Hello everyone. It's good to see you again. I mean, I see you often, but normally from the dark corner where I sit, not letting anyone see me.

As you may or may not be aware, I've been hard at work these last two years on novels (I have two of them on the go). They're coming together fairly well I think; the fourth draft of one of them, which is my current effort, is easily the best thing I've written to date.

However, I've encountered a fantasy math problem. I believe I've solved it, but the answer is counter-intuitive and I would be very grateful if you would let me go over it with you. It would be wonderful if anyone could either confirm my work or point out my fatal error.

The background -- my fantasy world consists of a very large city, and by large I mean large. This one city is only a little smaller than the entire nation of France. It's somewhere between a square and a circle in shape, and the farthest edge of the city is about 800 miles from the center. In the center is a very tall clocktower. Astonishingly tall. Impossibly tall. Don't worry about the physical logistics of such a tall tower; in reality I'm well aware that it would collapse under its own weight. In the story, people believe that God created this clocktower in the days before history, and therefore it doesn't have to follow the rules of mortal building materials. There's a deeper answer, but I'm not at liberty to spoil it. Suffice it to say that this clocktower exists.

Due to the nature of my world, it is possible to see the clocktower from even the farthest edge of the city, 800 miles away. Again, don't worry about that part. Within the context of the world's rules, this is possible.

The question is, how high should the tower be? Both so that it can be seen at that distance above other buildings and so that the highest clockface is clearly visible and comparable to the size of our moon? In addition, the top must be higher than airplanes can fly, because reasons.

That last part, from what I've read, means I have to be at least 50 miles or higher. Check. I went up to 70 miles to see how that worked. At 70 miles high (again, don't worry about how impossible that is in our world), a person standing at the edge of the city 800 miles away could clearly see it over other buildings, unless they were standing close to said other buildings.

The next problem is, how big would that clock face look from such a distance? From my research, a base to height ration of 1:6 is reasonable, so we can say that the clocktower is 12 miles wide at the base. From there, let's say the clock face at the top has a 10 mile diameter.

This becomes a question of angular diameter, which is how large an object appears to be given its true size and its distance away. Angular diameter is measured in degrees -- how many degrees an eye has to move to get from one side of the object to the other. The moon, for instance, has an angular diameter of 0.5 degrees; even though the moon is rather large, the human eye doesn't need to move very far to cover it.

Angular diameter can be determined by the quick and dirty method of taking the actual size and dividing by the distance away, then converting the answer from radians to degrees. In this case, actual size is 10, distance away is 800. Divide to get 0.0125 radians. To convert to degrees, multiply by 57.2957795, equaling roughly 0.7 degrees. This is 40% larger than the moon, which is close enough that I can live with it.

But I'm having trouble reconciling this math with how I imagine the reality. Would something ten miles wide really appear to be bigger than the moon when viewed from 800 miles away? The math seems to say that yes, it would. Is my math correct?

I'm thinking that my difficulty in imagining this comes from the fact that in reality we can only see something 800 miles away if it's in space. The curvature of the Earth hides anything 800 miles away on the surface, no matter how wide it is. But even so, it seems counter-intuitive that something 10 miles wide would still look so large from such a great distance.

Help me, Obi-wan RAFObi, you're my only hope.