Lewis Carroll Tries Changing The Way You See Trigonometry

Today’s On This Day In Math tweet was well-timed. I’d recently read Robin Wilson’s Lewis Carroll In Numberland: His Fantastical Mathematical Logical Life. It’s a biography centered around Charles Dodgson’s mathematical work. It shouldn’t surprise you that he was fascinated with logic, and wrote texts — and logic games — that crackle with humor. People who write logic texts have a great advantage on other mathematicians (or philosophers). Almost any of their examples can be presented as a classically structured joke. Vector calculus isn’t so welcoming. But Carroll was good at logic-joke writing.

Developing good notation was one of Dodgson/Carroll’s ongoing efforts, though. I’m not aware of any of his symbols that have got general adoption. But he put forth some interesting symbols to denote the sine and cosine and other trigonometric functions. In 1861, the magazine The Athanaeum reviewed one of his books, with its new symbols for the basic trigonometric functions. (The link shows off all these symbols.) The reviewer was unconvinced, apparently.

I confess that I am, too, but mostly on typographical grounds. It is very easy to write or type out “sin θ” and get something that makes one think of the sine of angle θ. And I’m biased by familiarity, after all. But Carroll’s symbols have a certain appeal. I wonder if they would help people learning the functions keep straight what each one means.

The basic element of the symbols is a half-circle. The sine is denoted by the half-circle above the center, with a vertical line in the middle of that. So it looks a bit like an Art Deco ‘E’ fell over. The cosine is denoted by the half circle above the center, but with a horizontal line underneath. It’s as if someone started drawing Chad and got bored and wandered off. The tangent gets the same half-circle again, with a horizontal line on top of the arc, literally tangent to the circle.

There’s a subtle brilliance to this. One of the ordinary ways to think of trigonometric functions is to imagine a circle with radius 1 that’s centered on the origin. That is, its center has x-coordinate 0 and y-coordinate 0. And we imagine drawing the line that starts at the origin, and that is off at an angle θ from the positive x-axis. (That is, the line that starts at the origin and goes off to the right. That’s the direction where the x-coordinate of points is increasing and the y-coordinate is always zero.) (Yes, yes, these are line segments, or rays, rather than lines. Let it pass.)

The sine of the angle θ is also going to be the y-coordinate of the point where the line crosses the unit circle. That is, it’s the vertical coordinate of that point. So using a vertical line touching a semicircle to suggest the sine represents visually one thing that the sine means. And the cosine of the angle θ is going to be the x-coordinate of the point where the line crosses the unit circle. So representing the cosine with a horizontal line and a semicircle again underlines one of its meanings. And, for that matter, the line might serve as a reminder to someone that the sine of a right angle will be 1, while the cosine of an angle of zero is 1.

The tangent has a more abstract interpretation. But a line that comes up to and just touches a curve at a single point is, literally, a tangent line. This might not help one remember any useful values for the tangent. (That the tangent of zero is zero, the tangent of half a right angle is 1, the tangent of a right angle is undefined). But it’s still a guide to what things mean.

The cotangent is just the tangent upside-down. Literally; it’s the lower half of a circle, with a horizontal line touching it at its lowest point. That’s not too bad a symbol, actually. The cotangent of an angle is the reciprocal of the tangent of an angle. So making its symbol be the tangent flipped over is mnemonic.

The secant and cosecant are worse symbols, it must be admitted. The secant of an angle is the reciprocal of the cosine of the angle, and the cosecant is the reciprocal of the sine. As far as I can tell they’re mostly used because it’s hard to typeset \frac{1}{\sin\left(\theta\right)}. And to write instead \sin^{-1}\left(\theta\right) would be confusing as that’s often used for the inverse sine, or arcsine, function. I don’t think these symbols help matters any. I’m surprised Carroll didn’t just flip over the cosine and sine symbols, the way he did with the cotangent.

The versed sine function is one that I got through high school without hearing about. I imagine you have too. The versed sine, or the versine, of an angle is equal to one minus the cosine of the angle. Why do we need such a thing? … Computational convenience is the best answer I can find. It turns up naturally if you’re trying to work out the distance between points on the surface of a sphere, so navigators needed to know it.

And if we need to work with small angles, then this can be more computationally stable than the cosine is. The cosine of a small angle is close to 1, and the difference between 1 and the cosine, if you need such a thing, may be lost to roundoff error. But the versed sine … well, it will be the same small number. But the table of versed sines you have to refer to will list more digits. There’s a difference between working out “1 – 0.9999” and working with “0.0001473”, if you need three digits of accuracy.

But now we don’t need printed tables of trigonometric functions to get three (or many more) digits of accuracy. So we can afford to forget the versed sine ever existed. I learn (through Wikipedia) that there are also functions called versed cosines, coversed sines, hacoversed cosines, and excosecants, among others. These names have a wonderful melody and are almost poems by themselves. Just the same I’m glad I don’t have to remember what they all are.

Carroll’s notation just replaces the “sin” or “cos” or “tan” with these symbols, so you would have the half-circle and the line followed by θ or whatever variable you used for the angle. So the symbols don’t save any space on the line. They take fewer pen strokes to write, just two for each symbol. Writing the symbols out by hand takes three or four (or for cosecant, as many as five), unless you’re writing in cursive. They’re still probably faster than the truncated words, though. So I don’t know why precisely the symbols didn’t take hold. I suppose part is that people were probably used to writing “sin θ”. And typesetters already got enough hazard pay dealing with mathematicians and their need for specialized symbols. Why add in another half-dozen or more specialized bits of type for something everyone’s already got along without?

Still, I think there might be some use in these as symbols for mathematicians in training. I’d be interested to know how they serve people just learning trigonometry.


Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

16 thoughts on “Lewis Carroll Tries Changing The Way You See Trigonometry”

    1. It really stands out how much Lewis Carroll liked the playful side of mathematics. If I have a slow stretch I might just pull out various puzzles and games he developed — there were a lot of them — to show how much recreational mathematics he had to share.


  1. There is a crying need to get rid of sin x, along with sin^2 x (write it with superscript).
    Why don’t we write y = fx for any old function, or log^2 x for the square of log x ?
    Consistency is a non-feature of elementary math, and much confusion is a result.

    PUT THE BRACKETS IN ! (parentheses)
    sin(x), cos(x), sin(x+a), (sin(x))^2, or maybe ok with sin(x)^2
    and as for \sin^{-1}\left(\theta\right) then 1/sin(x) and arcsin(x) are far far better.

    Liked by 1 person

    1. There are some fields of analysis in which writing y = fx is the thing to do. But that does require working out how you want the composition of functions to look, and what f2 ought to mean in that context.

      There’s probably no way to be perfectly consistent in notation throughout mathematics. There’s just too much and some stuff is useful in some contexts that isn’t in others. That parentheses mark a group of symbols as a common unit is pretty nearly universal. We always need some aggregation term, after all, and just drawing in a vinculum won’t always cut it.


  2. There are several notations that irritate students, often, to be handled while writing or speaking and they often ask why we can’t write a certain notation this way or a that…! I feel there’s a particular reason. Which is, if you’ve done some extraordinary work in your discipline, and you’ve conquered some big problem of your world. Only after that you can challenge the previously defined notations. I think, except doing that, nobody is going to give value to your newly defined notation/symbols, doesn’t matter if the new notations are better.

    I hope you people know the Kumar Jan’s story.


    1. Most notation does have a history behind it. For the prominent symbols everyone uses there’s typically great names attached, too, and a string of attempted or possible alternate symbols. Sometimes they give hints about how the understanding of an idea developed. The decimal point, for example, is the result of a lot of uncertainty about how to express this radical idea, and how to make it clear, versus how to make it compact. We’ve got the idea down now to where it’s hard to imagine it being made much better. I think the only thing we really need to fix yet is how to aggregate digits after the decimal: 2,038.12834200 ends in a bit of a mess, but how to improve it?

      I think the problem in taking on well-established notation is mostly one of inertia. If we’ve settled on “sin(x)” as a tolerable way to express “the sine of the number called x” then to overthrow it we have to put forth something that’s somehow clearer enough to be worth changing. “sin(x)” has a lot going for it, though. In shape it parallels the “f(x)” notation that’s the most familiar way to call functions. By being several letters long you make clear you mean one of a common set of special functions. Of the common special functions — the trig functions, logarithms, exponentials, the Gaussian error function and so on — the letters “sin” pick out “sine” with very little room for alternate interpretations. The notation has a lot going for it. It’s difficult to picture something that would do better.

      I am not familiar with the Kumar Jan’s story; could you share it?


      1. Yeah…! I don’t know where you can find his story. But I heard it from my lecturer in my university. Kumar jan was an Indian mathematician, also he was very young, perhaps below age of 19 years. His could only complete his high school education and couldn’t afford expenses for graduation. But he did great work on a topic in mathematics (pardon me, I forgot what it was, but it was some new development in the form of a theorem). He sent his work to a lot of P.HD doctors across the world but no one gave him any importance. A professor named Howdy also got his research paper. But after reading, he just put it somewhere between files and folders. After one year, he again opened that paper and after reading it again, he was amazed a lot and he thought, the author of this paper is really a great mathematician. He replied to Kumar jan, and offered him a free P.HD education under his supervision. But till that, Kumar Jan was died by a natural death. Also, he couldn’t complete his p.hd because he didn’t had a graduation degree. So much big unfortunate he was. I told this story to explain my last message, that if you have to change something, you’ve to do something like Kumar Jan did.


        1. I am not certain, but is it possible that you’re thinking of Srinivasa Ramanujan? The mention of a genius Indian mathematician discovered by a contact with “Howdy” makes me think of Ramanujan, brought out of obscurity by his contact with G H Hardy. Ramanujan did have a sadly short life, but he did receive his doctorate and world renown for his work. His birthday was declared India’s National Mathematics Day (although it’s not clear to me if that was a one-time event, for the 125th anniversary of his birth, or an annual occurrence).

          Liked by 1 person

          1. Hi joseph, as you’re telling, I assume that you’re quite right. As I already mentioned, I heard this story in my classroom by my lecturer, that’s why I was pronouncing Dr. Hardy as Howdy haha, Pardon me for that. And that’s why I already stated you people knew this story :) . Thank you for sharing this info.


  3. Trying to answer your question: Learning theoretical physics I was conditioned to use the exponential function instead of cosine and sine, so I hardly used even those, let alone secant and cosecant (or sinh and cosh). I am in favor of using the lowest number of special functions as possible; and I think it also helps to to recognize that grand unification behind all that stuff.


    1. Generically, I like using as few tools as possible. But that has to yield sometimes. It can be that you spend so much time getting from the primordial tool to the thing you actually mean to use that it’s counterproductive. The prime example there is how yes, you can do every common binary logical operation as a string of NAND operations. But that turns into obscurantism too easily.

      So … sines and cosines. In analytical work it usually feels easier to me to use exponential functions. That’s surely because they make integration and differentiation such effortless things. But I don’t think I could do away with them, especially when studying the relationships between angles and arcs and lengths of things. I grant that I haven’t tried, though.

      Liked by 1 person

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