Circles in the sand

By shash

11 June 2011


I'd like to begin with something that's not particularly difficult, or unfamiliar - it's just an interesting factoid, and something that makes for a good story.

Understanding the history of a concept, sometimes even the very name of something can really push you to understanding the concept itself, where a dry explanation of it would just confuse you.

In school, we all learn trigonometry in two ways:

Measurement gimmicks:

By this, I mean the kind of tricks that Sherlock Holmes in The Adventure of the Musgrave Ritual - using the shadow of a tree, a man's height and the length of the man's shadow to measure the height of the tree and so on.

Identities:

Where we learn to prove things like

sin^2 Theta + cos^2 Theta = 1

This is usually extremely rigorous, extremely boring, and nobody involved has the least idea what it's useful for.

The first is far too simple, and is basically the application of a set of formulae to the same or similar problems almost mechanically. The second, on the other hand, is just confusing, and most of us gave up and learned those things by rote, to vomit out on exam day and forget (at most) an hour after 1Just to make things clear, my teachers in school tried their best, and in a way, they succeeded; that's why I'm even able to appreciate these things at all - they were up against an oppressive syllabus and my own hard head…. Later, in more advanced settings, we use these formulae - again without really understanding, on more and more complex problems, sometimes never really involving measurement of lengths and heights, and finally end up with the idea that “Maths is difficult”, and therefore any subject derived from maths - most interesting to me is signal processing, but pretty much anything else in science and engineering is even more opaque. In the “Not helping” category here, I'd add all the drama on (big/small)-screen2 Eureka, I'm looking at you where the scientist is a pencil-necked geek who “talks in math” as Radiohead put it, and pretty much every “Local author” book I've had in college, . Note for the non-Indians who came here by accident, “local authors” are the authors from within India who write sort of Cliff's Notes versions of the syllabus, making things easier for students, or so the theory goes. In reality, the books are generally pages of definitions and derivations with no context or explanation, which on the other hand, are easier to learn by rote and vomit in exams as described above. Note, this doesn't mean that all Indian authors are bad - “Local author” is merely a sub-genre that's designed for cramming. This is so in Engineering, and I'm sure it's the case in other disciplines also.

In all of this, nobody really explains where the terms “sine”, “cosine” and “tangent” come from. We can figure out that “tangent” probably has something to do with the tangent of a circle, but what manner of beast is a “sine”? Learning what these names mean (or don't mean for that matter) can really go a long way.

So anyway, trigonometry has an incredibly long history - all the ancient structures we know of - pyramids, ziggurats, the Indus Valley cities and every ancient temple, church, mosque and other big monument - couldn't have been built without this basic knowledge. The ancient Egyptians described it (in the Rhind manuscript, I think), but my story starts in India, with this guy:

Aryabhatta statue at IUCAA Figure 1. Aryabhatta statue at IUCAA

Aryabhatta is the most well-known of Indian mathematicians of the before-modern era, because he was one of the most prolific and insightful philosophers of his time. Later, his books were read by (among others) Arabs, who took this knowledge with them (along with the decimal system and so many other things) to the Middle East, from where Europeans picked it up. That is to say, the Arabs translated Aryabhatta's and other Indian mathematicians' works from Sanskrit to Arabic, the Europeans first from Arabic to Latin, and finally, from Latin to their vernaculars. Oddly enough, the same concepts were then brought back to India in the 19th and 20th centuries, and translated from English to our vernaculars, coming almost full circle, losing all meaning in the process.

And it's Aryabhatta's circles that give us the story…

Take a circle - any circle..

Now, a circle has chords, of which the diameter is a special case - it's the longest chord, and it always divides the circle into two equal halves. So, let's draw a diameter

and a chord.

If the chord is perpendicular to the diameter, it's always bisected by the diameter3Proof is left as an exercise to the reader because it gets in the way of my story. Let's define half-chord to mean (surprise surprise) half a chord.

Now, join this half-chord back to the centre of the circle

Now, no matter how you resize the circle and its components, the ratio between the half chord and the line joining its end to the center (in this diagram) is always the same. This line just happens to be a radius of the circle, because it joins a point on the circle to its centre.

Look at the centre of the circle, where the radius from the end of the half-chord comes and joins. There's an angle formed between this line and the first diameter we drew. This angle also remains the same, as you resize the circle. So, we have two things that remain constant:

  1. The ratio between the length of the half-chord and the radius
  2. The angle

This, in turn, means that we can relate the angle to this ratio, or rather, we can relate a function of the angle to this ratio:

f(angle) = halfchord / length

This function is equal to the half-chord, if the radius is 1. In Sanskrit, “chord” is “jya”, and “half” is “ardha”, so Aryabhatta called this function the “ardha-jya” of the angle. The word "ज्य" (jyaa) actually means “bow-string” - imagine the arc of the circle to be a bow, and the chord to be its bow-string. He also defined a function that he called “koti-jya”, which means something like other-half-chord. In our earlier diagram, this would be the half-chord drawn horizontally, instead of vertically.

For brevity, Aryabhatta often just said “jya” instead of “ardha-jya”. When the Arabs took his works and translated them into Arabic, instead of translating the word “jyaa” as “bow-string”, they ended up just transliterating it instead. The closest they could come to this word in Arabic was “jiva” or “jiba”, which is meaningless in Arabic. Over time, it got reworked as “jaib”, which means or “bay”. In Latin, “bay” is “sinus”, and thus, when the Europeans took the transliterated-and-mistranslated term from the Arabs, they (probably with much bewilderment) again translated it to “sinus”, and when translating from Latin to other languages, just droppedthe -us prefix to get “sine”. The Arabs also took the “koti-jyaa”, and wrote it as ko-jiba, which came down to us (quite faithfully) as “cosine”.

And so, a very appropriate term, “half the string of a bow” became “a bay”, which is pretty much meaningless.

Comments

1 kathie B. says...

I was puzzled by the word [HTML_REMOVED]chord[HTML_REMOVED]. Doesn[HTML_REMOVED]t seem to connect w the musical meaning at all, but I looked it up and see that the spelling is correct. Never took Trig so this is mostly beyond me, but you explain things really well.

Posted at 9:53 p.m. on July 1, 2011

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