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Aristarchus of Samos : The Ancient Copernicus : A History of Greek Astronmy to Aristarchus Together with Aristarchus's Treatise on the Sizes and Distances of the sun and Moon. A New Greek Test with Translation and Notes By Sir Thomas Heath

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This old (early twentieth century) book is not just about Aristarchus, the "Greek Copernicus" -- that comes in the final chapters. Rather, it is the best survey I know of ancient Greek astronomy, starting from the very beginning. I kept coming across references to this book by Sir Thomas Heath in books on the Copernican revolution, e.g. those by Angus Armitage and Thomas Kuhn, and finally decided to read it -- after having bought his smaller volume on Greek astronomy, also reprinted by Dover.

Heath really goes into detail on each of the ancient schools of astronomy in Greece. There were in fact a good many points of view -- the earth-centered view, the view that the earth rotates daily about its axis, the view of everything going around a fiery center, the "concentric spheres" variant of Eudoxus of the earth centered view, the Aristotelian variant of that, the eventual Ptolemaic view of epicycles and all that ... and finally the view of Aristarchus, which was essentially the Copernican sun-centered view (but without Copernicus' marvelous insights into how such a view simplified everything).

Heath not only goes into detail about each of these schools of thought -- it can be pretty rough going trying to follow all of this, especially because of the esoteric modes of speaking they often employed -- he also gives pithy summaries of what each school was saying, and a critical evaluation of their worth and influence.

If you really want to gain an appreciation of the variety of the ancient Greek schools of thought, of their struggle and progress over seven centuries in describing the motions in the heavens, of the reasonableness of much of their thinking, of the magnificence of their achievement -- I heartily recommend this book, then I recommend, either before or after, exploring the Copernican revolution and the rest of the scientific revolution, in the books by Armitage, Kuhn, Hall, and others.

Thomas Heath published this book in 1913. As he explains, he was first motivated by a request from a friend to produce a translation of Aristarchus's treatise "On the sizes and distances of the sun and the moon." In that book Aristarchus does not say anything about heliocentrism or the motion of the earth, but we know from ancient reports that he was an early heliocentrist, probably the first. However, at the time Heath wrote this, the definitive studies of ancient heliocentrism and the earth's motion had been written by Schiaparelli, who insisted that Aristarchus had been anticipated by Heraclides Ponticus. (Yes, Schiaparelli was the astronomer who found "canali" on Mars!) Heath took the opportunity to set the record straight, and ended up writing a fantastic study of Greek astronomy up to Aristarchus in the 3rd century BC. The book is nearly a hundred years old, so don't rely on it as your only source if you want a detailed up-to-date picture of early astronomy. (If that's what you want, start with James Evans's "History and Practice of Ancient Astronomy" and work from the footnotes.) But this is a real classic, constantly cited by later authors, and more readable than a lot of what you will find written on Greek astronomy. Heath's translation of Aristarchus's work, at the back of the book, became the definitive English translation.

If you want to know more about what the Greeks did after Aristarchus, I recommend, in addition to Evans's book mentioned above, JLE Dreyer's "A History of Astronomy from Thales to Kepler," another classic survey from about the same time.

Although this book contains much historical background, the highlight, in my opinion, is the complete translation of Aristarchus' only extant work, On the distances and sizes of the sun and moon (c. -270). I am going to summarise this work here. My summary will contain all the main results of the treatise, and indicate what is essentially Aristarchus' derivations. However, my summary will be a simplification in that I will assume that the sun lights up exactly half the moon, whereas Aristarchus knows that this is not quite true and so has a small correction factor in the calculations to account for this.

I will use this notation: E, M, S are the centers of the earth, moon and sun respectively, and E', M', S' are points on their apparent perimeters.

The ratio of the distances from the earth to the moon and from the earth to the sun can be determined by measuring the angle MES at half moon. For at half moon the angle EMS=90° and the angle MES is measurable, so we know all angles of this triangle and thus the ratios of its sides.

The ratio of the sizes of the moon and the sun can then be inferred at a solar eclipse. For at a solar eclipse, the moon precisely covers the sun. Thus EMM' is similar to ESS', with the scaling factor discovered above, i.e. SS':MM'::ES:EM.

The ratio of the distance of the moon to its size can be inferred from its angular size. For the angle EMM'=90° and the angle MEM' is measurable, so we know all angles of this triangle and thus the ratios of its sides.

These distances can be related to the radius of the earth at a lunar eclipse. For the shadow that the earth casts on the moon is about two moon-diameters wide. To incorporate this information into a similar triangles setup, let O be the point beyond the moon from which the earth has the same angular size as the sun (i.e. precisely blocks out the sun). Then SS':EE'::OS:OE. Now the algebra gets a little bit involved. We want to know the LHS so we have to reduce the RHS to a number, which we will do by expressing both OS and OE in terms of OM. From above we know SS':MM', and now we have OS:OM::SS':2MM', which enables us to express OS in terms of OM. To express OE in terms of OM we first note that OE=OM+EM. From above we know how to express EM in terms of ES, or, if we prefer, MS. But again from OS:OM::SS':2MM' we know OS=OM+MS in terms of OM, so we know MS in terms of OM, so we are done. OS:OE is now some multiple of OM over some multiple of OM, i.e. a number, so we have found SS':EE', i.e. we have expressed the size of the sun (and thereby the size of the moon, of course) in terms of the size of the earth.