Yesterday I saw Agora, and I was surprised (and pleased) that it had so many references to astronomy.

I knew there were already ancient Greek philosophers speculating about a heliocentric model.
What I didn't know is that they could measure differences in the observed size of the Sun's disk, depending on the time of the year
(and deduce that therefore Earth's orbit must be slightly elliptical).

So that's my question:
without modern instruments, can you really measure the periodic variations in the distance of the Sun due to our elliptical orbit?

Well, the change in distance is about 5 million km.
The diameter at the closest is: 32' 31.84"
The diameter at the furthest is: 31' 27.69"

Could a difference of 1 arcminute be determined then?

The difference between a true total, and an annular eclipse is pretty clear evidence that the sun, moon, or both change angular size.

A camera obscura type of device could have been used to make a tracing of the disk of the sun on two different dates, then the sizes compared. That's still a really small difference though.

Another question to be asked is that even if they were able to monitor the changes in size, what would have stopped them from assuming that the sun is "breathing"; growing larger and smaller as it does so? What would have led them to a change in distance as the cause?

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Thanks for the details!

That is a very simple-to-obtain evidence indeed, hadn't thought of it.

Yeah, I also assumed that some 'indirect' form of observation would have to be involved, for greater precision and also to avoid damage to the eyes.

Due to the epicycles of the Ptolomaic model, ancient astronomers were probably used to the notion that the distance of heavenly bodies could change.
Periodical variations of the Sun's distance would fit into that model.

What troubles the movie's main character, Hypatia,
is that in a heliocentric model with perfectly circular orbits, the Sun's distance from Earth should be constant.

It had been a long time since I'd enjoyed a film so thoroughly as Agora. Seeing a heroin whose motivation is reason instead of some overblown emotion is such a breath of fresh air! I highly recommend it.

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Most of that is due to the moon though, I think.Three percent? I dunno, that seems huge to me.

If they were able to get a decent drawing from a camera obscura (likely), the difference at five meters is 1.5 mm, right? ( 5m x 2pi / (360*60) ) The diameter itself would be 4.5 cm. Maybe they used longer distances than 5m even.

heroine

Oops, that was embarassing!

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

It seems that you're right. According to Wikipedia, the angular diameter of the moon varies between 29.3′ and 34.1′.

I am not aware of any historical record of measurements of the solar diameter which were made by the Greeks and which were accurate enough to show the periodic variation due to the ellipticity of the Earth's orbit.

Can you provide references, please? References to literature from the Greeks themselves, of course, not modern guesses about their achievements.