In praise of the slide rule

Slide rules: work under water, in sand, and are always on. Slide rules can be held and operated with the same hand, either hand. Slide rules don’t need batteries or solar for their cells; they can be used in bright sunlight. Slide rules can be dropped from a moving car and still work perfectly, don’t care if you throw them across the room and have an indestructible display. Slide rules round off automatically and give exactly the right number of decimals and never give you an error message. In six inches they give the ability to multiply and divide any number, take the square and cube roots, give the inverse of a number, give the log and ln, and basic trig functions. What’s not to like? Holding one in the hand is like holding mathematics. Everybody should own one.

My collection is shown in the photo. The top two rules  are starters for the young I suppose as they only multiply and divide. Third from the top is also a simple introductory rule of about the 1960s-era when aluminum was introduced. Fourth and fifth down are mostly the same, duplex models, with scales on both sides, the second rule being the reverse side of the first; as is obvious this is the one for the serious calculator. It was one of these I toted to my senior high school physics class. My teacher, Mr Eich, had a huge 6 foot long rule hanging over the blackboards. He could move the slide strip and cursor to demonstrate a calculation for all to see. Finally the last two are just compact shirt pocket versions of the big duplex ones, again the second being more or less the reverse side of the first. I still carry the bottom one, the last one I ever bought while in college in the late ‘60s, on trips and for quickly checking a calculation when reading; it’s just so much faster than key-stroking numbers into a digital calculator.

For some reason slide rule are considered devilishly hard to learn how to use. Or are totally misunderstood. For instance in the movie “Apollo 13” when a ground control engineer is told to add two numbers together and he furiously cranks it out on his slide rule, something, one of the only things, a slide rule can not do.

My friends, if you will simply take a slide rule in hand and play with it a few minutes you will discover how very easy they are to use. You will see the product of two numbers jump right out at you and division being the reverse of multiplication will happen just as easily. You can be a self-taught slide rule user with only a few minutes work. And soon you will realize the usefulness of the answers.

My point then is that the joys of holding and operating a slide rule are palpable and should not be missed.

Higgs announcement

I stayed up until midnight to catch the live video from CERN in Geneva at 9 A.M. their time. It was the conference to reveal the latest data from the LHC and its hunt for the Higgs boson. The presenter started talking and for 35 minutes it was non-stop ultra dense particle physics jargon. I could keep up with it, sort of. About 12:37 A.M.(California time) or so this morning he shows the slide of the experimental data: right where it was expected were hits, so the Higgs exists! Everybody started clapping and such tomfoolery then the lecture continued for a little bit more with additional results. Live blogging from the lecture hall in Geneva by Sean Carroll, professor of theoretical physics at Caltech, follows:

And the “money” plot:

V. I. Arnold and the ultimate calculus textbook?

V. I. Arnold passed away suddenly, on June 3rd, 2010, in a French hospital while being operated on for peritonitis. This was sudden and unexpected and took away a delightful teacher. It is most regretful.

On March 7th, 1997 He gave an address, “On teaching mathematics” at the Palais de Découverte in Paris. The entire transcript can be found online.

This speech was very forthright and direct. Reading the text one soon appreciates the fact that the audience possibly had one of three reactions: standing on their chairs cheering Arnold on; sitting in embarrassed silence; or rioting in the aisles. Since I have never read of any harsh reaction to his views rioting is out of the question. Perhaps a humiliation of the academic pedagogic mathematics community resulted. Professor Arnold made a clear denunciation of the then, and I suppose even now, technique of teaching mathematics divorced from the realities of physics and insight of geometry.

Towards the end he recommends some books he feels are well within the expectations of the general reader who is interested in becoming more aware of the subtleties of the science of mathematics to have read:

“Among these are Numbers and figures by Rademacher and Töplitz, Geometry and the Imagination by Hilbert and Cohn-Vossen, What is Mathematics? by Courant and Robbins, How to solve it and Mathematics and Plausible Reasoning by Polya...”

These congruency of mathematics with geometry is apparent to any who have opened these books.

Then he makes a remarkable statement:

“I remember well what a strong impression the calculus course by Hermite (which does exist in a Russian translation!) made on me in my school years.

“Riemann surfaces appeared in it, I think, in one of the first lectures (all the analysis was, of course, complex, as it should be). Asymptotics of integrals were investigated by means of path deformations on Riemann surfaces under the motion of branching points (nowadays, we would have called this the Picard-Lefschetz theory; Picard, by the way, was Hermite's son-in-law - mathematical abilities are often transferred by sons-in-law: the dynasty Hadamard - P. Levy - L. Schwarz - U. Frisch is yet another famous example in the Paris Academy of Sciences).

“The "obsolete" course by Hermite of one hundred years ago (probably, now thrown away from student libraries of French universities) was much more modern than those most boring calculus textbooks with which students are nowadays tormented.

“If mathematicians do not come to their senses, then the consumers who preserved a need in a modern, in the best meaning of the word, mathematical theory as well as the immunity (characteristic of any sensible person) to the useless axiomatic chatter will in the end turn down the services of the undereducated scholastics in both the schools and the universities.

“A teacher of mathematics, who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz, will then become a relict like the one nowadays who does not know the difference between an open and a closed set.”

So, here then, in a few brief paragraphs, we have a recommendation of books that should be read, then a book that should be read if it could be found, literally, and finally a series of books that could be read if one wants to consider himself or herself a serious student.

It is easy to obtain the first mentioned books 

and the last books, the famous Landau and Lifshitz series,

but where does one find the much older text by Hermite? Indeed, what exactly is it’s title?

Giving Professor Arnold full credit for his insight on calculus pedagogy, the problem becomes: What text is Arnold referring to and how can it be brought to life in English if it isn’t already?

I have done some investigation and narrowed down the search to Hermite’s Cours d’ Analyse de L’Ecole Polytechnique, from 1873.

So the chase is on: identify, obtain Hermite or the Russian translation of Hermite, or both, ideally, and initiate an attempt to translate one or both into English! In the meantime one should immediately hie themselves to the bookseller and obtain all of the books in the first list and start reading.

Wolfram Alpha

If you aren’t using “Wolfram|Alpha,” why not? It’s free, doesn’t require any software be loaded on your computer, and takes English language inputs! It’s a “computational knowledge engine.” You can ask it anything! The average weight of all the players on the New England Patriots football team; the distance to the moon in millimeters; crossword puzzles; airline flight info; the tides at your destination--anything you might want to know can be found and then computed to meet your request. Absolutely anything! It’s amazing!! It solves math problems and now can even find the next number in a random-looking sequence. Just type “wolframalpha” in the “http” bar on your computer and there you are! Try it now, think about it, you’ll be hooked! You are welcome!


Note added after initial posting. As I sat weak and weary pondering many things there came a tapping an incessant rapping: my brain was asking is iPhone 4S's Siri with Wolframalpha multi tasking? A check of all things known to man, Wikipedia, was the plan where should I espy Wolframalpha one of Siri's resource consorts by and by!

Fermi Problems--Part 2

Fermi Problems--Saturn’s Cassini rings, rocks and size

On April 8, 2009, some people were looking at photographs just returned from the Cassini spacecraft in orbit around Saturn and they noticed something extraordinary and heretofore never seen: rocks in the rings! What follows is my shameless plagiarization of Emily Lakdawalla’s analysis, paraphrased in Fermi problem format, found on “The Planetary Society Blog” for April 13, 2009 (“Moon shadow, moon shadow” at planetary.org/blog/article/00001909/). 

The photo from The Cassini Equinox Mission shows a long, dark, pointed shadow cast by the moon Mimas.  Look along the left edge of this shadow at the edge of the bright ring where it abuts the black ring.  Look closely!  See the ragged little features like snow plowed up along a street?  Those are the rocks.

The Fermi problem of interest is: “How big are these rocks?”

NASA gives a distance of 1,141,262 kilometers from the rings when Cassini took the photo.  They also tell us the camera resolution is 60 micro radians.  The product of these two numbers is roughly 7 km/pixel. 

It is well known from astronomical tables that in a couple of months Saturn’s rings will align with the sun, so our second piece of Fermi data is that the shadows cast by the rocks on the bright ring are very long because the rings are only 1.9 degrees away from pointing directly at the sun. 

Counting the number of pixels in the longest shadow (this is something that can be done, but we’ll just assume the given value is right) we get 15 pixels which works out to 15 pixels times 7 km/pixel or 105 km.

Now the Fermi calculation. If the sun is 1.9 degrees up in the ring’s sky and the shadow is 105 km long, then the trigonometric tangent of 1.9 degrees times 105 kilometers is 3 km.  The height of the rock is 3 kilometers…that’s a pretty significant rock!

NASA photo reference and caption:

(http://saturn.jpl.nasa.gov/photos/raw/rawimagedetails/index.cfm?imageID=188106)

Integral calculus, v.0.1

(1) “Hobbes objected strenuously to “”the whole herd of them who apply their algebra to geometry.”” The History of the Calculus and its Conceptual Development, p.175, C.B. Boyer.