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I asked the following question on the Physics stackexchange, only to be notified that it was more suitable for this forum. The link to original post can be found at: https://physics.stackexchange.com/questions/333125/alexandrian-knowledge-on-distance-of-sun-from-earth

The question is as follows. In the book "Cosmos", Carl Sagan describes the methods used by Eratosthenes in determining the radius of earth (Chapter 1). Going through the steps of the method, he makes the statement "The Sun is so far away that its rays are parallel when they reach the Earth", which is crucial for Eratosthenes' conclusion. Sagan does not explain how Alexandrian scholars knew this quoted fact.

Is it true that the Alexandrian scholars already knew this fact? If yes then how did they determine this or how might they have determined this? Did they know the distance of Sun from Earth? If yes then how?


Following link was given as an answer in Physics stackexchange: https://en.wikipedia.org/wiki/Astronomical_unit#History

Unfortunately, this link does not describe the method used by Eratosthenes, rather only by Ptolemy, who came into the scene much later. I am interested in the method used by or before Eratosthenes.

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Aristarchus showed how to determine the relative distances and sizes of earth, sun and moon based on eclipses: https://www.youtube.com/watch?v=ozuEb_qLNys

But the fact the sun is so far away that its rays can be considered parallel is obvious already without such calculations. When we travel large distances, the change in appearance of the night sky is consistent with the earth being a tiny dot and the sun and stars infinitely far away. If the sun was close in relation to the distance we travelled we would instead see drastic parallax effects.

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Aristarchus "method" mentioned in the answer of @Viktor Blasjo was impractical. It is clear from Aristarchus's book that he never tried to actually measure anything. His book has to be considered a pure mathematical exercise.

The simplest method of finding distances is measuring parallax (see Wikipedia). The ancients knew this and they also knew that parallax of the Sun is zero to the degree of the accuracy they could measure. Therefore they could only very crudely estimate this distance from below. The parallax of the Sun is too small to be measured in antiquity, or even much later. So they knew essentially nothing about this distance. The story of measuring Sun's parallax is explained here, for example:

https://www.eso.org/public/outreach/eduoff/aol/market/collaboration/solpar/#chap2

The first meaningful measurement of the distance to the Sun was made by Cassini in 1672. But even at that time it was impossible to measure Sun's parallax directly: he measured parallax of Mars instead (which is closer, and easier to measure) and used the known RELATIVE distances which were known correctly without the knowledge of absolute distances since Copernicus. So it is sufficient to measure just one distance in the solar system to calculate all the rest. A century later an ingenious method based on observation of transit of Venus was developed and more precise measurement was made as a result.

Remark. I tried myself to follow Aristarchus:-) With much better instruments than those available to him. And I confirm by my own experience the common opinion of serious historians of astronomy that his method is useless in practice.

EDIT. Aristarchus proposed several methods. One of his ideas is to use a very large base for the Sun parallax, namely the distance to the Moon. Consider the triangle SME. To find it up to similarity we need two angles. Angle E is measured directly, and or the angle M, Aristarchus had an ingenious ides: just wait until it equals 90 degrees. At this moment you will see exactly half of the Moon. There are two problems. First, how accurately you can fix the moment when exactly half of the Moon is visible? Second, the angle E you measure is close to 90 degrees, say $90-\epsilon$, where $\epsilon$ is small. It is this $\epsilon$ which you really need and have to measure. But it is much smaller than the accuracy of any instrument Aristarchus could possibly make. (Compute it yourself, using Wikipedia data!) So any measurement will give some number which can be slightly greater than 90 or slightly smaller, and you can make no conclusion whatsoever from such a measurement. This is why this method is impracticable. His other methods are of the same nature.

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  • $\begingroup$ Imprecise perhaps, but not impractical. Hipparchus used a similar method to estimate distance to the Moon. $\endgroup$ – Conifold May 17 '17 at 21:15
  • $\begingroup$ Distance to the Moon is a different matter: the parallax is large enough to be measured in antiquity. If you do not trust me, try yourself to perform the measurement which Apollonius proposes:-) $\endgroup$ – Alexandre Eremenko May 17 '17 at 22:20
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The best answer I can think of is to refer you to a 1985 book by Albert van Helden, "Measuring the Universe, Cosmic Dimensions from Aristarchus to Halley" (1985, Univ of Chicago Press). The overall history is intricate, it would not be useful for me to try to summarize anything of it: just read the book. Van Helden does refer to what little is known of Eratosthenes' work in this connection. It seems likely that assumptions as well as measurement figured in that work.

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An increasing number of modern scholars agree that the Alexandrian scholars knew the distance of the Earth from the Sun. The knowledge is found in the dimensions of the Great Pyramid, but how they knew this remains a question for many. More on the ancient Egyptian methods, sources for Eratosthenes' method and a detailed critique of Sagan's analysis from Christian Irigaray's

"How the Ancient Egyptians had Calculated the Earth’s Circumference between 3750-1500 BC: a revision of the method used by Eratosthenes"

We want to make it clear that we have nothing against Eratosthenes: on the contrary, after reading from Strabo it is clear that he was a great geographer which gave the Greeks a great number of details concerning the dimensions of the known world. All that we know from Eratosthenes comes from second hand accounts, and most of them speak very highly of Eratosthenes. The problem is that modernists and many Greeks congratulate Eratosthenes as “the first” to measure the circumference of the Earth. Many academics and scholars consider this true, but the reality is that this congratulation is unwarranted and based on biased accounts like the one given by Sagan.

...

Eratosthenes (276-194 BC) is credited as "the first" person to have measured the circumference of the Earth in our modern history books, but a serious study of the circumstances shows that Eratosthenes did not actually measure anything, and that he simply copied an earlier method of the Egyptians. Eratosthenes was a Greco-Egyptian director of the Library of Alexandria at the time, and he had access to the ancient records of science collected during the Alexandrian era which contained not only Egyptian but Mesopotamian and Persian knowledge of a variety of kinds. Among these documents was one from which Eratosthenes copied a method of deriving the Earth’s circumference, but the method itself (as we will see) shows that the actual experiment and record of this in Egypt occurred as early as 3750 BC, making “the first” to actually measure a portion of the Earth’s circumference and calculating were most probably Predynastic Egyptians living up to 3500 years before Eratosthenes’ time.

Related to our answer on this question from the hsm forum here:

In ancient times, how did people conclude that the shape of Earth is a sphere?

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