A matter of perspective - part I

Image by Luc Viatour.

A very long time ago, I read an argument that the peculiar nature of solar eclipses might be seen as a proof of God's existence. The argument went as follows: the dramatic effect of a full solar eclipse is the result of the Moon appearing to be almost precisely as large as the Sun from the surface of the Earth. This is incredibly improbable, and wouldn't work if the sizes of the bodies involved, or their distances from each other, were even just a smidge different. The argument continues that, because this phenomenon is very improbable and also very beautiful, it must have been designed by an all-powerful being who created and rules the whole universe and has also found the time to be very picky about who I have sex with and the shape of my food's feet.

The alert reader will have noticed that I find the eclipse->God logic somewhat less than compelling (though rather superior to the "no, really, I read it in a really old book" line of reasoning), but the math associated with the perceived sizes of the two bodies really fascinated me. The fact that particularly stuck, and that has been floating around in my head for years, was that the Moon and Sun seem to be the same size from here because the ratio of the distances was nearly identical to the ratio of their sizes. A few weeks ago, though, it occurred to me that, though I'd memorized the fact, I really didn't understand why it should be the case.

As is so often the case, once my brain had noticed this hole in its understanding of the world, it wouldn't leave me alone until I had filled it. It started harassing me at the most inopportune times: waking me up at midnight, compelling me to doodle on paper stolen from waitresses at restaurants, boring my poor wife with musings on trigonometry at the dinner table. Over the course of a few weeks, though, I was able to summon up enough of my high school geometry to grok the basics of why perspective works the way it does, and I thought it might be fun to share my learning process with all of you here.

Thus, over the next few days, I will be uploading posts that explain my learning process. Putting this together has been a really silly amount of work for what will probably end up being three or four posts at most, but forcing myself to describe and illustrate my logic has given me a much better understanding of it, too, so it was definitely worth it. I hope some of you enjoy reading this series as much as I've enjoyed preparing it for you!

Step 1: Checking my numbers

The brain is a marvelous thing, capable of storing vast quantities of data. Unfortunately it tends to use pretty lossy compression algorithms, so before I went off on a wild goose chase, attempting to prove a "fact" that I'd misremembered from years before, I thought it wise to do a bit of research to confirm that the ratios of the Moon's and Sun's sizes and distances really are about equal. A bit of google-fu gives us the following numbers:

Distance to the Moon: ~384,000 km
Distance to the Sun: ~150,000,000 km
Diameter of the Moon: ~3470 km
Diameter of the Sun: ~1,390,000 km

So the ratio of the Sun's size to the Moon's size is ~401:1, and the ratio of the distances to the two bodies is ~390:1. That's close enough that it's plausible to assume that my memory is right, especially since the Moon actually wobbles a bit in its orbit (which means that some eclipses don't completely obscure the Sun, when the Moon is a bit further away than usual). Let's accept the rule as a working hypothesis and then try to prove it; hopefully proving it true will help us understand why it's true.

Step 2: Formulating our hypothesis

Before we can start testing whether our hypothesis is right, it would be helpful to define it more formally. That makes it easier to check whether we've proven or disproven it later. I came up with this: Consider objects a and b. We'll call their sizes S_a and S_b, and their distances from an observer D_a and D_b. Our hypothesis is that the observed sizes of the two objects will be the same if S_a / S_b = D_a / D_b. If S_a / S_b > D_a / D_b, then a will seem larger than b, and if S_a / S_{b <} D_a / D_b, then a will seem smaller than b.

I hope that's fairly understandable. In Part II, we'll use awesome doodles and the power of trigonometry to test our hypothesis. See you then!