One of things I like to think about in science is “how do we know that?” It is interesting how one thing builds on another. This is a story of how the Greeks estimated the distance from the Earth to the Sun (an important idea in the development of the model of the solar system). I like this story because it is not too complicated. In fact, one could easily reproduce these measurements themselves. So, here is what I will talk about:

- Measuring the size of the Earth.
- Determining the distance from the Earth to the moon and the size of the moon.
- Calculating the distance (and size) to the Sun.

Now, I am not absolutely sure of the truthiness of these descriptions as I was not present during these times. However, they seem like plausible methods for calculating these things. Also, I do not contradict wikipedia.

### Shape of the Earth

To measure the size of the Earth, one first needs to know its shape. The Earth is roughly spherical. This was well known in the time of the Greeks (~500 BC). What evidence was there for a spherical Earth?

- First (not really evidence), the Greeks sure liked spheres. They thought they were awesome. So, why shouldn’t the Earth be a sphere? (yes, I simplified this whole argument, but I am ok with that).
- Next, when you see a ship coming in from the distance, you see the top of the ship first. This suggests that the surface is curved. Actually, I wish I had a picture of this but there is this long bridge going across lake Pontchartrain in Louisiana. When you are coming near the New Orleans side of the lake, you see the tops of the buildings firsts. I have always wanted to take some pictures and use them to measure the curvature of the Earth but it is dangerous to drive and photograph.
- If someone travels not too far towards the equator, that someone would be able to see stars in the sky that he or she had never seen before. I know most people now don’t really recognize the sky anymore, but before the internet people did. The picture below shows why you would see new stars. Also, moving East-West doesn’t really do anything since the Earth rotates that way already.

Here is this picture, the lower guy (or gal) can see different stars because the ground is not in the way. So, the Earth is round. This was not really a big mystery. Even the people during Christopher Columbus’s time knew the Earth was round (but that is a different story).

### Size of the Earth

The story is (don’t know if it is true) that Eratosthenes first measured and calculated the circumference of the Earth. He did this by measuring the angle of a shadow from a vertical stick at two different places. This picture should help:

Here are two cities. One is North of the other (Alexandria and Syene). One important observation (that modern people aren’t always aware of) is that the Sun reaches its highest point during the day. The highest point of the sun depends on the day of the year. In Syene, on June 21 the sun is at its highest point all year which is directly overheard. The same day of the year in Alexandria, the sun is at its highest point all year, but that is NOT directly overhead. So, by measuring the angle of the shadow in Alexandria compared to Syene AND by knowing the distance between these two, the radius of the Earth can be determined.

The thing that always confused me about this was “how did he take the measurements at the same time?” This may be obvious to many, but he could just take the measurements on the same day of the year, 1 year apart. I don’t know how he obtained a measure for the distance between the two cities. Too bad he didn’t have google maps. Perhaps he hired someone to walk and count steps. I suspect that these distances were roughly known from travelers between the two cities. Let me go ahead and do this calculation. I will assume a distance of 800 km between the two cities and an shadow angle of 7.5 degrees. From the picture above, you can see that the distance between the two cities is an arc length. The angle corresponding to this length is 7.5 degrees. The relationship between arc length and angle is:

and solving for r and then the circumference:

Using the values from above, I get:

This is a decent value – an accepted value of around 40,000 km is what google uses as the answer. Curious question: What if he had been off in the measurements by even more? This would be a great exercise for the reader (that I will probably do in the future) **Repost note:** I did this – error propagation and the distance to the Sun.

### Distance to the Moon

Once the size of the Earth is known, the distance (and size) of the moon can be found. The size can be found using the angular size and the distance. The further away something is, the smaller it appears. So, how was this done? The story I used to go by was that the size of the moon was determined by the size of the Earth’s shadow on the moon during a lunar eclipse. This may be true, but I like the following story a little better (because it is easier to understand).

Suppose that the moon moves around the Earth in a circle at a constant speed (not true). If that WAS true, then you could easily calculate where the moon would be at any time/day. The only problem with that calculation is that it assumes that you are at the center of the Earth or that the Earth is extremely small compared to the distance to the moon. The story is that Hipparchus used the difference between the calculated position of the moon and the actual position to determine the distance. Perhaps this picture will help (not drawn to scale):

With the angle between the actual and calculated positions of the moon and the radius of the Earth, there is a right triangle. One side and an angle can be used to calculate the distance to the moon. I like this method because it is easy to understand (didn’t I already say that?). However, this seems like a difficult thing to do especially since the moon does not move at a constant speed.

### Distance to the Sun

Now, the Greeks could use the distance to the moon to find the distance to the Sun. The way this was done (by Aristarchus) using the angle between a quarter moon and the Sun.

Again, this calculation uses a right triangle with one side distance known and a measured angle (as seen from the not-to-scale picture). There are two problems with this calculation. First, the angle between the sun and the quarter moon is very close to 90 degrees. Second, it is difficult to measure angles in the sky (with the Greek technology of the time). And a bonus difficulty – the Sun is really bright. You should never look at the Sun (just saying). With these difficulties, Aristarchus determined that the distance to Sun was 40 times farther than the moon. This is wrong (it is more like 400 times farther). Still, with this Aristarchus said that the Sun was ginormous (the Sun has the same angular size as the moon as seen from Earth).

Aristarchus used the idea of a ginormous Sun to say that it seems silly for the Sun to go around the Earth. Perhaps the Earth should orbit the Sun. The other Greeks laughed at him, called him names and wouldn’t let him play in any Greek games. Here is what the other Greeks said:

- It doesn’t FEEL like the Earth is moving.
- If the Earth WAS moving around the Sun, shouldn’t there be stellar parallax? Paralax is the phenomena of closer objects appearing to shift position with respect to the background when the viewing position is changed.

In fact the other Greeks were somewhat correct. It sure doesn’t feel like we are moving. Also, it is very difficult to detect stellar parallax because the stars are so far away.