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.