The subject of Aristotle's _On the Heavens_, Books 3-4, is the four elements of earth, air, fire and water, which exist below the heavens. Book 3, in chapters 1 to 7, frequently criticizes the Presocratic philosophers. Because of this, Simplicius' commentary is one of our main sources of quotations of the Presocratics. Ian Mueller's translation of this commentary gains added importance by enabling us to see the context which guided Simplicius' selection of Presocratic texts to quote. Simplicius also criticizes the lost commentary of the leading Aristotelian commentator, Alexander, and thereby gives us important information about that work. The English translation in this volume is accompanied by a detailed introduction, extensive commentary notes and a bibliography.