# Ancient Astronomy, Distances, and Parallax measurements

Andrew Michael Chuggs in his recent book The Pharos Lighthouse in Alexandria (Routledge 2024) goes into an excursus on the ancient astronomy of Eratosthenes. The author is a published historian whose previous books were met with considerable scepticism by experts for their outlandish claims. By training and vocation he's a professional engineer / scientist specialising in EM and ionising radiation. That's to say that he's not afraid of using some maths (unlike most historians), and I have no reason to think that there are any mistakes in his calculations, but there are reasons to be cynical of his more outlandish claims.

I want to ask about the methods he claims were probably used by Eratosthenes to calculate the Earth-Moon and Earth-Sun distance in 3rd Century BCE Alexandria. To the best of my knowledge, there is no direct evidence for any lenses in this period, except some very indirect hints that there might have been some primitive magnifying glasses. Mirrors of course existed, but there is no evidence for parabolic ones with a high enough polish to be optically useful. All measurements can therefore be assumed to be done with the naked eye.

My question is whether ancient astronomers would have been able to make the following measurements?

1:

Had Eratosthenes employed observations at two locations on the same meridian (line of longitude) like Alexandria and Syene, then simultaneity could have been confirmed by some particular star reaching its maximum elevation above the horizon (in the same way that the maximum elevation of the Sun in the skies at Syene and Alexandria indicated midday in the context of the measurement of the Earth’s circumference). These two locations were 5000 stades or 787.5km apart. Assuming that the Moon were somewhere near the zenith, it would have exhibited a parallax of about 0.002 radians or 0.12 degrees. Eratosthenes would then have calculated the distance to the Moon as 5000/0.002 = 2,500,000 stades or 390,000km, which is roughly correct.

Is measuring a parallax of 0.12 degrees with the naked eye plausible? This is harder than measuring an angular separation of 0.12 degrees: it requires two separate set of measurements to be made with respect to some other reference system, and the difference later computed.

2:

The solar parallax between the Earth and the Moon is about 0.0026 radians or 0.15 degrees, bigger than the parallax of the Moon between Alexandria and Syene. But how could Eratosthenes have accurately measured the direction of the Sun from the perspective of the Moon? The answer is, of course, that the line of division, known as the limb, between the hemisphere of the lunar surface that is sunlit and the hemisphere that lies in darkness does depend sensitively on the direction of the Sun with respect to the Moon (see Figure). There is a minor problem in that the solar disc has an angular width of half a degree as viewed from the Moon (just as from the Earth), but the Sun is so bright that there is still a fairly sharp drop off in the surface brightness of the Moon across the limb as the last sliver of the solar disc slips below the lunar horizon. It is just a question of allowing for the known angular width of the solar disc in the calculations. Another problem is that there is significant roughness or height variation and topography across the lunar surface due to craters and mountain ranges. However, fortunately, there are also some large and smooth ancient lava plains on the Moon that we call seas, so the location of the limb might be defined quite accurately where it crosses a lunar sea.

(There's a formula that goes with this in the text, maybe the same as in the image, but it's garbled beyond recognition in my ebook version)

Is measuring the angle defined by the limb on the Moon to within 0.15 degrees with the naked eye plausible? This seems extremely tenuous to me, not least of all for the reasons the author himself gives.

• FYI. The limb is the edge of the Moon. The line between the lit hemisphere and unlit hemisphere is the terminator. I'm not sure if the angle being measured to within 0.15 degrees is A or B, but I think the answer is no. Commented Aug 3 at 23:49
• Sorry I missed it - my bad. I've added a link to the book. It's not at all necessary/required but it helps with visibility for people like me who from time to time miss things. Upon reading more carefully I also learned a new word - excursus!
– uhoh
Commented Aug 4 at 21:21
• More than 300 years later, Ptolemy wrote in his Almagest that the solar parallax was too faint to be noticed and downright ignored it. However, he did mention lunar parallax and even provided tables for calculating it. French version of the Almagest at ecliptiqc.ca/Almageste.php and English version at archive.org/details/ptolemys-almagest-toomer/mode/2up Commented Aug 6 at 22:33

The naked eye has an angular resolution of about 1 arcminute (~0.017°).

So, the parallax measurement of 0.12 degrees (7.2 arcminutes) is certainly possible, but still difficult for the reasons you mention. The reference system used could be the background stars, which are essentially stationary for a measurement like this.

The measurement of the angle of the terminator on the moon (assuming that's what the author means by limb) is not possible with the naked eye.

The moon's angular size on the sky is about 30 arcminutes (0.5°). So, an observer could only possibly distinguish differences in the angle of the terminator when the angle appears larger than 1/30 the apparent diameter of the moon.

We see 180° of the moon at any given time, so the angle that corresponds to 1/30 the diameter is ~6° (assuming we're looking near the center of the moon). This is 40 times larger than the 0.15 degrees claimed, making this an impossible feat without magnification of some kind.

As a visual argument, the angle between longitude lines on this image is 15°, so the claimed interval of 0.15° is 100 times smaller than this. Go look out at the moon the next chance you get. Would you be able to tell by eye that the terminator had rotated by that tiny amount?

Credit: https://astrostrona.pl/moon-map/ (Creative Commons)

All said, history is not clear about how Eratosthenes made these measurements (and because of some language ambiguity, even the exact Earth-Sun/Moon distances that he calculated are somewhat lost to history), so the author's claims are speculative at best.

Hmm... I'll take a crack at it.

The first block quote from Chuggs feels contrived to me at least:

Had Eratosthenes employed observations at two locations on the same meridian (line of longitude) like Alexandria and Syene, then simultaneity could have been confirmed by some particular star reaching its maximum elevation above the horizon (in the same way that the maximum elevation of the Sun in the skies at Syene and Alexandria indicated midday in the context of the measurement of the Earth’s circumference).

That part seems hard - recognizing the time of maximum elevation is a challenge visually as elevation is a very slowly varying quantity near maximum. It's a lot easier to note the azimuth of a sunrise and sunset and say the meridian is half-way in between. But that's just me.

The quote continues:

These two locations were 5000 stades or 787.5km apart. Assuming that the Moon were somewhere near the zenith, it would have exhibited a parallax of about 0.002 radians or 0.12 degrees. Eratosthenes would then have calculated the distance to the Moon as 5000/0.002 = 2,500,000 stades or 390,000km, which is roughly correct.

Well the arithmetic checks out, but how about the feasibility of the measurement?

As a matter of practice, parallax is usually measured with respect to other nearby stars, not the zenith or horizon.

If on that night the Moon was passing north or south of a star bright enough to see with the Moon in close proximity, then one could probably visually estimate the ratio of the distance from lunar limb to moon to the moon's diameter.

The lunar diameter is about 0.5 degrees. If simple sand or water-based clocks existed, one can measure that from its 0.55 degrees per hour motion against the stars using its 27.3 day period. I'm a non-historian. Maybe there were also other ways to measure the diameter of the Moon.

Anyway - I think the parallax would be easily notable - a fairly bright star or planet would pass the Moon's top or bottom about 0.29 Moon diameters (~0.15 degrees) closer seen from one place than from the other. That should be easily estimable visually. If you had some friends along with you for company, then each person could make an estimate and one might take some average or median value.

You might have to wait for an opportunity but such opportunities are not so rare; answers to Does a lunar occultation of Mars happen twice a year? inform us that since the Moon moves around in the general proximity of the ecliptic, it passes near bright planets frequently (on time scales of years at least).

Conclusion: Seems plausible to me. I have not read the book to see if referencing nearby stars is covered by Chuggs, but it would have not been hard for an ancient thoughtful astronomer to figure out and execute.

1: Tycho Brahe also did not have a telescope and was able to achieve measurements of about 2' or about 0.03°, so 0.12° is possible. As far as I know the Greek scientists of 200 BC knew as much about geometry as Tycho did. But, we don't know what kind of instruments they built to make this measurement. One would expect that they measured the distances from the limb of the Moon to a few nearby stars to keep the angles small. If the time of measurement is off between the two locations, the differences in east-west position could be used to estimate it and correct for it.

2: The Earth-Moon angle at the Sun when the Moon reaches its maximum value at first or third quarter (when the terminator appears as a straight line) and is about 0.15°, meaning the Moon - Sun angle from Earth is 89.85°. This is a difficult measurement because it is such a wide angle, nearly 90$$^\circ$$, there will be differences in the amount of atmospheric aberration, and it varies with time. I would guess that they tried to measure the angle when the terminator was exactly straight, even though this is a bit early, because it is a well-defined and easy to identify moment of the moon phases. They could use geometry to correct for the fact that the Sun is not a point and therefore can light up 0.25° more than half the moon at the quarter. Aristarchus (c.$$~$$310 – c.$$~$$230 BC) is known to have made this measurement and got 87°.

We can infer from Eratosthenes' (c.$$~$$276 BC – c.$$~$$195 BC) calculation of the Sun's size — four times too small — while accurately determining the Earth's size, that he measured the Moon-Sun angle to be around 89.4°. This implies Eratosthenes was able to measure the angle (or get the timing right) with an accuracy of about half a degree or worse.

Some historians suggest the measurements reported were intended to provide lower limits for the distance to the Sun.

• Thanks! AFAIK, it's not clear if there was much trigonometry early in the 3rd century BC, which would be a difference from Tycho. In the book from the OP Chuggs claims that Eratosthenes got almost exactly the right figure and the error of ~3 is due to him giving distances relative to Earth's circumference which later manuscript copiers mis interpreted as being Earth diameters. Is this a commonly held view, or a unique one for Chuggs? Lastly, do you have a reference for Aristarchus' measurement? Commented Aug 6 at 14:54
• Well, Hipparchus is known to have tables of trigonometric functions, but he was born just after Erastosthenes died. The Egyptians seem to have known a lot about it also. You can read about Aristarchus in Wikipedia. But, what is the evidence that later manuscript copiers got it wrong if the original is still missing? Commented Aug 6 at 16:32
• Thanks for the extra information. As for your last question, as far as I can tell, it's wishful thinking from the authors part. Something which he seems prone to in general... Commented Aug 6 at 17:08
• I think it would be difficult to get διάμετρος mixed up with περιφέρεια. If you can't read Greek, they are pronounced diametros and peripheria. Commented Aug 6 at 20:22