another using the Cauchy-Schwartz inequality, and one using projective geometry. A few months ago, I was reading some fiction on the Vietnam War - One To Count Cadence, by James Crumley - and one of the characters mentions in passing that he was able to solve the ladder-around-a-corner problem (you know the one) without calculus. I struggled with that for a long time before a camper provided the key insight. A quick application of Holder’s Inequality, and the result falls right out. (Except that now I’m trying to recreate it. Damn; this is going to keep me up again.)

I’m sure there’s a lot more to say on the topic, and I’d like to write a (short) book on the topic. What I’m looking for: book recs. In particular, recommendations for good books on problem-solving, which tend to spend a lot of time on funky inequalities. I’m especially interested in geometric and otherwise intuitive proofs for the old standbys (AM-GM, Holder’s, C-S…) , and lots of examples - both of single variable problems that one would see in calculus texts, as well as multivariable ones that are a lot easier to solve without calculus. One of my favourite books of this sort is Problem Solving Through Problems, by Loren Larson, the talented mathematician and educator who first introduced me to this stuff. If you’ve got any recs, or anything even slightly pertinent, I’d love to hear them - post them below so I’ll have something to check out when I return to this big island.

My other reason for posting this, of course, is that now I’ll feel horribly guilty if I don’t actually have anything to show for this in a few weeks.

And, if [above] isn’t your thing, perhaps the Phallic Logo Awards can keep you busy for the next week. (What’s this you want, a better segue? Here: Galiano Island.)


  1. Have fun on the Gulf Islands there. I’ve spent a bit of time on Mayne Island (I have family there), and it is one of the most beautiful, relaxing places I have ever visited.

    Good luck on the book thing too, although I can’t offer any recs.

    - Simon — 6/6/2005 @ 5:27 am

  2. Try the following:
    Introduction to Inequalities (http://www.amazon.com/exec/obidos/tg/detail/-/0883856034/ref=pd_sim_b_1/002-3151223-0427225?%5Fencoding=UTF8&v=glance)

    Geometric Inequalities

    - Foo Won Yu — 6/6/2005 @ 6:05 am

  3. One of my colleagues here teaches a version of “Honors Calculus” where he spends a whole lot of time on just this sort of thing. I can see about getting you a copy of his course notes, if you like, and find out what books he uses.

    - Dr. Matt — 6/6/2005 @ 9:15 am

  4. Hee! Calculus problems + phalluses = fun post

    No help for you on the book, but this sounds like a really good idea. Beyond questions of what material should be taught and so forth, it sounds like what you are talking about is really teaching people problem solving techniques, independent of the mathematical formalisms involved. Getting students to think about problems in different ways is always a good thing.

    - sheepish — 6/6/2005 @ 9:30 am

  5. Foo Won Yu - thanks for the recs. I actually have (had…) a copy of ItI at some point - where the hell did I put it? - I bought an ancient copy for fifty cents at a library sale. It’s where I found the C-S method of finding tangents to ellipses. But for $10, I might order the new copy, and toss in GI.

    Dr. Matt - yes, if you could get me the notes, that would be wonderful - if not, a list of the books would be great.

    - Moebius Stripper — 6/6/2005 @ 9:37 am

  6. heh, I taught myself grades 9 and 10 math with old textbooks without getting the memo about no more geometry. And then in second year, on the surveying final exam, nobody else I knew was able to solve a particular problem involving a cicular curve.

    - Jen — 6/6/2005 @ 10:15 am

  7. They’ve taken geometry out of high school math in Canada? So much for my emigration plans. It would have been too cold for me anyway. New Zealand, that’s the ticket.

    - vito prosciutto — 6/6/2005 @ 10:59 am

  8. I know of three calculus-free methods of finding tangents to ellipses…

    From any point on the ellipse, draw a line to each focus. Bisect the angle thus formed; the bisecting line is normal to the ellipse (therefore perpendicular to the tangent). That must be the “projective geometry” method you were talking about.

    I actually had to find tangents to ellipses for a real-world job once; I was trying to program a CNC gear shaper to cut elliptical gears. I planned to use the method above to get tangents, because I knew no calculus. But there was still the question of length-of-curve, so I got a “teach yourself calculus” book and soon discovered it was easier to take a differential than go through the geometric business. I proved a method to rotate an arbitrary ellipse and translate an arbitrary circle, keeping the two surfaces in contact. It could have been done without calculus.

    I never did get the length-of-curve part sorted out. It involves what they call an elliptic integral, if I’m not mistaken. Nobody in the shop knew what to do with it. Since I was skunk-working anyway, I abandoned the project.

    - dipnut — 6/6/2005 @ 11:44 am

  9. well, don’t know about now, but even when I was in school they hadn’t taken out all geometry, but it would appear that there was less of it, and I don’t think my peers got the circle geometry that I covered on my own.

    - Jen — 6/6/2005 @ 5:25 pm

  10. I mourn the loss of geometry (and the dumbing down thereof). What I’d really like to teach would be an advanced Euclidean geometry class. Something that assumes a year of old-school HS Geometry as its starting point so you can start doing really interesting proofs. But the push to Calculus is at least partly responsible for this class not being possible. The students who would be good candidates for this wouldn’t be willing to pay the price of not taking Calc in High School (or only taking Calc AB) in exchange for more geometry.

    - vito prosciutto — 6/7/2005 @ 7:57 am

  11. i would love to teach an advanced geometry class…i’m working through Postamentier’s “Advanced Euclidean Geometry” right now. great book, by the way, i highly recommend it. but i agree that it’s hard to find time in the high school curriculum for it.

    i get to teach snippets of it to my talented 1st year geometry students…some triangle centers, euler’s line (which i’ve found an AMAZINGLY elegant synthetic proof of…online, not discovered myself) the simson line, etc.

    i’m afraid i can’t think of any good calculus problems solvable without calculus at the moment, though…i’ll keep my eye out.

    - TIAB — 6/7/2005 @ 4:07 pm

  12. I recall that at MOP, Coxeter’s “Geometry Revisited” was highly acclaimed, but given the previous comments you’ve made about Coxeter, I’m guessing that it was unnecessary to mention this.

    - Dog of Justice — 6/7/2005 @ 5:22 pm

  13. Paul Zeitz’s The Art and Craft of Problem Solving is awesome overall and has good problems. I don’t remember how good it is on inequalities, but it definitely has some stuff. Kiran Kedlaya’s inequalities notes (which you can find online) are also good, but lean more towards the sort of technical and unmotivated stuff found in olympiads these days (which I hated: if you do look among books that are primarily high school olympiad training materials, go for the older stuff). I think the Putnam exam also has better inequalites which don’t always require calculus: there was a good one a year or two back about minimizing(?) the absolute value of the sum of the six trig functions).

    Good luck!

    - Alison — 6/7/2005 @ 10:41 pm

  14. I’ve got the Posamentier Advanced Euclidean Geometry on my amazon wishlist. I’ve been working through his Challenging Problems in Euclidean Geometry off and on for about a year, although mostly off recently.

    I get to do 12 sessions of pre-Geometry prep for high school students later this summer. I’m going to focus almost exclusively on constructions and proofs. I also get to do 12 sessions of pre-Algebra II prep which I’m not sure what we’re going to do.

    - vito prosciutto — 6/8/2005 @ 8:56 am

  15. For general problem-solving, Polya’s “How to Solve It”.

    Aside note about geometry: Since Abraham Lincoln was able to work his way through Euclid (all of it) in front of a fireplace in the woods, I don’t see why today’s students can’t get at least through Book I.

    - Mike — 6/8/2005 @ 9:22 am

  16. There’s a standard piece of proto-calculus whose name I cannot recall. It’s a theorem that takes you as close as you can get to integral calculus without actually introducing limits of sums of increasingly-tiny rectangles.

    The theorem starts with two plane figures S and T, and a line L. Suppose it turns out that any line L’ parallel to L cuts S and T in line segments that bear a fixed proportion to each other. Then, concludes the theorem, S and T have areas in the same proportion.

    There is an analogous theorem in three dimensions, replacing the reference line with a plane and the plane figures with solids.

    It turns out that an astonishing variety of plane and space figures can be mensurated with this theorem; this includes many of the area and volume problems that are used triumphally in calculus textbooks to proclaim the utility of the discipline.

    Some reader of yours who is less lazy than I will give this theorem its proper name and provide references. I hope.

    - ACW — 6/8/2005 @ 1:18 pm

  17. This is childish, i know, but here is another one. (requires a rotation)


    - Marni — 6/8/2005 @ 4:29 pm

  18. ACW is referring to Cavalieri’s Principle.

    - Dog of Justice — 6/8/2005 @ 4:30 pm

  19. I believe there’s something in Proofs Without Words, or perhaps More Proofs Without Words. I’ll see what I can find in my copy. Lots of geometry, natch.

    - meep — 6/12/2005 @ 7:03 am

  20. “Calculus without calculus” is a great title! Write this book so I can buy a copy.

    - sam_nead — 6/14/2005 @ 10:13 am

  21. One more rec: “Maxima and Minima Without Calculus,” by Ivan Niven.

    - Foo Won Yu — 6/22/2005 @ 6:48 pm

  22. Oh my - that might be the book I want to write.

    Though, one of my references for Calculus Without Calculus is a Vietnam war novel, so maybe not.

    - Moebius Stripper — 6/23/2005 @ 10:42 am

  23. It’s occurred to me while reading Visual Complex Analysis (which might be a good reference, btw), that you really should look at the granddaddy of Calc books, Newton’s Principia. It’s all geometry there.

    - vito prosciutto — 6/24/2005 @ 7:40 am

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