# Curve of the shadow on the Moon

So I found this quote the other day:

And I can’t quite work out whether it makes sense. Yes the shadow on the moon is curved. But though that proves the Earth isn’t a flat square or a cube, does it prove the Earth isn’t a flat disc?

If you take a disc and project it’s shadow onto a wall then as you rotate the disc the shape of the shadow will change to an oval. But will the curve projected onto a small area (i.e. the Moon) change so that it’s clearly not from a sphere?

If the earth were a flat disc, near sunset or sunrise the earth is lit edge-on by sunlight and the shadow is a thin ellipse. If a lunar eclipse happens just after sunset (which isn’t all that unusual), you should see a thin elliptical shadow on the moon. Instead the shadow is always circular regardless of when the eclipse happens. That’s a pretty convincing proof that the earth is spherical.

Just to ensure nobody is confused, I assume we all understand Magellan was tlaking about a lunar eclipse, right?

Anyway, scr4 nailed it. Take a disc - a dinner plate will do - and project its shadow while turning the disc around. When the plate is edge-on to the light source you’ll just get a flat line of a shadow.

Now try the same experiment with a baseball. The shadow will always be round - just as the shadow of a lunar eclipse is.

The only thing I would wonder is that Magellan and any other sailor would have been familiar with the phenomenon of ships (or the shore) disappearing below the horizon and wouldn’t that have been even more convincing a demo? For one thing, how obvious is it that eclipses are caused by the earth’s shadow on the moon? You have to realize that the moon shines by reflected light, that the light comes from the sun, and that eclipses come only when the moon is on the ecliptic and that most of the time it is above or below it by a few degrees. All this seems clear enough to us today, but how obvious was it then?

Yeah, but what I can’t work out is whether hypothetically you could have a really big Earth, so that even when it’s nearly on edge the Moon takes up only a small part of the shadow. Thus meaning that you’d never see both sides of a thin shadow, and the curve might look pretty much the same on a distant object which isn’t obviously a sphere itself. Given that lunar exlipses aren’t that common, is the difference gonna be obvious?

And Hari Seldon, yeah, but the whole shadow on the moon thing is a really nice phrase.

I was under the impression that that quote from Magellan was thought to be a work of fiction.

Lunar eclipses happen once or twice a year, pretty much.

Magellan is almost certainly referring to Aristotle’s argument in De Caelo (On the Heavens) that the earth must be spherical. He refers to several forms of evidence of the earth’s shape, including the fact that Hari cites of ships disappearing over the horizon. However, he points out that this only proves that the earth’s surface is curved. On this evidence alone, the earth might be a hemisphere or some other portion of a sphere. He also refutes the possibilities of a flat disc, a cylinder and other shapes. As scr4 points out, the evidence of a circular shadow on the moon regardless of the orientation of the earth is the only point that proves the spherical theory.

The reasons for lunar eclipses were well known to the classical Greeks four or five centuries before the Common Era. And, therefore, to Magellan.

## The church says the earth is flat, but I know that it is round, ** for I have seen the shadow on the moon,** and I have more faith in a shadow than in the church. - Ferdinand Magellan

To me, the above fragment is ambiguous. Is he speaking of the actual shadow cast (only) during a lunar eclipse or the apparent shadow (terminator) seen any evening except on the full or new moon? The two have entirely different explanations, of course. The curved terminator results from the angular difference between the earth and the sun, with the moon at the apex of the angle. The terminator can be either concave (as the phase approaches new) or convex (as the phase approaches full.) Is there evidence that these effects (eclipse vs. uneclipsed terminator) were well understood to have different causes? If Magellan (or whomever) thought it was always a true shadow, then a convex shadow could only be explained by a concave earth, which would have been a pretty unlikely theory, or at least one of which I have never heard.

And at the quarter moon, the terminator is a straight line. The correct reasons for these effects (unrelated to the shadow of the earth) are mentioned by Aristotle in the same work I mentioned above.

Yes, but navigators of that time (like Magellan) DID have a fairly accurate knowledge of the actual size of the earth. (Actually, 2 sizes – the one Columbus used (without the American continent) and the rather more accurate post-Columbus one.)

Any “really big Earth” that might have made this shadow shape possible would have to have been way bigger than they knew the earth to be.

Isn’t the rather more accurate one pre-Columbus?

Aren’t you assuming that they also knew the size and distance of the moon?

There can be either two or three lunar eclipses in any given year.

Each year has a total of seven eclipses, either five solar eclipses and two lunar eclipses, or four solar eclipses and three lunar eclipses. The lunar eclipses are visible from the entire hemisphere of the earth that’s facing the moon, whilst any particular solar eclipses is only visible from a narrow strip of the earth’s surface.

Of course, not everyone on earth is going to see all of the lunar eclipses that occur in any given year, so in a given year you might only see one or two, as you said, even though either two or three actually occurred.

A rough estimate would have been possible based on the fuzziness of the shadow (i.e. size of the umbra vs. penumbra). I don’ t know if they actually did it or not.

What I’m wondering is, did the church in Magellan’s time still insist that the earth is flat?

I don’t think the church ever insisted that the earth was flat.

And while I was looking for something to back up what I said to scr4, I found this, from Cecil:

So it looks to me like someone took Aristotle’s quote, and falsely attributed it to Magellan, whilst confusing the Church’s position on heliocentrism (vis-a-vis Galileo) with some imaginary flat-earth position.

My apologies to commasense, for not noticing that commasense had already posted the stuff about Aristotle. :smack:

Could you work out the size/distance of the Moon based on the shadow and penumbra, if you didn’t already know the size/distance of the Sun? They knew the size of the Earth, and the angular sizes of the Sun and Moon, but I can’t figure out how to then calculate the absolute distance/size of the Moon or Sun from there based on shadows.

0, 1, 2, or 3 according to Mr. Eclipse

Mr. Eclipse says that there will have been 14,263 solar eclipses during the 6000 years from 2000BC to 4000AD, a little more than two per year on average. 2003 only has two lunar eclipses, both total, and two solar eclipses, one total, one annular. Same numbers for 2004 except both solar eclipses are partial.

Yes, I was responding to Shortie’s comment that lunar eclipses weren’t that common, with respect to European viewing.

The one from Greek times was indeed the better estimate. What happened was that once they figured that was a whole continent and TWO oceans between Spain and China, everyone ditched the Columbus estimate and went back to the older one. (In fairness to Chris, if as it is suspected he had intelligence sources about the existence of lands around where our East Coast is, he must have figured that must be Asia, and if so then the Earth has to be smaller than the Greeks said or else it would be 5 times as far, so let’s fudge the computations so that it squares…)

You just need to know the angular (apparent) size of the sun as seen from the earth, which you can measure by simply looking up. That angle is equal to the angle between the edge of the umbra and the edge of the penumbra.

Again, I don’t know if this method was used, or whether they had more accurate methods.