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Summary: The ubiquitous concept of the universe, in which mankind could feel safe, is that of a flat earth with the cupola of the celestial vault above it. The Presocratics replaced this world-picture by that of a free floating earth, surrounded by celestial bodies at different distances. No other culture has ever conceived this world-picture, to which we are so accustomed. The different outcomes of the same experiment by Eratosthenes (who believed in a spherical earth) and Chinese astronomers, (who believed that the earth was flat) illustrate these two world-pictures. The price which the ancient Greeks had to pay for their paradigm shift was that they had to cope with the fear of falling: why does the unsupported earth not fall, why do we not fall off the earth, why do the celestial bodies not fall upon the earth? They even envisaged the possibility of an infinite universe. The existential question behind this discussion was: how can mankind feel safe again within this new conception of the universe. Aristotle settled the battle among the Presocratics with metaphysical rather than empirical arguments. His theory of natural motion, which takes falling not as the problem but as the solution, furnishes the answer to three main questions: why does the earth not fall, what is the earth’s place in the universe, and what is the shape of the earth. The theory also explains why we do not fall off the earth, and why the celestial bodies do not fall upon the earth. Moreover, Aristotle closed the universe again by arguing that it is finite. In this way he could for many ages to come suppress the fear of falling. In a recent study, O’Grady shows little understanding for this discussion, as she holds that already Thales should have taught the sphericity of the earth. In order to achieve this, he should have used arguments which are known only since Aristotle. O’Grady’s theory is a typical example of the anachronistic fallacy in interpreting the Presocratics. In the sixteenth and seventeenth centuries A.D., by the work of Copernicus, Kepler and Digges, three pillars of Aristotle’s building were pulled down: the earth is no longer the center of the universe, the planets move in ellipses and not in circles, and the universe is considered as infinite. The new conception of the world which resulted from these mortal blows renewed the fear of falling. Newton’s gravitational laws tried to remove this fear of falling again. However, as is shown by questions asked by Bentley, Newton was not able to remove this fear completely. Pascal has uniquely expressed the existential fear of falling of modern man within this newly conceived universe. The history of our civilization in the light of this fear of falling still has to be written.


This picture, called "Ubermensch" (sic) is of a sculpture by Jake and Dinos Chapman. It shows Stephan Hawking, who plays a certain role in the article, sitting in his electric chair on the top of a rock. The image is representative for the title of our article.

·  "’Hatte die Welt ein Ziel, […] so wäre es […] mit allem Werden längst zu Ende’. Ein Beitrag zur Geschichte einer Argumentation”, in: Nietzsche-Studien 27.1998(1999), pp.107-118.

For the full text of the article (in German) click here


· “How Thales Was Able to "Predict" a Solar Eclipse without the Help of Alleged Babylonian Wisdom”, in: Early Science and Medicine 2004, vol. 9/4, p.321-337.

Summary: The first part of this article examines Patricia O’Grady’s recent attempt to identify the method by which Thales might have successfully predicted a solar eclipse. According to O’Grady, some 60% of the potentially visible lunar eclipses were followed 23½ months later by potentially visible solar eclipses. It is shown that this ratio is no more than 23%, and that the method fails to predict after which specific lunar eclipse a solar eclipse will appear. In the second half of the article it is argued that on the basis of his own observations of major solar eclipses, Thales could have concluded that solar eclipses come in clusters of three, the second appearing 17 or 18 months, and the third 35 months, after the first one. In the years after the “predicted” eclipse of 28 May 585 BC, this apparent pattern disappeared again, which would explain why Thales managed to “predict” no further eclipse.

This is, approximately, what Thales saw on May 25, 585 B.C.



Summary: When Plato in the Phaedo explains the shape of the earth, he uses the image of a dodecahedron. Some misunderstandings seem to impede a real appreciation of this story. In this article it is argued that we have to take the image seriously. Each of the pentagonal faces of the dodecahedron can be seen as a cavity in the earth, of which the Mediterranean basin is one. Such a cavity, according to Plato, is only an apparent earth, and has a cover of air. The sphere of the real earth can be considered to be the sum-total of the cavities with their covers. Plato compares the inhabitants of a cavity with a person who lives at the bottom of the sea. In this text an emendation is proposed (to skip the words oioito te epi tès thalattès oikein kai  at 109c5-6), in order to make it better understandable. The story of the real and the apparent earth shows parallels with the allegory of the cave in the Republic. For a good understanding of the text one has to realize that the people in other cavities than ours in the Phaedo correspond to the people behind the prisoners in the allegory of the cave.


Summary:  Plutarch, Hippolytus, and Diogenes Laërtius report that Anaxagoras compared the size of the sun with the Peloponnesus. It is the aim of this article to show that Anaxagoras was not mad when he said this, but that it was a fair estimate, from his point of view, which is that of a flat earth. More precisely, it is shown that, with the instruments (gnomon, clepsydra, sighting tube) and with the geometrical knowledge (the properties of similar triangles, simple equations, Pythagoras’ theorem) available, Anaxagoras must have been able to use the procedures and perform the calculations needed to obtain approximately his result.


Summary: In the ancient Greek debate on the shape of the earth Anaxagoras’ argument in defense of a flat earth - the rising or setting sun is cut off by the horizon with a straight line - is more sophisticated than it seems at first sight. What he must have hinted at is, as Simplicius indicates, the mathematical issue of the circumference of an infinite circle. Aristotle, on the other hand, intends to show that an optical illusion is at stake. His first counter-argument, that the sun is far away, is shown to be mistaken, whereas the second, that the earth is big, is correct. The result is that the issue of the shape of the earth cannot be settled by the argument of the sun at the horizon.

This is one of the pictures from the article: Sunset at San Diego.


Summary: Both the doxography on the tilting of the celestial axis in Presocratic cosmology and its reception in recent times are full of misunderstandings. This is due to the fact that ancient authors and modern scholars alike did not distinguish clear enough between the phenomena as seen by someone who thinks that the earth is flat and someone who knows that the earth is spherical. The article tries to disentangle the ways in which the phenomenon of the tilted axis of the heavens has been misrepresented and misinterpreted. It is argued that the Presocratics, including Leucippus and Democritus, taught a tilting of (the axis of) the heavens, and that the accounts on a dip of the earth are mistaken. The direction of this tilt of the celestial axis must be thought to the north, not to the south, and downwards, not upwards. This tilting of the celestial axis was not meant as an explanation for the obliquity of the ecliptic.