Exact sciences (The) in Hellenistic times: texts and issues


Exact sciences (The) in Hellenistic times: texts and issues
The exact sciences in Hellenistic times: Texts and issues1 Alan C.Bowen Modern scholars often rely on the history of Greco-Latin science2 as a backdrop and support for interpreting past philosophical thought. Their warrant is the practice established long ago by Greek and Latin philosophers, of treating science as paradigmatic in their explanations of what knowledge is, what its objects are, how knowledge is obtained, and how it is expressed or communicated. Unfortunately, when they turn to the history of ancient science, these same scholars usually remain too much under the spell of the ancient philosophers. Granted, it is true that Greco- Latin science often served as a model and touchstone for philosophy and that, on occasion, this philosophy may have inspired science. But the marked tendency to follow Greek and Latin writers in viewing ancient science through the complex, distorting lens of ancient philosophy has hindered recognition that the various sciences of antiquity sometimes differ significantly from one another as well as from philosophy in their intellectual, literary, and social contexts. Moreover, it has encouraged scholars to ignore or even disparage clear indications that some of these sciences were deeply indebted in the course of their history to work outside the Greco-Latin tradition, in Akkadian, for example. And, what is worse, out of ignorance and neglect of the various contexts of ancient science, modern scholars have misrepresented the past fundamentally in numerous ways by resorting to alien predilections and concerns when trying to explain the origins, character, and development of Greek and Latin science.<sup>3</sup> In sum, the amorphous system of learned belief expressed now in handbooks on ancient science and currently underlying the modern interpretation of ancient philosophy, for instance, is largely inadequate and erroneous. This failure of previous scholarship challenges historians of ancient science today to re-think the entire project from its beginning. In effect, it compels one to start afresh by imagining oneself the first modern scholar confronted with all the extant literary documents (papyri, inscriptions, manuscripts) and material artifacts (instruments) that come to us from the ancient Mediterranean and Near Eastern worlds. Such a prospect is admittedly daunting and brings to mind a variant of Meno’s question (cf. Plato, Meno 80d–e): how can you seek to understand ancient science if you do not already know what it is, and how will you know that you have understood it? There are, of course, several well known ways to answer this in the abstract. But the real task is to work out a credible solution in the particular, that is, in the process of analyzing historical data. And, as I have found in studying ancient astronomy and harmonic science, this process involves a vital, corrective interplay between historical analysis and reflection on how this analysis proceeds. In fact, the process is, I think, heuristic in the sense that medicine was said to be heuristic on the grounds that the goal of the physician’s craft, health, is articulated and known only through treating specific patients. Given that the standard accounts of Greco-Latin science are at best controversial and should be abandoned in most part, and since the development of alternative accounts is still in its earliest stages, I must decline in what follows to attempt a survey. Instead, I propose to confine my remarks to a few sample texts in Greek written in the interval between the death of Alexander the Great in 323 BC and the beginning of the third century AD. There is no great significance to this period so far as the exact sciences themselves are concerned: it simply covers the range in time of the documents I have chosen to discuss. And I select these texts because they provide the earliest direct evidence of certain features of ancient science that will, I trust, be of interest to historians of science and philosophy. In describing a text as direct evidence for some claim or other, I mean that the text itself is a sufficient basis for verifying the claim. Such direct evidence stands in sharp contrast to indirect evidence in the form of citations (that is, quotations, translations, paraphrases, and reports). For, one cannot verify a claim on the strength of indirect evidence alone; what one needs in addition is independent argument for maintaining that the citation is accurate and reliable. Since there are no general rules validating the accuracy or reliability of indirect evidence, such argument must be made case by case and is, in my experience, both difficult and rarely successful. Restricting attention primarily to direct evidence may seem unduly cautious at first. But it is, I submit, the only policy that makes sense at the outset of any radically critical, historical investigation of the sort now called for. In any case, this policy does offer substantial advantages. To begin with, confirmation by recourse to direct evidence introduces an order of certainty that cannot be attained on the basis of indirect evidence or citations. The reason is that much of our indirect evidence concerns documents no longer available for inspection; thus, the most one may hope for in justifying reliance on this evidence is an argument for the probability of its accuracy. Such arguments, however, usually fail because they involve reading the historian’s own expectations into the past, expectations often concerning empirical matters about which there may be considerable uncertainty and reasonable doubt. Next, if one is strict about how evidence is used and does not introduce indirect evidence except when it is demonstrably credible—and even then one should decline to build on it, since probabilities diminish when multiplied—the preference for direct evidence will counteract a major failing in traditional histories, the valorization of certain texts and authors at the expense of others. Finally, in dating the occurrences of concepts, theories, and the like, the historian may rely on direct evidence to identify the latest (that is, most recent) date possible for their introduction. This will seem a small gain, particularly to those who think it the proper business of historians to conjecture earliest possible dates. But such a program of conjecture is an enterprise to which there is no end except by convention. Moreover, by discouraging full appreciation of the documents we actually have, this fascination with the earliest dates assignable for the occurrences of concepts and theories in Greco-Latin science underlies in part the scholarly neglect of the Akkadian and Egyptian scientific traditions which, in various forms and sometimes through intermediaries, interacted with the Greek and Latin traditions. The preceding will have to suffice as an apologia for my deciding to present the history of the exact sciences in Hellenistic times by way of narrowly defined case-studies. Though such an approach is not without precedent (cf., for example, Aaboe 1964), it is admittedly a departure from the great number of general surveys and narrative accounts currently available.<sup>4</sup> The texts I have selected are: Archimedes, De lineis spiralibus dem. 1; Geminus, Introductio astronomiae ch. 18; and Ptolemy, Harmonica i 1–2. These texts have no explicit connection. Nevertheless, they raise fundamental issues in the history of ancient science that are well worth pursuing (in studies that are, of course, suitably cognizant of historiographic matters). Indeed, there are running through these texts thematic concerns about the conception and mathematical analysis of (loco) motion, the nature of scientific communication, and the role in such communication of observation and mathematical theory. MOTION IN MATHEMATICS: ARCHIMEDES In his De lineis spiralibus, Archimedes (died 212 BC) analyzes fundamental properties of a curve of his own invention, now called the spiral of Archimedes. In the letter prefacing this treatise, the statement of the conditions under which this curve is produced comes first in a list of propositions about the spiral that are proven in the treatise proper (cf. Heiberg 1910–23, ii 14–23). Later, immediately after the corollary to dem. 11, this same statement reappears virtually unchanged as the first of a sequence of definitions. According to the latter formulation, if a straight line is drawn in a plane and if, after being turned round as many times as one pleases at a constant speed while one of its extremities is fixed, it is restored again to the position from which it started, and if, at the same time as the line is turned about, a point moves at a constant speed along the straight line beginning at the fixed extremity, the point will describe a spiral in the plane. (Heiberg 1910–23, ii 44.17–23; cf. 8.18–23) The first eleven demonstrations of the De lineis spiralibus establish what is necessary for the subsequent theorems on the spiral itself. The first two of these auxiliary demonstrations are devoted to properties of the motion of points on straight lines at constant speeds. In dem. 1 (Heiberg 1910–23, ii 12.13–14.20), Archimedes proposes to show that if a point moving at a constant speed travels along a line and two segments are taken in the line, the segments will have the same ratio to one another as the time-intervals in which the point traversed the segments. The argument opens by specifying the task as follows (see Figure 9.2 ): let a point move along a line AB at a constant speed, and let two segments, CD and DE, be taken in the line. Let the time-intervals in which the point traverses CD and DE be FG and GH respectively. It is required to prove that the segment CD will have the same ratio to the segment DE as the time-interval FG will have to the time-interval GH. Next, Archimedes makes some assignments: let AD, DB be any multiples of CD,DE respectively, So that AD>DB; (1) let LG be the simple multiple of FG as AD is of CD, and (2) (let) GK be the same multiple of GH as BD is of DE. (3) The assignment in (1) is based on a lemma that Archimedes has stated in his covering letter to the treatise: That is, in modern terms, if a and b are magnitudes and a>b, there is a whole number n such that n(b−a)>c,where c is any magnitude of the same kind as a and b<sup>5</sup> The reasoning behind the assignment in (1) seems to be as follows. Any magnitude such as a line or area is divisible into a whole number of smaller magnitudes of the same sort. Thus, given any point D in AB and any whole numbers p and q, it is always possible to specify CD and DE such that (5) (This holds true, of course, regardless of whether AD and DB are commensurable or incommensurable, or whether AD>DB or AD=DB or AD< DB.) Now, since CB>DB and given the lemma in (4), there is, then, a whole number n such that (6) (7a) which is the case Archimedes considers, (7b) Figure 9.2 Archimedes, De lineis spiralibus dem. 1 if one assumes also that n is the least number to satisfy (6). After stipulating (1), (2), and (3), Archimedes draws attention to the fact that the point moves at a constant speed along AB. Obviously, he says, this point will then traverse each of the segments of AD that is equal to CD in the same time that it takes to traverse CD. Thus, given (2), he infers that LG is the time-interval in which the point traverses AD; and, similarly, given (3), that GK is the time-interval in which the point traverses DB. Accordingly, he maintains, since AD is greater than DB, the point will take more time to traverse AD than DB; that is, (8) Likewise, he says by way of generalization, if one takes any multiple of FG and any multiple of GH so that one of the resultant time-intervals exceeds the other, it will be proven that the line-segment corresponding to the greater time-interval will be greater, because these line-segments are to be produced by taking the corresponding multiples of CD and DE. In other words, (9a) or (9b) where r and s are whole numbers. Finally, Archimedes concludes that (10) This conclusion that CD:DE :: FG:GH rests on an unstated condition for asserting that magnitudes are in the same ratio, a condition of the sort given by Euclid in Elementa v. def. 5: magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples of the second and fourth, the former multiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. (Heath 1956, ii 114: cf. 120–6) Thus, (10) specifically requires (9a) and (9b) as well as that (9c) all which follow readily from the basic fact that the motion is constant. A striking feature of the De lineis spiralibus is that Archimedes nowhere gives an explicit mathematical or quantitative definition of constant speed. The locutions he uses to express this concept suggest that he is starting instead from the qualitative notion that a body moves at the same or constant speed (isotacheôs) if it changes place at the same speed as itself (Heiberg 1910–23, ii 12.13–14: cf. 8.21, 44.21–2), that is, if it is unchanging in its swiftness or speed (tachos). This point will, of course, be lost if one insists on modern convention and supposes that, for Archimedes too, the speed of a body is the quotient of the distance it travels divided by the time taken to travel that distance. But this is not, in fact, how Archimedes presents speed: for instance, he characterizes sameness of speed not as an equality of quotients obtained when distances are divided by time-intervals, but by identifying the ratio of line-segment to line-segment and the corresponding ratio of time-interval to time-interval. This may, admittedly, be an artifact of the ‘rules’ of mathematical exposition during his time, in particular, of the formal condition that ratios be defined only between magnitudes of the same kind (cf. Euclid, Elementa v defs. 3–4); and so it may not be a sure guide to the way Archimedes actually conceived speed. (In the next section, I consider a text from the first century AD in which quotients of unlike quantities are in fact computed, though it is still not said that these quantities stand in a ratio to one another.) Accordingly, let us leave open the question of how Archimedes thinks of speed or swiftness and concentrate instead on how he expresses it. And, on this count, I find in dem. 1 and the rest of the De lineis spiralibus that he talks of speed as a quality of bodies that is quantifiable only in relation to other instances of this quality; and, moreover, that constancy of speed is to be understood as the sameness of this quality over time in a given body. The next question, however, is whether such talk is supplanted by a quantitative definition in dem. 1. That is, does Archimedes, as some suppose, posit that traversing line-segments in equal times is just what motion at a constant speed is; or does he infer that a point will traverse equal line-segments in equal times from the fact that it moves with a constant speed (cf. Dijksterhuis 1987, 140–1)? The critical passage now, since it is posited that the point moves at a constant speed along the line AB, it is clear that it travels CD in the same amount of time as it also traverses each of the segments equal to CD. (Heiberg 1910–23, ii 12.30–14.4) is, regrettably, not decisive. Nevertheless, there is, I think, compelling reason to maintain that Archimedes does not in fact identify motion at a constant speed with traversing equal segments of a straight line in equal times. For, as he is well aware, in the course of each revolution, though the generating point of the spiral always describes equal angles in equal times about the spiral’s origin, that is, about the fixed extremity of the generating line, and though it always traverses equal segments of the generating line in the same equal times as well, this same point traces out arcs of the spiral itself that are not equal to another (cf., for example, dem. 12). In other words, the very construction of the Archimedean spiral entails that the generating line (and, hence, any point on it) will by virtue of its constant revolution define equal angles in equal times about the fixed extremity; and that the generating point will by virtue of its constant motion on the generating line traverse equal segments of this line in equal times. Yet, the combined motion of the generating point and line has the result as well that this point will not describe equal arcs of the spiral in equal times. Thus, from the vantage point of Archimedes’ De lineis spiralibus, the qualitative idea of motion at a constant speed has to be more fundamental than the quantitative ideas of traversing equal segments of a straight line or equal angles of a circle in equal times. In fact, since these are the relevant ways of quantifying the motion of the generating point, and since they are not equivalent here, it would be a serious blunder to open a treatise on spirals with a demonstration presupposing that motion at a constant speed is to be defined simply as traversing equal segments of a straight line in equal times. Let us now consider briefly the preface to Autolycus’ De sphaera quae movetur. Autolycus begins by declaring that a point is said to move smoothly (homalôs) when it traverses equal or similar magnitudes in equal time-intervals. If a point moving smoothly along some line6 traverses two segments, the ratio of the time-intervals in which the point traverses the corresponding segments and the ratio of the segments will be the same. (Mogenet 1950, 195.3–8) The first sentence gives clear indication that smooth motion has been defined mathematically in terms of line-segments and time-intervals, albeit not as a quotient. In short, though this treatise and the De lineis spiralibus agree that motion can be characterized quantitatively, Autolycus’ treatise alone stipulates that smooth motion is just traversing equal line-segments in equal times. Indeed, that Autolycus calls the point’s motion smooth (homalês) rather than constant (isotachês) may signify that this definition was seen to obviate any need to present such motion in terms of a point’s moving at the same speed as itself. Yet, while Autolycus explicitly defines (or reports a definition of) smooth motion, he simply states the theorem about the proportionality of time-intervals and line-segments. That is, he does not offer a proof covering the case of straight lines (such as Archimedes does in De lineis spiralibus dem. 1) or the case of circular arcs on a sphere, the latter of which is crucial for his treatise. Two points emerge from this. First is that our understanding of the history of the exact sciences can only advance if proper attention is given to the language in which it is expressed. For example, any interpretation that renders both Archimedes’ ‘at a constant speed’ and Autolycus’ ‘smoothly’ by ‘uniformly’ will obliterate the complexity in the conceptual and linguistic apparatus that underlies the difference in their terminology. Indeed, to see that the very idea of ‘uniform’ motion is itself problematic in ancient texts, the reader should consult Aristotle, Physics 228b1–30. Second is that, so far as I can tell given the documentary evidence available, Archimedes was first to appreciate the complexity of the ‘equal segments of a straight line in equal times’, the ‘equal arcs in equal times’ and the ‘equal angles in equal times’ formulae in curvilinear motion and to ground them all in the qualitative idea that a body moving at a constant speed changes place at the same speed as itself. Was Autolycus, then, the first to realize that, in the special case of a point’s circular motion at a constant speed, the first formula was irrelevant, that the latter two formulae were equivalent, and that such motion (here called smooth) could be defined in terms of either? This question is difficult to answer. Perhaps he was, but the same ideas figure in the Phaenomena attributed to Euclid. Now, this treatise itself can be dated only to the period from the third to the first centuries BC.<sup>7</sup> Thus, to affirm priority for Autolycus in the mathematical definition of constant circular motion would require knowing his dates relative to Euclid’s8 and whether Euclid actually wrote the Phaenomena,<sup>9</sup> since the case for assigning Autolycus to the period from 360 to 290 BC (Aujac 1979, 8–10; cf., for example, Mogenet 1950, 5–7, 8–9) is nugatory.<sup>10</sup> Apart from these concerns about the history of the idea of constant motion, it is important to realize that Archimedes’ very inclusion of motion of any sort in the definition of his spiral is also remarkable. In the works of Euclid, for example, motion is limited to the construction of figures defined statically (cf. for example, Elementa i defs. 15–22, dem. 46) and to serving as a hidden assumption in proofs of such relations among figures as congruence (cf. Euclid, Elementa, i not. com. 4, with Heath 1956, i 224– 31). Now, a common way of interpreting this contrast is to suppose that Euclid belongs to a stage in the history of Greek mathematics earlier than Archimedes. The case offered thus far for this view, however, is unavailing, resting as it does on no more than an ancient inference concerning an anecdote told also of Menaechmus, a mathematician of the fourth century BC, and Alexander the Great, as well as on two suspect citations in Archimedes’ De sphaera et cylindro (see Bowen and Goldstein 1991, 246 n30). But, if Euclid’s work is not demonstrably earlier than Archimedes’, should one continue to view it as earlier in substance or form? This too is a difficult question, in part because it requires what has yet to be undertaken in any serious way, a critical study of the ancient testimonia about Euclid and the early history of Greek mathematics. In such a study of Proclus’ reports, for example, the alternatives against which the claims are judged will have to be founded on more than the simple-minded dichotomy that Proclus is either lying or telling the truth. Indeed, it will have to be rooted in a full examination of Proclus’ historiography, an examination informed by awareness of the numerous ways in which the ancients use history to make their cases and persuade their contemporaries. And should it turn out that Euclid’s work draws on and even recasts earlier mathematical theory, it will still be valuable to discover its intellectual and cultural context, as this context was defined in the third century when Archimedes was active. This will, of course, require paying attention to philosophical, technical, and social issues bearing on the understanding and treatment of motion that have too long been ignored in the scholarly haste to locate Euclid in relation to Aristotle and the Academy. THE ARITHMETICAL ANALYSIS OF LUNAR MOTION: GEMINUS The Introductio astronomiae by Geminus dates from the century or so prior to Ptolemy (c.100–c.170 AD; cf. Toomer 1978, 186–7).<sup>11</sup> It is, accordingly, one of a number of valuable witnesses to the character of the astronomical theory which Ptolemy inherited and transformed. Of particular interest is Geminus’ account of lunar motion in chapter 18. For it is here that Geminus not only shows some awareness of Babylonian astronomy, he undertakes to state its rationale. Granted, his account is historically incorrect, as we now know (cf. Neugebauer 1975, 586–7). But to focus on this is to miss the fundamental point that this chapter is the earliest Greco-Latin text available today that tries to explain the structure and derivation of a common Babylonian arithmetical scheme for determining the daily progress of a planet, the Moon.<sup>12</sup> So, let us turn to his account and examine it in detail. Chapter 18 (18.1–19) begins by introducing the exeligmos, which Geminus describes as the least period containing a whole number of days, months, and lunar returns.<sup>13</sup> By ‘month’ Geminus understands a synodic month, that is, the period from one coincidence of the Sun and Moon at the same degree of longitude on the ecliptic (conjunction) to the next, or from one full Moon to the next. As for ‘lunar return’, Geminus explains that the Moon is observed traversing the ecliptic unsmoothly (anômalôs) in the sense that the arcs of the ecliptic which it travels increase day by day from a minimum to a maximum and then decrease from this maximum to the minimum. Thus, a lunar return—nowadays called an anomalistic month —is the period from one least daily lunar motion or displacement (kinêsis) to the next. After claiming that, according to observation, (1) and (2) Geminus remarks that the problem was to find the least period containing a whole number of days, months, and lunar returns, that is, to discover the exeligmos. This period, he says (Introductio astronomiae 18.3; cf. 18.6), has been observed to comprise 669 (synodic) months, or 19,765 days, (3) in which there are 717 lunar returns (anomalistic months), or 723 zodiacal revolutions plus 32° by the Moon. According to Geminus, since these phenomena, ‘which have been investigated from ancient times’ are known, it remains to determine what he calls the Moon’s daily unsmoothness (anômalia) in longitude. Specifically, he continues, this means finding out what is its minimum, its mean (mesê) and its maximum daily displacement, as well as the daily increment by which this displacement changes, taking into account the additional observational datum that (4) where m is the minimum daily displacement and M the maximum daily displacement. From (3), Geminus reckons that the Moon’s (5) a value which he suggests the Chaldaeans discovered in this way, and that (6) My insertion of ‘periodic’ in parentheses in (5) is for the reader’s benefit, because throughout this chapter Geminus writes of two different and independent sorts of mean motion or mean daily displacement without making any terminological distinction. Thus far, he has computed the Moon’s mean motion by taking the periodic relation stated in (3), converting the number of sidereal cycles to degrees, and dividing the resultant number of degrees by the number of days. As for the computation in (5) itself, it actually yields 13; 10, 34, 51, 55… ° as the value for the periodic mean daily displacement of the Moon; but Geminus’ 13; 10, 35 may be excused as a rounding (cf. Aujac 1975, 95 n1). More puzzling, however, is the computation of the length of the anomalistic month in (6), since (6a) which differs somewhat from Geminus’ 27; 33, 20 days. (The difference amounts to 1; 20 days in one exeligmos.) One possibility is that Geminus has wrongly taken it for granted that his computation of the length of the anomalistic month in (6) yields the value stated in (2), namely, . Another possibility is that Geminus is here ‘telescoping two different (Babylonian) methods into one’ (Neugebauer 1969, 185: cf. 162). For, if one follows Neugebauer (1975, 586) and focuses only on the parameters of Geminus’ account, it seems that Geminus is drawing on two different Babylonian text-traditions, namely, on texts from Uruk presenting a scheme in which the lunar displacement is 13; 10, 35o/d and the length of the anomalistic month is 27; 33, 20d, and on Babylonian Saros-texts, the Saros being a cycle in which the length of the anomalistic month is 27; 33, 13, 18, 19…days. A third possibility, and perhaps the most charitable, is that Geminus actually understands the number of anomalistic months in the exeligmos to be derived from the length of the anomalistic month by computing (6b) Next, Geminus divides the anomalistic month of 27; 33, 20 days into four equal subintervals such that (7) where, for example, I(m, μ) is the interval from the day of minimum lunar displacement to the day of (arithmetic) mean lunar displacement (μ). Then, he argues, since the Moon’s (8) (9) This argument introduces a second type of mean motion. For here, Geminus presents μ as what I propose to call an arithmetic mean daily displacement, that is, the simple average of two extreme values for daily lunar displacement in longitude. Now, according to Geminus, the sum of the maximum and minimum daily displacements is known from observation to be only 26°; the fractional part, 0; 21, 10°, apparently escapes observation by instruments. This means, he says, that one has to assign 0; 21, 10° to M and m in a way that meets three conditions (see (4), (9)): (10) To do this, Geminus first reiterates that in each of the four subintervals of the anomalistic month (see (7)), the daily difference (d)—which is either incremental or decremental—is the same; this means, he remarks, that one has to find d such that (11) (see (9)), where k is the Moon’s total displacement in longitude in 1/4anomalistic month. The value for the daily difference, he flatly declares in conclusion, is 0; 18°. For, (12) This declaration of the scheme’s basic parameters is, however, a non sequitur. Geminus does not supply enough information to deduce the value for the daily difference in the Moon’s longitudinal motion. In fact, what he gives suffices only to specify a range of values for d. To see this, consider the values for d when m(=μ−k) and M(=μ+k) take on the extreme values of the range of possible values indicated in (11). Suppose, for instance, that From (8) and (11) it would follow that Similarly, if then Accordingly, given (4), (13) Likewise, if it would follow from (8) and (11) that Again, given (4), (14) Therefore, from (13) and (14), it follows that (15) Obviously, one could select a value for d by rounding the lower bound in (15) upwards to 0; 16o/d or by truncating the upper bound to 0; 18o/d (cf. Neugebauer 1975, 587). It is, of course, not possible to decide in light of the text alone whether Geminus’ claim that d is 0; 18o/d was reached by truncation. Indeed, one should not discount the possibility that the value Geminus assigns d was given at the outset or entailed in the information he had. Chapter 18 is the earliest text extant in Greek or Latin to present an account of an arithmetical scheme of a type now associated with the Babylonians. Since Geminus mentions the Chaldaeans, and since he ascribes this account to no one else, it would seem that he is in fact reconstructing what he takes to be the theory underlying information that ultimately came to him from Mesopotamia. So one may reasonably ask, what did he actually have? Presumably, he had access either to tabular data itself, to a set of procedures for entering the data, or to an account of how this data was organized data. Unfortunately, there is no way to determine which was the case. Still, it is true that in chapter 18 Geminus describes the arithmetical principles and parameters underlying tables for daily lunar motion, of a sort we now have from Uruk (Neugebauer 1969, 161–2; 1975, 480–1). In modern terms, these ephemerides are said to be structured according to a linear zigzag function (see Figure 9.3) in which (16) and, thus, (17) At the same time, Geminus’ account of the exeligmos derives from a Babylonian eclipse-cycle now called the Saros (cf. Neugebauer 1969, 141– 2). According to the Saros-cycle,<sup>16</sup> in a period of 223 synodic months, as the New or Full Moon returns 242 times to the same position relative to the same node, the Moon completes 239 cycles of its unsmooth motion in longitude, and travels through the zodiac 241 times and 10; 30° (see Britton and Walker 1996, 52–4). In other words, 223 synodic months = 239 anomalistic months=242 draconitic months<sup>17</sup> = 241 zodiacal revolutions by the Moon and 10; 30°<sup>18</sup> = 6585; 20 This cycle was certainly known in some form to Greco-Latin writers in Geminus’ time. Pliny (Historia naturalis ii 56), for example, affirms that eclipses recur in cycles of 223 months.<sup>19</sup> The Introductio astronomiae, however, would seem to be the oldest surviving Greek or Latin text to introduce the exeligmos, an eclipse-cycle three times as long as the Saroscycle, albeit without giving any indication of its purpose or its essential structure.<sup>20</sup> Geminus does not, for instance, connect the exeligmos with eclipses explicitly, and he does not mention the critical correlation of 669 (=3•223) synodic months with 726 (=3•242) draconitic months. Indeed, for a full statement of the exeligmos by a Greco-Latin writer, one must turn to Ptolemy, Almagest iv 2.<sup>21</sup> Geminus is silent about the relation between his exeligmos and the Babylonian Saros. Now, it is possible that this is due to his ignorance of the fact that Saros is an eclipse-cycle and that the exeligmos is a longer version of the Saros. Yet, at the same time, it is also possible that he has suppressed this information in order to present the exeligmos as just another calendrical cycle of the sort he describes in Introductio astronomiae ch. 8. So, his silence permits no conclusions about the condition and form of the data that he reconstructs in chapter 18. Still, it is clear that, at the very least, he had Babylonian values for the Moon’s mean daily displacement in longitude (13; 35, 10°), the daily difference in the Moon’s displacement in longitude (0; 18°), and the mean anomalistic month (27; 33, 20d), as well as the equation, 19756d = 669 (synodic) months = 717 anomalistic months = 723 revolutions by the Moon+32°. Though Geminus is right that the Babylonians had long ago identified the exeligmos, his claim about how they did it is unwarranted and implausible. Indeed, when one considers Babylonian lunar ephemerides of the sort that lie behind his account (cf. Neugebauer 1955, nos 190–6), it is difficult not to conclude that he was either unfamiliar with them or that he failed to realize that their schematic character makes it virtually impossible to determine their observational basis. Figure 9.3 A linear zigzag function for lunar motion in longitude Nevertheless, on its own terms, Geminus’ account in chapter 18 of the exeligmos and of lunar motion in longitude is noteworthy, in the first place, because he seeks to derive the scheme by which the data in these ephemerides are organized from a few parameters. Granted, this derivation does not come with an epoch or starting-point for the anomalistic month: Geminus neither gives such a date nor indicates how to determine one. Thus, he does not recognize, or allow for, any interest there might be in actually determining the Moon’s position in longitude at a given time. Next, Geminus’ account is also notable because he identifies fundamental parameters as observational data. Admittedly, this is scarcely credible even on Geminus’ own terms, if, as he reports (Introductio astronomiae 18.14), the best observation can do (with the aid of instruments) is to determine the sum of M and m to the nearest degree, a remark which is at odds with his claim that the values for the synodic and anomalistic months reported in (1) and (2) have been observed. And, as I have said, so far as history is concerned, though there is certainly some observational basis to the Babylonian Saros-texts and to the lunar tables from Uruk, there is no warrant for supposing that it consisted in observing the fundamental parameters of the arithmetical schemes structuring these tables. Still, Geminus’ assumption that these basic parameters were observed is important as an indication of how he understands astronomy and its use of mathematics. For Geminus, apparently, his arithmetical scheme actually describes the Moon’s unsmooth motion in longitude, and the accuracy of this description is guaranteed by the fact that it derives from arithmetical manipulation of observed parameters. Regrettably, he leaves unanswered pertinent questions about what counts as an observation, how observations are made, and so on. Moreover, within the context of the Introductio astronomiae, Geminus’ arithmetical account of lunar motion in longitude is also remarkable for two reasons. First is that it contrasts sharply with the rest of the treatise. Only in chapter 18 (and chapter 8, which concerns calendrical cycles) does Geminus introduce quantitative argument. Elsewhere, his remarks are qualitative and geometrical. Thus, in his account of the Sun’s unsmooth motion in longitude (Introductio astronomiae 1.18–41), for example, though he supposes that it is only apparent because the Sun moves at a constant speed on a circle eccentric to the Earth, Geminus does not use his values for the lengths of the seasons (cf. 1.13–17) to specify the eccentricity of this circle and so on (cf. Neugebauer 1975, 581–4). Second, and more striking, is that Geminus’ account of lunar motion in chapter 18 is at odds with principles laid down earlier in the treatise. For, as he writes: It is posited for astronomy as a whole that the Sun, Moon, and five planets move at a constant speed [isotachôs], in a circle, and in a direction opposite to [the daily rotation of] the cosmos. For the Pythagoreans, who first came to investigations of this sort, posited that the motions of the Sun, Moon, and five planets were circular and smooth [homalas]. Regarding things that are divine and eternal they did not admit disorder of the sort that sometimes [these things] move more quickly, sometimes more slowly, and sometimes they stand still (which they call stations in the case of the five planets). One would not even admit this sort of unsmoothness [anômalian] of motion regarding a man who is ordered and fixed in his movements. For the needs of life are often causes of slowness and speed for men; but as for the imperishable nature of the celestial bodies, it is impossible that any cause of speed and slowness be introduced. For which reason they have proposed [the question] thus: How can one explain the phenomena by means of circular, smooth motions? Accordingly, we will give an explanation concerning the other celestial bodies elsewhere; but just now we will show concerning the Sun why, though it moves at a constant speed, it traverses equal arcs in unequal times. (Introductio astronomiae 1.19–22) This means that the arithmetical scheme presented in chapter 18 does not describe the Moon’s real motion in longitude; at best, it can represent the Moon’s apparent motion—assuming, for the moment, with Geminus that the daily variations in the Moon’s longitudinal displacement are indeed observable. But, if so, Geminus has yet to supply the account of the Moon’s real motion in longitude that he has promised. Such an explanation would, of course, have to overcome a serious problem; namely, that there is no way, using resources presented in the treatise thus far, to construct a coherent argument that begins with the qualitative geometry of the Moon’s real motions and concludes with the arithmetical detail of his scheme for the Moon’s apparent motion. In short, by introducing the sort of arithmetical detail that he does in his account of the Moon’s ‘observed’ variable motion in longitude, Geminus undermines his ostensible project of explaining this motion in terms of the smooth circular motion(s) that it supposedly makes in reality. In effect, chapter 18 exposes a problem at the heart of Greco-Latin astronomy of the time that becomes evident once it attempts to incorporate in its explanatory structure arithmetical procedures and results from Babylonian astronomy. Geminus’ mean daily displacement can only be an apparent lunar motion in longitude and not one the Moon really makes, if the mean in question is arithmetic. If the mean is periodic, however, the Moon’s mean daily displacement can become a basis for specifying its true or real motion. But it would take Ptolemy to straighten out Geminus’ conflated notion of mean motion and its relation to real and apparent planetary motion. Indeed, part of Ptolemy’s genius lay in seeing that texts such as Geminus’ Introductio astronomiae were typical of what was wrong with the astronomy of his time; that, in assimilating Babylonian astronomy, earlier and contemporary Greco-Latin writers betrayed a confused, inconsistent, and insufficiently sophisticated grasp of the proper role of arithmetic, geometry, and observation in astronomical argument (see Bowen 1994). HEARING AND REASON IN HARMONIC SCIENCE: PTOLEMY<sup>22</sup> In the opening chapter to his great astronomical work, the Almagest, Ptolemy presents himself as a philosopher. What this actually means to Ptolemy is a question that involves understanding not only his literary and scientific context but also how he appropriates and transforms this context in his own highly technical work.<sup>23</sup> Granted, there are scattered throughout Ptolemy’s treatises tantalizing passages in which he talks of method and indicates a conceptual framework in which the sciences discussed somehow fit. There is even a treatise, the De iudicandi facultate, in which Ptolemy sets out an epistemology that is intended to explain and justify what one finds in his scientific works (cf. Long 1988, 193–6, 202– 4). But research on these issues is still at a primitive stage primarily because scholars have yet to interpret this treatise and the related passages found in Ptolemy’s other works in the light of the technical, scientific matters which give them their real meaning.<sup>24</sup> Yet the promise of such research is great, since Ptolemy is a pivotal figure in the history of western science. Accordingly, in this final section, I will make a preliminary assault on the question of Ptolemy’s philosophical views by examining the first two chapters of the first book of his Harmonica (Düring 1930, 3.1–6.13) with occasional reference to the De iudicandi facultate.<sup>25</sup> In these chapters, Ptolemy focuses on the question of criteria in the domain of music and on the related matter of the goal of the harmonic theorist, though he does mention astronomy and astronomers as well. By Ptolemy’s time, argument about the criteria of truth was prominent in intellectual circles: in fact, by then, the problem was to explain the contributions of reason and the senses to knowledge of external objects, and to determine what infallible means there are for distinguishing particular truths about these objects from falsehoods (cf. Long 1988, 180, 192). But Ptolemy recasts the problem. To begin, he decides to ignore the technical vocabulary current among philosophers of his time in favor of a simpler vocabulary that suffices to aid non-experts and to clarify reflection on the realities signified (cf. De iudicandi facultate 4.2–6.3). Accordingly, he proposes to use ‘criterion’ (kritêrion) to designate (a) the object about which one makes judgments, (b) the means through which and the means by which judgments about such objects are made, (c) the agent of judgment, (d) the goal of the judgments made, as well as the more usual sense, (e) the standard(s) by which the truth of judgments is assessed (cf. Blumenthal 1989, 257–8). Thus, given that in his view truth is a criterion qua goal of judgment (cf. De iudicandi facultate 2.1–2), Ptolemy represents the general problem as one of discovering the criterion of what there is (cf. 1.1). In the context of harmonic science (harmonikê), this becomes the problem of determining the criteria of what there is in the domain of music, that is, the criteria of harmonia, where harmonia is ultimately tunefulness or the way pitches should or do fit together properly. Chapter 1 of the first book of the Harmonica opens with the assertion that Harmonic science is a capacity for apprehending intervals of high and low pitch in sounds,<sup>26</sup> while sound is a condition of air that is struck —the primary and most general feature of what is heard—and hearing and reason are criteria of tunefulness [harmonia] though not in the same way. (Düring 1930, 3.1–4) I take this to mean that harmonic science is a branch of knowledge by which one is able to account systematically for intervals among pitches and to determine their harmonia. Obtaining and exercising this knowledge, however, is to draw on two faculties, hearing and reason, that serve as criteria in different ways:<sup>27</sup> as Ptolemy says, ‘hearing is [a criterion] in relation to matter and experience (pathos); but reason [is a criterion] in relation to form and cause’.<sup>28</sup> To explain why hearing and reason are united in this way in developing or using harmonic science, Ptolemy first points out that even in general that which can discover what is similar [to suneggus] and admit precise detail [to akribes] from elsewhere is characteristic of the senses; while that which can admit from elsewhere what is similar and discover precise detail is characteristic of reason. (Düring 1930, 3.5–8) That is, he continues, since matter is defined and delimited only by form whereas experiences [are defined and delimited] by the causes of motions, and since [matter and experience] are proper to sense-perception but [form and cause] are proper to reason, it follows fittingly that our sensory apprehensions are defined and delimited by our rational apprehensions, in that, at least in the case of things known through sense-perception,<sup>29</sup> the sensory apprehensions first submit their rather crudely [holoscheresteron] grasped distinctions to the rational apprehensions and are guided by them to distinctions that are precisely detailed and coherent. (Düring 1930, 3.8–14) The key to understanding Ptolemy’s account thus far of the roles of reason and the sense of hearing in harmonic science is the distinction between to suneggus and to akribes. One possibility is that Ptolemy is concerned with truth, that he means to affirm the approximate character of perception and the accuracy of reason. Thus, as Barker translates the critical lines: …it is in general characteristic of the senses to discover what is approximate and to adopt from elsewhere what is accurate, and of reason to adopt from elsewhere what is approximate, and to discover what is accurate. (Barker 1991, 276) The problem here, in the first place, is that Ptolemy does not actually say that the senses characteristically discover to suneggus and so forth. What he maintains instead is that some thing, which is capable of discovering to suneggus and admitting to akribes from elsewhere, is characteristic of the senses. Likewise for reason, he does not say that it characteristically discovers to akribes and so forth, but that some thing, which is capable of discovering to akribes and admitting to suneggus from elsewhere, is characteristic of it. Now these items are, I submit, the perceptum and thought, respectively. Second, it is important to realize that Ptolemy is not here directly concerned with truth but with how information is transformed into knowledge. In short, as his talk of matter and form suggests, the contrast he has in mind is one between sensory information before and after it has been articulated as knowledge, and not one between the approximate and the accurate. Thus, I take Ptolemy’s point to be that the scientific analysis of pitch requires hearing to discern similarity and difference among pitches and intervals, and reason to articulate this similarity or difference by quantifying it numerically according to a theoretical system. This process of informing or articulating and thereby appropriating what hearing discerns into a system of knowledge involves introducing precision or numerical detail. Thus, for Ptolemy, what hearing grasps is rather crude (holoscheresteron), either because it is not numerically quantified at all or because it is quantified in a way not involving theory (as when someone hears an interval and simply says that it is a fifth, for instance). Thus, what is holoscheresteron is rather crude because it lacks the sort of precise detail it must have to be scientific knowledge, which does not mean of itself that it cannot be exact or accurate.<sup>30</sup> Reason and the senses are criteria of science in the ways they are because, as Ptolemy says, it happens that reason is simple, unmixed, and, thus, complete in itself, fixed, and always the same in relation to the same things; but that sense-perception is always involved with matter which is confused and in flux. Consequently, because of matter’s instability, neither the sense-perception of all people nor even that of the same people is ever observed to be the same in relation to objects similarly disposed, but needs the further instruction of reason as a kind of cane. (During 1930, 3.14–20; cf. De iudicandi facultate 8.3–5, 9.6). In other words, assuming the principle that cognitive faculties are like their objects, reason alone is fit for articulating consistently what is grasped by the senses: the senses themselves cannot do this. In saying this, Ptolemy has raised the related issues of disagreement among listeners and error. If hearing does not on every occasion discern the same distinction in what is heard though the circumstances are such that it should, it is important to discover whether reason may ever rely on hearing and in what way, if one is to account fully for the roles of reason and hearing in harmonic science. The question is, then: does hearing ever discern similarities and differences among pitches correctly and, if so, whose hearing is it? Ptolemy answers by pointing out first that hearing may be brought to recognize its errors by reason.<sup>31</sup> So, under some circumstances at least, it is possible for hearing to discern things accurately. Then, Ptolemy affirms the stronger thesis that sometimes what hearing presents to reason does not need any correction at all. Let us consider these claims in turn. Ptolemy maintains that reason can bring hearing to a knowledge of errors in its apprehensions, by way of an analogy: So, just as the circle drawn by the eye alone often seems to be accurate until the circle made by reason brings [the eye] to the recognition of one that is in reality accurate [akribôs echein], thus when some definite interval between sounds is taken by hearing alone, it will initially seem sometimes neither to fall short nor to exceed what is appropriate, but is often exposed as not being so when the interval selected according to proper ratio is compared, since hearing recognizes by the juxtaposition the more accurate [one] as something genuine, as it were, beside that counterfeit. (Düring 1930, 3.20–4.7) Evidently, hearing is corrigible if reason, on the strength of theory, produces in sound what is correct (i.e., an interval defined by the proper ratio) so that hearing may apprehend it and thereby come to discern error.<sup>32</sup> Obviously, reason will be obliged to be equip itself with an instrument that it can employ in a way consistent with theory in order to produce the correct sounds—a point Ptolemy makes explicit later. In any case, Ptolemy clearly holds that both reason and hearing can detect errors in the apprehensions of hearing. But, though this entails that the apprehensions of hearing are sometimes accurate, it does not yet follow that reason may rely on hearing for information about differences among sounds. The analogy illustrating how reason can bring the eye and hearing to discern error when none was recognized previously also suggests that the senses are better as judges than as producers of percepta. But to establish that hearing may apprehend distinctions correctly unaided by reason, Ptolemy must evaluate the capacity of the senses to make distinctions on their own. He begins by affirming that it is in general easier to judge something than it is to do it. His elaboration of this premiss makes it clear that hearing will have better results in recognizing that an interval or melody is out of tune than when it guides the production of the interval or melody by means of some instrument such as the voice or aulos. Indeed, he explains, this sort of deficiency of our sense-perceptions does not miss the truth by much in the case of [our] recognizing whether there is a simple difference between them [sc. our sense-perceptions] nor, again, in the case of [our] observing the excesses of things that differ, at least when [the excesses] are taken in greater parts of the things to which they belong.<sup>33</sup> (Düring 1930, 4.10–13) The locution here may strike the modern reader as odd. The deficiency in question is, I think, the deficiency of the senses in apprehending what is in reality accurate, which he has just described. Now, as I understand it, the claim that this deficiency ‘does not miss the truth by much’ is a figure of speech: Ptolemy actually means that, when the senses discern a mere difference or report the amount of this difference (providing that the amount is suitably large), they do not under these circumstances miss the truth at all.<sup>34</sup> In other words, I maintain that, for Ptolemy, the apprehensions of hearing are in fact correct and accurate, when hearing attends to the mere occurrence of an interval between sounds or when it reports the amount of this interval (if the amount is large enough). It is important that the amounts of the differences between the sounds be large in comparison to the sounds, that is, for example, that the difference between two intervals be large in comparison to the two intervals. As Ptolemy says of the senses in general, if the amounts of the differences they apprehend are a relatively small part of the things exhibiting them, the senses may not discern any difference at all; yet, when such apprehensions are iterated, the error or difference accumulates and eventually becomes perceptible. The upshot is that Ptolemy assigns the senses a well-circumscribed reliability: sensory apprehensions of sameness and difference are for the most part deemed unreliable. Only in apprehending the fact of difference or the amount of this difference (when the amount is suitably large) do the senses such as hearing provide a reliable and fitting empirical basis for science (cf. De iudicandi facultate 12.4),<sup>35</sup> The question of whose hearing it is still remains, and Ptolemy approaches it by considering the class of those instances when hearing by nature goes astray. After all, what hearing apprehends or reports can become scientific only when integrated by reason in an explanatory system;<sup>36</sup> and this means that reason will often have to deal with error in what hearing reports. As he says, just as for the eyes there is a need for some rational criterion through appropriate instruments—for example, for the ruler in relation to straightness and for the pair of compasses in relation to the circle and the measurements of parts—in the same way as well there must be for the ears, which are with the eyes especially servants of the theoretical or reason bearing part of the soul, some procedure [ephodos] from reason for things which [the ears] do not by nature judge accurately, a procedure against which they will not testify but will agree that it is correct. (Düring 1930, 5.3–10) Ptolemy begins chapter 2 by identifying the instrument for correcting aural apprehensions as the harmonic canon or ruler (kanôn), adding that the name is taken from common usage and from its straightening (kanonizein) things in the senses that fall short regarding truth (cf. Düring 1930, 5.11– 13). But what is this rational criterion of harmonic science, the third criterion that Ptolemy has designated as such thus far in the opening chapters of the Harmonica? According to Ptolemy, it should be the goal of the harmonic theorist to preserve in every way the rational hypotheses [hupotheseis]<sup>37</sup> of the canonas never conflicting in any way with the senses in the judgment of most people, just as it should be the goal of the astronomer to preserve the hypotheses of the celestial motions as in agreement with their observed periods, hypotheses that while they have themselves been taken from obvious and rather crude [holoscheresteron] phenomena, find things in detail accurately through reason so far as it is possible. For in all things it is characteristic of the theorist or scientist to display the works of nature as crafted with a certain reason and fixed cause, and [to display] nothing as produced [by nature] without a purpose or by chance especially in its so very beautiful constructions, which sorts of things the [constructions] of the more rational senses, seeing and hearing, are.<sup>38</sup> (Düring 1930, 5.13–24) This third criterion, to which reason may appeal in distinguishing truth from falsehood in musical sound and on which it may rely, turns out, in fact, to be the consensus of the majority about what is heard when the canon is properly set up according to theory and actually struck.<sup>39</sup> For this criterion entails that, on such occasions, the standard of accuracy in determining not only the fact of differences among sounds but also, under certain circumstances, how great these differences are, is what most people hear. In sum, the hearing that stands as the reliable counterpart of, and standard for, reason in harmonic science is that of the majority.<sup>40</sup> In the remainder of chapter 2, Ptolemy explains how rival schools fail to pursue this basic goal of the harmonic theorist (cf. Bowen and Bowen 1997, 111–12). But rather than pursue this, by way of conclusion I will now briefly address the question ‘What does the harmonic theorist actually know?’ In the first place, the harmonic theorist understands harmonia, that is, the organization of differences in pitch. To say more than this, however, it is necessary to discover just what it is that the majority reports about such differences. In particular, one should at least ask whether the consensus of the majority concerns a subjective experience or the objects underlying this experience. Now, Ptolemy’s answer to this question comes in the next chapters (Harmonica i 3–4), which discuss (a) the causes of high and low pitch in sound (psophos), and (b) musical notes (phthoggoi) and their differences. But this very distinction between sounds and musical notes suggests another feature of harmonia that one must not neglect, namely, that harmonia is fundamentally an aesthetic phenomenon, that the differences in pitch have an intrinsically aesthetic character. That is, implicit in Ptolemy’s account of the third criterion is the view that harmonia is ultimately defined by the musical sensibilities or tastes of a community—no matter whether one assumes (as I do) that the phrase kata tên tôn pleistôn hupolêpsin means ‘in the opinion of most people’ or that it means ‘in the opinion of most experts’:<sup>41</sup> in either case, the harmonic theorist is to appeal to, and to rely on, a shared sense of differences in pitch and their melodic propriety. It would seem, then, that in answer to the question ‘What does the harmonic theorist know?’, one might point out that harmonic science articulates systematically by means of number a communal sense of musical propriety. But, if so, does this science change over time? There is, after all, a tension in Ptolemy’s account between reason and what most people hear, and I suspect that it is essential to his understanding of harmonic science itself: such tension is certainly built into his third criterion to the extent that agreement or consent is an issue. One way to cast the problem is to ask, does hearing ever correct or bear witness against theory? Obviously, it must as the theoretical account of the music characteristic of a culture becomes more scientific and accurate, a possibility implicit in Ptolemy’s criticism of contemporary and earlier theorists at the close of chapter 2, for instance, and elsewhere. But does theory ever have to adapt to changes in what most people hear when the canon is set up according to theory and struck? This is a question to bring to a careful reading of the Harmonica. For if Ptolemy denies that musical sensibility changes over time, harmonic science has a perfection it can reach in articulating the sense of musical propriety shared by most people. But, if he allows that it does change, then harmonic science must too and so it cannot have a final form. In this case, then, what most people hear when the canon is set up according to theory and then struck will serve not only to confirm theory and to correct practice, it will on occasion serve also to confirm practice and to correct theory.<sup>42</sup> And if observation may take on such a role in harmonic science, may it do the same in astronomy?<sup>43</sup> CONCLUSION It is perhaps appropriate to finish with a question, since a series of casestudies will hardly generate global results. To philosophers it is often given that one may grasp the universal in the particular; but rarely is this granted to historians of ancient science. Thus, for now, I content myself with the more mundane hope that the preceding studies of particulars in detail will at least raise questions leading to other particulars in a fruitful way. NOTES 1 I take the exact sciences to include arithmetic, geometry and all those sciences involving arithmetic and geometry in a significant way (for example, astronomy, astrology, harmonics, mechanics, and optics). Isolating these sciences as a class is not a uniform characteristic of Greco-Latin thought. Still, it is a useful starting-point, particularly if one considers the various exact sciences throughout their histories and inquires of each whether it was in fact (always) viewed as scientific by the ancients, and to what extent the role of mathematics affected this decision. 2 When writing of Greek and Latin science, philosophy, and so on, I refer only to the respective languages in which the relevant texts are written. 3 See, for example, the critical studies by von Staden (1992) and Pingree (1992). 4 Of these, Lloyd 1984 is a useful and instructive contribution. 5 Following Dijksterhuis 1987, 147–9. According to Dijksterhuis, though this lemma bears an obvious formal resemblance to Euclid, Elementa v def. 4 (which posits that, if a and b are magnitudes and a<b, there is a whole number n such that n•a>b), it is essential to Archimedes’ indirect calculations of magnitudes by means of infinite processes (cf. Dijksterhuis 1987, 130–3), because, in so far as it entails that the difference between two magnitudes is a magnitude of the same kind, it excludes the possibility of infinitesimals such as one would admit if the difference between two lines, say, were a point. 6 In the present context, the lines will be circular arcs on a sphere. Such arcs may be equal or similar: they are similar if they lie on parallel circles and are cut off by the same great circles (cf. Aujac 1979, 41 nn2–3). 7 The use of the names of the zodiacal constellations to designate the twelve equal arcs of the ecliptic in the Phaenomena would seem to place it after the fourth century and perhaps in the third (cf. Bowen and Goldstein 1991, 246– 8). But note, however, that according to Berggren and Thomas (1992: cf. Berggren 1991), the aim of this treatise is to account qualitatively for the annual variations in the length of daytime, a concern characteristic of the second and first centuries BC. (HypsiclesAnaphoricus, a treatise presenting a Babylonian arithmetical scheme for determining the length of daytime throughout the year, is commonly thought to belong to the second century BC.) 8 It is difficult to determine the relative dates of Autolycus and Euclid. The argument from evidence internal to their treatises (cf. Heath 1921, i 348–53) that Autolycus is prior to Euclid is, as Neugebauer (1975, 750) points out, ‘singularly naive’: there is no reason to dismiss the possibility that Autolycus and Euclid were contemporary. 9 See Bowen and Goldstein 1991, 246 n30. 10 See Bowen and Goldstein 1991, 246 1129. 11 Geminus’ dates are uncertain. Scholars have traditionally supposed that he was active in the first century BC; but Neugebauer (1975, 579–81) has argued for a date in the first half of the first century AD. 12 So far as I am aware, P.Hibeh 27 (third century BC) is the earliest Greek text which organizes information according to a (modified) Babylonian scheme of the sort which Geminus attempts to explain: cf. MUL.APIN 1.3.49–50, 2.2. 43–2.3.15; Bowen 1993, 140–1. 13 There are periods shorter than the one Geminus actually identifies as the exeligmos: see pp. 300–1 below, on the Saros. 14 Geminus represents numbers in two ways, either as whole numbers plus a sequence of unit-fractions (in decreasing order of size) or as sexagesimals. I will follow convention by writing unit-fractions by means of numerals with bars over them: thus , stands for 1/n and for 2/3 (cf. Neugebauer 1934, 111). Moreover, I shall use the semicolon to separate sexagesimal units and the sixtieths, and commas to separate sexagesimal places to the right of the semicolon. See Introductio astronomiae 18.8, for an explanation—Manitius views this as a marginal gloss that has been moved into the text—of Geminus’ nomenclature for sexagesimal fractions of a degree: first sixtieths are units of ; second sixtieths, units of •; and so on. 15 Since (4) rules out and , it follows that and 16 For texts and analysis, see Aaboe, Britton, et al. 1991. 17 The draconitic month is the period of the Moon’s return to the same node or point where its orbit crosses the plane of the ecliptic in the same direction. Determining the length of the draconitic month is useful in understanding eclipses, since they occur only when the Moon is at or near the nodes. 18 That is, 241 returns to the same star or sidereal months plus 10; 30°. The Sun completes 18 zodiacal revolutions (sidereal years) and 10; 30° in the same period. 19 Not all the manuscripts of Pliny, Historia naturalis ii 56 have 223 as the number of synodic months: cf. Mayhoff 1906, 144; Neugebauer 1969, 142. 20 In Geminus’ version of the exeligmos—as in Ptolemy’s (Almagest iv 2)—the Moon makes 723 zodiacal revolutions and then travels 32° farther, whereas, if one triples the Babylonian Saros-cycle, the Moon circles the zodiac 723 times but then travels only 31; 30° farther. 21 Ptolemy’s accounts of the shorter cycle (the Babylonian Saros) and of the exeligmos are consistent: he posits that in one Saros the Moon makes 241 zodiacal revolutions and 10; 40°,and that the Sun makes 18 such revolutions and 10; 40°. 22 I take the opportunity in what follows to revise and develop the analysis given in Bowen and Bowen 1997, 104–12. 23 See Grasshoff 1990, 198–216 and Taub 1993 for two recent attempts to discover Ptolemy’s philosophical views. 24 See Bowen 1994 on Taub 1993: cf. Lloyd 1994. 25 Barker’s translation (1989, 276–9) of Harmonica i 1–2 is helpful, albeit misleading in critical matters of philosophical and technical detail. 26 Cf. Gersh 1992, 149. See Bowen and Bowen 1997, 137 1122 for criticism of Solomon’s analysis (1990, 71–2) of Ptolemy’s definition of harmonic science. 27 Hearing and reason are both instrumental, though the mode of their instrumentality differs, as Ptolemy’s use of different instrumental constructions at De iudicandi facultate 1.5, 2.2–4 indicates: hearing, like any other sense, is ‘the means through which’ one makes judgments and reason is ‘the means by which’ one does this. Note that one of the basic meanings of ‘kritêrion’ is ‘instrument’: cf. De iudicandi facultate 2.3; Friedlein 1867, 352. 5–6. 28 In this analogy, matter is, I presume, to be taken in relation to form and pathos in relation to explanation. Barker renders pathos by ‘modification’; but it makes little sense to compare hearing to a modification (that is, to a change or enmattered form) in the present context. So, I propose instead to render pathos by ‘experience’: cf. Ptolemy, De iudicandi facultate 8.3, 10.1–3; Barker 1989, 280 n20. 29 Cf. Düring 1930 3.13: Barker (1989, 276) has ‘at least in the case of things that can be detected through sensation’. See Ptolemy, De iudicandi facultate 10.5 which allows that there are things known by reason without the aid of the senses. 30 This is consistent with Ptolemy’s usage in the Almagest (cf., for example, Heiberg 1898–1907, i 203.12–22, 270.1–9; ii 3.1–5, 18.1–5, 209.5–7). In most cases, the astronomical observations criticized by Ptolemy involve measurement; so number is already present in what the eyes report to reason: the problem is that the means by which these measurements were made is not known. One should compare Geminus’ use of akribes and holoscheresteron at, for instance, Manitius 1898, 100.16–20, 114.13–18, 116.20–3, 118.10–12, and 206.17–21. 31 At this point, when Ptolemy turns to the problem of error, the meaning of akribes changes from ‘detailed’ to ‘accurate’ or ‘true’. 32 The force of this analogy is not, as Barker (1989, 277 n9) supposes, that hearing alone can detect its unreliability (cf. De iudicandi facultate 8.3–5, 10. 1–3). Reason, for example, may well avail itself of visual information to determine (again on the basis of theory) that the sound produced is incorrect. In De iudicandi facultate 10.4–5, Ptolemy writes that on certain occasions reason may choose to correct sense-perception through the means of senseperceptions. Thus, if a sense is affected in a way inappropriate to the object sensed, reason may determine the error either through similar, unaffected or uncorrupted sense-perceptions when the cause of error involves the senseperceptions, or through dissimilar sense-perceptions of the same object when the cause does not involve them but something external. 33 On Solomon’s version (1990, 73–4) of these lines, see Bowen and Bowen 1997, 140 n33 and Barker 1989, 277. 34 Barker (1991, 118) does not recognize a rhetorical figure here, though the fact that hearing does sometimes disclose the truth is assumed in the next lines, when Ptolemy considers how imperceptible error in sense-perception may accumulate and eventually become perceptible. Cf., for example, Düring 1930, 23.19–24.8. 35 Long’s claim (1988, 193) that, for Ptolemy, ‘sense-perception is limited to the immediate experiences it undergoes and it cannot pass judgment on any external objects as such’, though it neglects kai epi poson apallagentôn at De iudicandi facultate 8.5, does draw attention to an important puzzle. According to 8.4, sense-perception judges only its experiences (pathê) and not the underlying objects. The same is said at 10.1–3 (cf. 11.1), except Ptolemy here remarks that sense-perception sometimes reports falsely about the underlying objects perceived. This latter claim makes sense, however, only if the senses may sometimes report truly about these objects as well. In any case, the question raised by Ptolemy’s argument thus far in the Harmonica becomes, ‘when hearing reports veridically about the occurrence of sounds and certain of their differences, does it simply report truly its experiences or does it somehow disclose true information about the physical state of affairs producing these experiences?’ At issue is Ptolemy’s idea of what sound and, in particular, musical sound, is (see p. 310). 36 In De iudicandi facultate 2.4–5, Ptolemy distinguishes phantasia, which is the impression and transmission to the intellect of information reached by contact through the sense-organs, and ennoia (conception), which is the possession and retention of these transmissions in memory. The conceptions are what may become scientific if integrated into theory (cf. 10.2–6). 37 In the Almagest, Ptolemy uses hupotheseis in reference to his planetary models: cf. Toomer 1984, 23–4. 38 Where I have ‘in its so very beautiful constructions, which sorts of things the [constructions] of the more rational senses, hearing and vision, are’, Barker (1989, 279) proposes ‘the kinds [sc. constructions] that belong to the more rational of the senses, sight and hearing’. It is not clear just what these constructions belonging to sight and hearing are supposed to be. They are most likely not the objects of these senses: Ptolemy’s meaning here is that the theoretician is duty-bound to maintain the reliability of sight and hearing, not the intelligibility of their objects. Cf. During 1934, 23. 39 Cf. Long 1988, 189. The claim made by Blumenthal, Long, et al. (1989, 217) that Ptolemy nowhere uses ‘kritêrion’ to signify a standard is apparently mistaken. If one recalls the various meanings of ‘criterion’ that Ptolemy lists at De iudicandi facultate 2.1–2, the argument of Harmonica i 1–2 would seem to indicate that: harmonia is a criterion qua object of judgment; hearing and reason are criteria qua means through which and means by which, respectively; the harmonic theorist is a criterion qua agent of judgment; what most people hear when the canon is set up according to theory and struck is a criterion qua standard; and preserving truth, that is, the concordance of the hypotheses of harmonic science with this standard, is a criterion qua goal. 40 Thus Ptolemy counters scepticism that harmonic science does in fact constitute knowledge, because of the acknowledged variations in hearing from person to person, and so on (cf. Düring 1930, 3.17–21). 41 There is, after all, no evidence in Ptolemy’s preceding remarks that he limits this criterion to experts or cognoscenti. 42 Until such matters are settled, I hesitate to follow Long (1988, 194) in inferring that the De iudicandi facultate presents science as ‘a stable and incontrovertible state of the intellect, consisting in self-evident and expert discrimination’, especially since there is, so far as I can tell, nothing in this treatise that favors this view and excludes the role of third criterion I have just indicated. 43 On progress in astronomy, see Almagest i 1 with Toomer 1984, 37 n11. At its most general, the question is, How does Ptolemy understand the Almagest and its place in the history of astronomy? BIBLIOGRAPHY PRIMARY LITERATURE Ancient documents In this subsection, translations are in English unless specified otherwise. Archimedes. De lineis spiralibus. See Heiberg 1910–23, ii (with Latin trans.); Mugler 1970–72, ii (with French trans.). ——De sphaera et cylindro. See Heiberg 1910–23, i (with Latin trans.). ——Quadratura parabolae. See Heiberg 1910–23, ii (with Latin trans.). Aristotle. Physica. See Ross 1955. Astronomical Cuneiform Texts. See Neugebauer 1955. Autolycus. De sphaera quae movetur. See Mogenet 1950, Aujac 1979 (with French trans.). Boethius. De institutione musica. See Friedlein 1867, 175–371 (with Latin trans.). Euclid. Elementa. See Heiberg and Stamatis 1969–77. Trans. by Heath 1956. ——Sectio canonis. See Menge 1916. Geminus. Introductio astronomiae. See Manitius 1898 (with German trans.), Aujac 1975 (with French trans.). Hypsicles. Anaphoricus. See de Falco and Krause 1966. MUL.APIN. See Hunger and Pingree 1989 (with trans.). P.Hibeh 27. See Grenfell and Hunt 1906, 138–57 (with trans.). Plato. Meno. See Burnet 1900–7, iii; Bluck 1964. Trans. by Grube 1967. Pliny. Historia naturalis ii. See Mayhoff 1906. Proclus. In primum Euclidis elementorum librum. See Friedlein 1873. Ptolemy. Almagest. See Heiberg 1898–1907, i–ii. Trans. by Toomer 1984. ——De iudicandi facultate et animi principatu. See Blumenthal, Long, et al. 1989 (with trans.). ——Harmonica. See Düring 1930. Trans. by Barker 1989, 270–391; Düring 1934 (German). For a Latin version of Harmonica i 1–2, see Boethius, De institutione musica v 2 with Bowen and Bowen 1997. Saros Texts. See Aaboe, Britton, et al. 1991 (with trans.). Modern editions and translations Aaboe, A., Britton, J.P., Henderson, J.A., Neugebauer, O., and Sachs, A.J. (1991) Saros Cycle Dates and Related Babylonian Astronomical Texts, Transactions of the American Philosophical Society 81.6, Philadelphia: American Philosophical Society. Aujac, G. (1975) ed. and trans. Géminos, Introduction aux phénomènes, Paris: Les Belles Lettres. ——(1979) ed. and trans. Autolycos de Pitane: La sphere en mouvement, Levers et couchers, testimonia, Paris: Les Belles Lettres. Barker, A.D. (1989) trans. Greek Musical Writings: II. Harmonic and Acoustic Theory, Cambridge/New York: Cambridge University Press. Bluck, R.S. (1964) ed. Plato’s Meno Edited with an Introduction, Commentary and an Appendix, Cambridge: Cambridge University Press. Blumenthal, H., Long, A.A., et al. (1989) ed. and trans. ‘On the Kriterion and Hegemonikon: Claudius Ptolemaeus’. See Huby and Neal 1989, 179–230. Burnet, J. (1900–7) ed. Platonis opera, 5 vols, Oxford Classical Texts, Oxford: Clarendon Press. de Falco, V. and Krause, M.K. (1966) ed. and trans. Hypsikles, Die Aufgangszeiten der Gestirne mit einer Einführung von O.Neugebauer, Abhandlungen der Akademie der Wissenschaften in Göttingen, philologisch-historische Klasse, dritte Folge, Nr. 62. Göttingen: Vandenhoeck and Ruprecht. Düring, I. (1930) ed. Die Harmonielehre des Klaudios Ptolemaios, Göteborg Högskolas Ärsskrift 36, Göteborg: Elanders Boktryckeri Aktiebolag. (Reprinted, New York and London: Garland 1980). ——(1934) Ptolemaios und Porphyrios über die Musik, Göteborg Högskolas Ärsskrift 40, Göteborg: Elanders Boktryckeri Aktiebolag. (Reprinted, New York and London: Garland 1980). Friedlein, G. (1867) ed. and trans. Anicii Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de institutione musica libri quinque, Leipzig: Teubner. ——(1873) ed. Procli Diadochi in primum Euclidis elementorum librum commentarii, Leipzig: Teubner. Grenfell, B.P. and Hunt, A.S. (1906) ed. and trans. The Hibeh Papyri, Part 1, London: Egypt Exploration Fund. Grube, G.M.A. (1967) trans. Plato’s Meno, Indianapolis: Hackett Publishing. Heath, T.L. (1956) trans. Euclid’s Elements, 3 vols. New York: Dover. Heiberg, J.L. (1898–1907) ed. and trans. Claudii Ptolemaei opera, quae exstant omnia, 3 vols, Leipzig: Teubner. ——(1910–23) ed. and trans. Archimedis opera omnia cum commentariis Eutocii, 4 vols, Leipzig: Teubner. Heiberg, J.L. and Stamatis, E.S. (1969–77) ed. Euclides, Elementa, 5 vols. Leipzig: Teubner. Hunger, H. and Pingree, D. (1989) ed. and trans. MUL.APIN: An Astronomical Compendium in Cuneiform, Archiv für Orientforschung, Beiheft 24, Horn, Austria: Ferdinand Berger & Söhne. Manitius, K. (1898) ed. and trans. Geminus: Elementa astronomiae. Leipzig: Teubner. Mayhoff, C. (1906) ed. C.Plinii Secundi naturalis historia i, Leipzig: Teubner. Menge, H. (1916) ed. Euclidis phaenomena et scripta musica, Leipzig: Teubner. Mogenet, J. (1950) ed. Autolycus de Pitane: Histoire du texte, suivie de l'édition critique des traités, De la sphere en mouvement et Des levers et couchers, Louvain: Publications Universitaires de Louvain. Mugler, C. (1970–72) ed. and trans. Archimède, 3 vols, Paris: Les Belles Lettres. Neugebauer, O. (1955) ed. Astronomical Cuneiform Texts: Babylonian Ephemerides of the Seleucid Period for the Motion of the Sun, the Moon, and the Planets, London: Lund Humphries. Repr. Sources in the History of Mathematics and Physical Sciences 5, New York/Heidelberg/Berlin: Springer- Verlag. Ross, D. (1955) Aristotle’s Physics: A Revised Text with Introduction and Commentary, Oxford: Clarendon Press. Toomer, G.J. (1984) trans. Ptolemy’s Almagest, New York/Berlin: Springer-Verlag. SECONDARY LITERATURE Books Aaboe, A. (1964) Episodes from the Early History of Mathematics, New Mathematical Library 13, Washington, DC: Mathematical Association of America. Barbera, A. (1990) ed. Music Theory and its Sources: Antiquity and the Middle Ages, Notre Dame, IN: University of Notre Dame Press. Bowen, A.C. (1991) ed. Science and Philosophy in Classical Greece, Institute for Research in Classical Philosophy and Science: Sources and Studies in the History and Philosophy of Classical Science 2, New York/London: Garland Publishing. Dijksterhuis, E.J. (1987) Archimedes, trans. by C.Dikshoorn with a new bibliographic essay by W.R.Knorr, Princeton: Princeton University Press. Dillon, J.M. and Long, A.A. (1988) eds. The Question of ‘Eclecticism’: Studies in Later Greek Philosophy, Berkeley/Los Angeles: University of California Press. Gersh, S. and Kannengieser, C. (1992) eds. Platonism in Late Antiquity, Christianity and Judaism in Antiquity 8, Notre Dame, IN: University of Notre Dame Press. Gillispie, C.C. (1970–80) ed. Dictionary of Scientific Biography, New York: Scribners. Grasshoff, G. (1990) The History of Ptolemy’s Star Catalogue, Studies in the History of Mathematics and Physical Sciences 14, New York: Springer-Verlag. Heath, T.L. (1921) A History of Greek Mathematics, 2 vols, Oxford: Clarendon Press. Huby, P. and Neal, G. (1989) eds. The Criterion of Truth: Essays in Honour of George Kerferd, Liverpool: Liverpool University Press. Maniates, M.R. (1997) ed. Music Discourse from Classical to Early Modern Times: Editing and Translating Texts, Conference on Editorial Problems 26, Toronto/ Buffalo/London: University of Toronto Press. Neugebauer, O. (1934) Vorlesungen über Geschichte der antiken mathematischen Wissenschaften. I: Vorgriechische Mathematik, Berlin: Springer-Verlag. ——(1969) The Exact Sciences in Antiquity, 2nd edn, New York: Dover. ——(1975) A History of Ancient Mathematical Astronomy, 3 vols, Studies in the History of Mathematics and Physical Sciences 1, Berlin/Heidelberg/New York: Springer-Verlag. Taub, L.C. (1993) Ptolemy’s Universe: The Natural Philosophical and Ethical Foundations of Ptolemy’s Astronomy, Chicago/La Salle: Open Court. Walbank, F.W. et al. (1984) eds. The Cambridge Ancient History. VII.1: The Hellenistic World, 2nd edn, Cambridge/London/New York: Cambridge University Press. Wallace, R.W. and MacLachlan, B. (1991) eds. Harmonia mundi: Musica e filosofia nell’ antichità, Biblioteca di quaderni urbinati di cultura classica 5, Rome: Edizione dell’ Ateneo. Walker, C.B.F. (1996) Astronomy before the Telescope, London/New York: British Museum Press/St Martin’s Press. Papers and reviews Barker, A.D. (1991) ‘Reason and Perception in Ptolemy’s Harmonics’. See Wallace and MacLachlan 1991, 104–30. Berggren, J.L. (1991) ‘The Relation of Greek Sphaerics to Early Greek Astronomy’. See Bowen 1991, 227–48. ——and Thomas, R.S.D. (1992) ‘Mathematical Astronomy in the Fourth Century BC as Found in Euclid’s Phaenomena’, Physis 39:7–33. Blumenthal, H. (1989) ‘Plotinus and Proclus on the Criterion of Truth’. See Huby and Neal 1989, 257–80. Bowen, A.C. (1993) review of Hunger and Pingree 1989. Ancient Philosophy 13: 139–42. ——(1994) review of Taub 1993. Isis 85:140–1. ——and Bowen, W.R. (1997) ‘The Translator as Interpreter: Euclid’s Sectio canonis and Ptolemy’s Harmonica in the Latin Tradition’. See Maniates 1997, 97–148. ——and Goldstein, B.R. (1991) ‘Hipparchus’ Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime’, Proceedings of the American Philosophical Society 135:233–54. Britton, J.P. and Walker, C.B. F. (1996) ‘Astronomy and Astrology in Mesopotamia’. See Walker 1996, 42–67. Gersh, S. (1992) ‘Porphyry’s Commentary on the “Harmonics” of Ptolemy and Neoplatonic Musical Theory’. See Gersh and Kannengieser 1992, 141–55. Lloyd, G.E.R. (1984) ‘Hellenistic Science’. See Walbank et al. 1984, 321–52, 591–8. ——(1994) review of Taub 1993. Journal for the History of Astronomy 25:62–3. Long, A.A. (1988) ‘Ptolemy On the Criterion: An Epistemology for the Practicing Scientist’. See Dillon and Long 1988, 176–207. Pingree, D. (1992) ‘Hellenophilia versus the History of Science’, Isis 83:554–63. Solomon, J. (1990) ‘A Preliminary Analysis of the Organization of Ptolemy’s Harmonics’. See Barbera 1990, 68–84. Toomer, G.J. (1978) ‘Ptolemy’. See Gillispie 1970–80, xi 186–206. von Staden, H. (1992) ‘Affinities and Elisions: Helen and Hellenocentrism’, Isis 83: 578–95.

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