The BBC’s In Our Time program, and podcast, did a 50-minute chat about the longitude problem. That’s the question of how to find one’s position, east or west of some reference point. It’s an iconic story of pop science and, I’ll admit, I’d think anyone likely to read my blog already knows the rough outline of the story. But you never know what people don’t know. And even if you do know, it’s often enjoyable to hear the story told a different way.
The mathematics content of the longitude problem is real, although it’s not discussed more than in passing during the chat. The core insight Western mapmakers used is that the difference between local (sun) time and a reference point’s time tells you how far east or west you are of that reference point. So then the question becomes how you know what your reference point’s time is.
This story, as it’s often told in pop science treatments, tends to focus on the brilliant clockmaker John Harrison, and the podcast does a fair bit of this. Harrison spent his life building a series of ever-more-precise clocks. These could keep London time on ships sailing around the world. (Or at least to the Caribbean, where the most profitable, slavery-driven, British interests were.) But he also spent decades fighting with the authorities he expected to reward him for his work. It makes for an almost classic narrative of lone genius versus the establishment.
But, and I’m glad the podcast discussion comes around to this, the reality more ambiguous than this. (Actual history is always more ambiguous than whatever you think.) Part of the goal of the goal of the British (and other powers) was finding a practical way for any ship to find longitude. Granted Harrison could build an advanced, ingenious clock more accurate than anyone else could. Could he build the hundreds, or thousands, of those clocks that British shipping needed? Could anyone?
And the competing methods for finding longitude were based on astronomy and calculation. The moment when, say, the Moon passes in front of Jupiter is the same for everyone on Earth. (At least for the accuracy needed here.) It can, in principle, be forecast years, even decades ahead of time. So why not print up books listing astronomical events for the next five years and the formulas to turn observations into longitudes? Books are easy to print. You already train your navigators in astronomy so that they can find latitude. (This by how far above the horizon the pole star, or the sun, or another identifiable feature is.) And, incidentally, you gain a way of computing longitude that you don’t lose if your clock breaks. I appreciated having some of that perspective shown.
(The problem of longitude on land gets briefly addressed. The same principles that work at sea work on land. And land offers some secondary checks. For an unmentioned example there’s triangulation. It’s a great process, and a compelling use of trigonometry. I may do a piece about that myself sometime.)
Also a thing I somehow did not realize: British English pronounces “longitude” with a hard G sound. Huh.
Jacob Siehler suggested the term for today’s A to Z essay. The letter V turned up a great crop of subjects: velocity, suggested by Dina Yagodich, and variable, from goldenoj, were also great suggestions. But Siehler offered something almost designed to appeal to me: an obscure function that shone in the days before electronic computers could do work for us. There was no chance of my resisting.
A story about the comeuppance of a know-it-all who was not me. It was in mathematics class in high school. The teacher was explaining logic, and showing off diagrams. These would compute propositions very interesting to logic-diagram-class connecting symbols. These symbols meant logical AND and OR and NOT and so on. One of the students pointed out, you know, the only symbol you actually need is NAND. The teacher nodded; this was so. By the clever arrangement of enough NAND operations you could get the result of all the standard logic operations. He said he’d wait while the know-it-all tried it for any realistic problem. If we are able to do NAND we can construct an XOR. But we will understand what we are trying to do more clearly if we have an XOR in the kit.
So the versine. It’s a (spherical) trigonometric function. The versine of an angle is the same value as 1 minus the cosine of the angle. This seems like a confused name; shouldn’t something called “versine” have, you know, a sine in its rule? Sure, and if you don’t like that 1 minus the cosine thing, you can instead use this. The versine of an angle is two times the square of the sine of half the angle. There is a vercosine, so you don’t need to worry about that. The vercosine is two times the square of the cosine of half the angle. That’s also equal to 1 plus the cosine of the angle.
This is all fine, but what’s the point? We can see why it might be easier to say “versine of θ” than to say “2 sin(1/2 θ)”. But how is “versine of θ” easier than “one minus cosine of θ”?
The strongest answer, at the risk of sounding old, is to ask back, you know we haven’t always done things the way we do them now, right?
We have, these days, settled on an idea of what the important trigonometric functions are. Start with Cartesian coordinates on some flat space. Draw a circle of radius 1 and with center at the origin. Draw a horizontal line starting at the origin and going off in the positive-x-direction. Draw another line from the center and making an angle with respect to the horizontal line. That line intersects the circle somewhere. The x-coordinate of that point is the cosine of the angle. The y-coordinate of that point is the sine of the angle. What could be more sensible?
That depends what you think sensible. We’re so used to drawing circles and making lines inside that we forget we can do other things. Here’s one.
Start with a circle. Again with radius 1. Now chop an arc out of it. Pick up that arc and drop it down on the ground. How far does this arc reach, left to right? How high does it reach, top to bottom?
Well, the arc you chopped out has some length. Let me call that length 2θ. That two makes it easier to put this in terms of familiar trig functions. How much space does this chopped and dropped arc cover, horizontally? That’s twice the sine of θ. How tall is this chopped and dropped arc? That’s the versine of θ.
We are accustomed to thinking of the relationships between pieces of a circle like this in terms of angles inside the circle. Or of the relationships of the legs of triangles. It seems obviously useful. We even know many formulas relating sines and cosines and other major functions to each other. But it’s no less valid to look at arcs plucked out of a circle and the length of that arc and its width and its height. This might be more convenient, especially if we are often thinking about the outsides of circular things. Indeed, the oldest tables we in the Western tradition have of trigonometric functions list sines and versines. Cosines would come later.
This partly answers why there should have ever been a versine. But we’ve had the cosine since Arabian mathematicians started thinking seriously about triangles. Why had we needed versine the last 1200 years? And why don’t we need it anymore?
One answer here is that mention about the oldest tables of trigonometric functions. Or of less-old tables. Until recently, as things go, anyone who wanted to do much computing needed tables of common functions at many different values. These tables might not have the since we really need of, say, 1.17 degrees. But if the table had 1.1 and 1.2 we could get pretty close.
But trigonometry will be needed. One of the great fields of practical mathematics has long been navigation. We locate points on the globe using latitude and longitude. To find the distance between points is a messy calculation. The calculation becomes less longwinded, and more clear, written using versines. (Properly, it uses the haversine, which is one-half times the versine. It will not surprise you that a 19th-century English mathematician coined that name.)
Having a neat formula is pleasant, but, you know. It’s navigators and surveyors using these formulas. They can deal with a lengthy formula. The typesetters publishing their books are already getting hazard pay. Why change a bunch of references to instead?
We get a difference when it comes time to calculate. Like, pencil on paper. The cosine (sine, versine, haversine, whatever) of 1.17 degrees is a transcendental number. We do not have the paper to write that number out. We’ll write down instead enough digits until we get tired. 0.99979, say. Maybe 0.9998. To take 1 minus that number? That’s 0.00021. Maybe 0.0002. What’s the difference?
It’s in the precision. 1.17 degrees is a measure that has three significant digits. 0.00021? That’s only two digits. 0.0002? That’s got only one digit. We’ve lost precision, and without even noticing it. Whatever calculations we’re making on this have grown error margins. Maybe we’re doing calculations for which this won’t matter. Do we know that, though?
This reflects what we call numerical instability. You may have made only a slight error. But your calculation might magnify that error until it overwhelms your calculation. There isn’t any one fix for numerical instability. But there are some good general practices. Like, don’t divide a large number by a small one. Don’t add a tiny number to a large one. And don’t subtract two very-nearly-equal numbers. Calculating 1 minus the cosines of a small angle is subtracting a number that’s quite close to 1 from a number that is 1. You’re not guaranteed danger, but you are at greater risk.
A table of versines, rather than one of cosines, can compensate for this. If you’re making a table of versines it’s because you know people need the versine of 1.17 degrees with some precision. You can list it as 2.08488 times 10-4, and trust them to use as much precision as they need. For the cosine table, 0.999792 will give cosine-users the same number of significant digits. But it shortchanges versine-users.
And here we see a reason for the versine to go from minor but useful function to obscure function. Any modern computer calculates with floating point numbers. You can get fifteen or thirty or, if you really need, sixty digits of precision for the cosine of 1.17 degrees. So you can get twelve or twenty-seven or fifty-seven digits for the versine of 1.17 degrees. We don’t need to look this up in a table constructed by someone working out formulas carefully.
This, I have to warn, doesn’t always work. Versine formulas for things like distance work pretty well with small angles. There are other angles for which they work badly. You have to calculate in different orders and maybe use other functions in that case. Part of numerical computing is selecting the way to compute the thing you want to do. Versines are for some kinds of problems a good way.
There are other advantages versines offered back when computing was difficult. In spherical trigonometry calculations they can let one skip steps demanding squares and square roots. If you do need to take a square root, you have the assurance that the versine will be non-negative. You don’t have to check that you aren’t slipping complex-valued numbers into your computation. If you need to take a logarithm, similarly, you know you don’t have to deal with the log of a negative number. (You still have to do something to avoid taking the logarithm of zero, but we can’t have everything.)
So this is what the versine offered. You could get precision that just using a cosine table wouldn’t necessarily offer. You could dodge numerical instabilities. You could save steps, in calculations and in thinking what to calculate. These are all good things. We can respect that. We enjoy now a computational abundance, which makes the things versine gave us seem like absurd penny-pinching. If computing were hard again, we might see the versine recovered from obscurity to, at least, having more special interest.
Wikipedia tells me that there are still specialized uses for the versine, or for the haversine. Recent decades, apparently, have found useful tools for calculating lunar distances, and sight reductions. The lunar distance is the angular separation between the Moon and some other body in the sky. Sight reduction is calculating positions from the apparent positions of reference objects. These are not problems that I work on often. But I would appreciate that we are still finding ways to do them well.
I mentioned that besides the versine there was a coversine and a haversine. There’s also a havercosine, and then some even more obscure functions with no less wonderful names like the exsecant. I cannot imagine needing a hacovercosine, except maybe to someday meet an obscure poetic meter, but I am happy to know such a thing is out there in case. A person on Wikipedia’s Talk page about the versine wished to know if we could define a vertangent and some other terms. We can, of course, but apparently no one has found a need for such a thing. If we find a problem that we would like to solve that they do well, this may change.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.