Geometry as described in Euclid's Elements originated roughly at the same same time as Democritus described his atomic theory.

I wonder how close these two points of view were related at those times: mathematical geometry and physical atomism?

Is there evidence that mathematicians like Euclid considered physical theories like that of Democritus and/or vice versa?

Are there mentions in Euclid's Elements that possibly suggest that he was aware of and had thought about Democritus' atomic theory?

Might a mathematician like Euclid have said: "Consider the world surrounding us and neglect all the atoms in it (which master Democritus has shown to exist and describes): What remains is 'pure space' and these are the things existing in it and the laws governing them: ..."


Later atoms could come back into space:

  • At each point in time an atom occupies/defines a point in space.

  • The distance between two atoms is the distance between the points in space they occupy.

  • Atoms move along straight lines and circles.

(The same could have been said about celestial bodies – which from the distance look like atoms.)

Even later:

  • Atoms and/or celestial bodies move on epicycles.

  • Atoms and/or celestial bodies move on conic sections.

  • Historically the two appear to be diverging. Democritus might have been inspired by the letters of the phonetic alphabet which are the original stoicheia (i.e. elements). – sand1 Nov 2 at 23:02
up vote 5 down vote accepted

They were the opposite of close. Geometers and atomists were bitter ideological enemies since before Euclid. The reason for such high passions was that Greek view of geometry was different from the modern relativistic/formalistic view, to them the nature of real space was at stake, not one idealized fiction among others. The indivisibles (atoms) and the infinite divisibility (geometry) could not go together, only one had to be true. As for points, geometers treated them as marks placed at will rather than anything real, Aristotle explicitly rejects the idea that continuum is assembled from points. For a recent review see Aristidou Some Thoughts on the Epicurean Critique of Mathematics.

Epicurus postulated the least conceivable length called elahiston, declared that geometry is based on falsehoods, and banned it from the curriculum at his school, he similarly rejected Eudoxian mathematical astronomy because it was based on geometry. The rivalry between contemporary Epicurean and Eudoxian schools was quite intense, see Sedley's Epicurus and the Mathematicians of Cyzicus:

"That Epicurus believed in a minimal unit of measure out of which not only atoms but also all larger lengths, areas, and volumes are composed, is nowadays widely accepted; and most would also agree that it is not merely a physical minimum, contingent upon the nature of matter, but a theoretical minimum, than which nothing smaller is conceivable. Others both before and since Epicurus have been seduced by similar theories without being led to reject conventional geometry. Yet this is precisely the penalty which a theory of minimal parts should carry with it, for one of its consequences is to make all lines integral multiples of a single length and therefore commensurable with each other, whereas the incommensurability of lines in geometrical figures had been recognized by Greek mathematicians since the fifth century. Moreover the principle of infinite divisibility lay at the heart of the geometrical method commonly called the ‘method of exhaustion’, which was fruitfully developed by Eudoxus in the fourth century."

Euclid largely recorded geometry as presented by Eudoxus and other late Pythagoreans, so atomism and motion were alien to him. Conversely, "while there were some Epicureans who were — or previously had been — mathematicians, it seems that acceptance of the Epicurean world-view generally involved, for these individuals, a conversion from the practice of geometry" (White, What to Say to a Geometer, 1989). Some Epicureans sought to undermine geometric arguments, e.g. Zeno of Sidon, according to Proclus:

"Since some persons have raised objections to the construction of the equilateral triangle with the thought that they were refuting the whole of geometry, we shall also briefly answer them. The Zeno whom we mentioned above asserts that, even if we accept the principles of the geometers, the later consequences do not stand unless we allow that two straight lines cannot have a common segment. For if this is not granted, the construction of the equilateral triangle is not demonstrated."

Archimedes in his Method seems to be at least inspired by Democritus's computation of the volume of the pyramid by slicing it into the indivisibles, and writes:

"In the case of the theorems the proof of which Eudoxus was first to discover, that the cone is a third part of the cylinder, and the pyramid of the prism, we should give no small share of the credit to Democritus who was the first to make the assertion with regard to the said figure, though he did not prove it."

But he then explicitly distinguishes "discovering by mechanical methods" from "demonstrating by geometric methods", and gives clear priority to the latter.

  • s/ce/BC, right? – Francois Ziegler Nov 5 at 2:22
  • @FrancoisZiegler Thanks, I am not sure why the copy cut off "century", I fixed it. – Conifold Nov 5 at 21:27

The difference between a point (as in Euclid) and atom in Democritus is that an interval (or a plane or solid figure) in Euclid is infinitely divisible, every interval, no matter how small can be divided further. While in the atomic theory, there is a smallest part of matter which is not further divisible. If applied to the real world, these are two opposite points of view (infinite divisibility of matter against atomic theory) which were both discussed since antiquity. Another important difference is that Euclid did not deal with "real world", but with idealized mathematical objects, and he probably perfectly understood this.

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