apeiron Greek term meaning ‘the boundless’ or ‘the unlimited’, which evolved to signify ‘the infinite’. Anaximander introduced the term to philosophy by saying that the source of all things was apeiron. There is some disagreement about whether he meant by this the spatially unbounded, the temporally unbounded, or the qualitatively indeterminate. It seems likely that he intended the term to convey the first meaning, but the other two senses also happen to apply to the spatially unbounded. After Anaximander, Anaximenes declared as his first principle that air is boundless, and Xenophanes made his flat earth extend downward without bounds, and probably outward horizontally without limit as well. Rejecting the tradition of boundless principles, Parmenides argued that ‘what-is’ must be held within determinate boundaries. But his follower Melissus again argued that what-is must be boundless – in both time and space – for it can have no beginning or end. Another follower of Parmenides, Zeno of Elea, argued that if there are many substances, antinomies arise, including the consequences that substances are both limited and unlimited (apeira) in number, and that they are so small as not to have size and so large as to be unlimited in size. Rejecting monism, Anaxagoras argued for an indefinite number of elements that are each unlimited in size, and the Pythagorean Philolaus made limiters (perainonta) and unlimiteds (apeira) the principles from which all things are composed. The atomists Leucippus and Democritus conceived of a boundless universe, partly full (of an infinite number of atoms) and partly void; and in the universe are countless (apeiroi) worlds. Finally Aristotle arrived at an abstract understanding of the apeiron as ‘the infinite,’ claiming to settle paradoxes about the boundless by allowing for real quantities to be infinitely divisible potentially, but not actually (Physics III.4–8). The development of the notion of the apeiron shows how Greek philosophers evolved ever more abstract philosophical ideas from relatively concrete conceptions. See also ARISTOTLE , PRE — SOCRATIC. D.W.G.