Two main conceptual developments were required for the Greeks to achieve a scientific understanding of motion. First, they needed a coherent conception of nonbeing: if something moves then it is not somewhere at one time where it is at another time; and Parmenides showed that this causes problems if you think of it in the wrong way. Second, they had to conceive an appropriate relation between space and time, such that space and time could be combined in an understanding of speed. Before Aristotle, Greek thinkers could not with full generality think of one motion as faster or slower than another; they could only make this comparison in special cases (such as when two runners run the same course and one finishes first).So runs the organizing thesis of Barbara Sattler’s valuable and wide-ranging book. The book traces the problem of nonbeing with a focus on Parmenides, the ancient atomists Democritus and Leucippus, and Plato’s Sophist. Relations between space and time are examined with a focus on Zeno’s paradoxes, Plato’s Timaeus, and Aristotle’s Physics.Parmenides sets the program by articulating general criteria for rational inquiry. He is credited with the first explicit statement of a principle of noncontradiction, as well as the explicit introduction of a demand for “rational admissibility” (84–87) and a principle of sufficient reason (91–92). Sattler argues for the following interpretation of the Parmenidean principle of noncontradiction. First, Parmenides treats negation as applying to the predicate term of a proposition, as opposed to the copula or the proposition as a whole. Second, Parmenides treats negation as “extreme” negation, so that the negation of a predicate term is the polar opposite of the term. Finally, Parmenides’s principle of noncontradiction encompasses also a principle of the excluded middle. In sum, where x is a subject term, F is a predicate term, and neg-F is the polar opposite of F, Sattler’s Parmenides holds: Parmenidean PNC:Eitherxis F orxis neg-F, and not both.It should be emphasized that Sattler interprets this as a principle governing scientific investigation, not necessarily all human thought. Presumably, many predicate terms that figure in everyday speech will be absent from the language of Parmenidean science, since the principle would lead to incoherence if applied to terms such as white, black, and gray (supposing white and black are polar opposites of each other and neither is a polar opposite of gray). It’s not implausible that Parmenides in the Way of Truth would exclude such terms from science. But I wasn’t fully persuaded that Parmenidean negation should be understood as term negation (x is not-F) rather than copula negation (x is-not F) or that his negation is always “extreme.” Sattler rightly emphasizes Parmenides’s habit of forcing a binary choice between extremes, but I thought the cases might be explained by his all-or-nothing view of being in particular, rather than a logical principle applying to all predicates. Regardless, the issues are subtle, and Sattler’s discussion is rewarding.On the topic of speed, Sattler’s thesis is highly interesting and provocative. Other scholars have expressed views on the topic, with some saying that the ancient Greeks could not define or quantify speed (Sambursky 1956: 150, 239–40) and others suggesting that speed was a straightforwardly measurable quantity (e.g., Castelli 2018: 54). Yet apart from the present book, it is difficult to find detailed examination of the evidence, even though the question is clearly of the first importance for the history of science. Sattler’s textual arguments are thus very welcome. To give a flavor, I will try to summarize her main arguments about Plato’s Timaeus, and then outline some opposing considerations.Sattler argues that, for Plato in the Timaeus, motion is quantifiable solely in terms of time (253, 257), and that the temporal duration of a motion is identified with its speed (273). Thus, one motion is faster than another if and only if it takes less time. Plato does not consider distance as a second, independent factor determining the speed of a motion. Sattler gives two sorts of argument for this time-only conception of speed in Plato’s Timaeus.First, Sattler argues that when Plato talks about speeds and their measurement, he mentions no units of measure other than temporal units. “[The sun’s] movement is understood as a measurement unit for the speed of the planets.... And this unit is thought of as a temporal unit, a unit to measure the time needed by the planetary movements: the unit of one day and night” (259). When Plato mentions distance at all, he does not treat it as an independent factor. Rather, in Plato’s discussion of planetary speed, “the length of an orbit is merely translated into time” so that “larger or smaller circles … can be immediately related to different speeds” (269). A smaller circle is ipso facto traversed in less time and therefore traversed faster.Second, Sattler argues that Plato could not have had an understanding of speed in terms of both time and distance, because (a) in the Timaeus theory, space is not measurable, being grounded in the uncreated and chaotic receptacle (268), and (b) even if the created cosmos contains measurable distances, still the mathematical structure of space will be geometrical, whereas that of Timaean time is arithmetical. This difference in structure “does not allow for measuring [time and space] in the same way, and so they cannot be combined in an account of motion. Speed as we understand it, in terms of the distance covered over the time taken, cannot be conceptualised in this framework” (276).It’s impossible to discuss all the evidence here, and I am not certain where the truth lies, but let me sketch an argument that Plato at least could have had a precise understanding of speed in terms of both distance and time. I’m inclined to believe that he also did have such an understanding when he wrote the Timaeus, but the question is tricky and space is short.First, a bit of circumstantial evidence. An Old Babylonian cuneiform tablet (2100–1600 BCE) includes a number labeled “the going,” which when divided by a number of nindan (a unit of distance) yields the number of round trips made in a day by a worker carrying baskets of earth (Robson 1999: 79). The going apparently represents a speed, a number of nindan gone per day. Other numbers also seem to represent rates of some sort; they are used, for example, to allocate tasks, like digging up earth or carrying it a specified distance, among workers in such a way as to avoid any idle waiting (Goetze 1951: 142; cited in Robson 1999: 81). Babylonian mathematics had extraordinary features that were not all transmitted to Greece, so this doesn’t prove something about Plato. But it shows, I think, that the necessary concepts could be grasped a long time ago by people thinking about business and labor administration.Did Plato grasp the concepts? He speaks in the Statesman of “arts that measure number, lengths, depths, widths, and speeds” (284e; translations of Plato are my own), but he doesn’t say which art measures speeds or how. In the Gorgias, Socrates says that astronomy is “about the motion of the stars, sun, and moon, how they partake of speed in relation to each other” (451c), but he still doesn’t say what speed is or how it is measured.In the Laches we come close: “the power that accomplishes a lot in little time I call speed, both in speech and in running and in all the other cases” (192a). Here speed is a disposition of a person, not an attribute of motion, so it’s not exactly what we are looking for. But at least Plato here expresses a conception that combines the quantity of a task with the quantity of time in which the task is accomplished, and it is this combination that was supposed to be the main obstacle.To develop the nature of this combination further, it is helpful to think about money. Let Agathon purchase a three-cubit-long piece of linen for two drachmae, and let Callias purchase a five-cubit-long piece of linen for three drachmae. Would Socrates know who had purchased linen at the cheaper rate? The Greeks were commercial enough that I am sure the answer is yes.Drachmae are counted in numbers, whereas linen is measured according to its length, a continuous quantity. These disparate quantities can be combined in an account of purchasing, one that allows comparisons of rate across different purchases. How? By comparing a ratio of linen-lengths, on the one hand, with a ratio of numbers of drachmae, on the other. The ratio of lengths (five to three) is greater than the ratio of numbers (three to two). Therefore, Callias paid the cheaper rate. The comparison is possible because (a) lengths stand in ratios to lengths, (b) numbers stand in ratios to numbers, and (c) each purchase was the purchase of a length of linen for a number of drachmae.The resources for a strictly parallel account of speed were available in Plato’s time. Provided that two motions each take a quantifiable time and cover a quantifiable distance, the speeds of the motions could have been compared by comparing the ratio of distances to the ratio of times. There was no need to posit a ratio between a distance and a time, or to believe that a distance is comparable to (meaning greater, less than, or equal to) a time. Sattler’s claim to the contrary (391; likewise Sambursky 1956: 150, 239–40) should be rejected. Distance and time are combined by the fact that one motion both covers a distance and takes a time; no further combination or comparison is needed.I have argued that Plato could have understood speed in much the way we do now, or anyway as Galileo did on day three of his Two New Sciences (Galilei 1914: 154–60). I have not shown that Plato did understand speed in this way. Sattler is right that we should be cautious about projecting our own understandings into ancient texts. She has given a valuable discussion of the evidence on a topic that deserves further close study.