 We have very little evidence for Archytas’ cosmology, yet he was responsible for one of the most famous cosmological arguments in antiquity, an argument which has been hailed as “the most compelling argument ever produced for the infinity of space” (Sorabji 1988, 125). The argument is ascribed to Archytas in a fragment of Eudemus preserved by Simplicius (= A24), and it is probably to Archytas that Aristotle is referring when he describes the fifth and “most important” reason that people believe in the existence of the unlimited (Ph. 203b22 ff.). Archytas asks anyone who argues that the universe is limited to engage in a thought experiment (this is one of the first recorded thought experiments in antiquity): “If I arrived at the outermost edge of the heaven, could I extend my hand or staff into what is outside or not? It would be paradoxical [given our normal assumptions about the nature of space] not to be able to extend it.” The end of the staff, once extended will mark a new limit. Archytas can advance to the new limit and ask the same question again, so that there will always be something, into which his staff can be extended, beyond the supposed limit, and hence that something is clearly unlimited. Neither Plato nor Aristotle accepted this argument, and both believed that the universe was limited. Nonetheless, Archytas’ argument had great influence and was taken over and adapted by the Stoics, Epicureans (Lucretius I 968–983), Locke and Newton, among others, while eliciting responses from Alexander and Simplicius (Sorabji 1988, 125–141). For a discussion of the relation between Archytas’ thought experiment and other thought experiments in antiquity see Ierodiakonou 2011. Not all scholars have been impressed by the argument (see Barnes 1982, 362), and modern notions of space allow for it to be finite without having an edge, and without an edge Archytas’ argument cannot get started (but see Sorabji 1988, 160–163). Beyond this argument, there is only exiguous evidence for Archytas’ system of the physical world. Eudemus praises Archytas for recognizing that the unequal and uneven are not identical with motion as Plato supposed (see Ti. 52e and 57e) but rather the causes of motion (A23). Another testimonium suggests that Archytas thought that all things are moved in accordance with proportion (Arist., Prob. 915a25–32 = A23a). The same testimonium indicates that different sorts of proportion defined different sorts of motion. Archytas asserted that “the proportion of equality” (arithmetic proportion?) defined natural motion, which he regarded as curved motion (for a different account of the proportion of equality see De Groot 2014: 195–207). This explanation of natural motion is supposed to explain why certain parts of plants and animals (e.g. the stem, thighs, arms and trunk) are rounded rather than triangular or polygonal. Some scholars argue that it was the influence of Archytas that led Plato and Eudemus to emphasize uniform circular movement in explaining the heavens (Zhmud 2006: 97). An explanation of motion in terms of proportion fits well with the rest of evidence for Archytas, but the details remain obscure.