My 2018 Mathematics A To Z: Kelvin (the scientist)


Today’s request is another from John Golden, @mathhombre on Twitter and similarly on Blogspot. It’s specifically for Kelvin — “scientist or temperature unit”, the sort of open-ended goal I delight in. I decided on the scientist. But that’s a lot even for what I honestly thought would be a quick little essay. So I’m going to take out a tiny slice of a long and amazingly fruitful career. There’s so much more than this.

Before I get into what I did pick, let me repeat an important warning about historical essays. Every history is incomplete, yes. But any claim about something being done for the first time is simplified to the point of being wrong. Any claim about an individual discovering or inventing something is simplified to the point of being wrong. Everything is more complicated and, especially, more ambiguous than this. If you do not love the challenge of working out a coherent narrative when the most discrete and specific facts are also the ones that are trivia, do not get into history. It will only break your heart and mislead your readers. With that disclaimer, let me try a tiny slice of the life of William Thomson, the Baron Kelvin.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Kelvin (the scientist).

The great thing about a magnetic compass is that it’s easy. Set the thing on an axis and let it float freely. It aligns itself to the magnetic poles. It’s easy to see why this looks like magic.

The trouble is that it’s not quite right. It’s near enough for many purposes. But the direction a magnetic compass points out to be north is not the true geographic north. Fortunately, we’ve got a fair idea just how far off north that is. It depends on where you are. If you have a rough idea where you already are, you can make a correction. We can print up charts saying how much of a correction to make.

The trouble is that it’s still not quite right. The location of the magnetic north and south poles wanders. Fortunately we’ve got a fair idea of how quickly it’s moving, and in what direction. So if you have a rough idea how out of date your chart is, and what direction the poles were moving in, you can make a correction. We can communicate how much the variance between true north and magnetic north vary.

The trouble is that it’s still not quite right. The size of the variation depends on the season of the year. But all right; we should have a rough idea what season it is. We can correct for that. The size of the variation also depends on what time of day it is. Compasses point farther east at around 8 am (sun time) than they do the rest of the day, and farther west around 1 pm. At least they did when Alan Gurney’s Compass: A Story of Exploration and Innovation was published. I would be unsurprised if that’s changed since the book came out a dozen years ago. Still. These are all, we might say, global concerns. They’s based on where you are and when you look at the compass. But they don’t depend on you, the specific observer.

The trouble is that it’s still not quite right yet. Almost as soon as compasses were used for navigation, on ships, mariners noticed the compass could vary. And not just because compasses were often badly designed and badly made. The ships themselves got in the way. The problem started with guns, the iron of which led compasses astray. When it was just the ship’s guns the problem could be coped with. Set the compass binnacle far from any source of iron, and the error should be small enough.

The trouble is when the time comes to make ships with iron. There are great benefits you get from cladding ships in iron, or making them of iron altogether. Losing the benefits of navigation, though … that’s a bit much.

There’s an obvious answer. Suppose you know the construction of the ship throws off compass bearings. Then measure what the compass reads, at some point when you know what it should read. Use that to correct your measurements when you aren’t sure. From the early 1800s mariners could use a method called “swinging the ship”, setting the ship at known angles and comparing what the compass read. It’s a bit of a chore. And you should arrange things you need to do so that it’s harder to make a careless mistake at them.

In the 1850s John Gray of Liverpool patented a binnacle — the little pillar that holds the compass — which used the other obvious but brilliant approach. If the iron which builds the ship sends the compass awry, why not put iron near the compass to put the compass back where it should be? This set up a contraption of a binnacle surrounded by adjustable, correcting magnets.

Enter finally William Thomson, who would become Baron Kelvin in 1892. In 1871 the magazine Good Words asked him to write an article about the marine compass. In 1874 he published his first essay on the subject. The second part appeared five years after that. I am not certain that this is directly related to the tiny slice of story I tell. I just mention it to reassure every academic who’s falling behind on their paper-writing, which is all of them.

But come the 1880s Thomson patented an improved binnacle. Thomson had the sort of talents normally associated only with the heroes of some lovable yet dopey space-opera of the 1930s. He was a talented scientist, competent in thermodynamics and electricity and magnetism and fluid flow. He was a skilled mathematician, as you’d need to be to keep up with all that and along the way prove the Stokes theorem. (This is one of those incredibly useful theorems that gives information about the interior of a volume using only integrals over the surface.) He was a magnificent engineer, with a particular skill at developing instruments that would brilliantly measure delicate matters. He’s famous for saving the trans-Atlantic telegraph cable project. He recognized that what was needed was not more voltage to drive signal through three thousand miles of dubiously made copper wire, but rather ways to pick up the feeble signals that could come across, and amplify them into usability. And also described the forces at work on a ship that is laying a long line of submarine cable. And he was a manufacturer, able to turn these designs into mass-produced products. This through collaborating with James White, of Glasgow, for over half a century. And a businessman, able to convince people and organizations to use the things. He’s an implausible protagonist; and yet, there he is.

Thomson’s revision for the binnacle made it simpler. A pair of spheres, flanking the compass, and adjustable. The Royal Museums Greenwich web site offers a picture of this sort of system. It’s not so shiny as others in the collection. But this angle shows how adjustable the system would be. It’s a design that shows brilliance behind it. What work you might have to do to use it is obvious. At least it’s obvious once you’re told the spheres are adjustable. To reduce a massive, lingering, challenging problem to something easy is one of the great accomplishments of any practical mathematician.

This was not all Thomson did in maritime work. He’d developed an analog computer which would calculate the tides. Wikipedia tells me that Thomson claimed a similar mechanism could solve arbitrary differential equations. I’d accept that claim, if he made it. Thomson also developed better tools for sounding depths. And developed compasses proper, not just the correcting tools for binnacles. A maritime compass is a great practical challenge. It has to be able to move freely, so that it can give a correct direction even as the ship changes direction. But it can’t move too freely, or it becomes useless in rolling seas. It has to offer great precision, or it loses its use in directing long journeys. It has to be quick to read, or it won’t be consulted. Thomson designed a compass that was, my readings indicate, a great fit for all these constraints. By the time of his death in 1907 Kelvin and White (the company had various names) had made something like ten thousand compasses and binnacles.

And this from a person attached to all sorts of statistical mechanics stuff and who’s important for designing electrical circuits and the like.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link.

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Reading the Comics, September 1, 2017: Getting Ready For School Edition


In the United States at least it’s the start of the school year. With that, Comic Strip Master Command sent orders to do back-to-school jokes. They may be shallow ones, but they’re enough to fill my need for content. For example:

Bill Amend’s FoxTrot for the 27th of August, a new strip, has Jason fitting his writing tools to the class’s theme. So mathematics gets to write “2” in a complicated way. The mention of a clay tablet and cuneiform is oddly timely, given the current (excessive) hype about that Babylonian tablet of trigonometric values, which just shows how even a nearly-retired cartoonist will get lucky sometimes.

Dan Collins’s Looks Good On Paper for the 27th does a collage of school stuff, with mathematics the leading representative of the teacher-giving-a-lecture sort of class.

Olivia Walch’s Imogen Quest for the 28th uses calculus as the emblem of stuff that would be put on the blackboard and be essential for knowing. It’s legitimate formulas, so far as we get to see, the stuff that would in fact be in class. It’s also got an amusing, to me at least, idea for getting students’ attention onto the blackboard.

Tony Carrillo’s F Minus for the 29th is here to amuse me. I could go on to some excuse about how the sextant would be used for the calculations that tell someone where he is. But really I’m including it because I was amused and I like how detailed a sketch of a sextant Carrillo included here.

Jim Meddick’s Monty for the 29th features the rich obscenity Sedgwick Nuttingham III, also getting ready for school. In this case the summer mathematics tutoring includes some not-really-obvious game dubbed Integer Ball. I confess a lot of attempts to make games out of arithmetic look to me like this: fun to do but useful in practicing skills? But I don’t know what the rules are or what kind of game might be made of the integers here. I should at least hear it out.

Michael Cavna’s Warped for the 30th lists a top ten greatest numbers, spoofing on mindless clickbait. Cavna also, I imagine unintentionally, duplicates an ancient David Letterman Top Ten List. But it’s not like you can expect people to resist the idea of making numbered lists of numbers. Some of us have a hard time stopping.

Todd: 'If I'm gonna get a good job someday, I've decided I'm gonna have to buckle down and get serious with my studies!' 'Good for you, Todd!' 'When I get to Junior High and High School, I'm gonna take stuff like trickanometree, calculatorius and alge-brah! Hee hee! Snicker! Snicker!' 'What?' 'I said Bra! Hee! Hee!' 'Better keep buckling down, bub.'
Patrick Roberts’s Todd the Dinosaur for the 1st of September, 2017. So Paul Dirac introduced to quantum mechanics a mathematical construct known as the ‘braket’. It’s written as a pair of terms, like, < A | B > . These can be separated into pieces, with < A | called the ‘bra’ and | B > the ‘ket’. We’re told in the quantum mechanics class that this was a moment of possibly “innocent” overlap between what’s a convenient mathematical name and, as a piece of women’s clothing, unending amusement to male physics students. I do not know whether that’s so. I don’t see the thrill myself except in the suggestion that great physicists might be aware of women’s clothing.

Patrick Roberts’s Todd the Dinosaur for the 1st of September mentions a bunch of mathematics as serious studies. Also, to an extent, non-serious studies. I don’t remember my childhood well enough to say whether we found that vaguely-defined thrill in the word “algebra”. It seems plausible enough.

Reading the Comics, April 19, 2016: Mostly Reruns Edition


Though I believe all my commentary on this is new, most of the comic strips to mention mathematical subjects since last time were strips in reruns. It’ll pick up again.

Lincoln Pierce’s Big Nate: First Class for the 16th of April originally ran the 11th of April, 1992. (First Class is a day-by-day reprinting of the comic strip.) Nate can’t believe that Francis is enchanted by the shapes in geometry. I can believe it, although I have a certain selection bias in the matter. Many fields of mathematics offer beauty. Geometry offers one that even the untrained eye can see. The diagrams that help along a geometric proof can be works of art, or at least suggest art. They also can be links to the world of Platonic ideals. The idea that there are perfect circles and squares and dodecahedrons and such is a strong one, at least in the Western tradition. And even a shaky sketch of that seems to evoke this perfection and render it understandable, even understood. There is joy to be had in this.

Nate Fakes’s Break Of Day for the 16th is a name-drop strip. Arithmetic serves as an easy-to-understand bit of work any reader can imagine making a mistake on. Really any work in any field can produce a mistake. And sometimes a mistake can be productive. This is as true in mathematics as it is in any creative field, and for much the same reason. It can teach why to do things one way rather than another. It can suggest alternate approaches. It can make you notice things you hadn’t noticed. But it’s easy in arithmetic to conclude that a mistake is just a wrongheaded effort, to be cut as soon as possible.

'You're lousy at math. That's why we've been lost for twenty years!' 'That really hurts.' 'You're right. How can we make it up to you?' 'Well, all five of you could apologize.' There are three of them.
Bill Rechin’s Crock for the 18th of April, 2016. It’s a rerun, I assume, although I can’t say from when. Also the ‘L’ of the Lost Squadron’s flag is challenging.

Bill Rechin’s Crock for the 18th (a rerun, though I don’t know from when) does mention mathematics in an appropriate context. Possibly the most important use of mathematics, after bookkeeping, is navigation. To know where one is, and where one means to go, is of great value. Finding ways to turn the observations and calculations needed to find one’s position into something that could be done in the field was a great challenge to armies and navies. You may remember the slide rule scene in the movie Apollo 13. The calculations there were all about converting navigational data for the Apollo Command Module to that for the Lunar Module. The Lunar Module was, relative to the Command Module, upside-down and rotated a little bit. Good navigation does demand a good sense of numbers.

Julie Larson’s The Dinette Set rerun from the 19th is about an application of mathematics I hear about but never see. I’m told there are many people in the world who need to halve or double recipes. And further, that the traditional English units of measure — three teaspoons in a tablespoon, two cups in a quart, four quarts in a gallon, et cetera — makes this sort of recipe scaling particularly easy. I am unconvinced, but I do like the array of extra size- and mathematics-related jokes stuffed into the background.

'Where are we now?' 'According to my calculations, we're about to enter a supermarket in downtown Topeka. Of course that's just a rough calculation.' They're on the high seas, as they were through the last Boner's Ark strip ever.
Mort Walker’s Boner’s Ark originally run the 1st of June, 1970. Why is Boner’s nose a poker chip?

Mort Walker’s Boner’s Ark for the 20th of April, originally run the 1st of June, 1970, is a curious pre-echo of the rock mentioned above. It’s a joke along the same lines anyway.

Who Was Jonas Moore?


I imagine I’m not the only person to have not realized the anniversary of Jonas Moore’s death was upon us again. Granted he’s not in anyone’s short list of figures from mathematical history. The easiest thing to say about him is that he appears to have coined common shorthands for the trigonometric functions: cot for cotangent, that sort of thing. Perhaps nothing exciting, but it’s something that had to be done.

Moore’s more interesting than that. The Renaissance Mathematicus has a biographic essay. Particularly of interest is that Moore oversaw the building of the Royal Observatory in Greenwich, and paid for the first instruments put into it. And, with Samuel Pepys, he founded the Royal Mathematical School at Christ’s Hospital, to train men in scientific navigation. As such he’s got a place in the story of longitude, and time-keeping, and our understanding of how to measure things.

That won’t put him onto your short list of important figures in the history of mathematics and science. But it’s interesting anyway.

Reading the Comics, January 17, 2015: Finding Your Place Edition


This week’s collection of mathematics-themed comic strips includes one of the best examples of using mathematics in real life, because it describes how to find your position if you’re lost in, in this case, an uncharted island. I’m only saddened that I couldn’t find a natural way to work in how to use an analog watch as a makeshift compass, so I’m shoehorning it in up here, as well as pointing out that if you don’t have an analog clock to use, you can still approximate it by drawing the hands of the clock on a sheet of paper and using that as a pretend watch, and there is something awesome about using a sheet of paper with the time drawn on it as a way to finding north.

Dave Whamond’s Reality Check (January 12) is a guru-on-the-mountain joke, explaining that the answers to life are in the back of the math book. It’s certainly convention for a mathematics book, at least up through about Intro Differential Equations, to include answers to the problems, or at least a selection of problems, in the back, and on reflection it’s a bit of an odd convention. You don’t see that in, say, a history book even where the questions can be reduced to picking out trivia from the main text. I suppose the math-answers convention reflects an idea that there’s a correct way to go about solving a problem, and therefore, you can check whether you picked the correct way and followed it correctly with no more answer than a printed “15/2” as guide. In this way, I suppose, a mathematics textbook can be self-teaching — at least, the eager student can do some of her own pass/fail grading — which was probably invaluable back in the days when finding a skilled mathematics teacher was so much harder than it is today.

Continue reading “Reading the Comics, January 17, 2015: Finding Your Place Edition”

Could “Arthur Christmas” Happen In Real Life?


If you haven’t seen the Aardman Animation movie Arthur Christmas, first, shame on you as it’s quite fun. But also you may wish to think carefully before reading this entry, and a few I project to follow, as it takes one plot point from the film which I think has some interesting mathematical implications, reaching ultimately to the fate of the universe, if I can get a good running start. But I can’t address the question without spoiling a suspense hook, so please do consider that. And watch the film; it’s a grand one about the Santa family.

The premise — without spoiling more than the commercials did — starts with Arthur, son of the current Santa, and Grand-Santa, father of the current fellow, and a linguistic construct which perfectly fills a niche I hadn’t realized was previously vacant, going off on their own to deliver a gift accidentally not delivered to one kid. To do this they take the old sleigh, as pulled by the reindeer, and they’re off over the waters when something happens and there I cut for spoilers.

Continue reading “Could “Arthur Christmas” Happen In Real Life?”