EXACT SCIENCES (THE) IN HELLENISTIC TIMES: TEXTS AND ISSUES

The exact sciences in Hellenistic times: Texts and issues1Alan C.BowenModern scholars often rely on the history of Greco-Latin science2 as abackdrop and support for interpreting past philosophical thought. Theirwarrant is the practice established long ago by Greek and Latinphilosophers, of treating science as paradigmatic in their explanations ofwhat knowledge is, what its objects are, how knowledge is obtained, andhow it is expressed or communicated. Unfortunately, when they turn to thehistory of ancient science, these same scholars usually remain too muchunder the spell of the ancient philosophers. Granted, it is true that Greco-Latin science often served as a model and touchstone for philosophy andthat, on occasion, this philosophy may have inspired science. But themarked tendency to follow Greek and Latin writers in viewing ancientscience through the complex, distorting lens of ancient philosophy hashindered recognition that the various sciences of antiquity sometimes differsignificantly from one another as well as from philosophy in theirintellectual, literary, and social contexts. Moreover, it has encouragedscholars to ignore or even disparage clear indications that some of thesesciences were deeply indebted in the course of their history to work outsidethe 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 numerousways by resorting to alien predilections and concerns when trying toexplain the origins, character, and development of Greek and Latinscience.<sup>3</sup> In sum, the amorphous system of learned belief expressed now inhandbooks on ancient science and currently underlying the moderninterpretation of ancient philosophy, for instance, is largely inadequate anderroneous.This failure of previous scholarship challenges historians of ancientscience today to re-think the entire project from its beginning.In effect, itcompels one to start afresh by imagining oneself the first modern scholarconfronted with all the extant literary documents (papyri, inscriptions,manuscripts) and material artifacts (instruments) that come to us from theancient Mediterranean and Near Eastern worlds. Such a prospect isadmittedly 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 youdo not already know what it is, and how will you know that you haveunderstood it? There are, of course, several well known ways to answerthis in the abstract. But the real task is to work out a credible solution inthe particular, that is, in the process of analyzing historical data. And, as Ihave found in studying ancient astronomy and harmonic science, thisprocess involves a vital, corrective interplay between historical analysis andreflection 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 groundsthat the goal of the physician’s craft, health, is articulated and known onlythrough treating specific patients.Given that the standard accounts of Greco-Latin science are at bestcontroversial and should be abandoned in most part, and since thedevelopment of alternative accounts is still in its earliest stages, I mustdecline in what follows to attempt a survey. Instead, I propose to confinemy remarks to a few sample texts in Greek written in the interval betweenthe death of Alexander the Great in 323 BC and the beginning of the thirdcentury AD. There is no great significance to this period so far as the exactsciences themselves are concerned: it simply covers the range in time of thedocuments I have chosen to discuss. And I select these texts because theyprovide the earliest direct evidence of certain features of ancient science thatwill, I trust, be of interest to historians of science and philosophy.In describing a text as direct evidence for some claim or other, I meanthat the text itself is a sufficient basis for verifying the claim. Such directevidence stands in sharp contrast to indirect evidence in the form ofcitations (that is, quotations, translations, paraphrases, and reports). For,one cannot verify a claim on the strength of indirect evidence alone; whatone needs in addition is independent argument for maintaining that thecitation is accurate and reliable. Since there are no general rules validatingthe accuracy or reliability of indirect evidence, such argument must bemade case by case and is, in my experience, both difficult and rarelysuccessful.Restricting attention primarily to direct evidence may seem undulycautious at first. But it is, I submit, the only policy that makes sense at theoutset of any radically critical, historical investigation of the sort nowcalled for. In any case, this policy does offer substantial advantages. To beginwith, confirmation by recourse to direct evidence introduces an order ofcertainty that cannot be attained on the basis of indirect evidence orcitations. The reason is that much of our indirect evidence concernsdocuments no longer available for inspection; thus, the most one may hopefor in justifying reliance on this evidence is an argument for the probabilityof its accuracy. Such arguments, however, usually fail because they involvereading the historian’s own expectations into the past, expectations oftenconcerning empirical matters about which there may be considerableuncertainty and reasonable doubt. Next, if one is strict about how evidenceis used and does not introduce indirect evidence except when it isdemonstrably credible—and even then one should decline to build on it,since probabilities diminish when multiplied—the preference for directevidence will counteract a major failing in traditional histories, thevalorization of certain texts and authors at the expense of others. Finally,in dating the occurrences of concepts, theories, and the like, the historianmay rely on direct evidence to identify the latest (that is, most recent) datepossible for their introduction. This will seem a small gain, particularly tothose who think it the proper business of historians to conjecture earliestpossible dates. But such a program of conjecture is an enterprise to whichthere is no end except by convention. Moreover, by discouraging fullappreciation of the documents we actually have, this fascination with theearliest dates assignable for the occurrences of concepts and theories inGreco-Latin science underlies in part the scholarly neglect of the Akkadianand Egyptian scientific traditions which, in various forms and sometimesthrough intermediaries, interacted with the Greek and Latin traditions.The preceding will have to suffice as an apologia for my deciding topresent the history of the exact sciences in Hellenistic times by way ofnarrowly defined case-studies. Though such an approach is not withoutprecedent (cf., for example, Aaboe 1964), it is admittedly a departure fromthe great number of general surveys and narrative accounts currentlyavailable.<sup>4</sup> The texts I have selected are: Archimedes, De lineis spiralibusdem. 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 wellworth pursuing (in studies that are, of course, suitably cognizant ofhistoriographic matters). Indeed, there are running through these textsthematic concerns about the conception and mathematical analysis of (loco)motion, the nature of scientific communication, and the role in suchcommunication of observation and mathematical theory.MOTION IN MATHEMATICS: ARCHIMEDESIn his De lineis spiralibus, Archimedes (died 212 BC) analyzes fundamentalproperties of a curve of his own invention, now called the spiral ofArchimedes. In the letter prefacing this treatise, the statement of theconditions under which this curve is produced comes first in a list ofpropositions 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 asequence of definitions. According to the latter formulation,if a straight line is drawn in a plane and if, after being turned round asmany times as one pleases at a constant speed while one of itsextremities is fixed, it is restored again to the position from which itstarted, and if, at the same time as the line is turned about, a pointmoves at a constant speed along the straight line beginning at thefixed 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 isnecessary for the subsequent theorems on the spiral itself. The first two ofthese auxiliary demonstrations are devoted to properties of the motion ofpoints on straight lines at constant speeds. In dem. 1 (Heiberg 1910–23, ii12.13–14.20), Archimedes proposes to show thatif a point moving at a constant speed travels along a line and twosegments are taken in the line, the segments will have the same ratioto one another as the time-intervals in which the point traversed thesegments.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 twosegments, CD and DE, be taken in the line. Let the time-intervals inwhich the point traverses CD and DE be FG and GH respectively. Itis required to prove that the segment CD will have the same ratio tothe 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 thatAD>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 inhis 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 suchthat 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. Anymagnitude such as a line or area is divisible into a whole number of smallermagnitudes of the same sort. Thus, given any point D in AB and any wholenumbers 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 arecommensurable or incommensurable, or whether AD>DB or AD=DB orAD< DB.) Now, since CB>DB and given the lemma in (4), there is, then, awhole number n such that(6)(7a)which is the case Archimedes considers,(7b)Figure 9.2 Archimedes, De lineis spiralibus dem. 1if one assumes also that n is the least number to satisfy (6).After stipulating (1), (2), and (3), Archimedes draws attention to the factthat the point moves at a constant speed along AB. Obviously, he says, thispoint will then traverse each of the segments of AD that is equal to CD inthe same time that it takes to traverse CD. Thus, given (2), he infers thatLG 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 takemore time to traverse AD than DB; that is,(8)Likewise, he says by way of generalization, if one takes any multiple of FGand any multiple of GH so that one of the resultant time-intervals exceedsthe other, it will be proven that the line-segment corresponding to thegreater time-interval will be greater, because these line-segments are to beproduced by taking the corresponding multiples of CD and DE. In otherwords,(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 forasserting that magnitudes are in the same ratio, a condition of the sortgiven by Euclid in Elementa v. def. 5:magnitudes are said to be in the same ratio, the first to the secondand the third to the fourth, when, if any equimultiples whatever betaken of the first and third, and any equimultiples of the second andfourth, the former multiples alike exceed, are alike equal to, or alikefall short of, the latter equimultiples respectively taken incorresponding 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 nowheregives an explicit mathematical or quantitative definition of constant speed.The locutions he uses to express this concept suggest that he is startinginstead from the qualitative notion that a body moves at the same orconstant 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 isunchanging in its swiftness or speed (tachos). This point will, of course, belost if one insists on modern convention and supposes that, for Archimedestoo, the speed of a body is the quotient of the distance it travels divided bythe time taken to travel that distance. But this is not, in fact, howArchimedes presents speed: for instance, he characterizes sameness of speednot as an equality of quotients obtained when distances are divided bytime-intervals, but by identifying the ratio of line-segment to line-segmentand the corresponding ratio of time-interval to time-interval. This may,admittedly, be an artifact of the ‘rules’ of mathematical exposition duringhis time, in particular, of the formal condition that ratios be defined onlybetween 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 conceivedspeed. (In the next section, I consider a text from the first century AD inwhich quotients of unlike quantities are in fact computed, though it is stillnot said that these quantities stand in a ratio to one another.) Accordingly,let us leave open the question of how Archimedes thinks of speed orswiftness and concentrate instead on how he expresses it. And, on thiscount, I find in dem. 1 and the rest of the De lineis spiralibus that he talksof speed as a quality of bodies that is quantifiable only in relation to otherinstances of this quality; and, moreover, that constancy of speed is to beunderstood as the sameness of this quality over time in a given body.The next question, however, is whether such talk is supplanted by aquantitative definition in dem. 1. That is, does Archimedes, as somesuppose, posit that traversing line-segments in equal times is just whatmotion at a constant speed is; or does he infer that a point will traverseequal line-segments in equal times from the fact that it moves with aconstant speed (cf. Dijksterhuis 1987, 140–1)? The critical passagenow, since it is posited that the point moves at a constant speed alongthe line AB, it is clear that it travels CD in the same amount of timeas 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, compellingreason to maintain that Archimedes does not in fact identify motion at aconstant speed with traversing equal segments of a straight line in equaltimes. For, as he is well aware, in the course of each revolution, though thegenerating point of the spiral always describes equal angles in equal timesabout the spiral’s origin, that is, about the fixed extremity of the generatingline, and though it always traverses equal segments of the generating line inthe same equal times as well, this same point traces out arcs of the spiralitself that are not equal to another (cf., for example, dem. 12). In otherwords, the very construction of the Archimedean spiral entails that thegenerating line (and, hence, any point on it) will by virtue of its constantrevolution define equal angles in equal times about the fixed extremity; andthat the generating point will by virtue of its constant motion on thegenerating line traverse equal segments of this line in equal times. Yet, thecombined motion of the generating point and line has the result as wellthat this point will not describe equal arcs of the spiral in equal times.Thus, from the vantage point of Archimedes’ De lineis spiralibus, thequalitative idea of motion at a constant speed has to be more fundamentalthan the quantitative ideas of traversing equal segments of a straight line orequal angles of a circle in equal times. In fact, since these are the relevantways of quantifying the motion of the generating point, and since they arenot equivalent here, it would be a serious blunder to open a treatise onspirals with a demonstration presupposing that motion at a constant speedis to be defined simply as traversing equal segments of a straight line inequal times.Let us now consider briefly the preface to Autolycus’ De sphaera quaemovetur. Autolycus begins by declaring thata point is said to move smoothly (homalôs) when it traverses equal orsimilar magnitudes in equal time-intervals. If a point movingsmoothly along some line6 traverses two segments, the ratio of thetime-intervals in which the point traverses the correspondingsegments 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 beendefined mathematically in terms of line-segments and time-intervals, albeitnot as a quotient. In short, though this treatise and the De lineis spiralibusagree that motion can be characterized quantitatively, Autolycus’ treatisealone stipulates that smooth motion is just traversing equal line-segments inequal times. Indeed, that Autolycus calls the point’s motion smooth(homalês) rather than constant (isotachês) may signify that this definitionwas seen to obviate any need to present such motion in terms of a point’smoving at the same speed as itself. Yet, while Autolycus explicitly defines(or reports a definition of) smooth motion, he simply states the theoremabout the proportionality of time-intervals and line-segments. That is, hedoes not offer a proof covering the case of straight lines (such asArchimedes does in De lineis spiralibus dem. 1) or the case of circular arcson a sphere, the latter of which is crucial for his treatise.Two points emerge from this. First is that our understanding of thehistory of the exact sciences can only advance if proper attention is given tothe language in which it is expressed. For example, any interpretation thatrenders both Archimedes’ ‘at a constant speed’ and Autolycus’ ‘smoothly’by ‘uniformly’ will obliterate the complexity in the conceptual andlinguistic apparatus that underlies the difference in their terminology.Indeed, to see that the very idea of ‘uniform’ motion is itself problematic inancient texts, the reader should consult Aristotle, Physics 228b1–30.Second is that, so far as I can tell given the documentary evidenceavailable, Archimedes was first to appreciate the complexity of the ‘equalsegments 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 toground them all in the qualitative idea that a body moving at a constantspeed changes place at the same speed as itself.Was Autolycus, then, the first to realize that, in the special case of a point’scircular motion at a constant speed, the first formula was irrelevant, thatthe latter two formulae were equivalent, and that such motion (here calledsmooth) could be defined in terms of either? This question is difficult toanswer. Perhaps he was, but the same ideas figure in the Phaenomenaattributed to Euclid. Now, this treatise itself can be dated only to theperiod from the third to the first centuries BC.<sup>7</sup> Thus, to affirm priority forAutolycus in the mathematical definition of constant circular motion wouldrequire knowing his dates relative to Euclid’s8 and whether Euclid actuallywrote the Phaenomena,<sup>9</sup> since the case for assigning Autolycus to theperiod from 360 to 290 BC (Aujac 1979, 8–10; cf., for example, Mogenet1950, 5–7, 8–9) is nugatory.<sup>10</sup>Apart from these concerns about the history of the idea of constantmotion, it is important to realize that Archimedes’ very inclusion of motionof any sort in the definition of his spiral is also remarkable. In the works ofEuclid, for example, motion is limited to the construction of figures definedstatically (cf. for example, Elementa i defs. 15–22, dem. 46) and to servingas a hidden assumption in proofs of such relations among figures ascongruence (cf. Euclid, Elementa, i not. com. 4, with Heath 1956, i 224–31).Now, a common way of interpreting this contrast is to suppose thatEuclid belongs to a stage in the history of Greek mathematics earlier thanArchimedes. The case offered thus far for this view, however, is unavailing,resting as it does on no more than an ancient inference concerning ananecdote told also of Menaechmus, a mathematician of the fourth centuryBC, and Alexander the Great, as well as on two suspect citations inArchimedes’ De sphaera et cylindro (see Bowen and Goldstein 1991, 246n30). 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 adifficult question, in part because it requires what has yet to be undertakenin any serious way, a critical study of the ancient testimonia about Euclidand the early history of Greek mathematics. In such a study of Proclus’reports, for example, the alternatives against which the claims are judgedwill have to be founded on more than the simple-minded dichotomy thatProclus is either lying or telling the truth. Indeed, it will have to be rootedin a full examination of Proclus’ historiography, an examination informedby awareness of the numerous ways in which the ancients use history tomake their cases and persuade their contemporaries. And should it turn outthat Euclid’s work draws on and even recasts earlier mathematical theory,it will still be valuable to discover its intellectual and cultural context, asthis 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 motionthat have too long been ignored in the scholarly haste to locate Euclid inrelation to Aristotle and the Academy.THE ARITHMETICAL ANALYSIS OF LUNAR MOTION: GEMINUSThe Introductio astronomiae by Geminus dates from the century or so priorto 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 theastronomical theory which Ptolemy inherited and transformed. Ofparticular interest is Geminus’ account of lunar motion in chapter 18. Forit is here that Geminus not only shows some awareness of Babylonianastronomy, he undertakes to state its rationale. Granted, his account ishistorically incorrect, as we now know (cf. Neugebauer 1975, 586–7). Butto focus on this is to miss the fundamental point that this chapter is theearliest Greco-Latin text available today that tries to explain the structureand derivation of a common Babylonian arithmetical scheme fordetermining the daily progress of a planet, the Moon.<sup>12</sup> So, let us turn tohis account and examine it in detail.Chapter 18 (18.1–19) begins by introducing the exeligmos, whichGeminus describes as the least period containing a whole number of days,months, and lunar returns.<sup>13</sup> By ‘month’ Geminus understands a synodicmonth, that is, the period from one coincidence of the Sun and Moon atthe same degree of longitude on the ecliptic (conjunction) to the next, orfrom one full Moon to the next. As for ‘lunar return’, Geminus explainsthat the Moon is observed traversing the ecliptic unsmoothly (anômalôs) inthe sense that the arcs of the ecliptic which it travels increase day by dayfrom a minimum to a maximum and then decrease from this maximum tothe 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 containinga whole number of days, months, and lunar returns, that is, to discover theexeligmos. This period, he says (Introductio astronomiae 18.3; cf. 18.6),has been observed to comprise669 (synodic) months, or19,765 days,(3)in which there are717 lunar returns (anomalistic months), or723 zodiacal revolutions plus 32° by the Moon.According to Geminus, since these phenomena, ‘which have beeninvestigated from ancient times’ are known, it remains to determine what hecalls 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 bywhich this displacement changes, taking into account the additionalobservational datum that(4)where m is the minimum daily displacement and M the maximum dailydisplacement. 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 andindependent sorts of mean motion or mean daily displacement withoutmaking any terminological distinction. Thus far, he has computed theMoon’s mean motion by taking the periodic relation stated in (3),converting the number of sidereal cycles to degrees, and dividing theresultant 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; butGeminus’ 13; 10, 35 may be excused as a rounding (cf. Aujac 1975, 95n1). More puzzling, however, is the computation of the length of theanomalistic month in (6), since(6a)which differs somewhat from Geminus’ 27; 33, 20 days. (The differenceamounts to 1; 20 days in one exeligmos.)One possibility is that Geminus has wrongly taken it for granted that hiscomputation of the length of the anomalistic month in (6) yields thevalue stated in (2), namely, . Another possibility is that Geminus is here‘telescoping two different (Babylonian) methods into one’ (Neugebauer1969, 185: cf. 162). For, if one follows Neugebauer (1975, 586) andfocuses only on the parameters of Geminus’ account, it seems thatGeminus is drawing on two different Babylonian text-traditions, namely,on texts from Uruk presenting a scheme in which the lunar displacement is13; 10, 35o/d and the length of the anomalistic month is 27; 33, 20d, andon Babylonian Saros-texts, the Saros being a cycle in which the length of theanomalistic month is 27; 33, 13, 18, 19…days. A third possibility, andperhaps the most charitable, is that Geminus actually understands thenumber of anomalistic months in the exeligmos to be derived from the lengthof the anomalistic month by computing(6b)Next, Geminus divides the anomalistic month of 27; 33, 20 days into fourequal subintervals such that(7)where, for example, I(m, μ) is the interval from the day of minimum lunardisplacement 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 dailydisplacement, that is, the simple average of two extreme values for dailylunar displacement in longitude.Now, according to Geminus, the sum of the maximum and minimumdaily displacements is known from observation to be only 26°; thefractional 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 waythat meets three conditions (see (4), (9)):(10)To do this, Geminus first reiterates that in each of the four subintervals ofthe anomalistic month (see (7)), the daily difference (d)—which is eitherincremental or decremental—is the same; this means, he remarks, that onehas to find d such that(11)(see (9)), where k is the Moon’s total displacement in longitude in 1/4anomalisticmonth.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 nonsequitur. Geminus does not supply enough information to deduce the valuefor the daily difference in the Moon’s longitudinal motion. In fact, what hegives suffices only to specify a range of values for d. To see this, considerthe values for d when m(=μ−k) and M(=μ+k) take on the extreme values ofthe range of possible values indicated in (11). Suppose, for instance, thatFrom (8) and (11) it would follow thatSimilarly, ifthenAccordingly, given (4),(13)Likewise, ifit would follow from (8) and (11) thatAgain, 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 ofthe text alone whether Geminus’ claim that d is 0; 18o/d was reached bytruncation. Indeed, one should not discount the possibility that the valueGeminus assigns d was given at the outset or entailed in the information hehad.Chapter 18 is the earliest text extant in Greek or Latin to present anaccount of an arithmetical scheme of a type now associated with theBabylonians. Since Geminus mentions the Chaldaeans, and since heascribes this account to no one else, it would seem that he is in factreconstructing what he takes to be the theory underlying information thatultimately came to him from Mesopotamia. So one may reasonably ask,what did he actually have? Presumably, he had access either to tabular dataitself, to a set of procedures for entering the data, or to an account of howthis data was organized data. Unfortunately, there is no way to determinewhich was the case.Still, it is true that in chapter 18 Geminus describes the arithmeticalprinciples and parameters underlying tables for daily lunar motion, of asort we now have from Uruk (Neugebauer 1969, 161–2; 1975, 480–1). Inmodern terms, these ephemerides are said to be structured according to alinear zigzag function (see Figure 9.3) in which(16)and, thus,(17)At the same time, Geminus’ account of the exeligmos derives from aBabylonian 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, asthe New or Full Moon returns 242 times to the same position relative tothe same node, the Moon completes 239 cycles of its unsmooth motion inlongitude, and travels through the zodiac 241 times and 10; 30° (seeBritton and Walker 1996, 52–4). In other words,223 synodic months = 239 anomalistic months=242 draconiticmonths<sup>17</sup>= 241 zodiacal revolutions by the Moon and 10;30°<sup>18</sup>= 6585; 20This cycle was certainly known in some form to Greco-Latin writers inGeminus’ time. Pliny (Historia naturalis ii 56), for example, affirms thateclipses recur in cycles of 223 months.<sup>19</sup> The Introductio astronomiae,however, would seem to be the oldest surviving Greek or Latin text tointroduce the exeligmos, an eclipse-cycle three times as long as the Saroscycle,albeit without giving any indication of its purpose or its essentialstructure.<sup>20</sup> Geminus does not, for instance, connect the exeligmos witheclipses explicitly, and he does not mention the critical correlation of 669(=3•223) synodic months with 726 (=3•242) draconitic months. Indeed, fora full statement of the exeligmos by a Greco-Latin writer, one must turn toPtolemy, Almagest iv 2.<sup>21</sup>Geminus is silent about the relation between his exeligmos and theBabylonian Saros. Now, it is possible that this is due to his ignorance of thefact that Saros is an eclipse-cycle and that the exeligmos is a longer versionof the Saros. Yet, at the same time, it is also possible that he has suppressedthis information in order to present the exeligmos as just anothercalendrical cycle of the sort he describes in Introductio astronomiae ch. 8.So, his silence permits no conclusions about the condition and form of thedata that he reconstructs in chapter 18. Still, it is clear that, at the veryleast, he had Babylonian values for the Moon’s mean daily displacement inlongitude (13; 35, 10°), the daily difference in the Moon’s displacement inlongitude (0; 18°), and the mean anomalistic month (27; 33, 20d), as well asthe 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 theexeligmos, his claim about how they did it is unwarranted and implausible.Indeed, when one considers Babylonian lunar ephemerides of the sort thatlie behind his account (cf. Neugebauer 1955, nos 190–6), it is difficult notto conclude that he was either unfamiliar with them or that he failed torealize that their schematic character makes it virtually impossible todetermine their observational basis.Figure 9.3 A linear zigzag function for lunar motion in longitudeNevertheless, on its own terms, Geminus’ account in chapter 18 of theexeligmos and of lunar motion in longitude is noteworthy, in the first place,because he seeks to derive the scheme by which the data in theseephemerides are organized from a few parameters. Granted, this derivationdoes 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 inactually determining the Moon’s position in longitude at a given time.Next, Geminus’ account is also notable because he identifies fundamentalparameters as observational data. Admittedly, this is scarcely credible evenon Geminus’ own terms, if, as he reports (Introductio astronomiae 18.14),the best observation can do (with the aid of instruments) is to determinethe sum of M and m to the nearest degree, a remark which is at odds withhis claim that the values for the synodic and anomalistic months reportedin (1) and (2) have been observed. And, as I have said, so far as history isconcerned, though there is certainly some observational basis to theBabylonian Saros-texts and to the lunar tables from Uruk, there is nowarrant for supposing that it consisted in observing the fundamentalparameters of the arithmetical schemes structuring these tables. Still,Geminus’ assumption that these basic parameters were observed isimportant as an indication of how he understands astronomy and its use ofmathematics. For Geminus, apparently, his arithmetical scheme actuallydescribes the Moon’s unsmooth motion in longitude, and the accuracy ofthis description is guaranteed by the fact that it derives from arithmeticalmanipulation of observed parameters. Regrettably, he leaves unansweredpertinent questions about what counts as an observation, how observationsare made, and so on.Moreover, within the context of the Introductio astronomiae, Geminus’arithmetical account of lunar motion in longitude is also remarkable for tworeasons. First is that it contrasts sharply with the rest of the treatise. Onlyin chapter 18 (and chapter 8, which concerns calendrical cycles) doesGeminus introduce quantitative argument. Elsewhere, his remarks arequalitative and geometrical. Thus, in his account of the Sun’s unsmoothmotion in longitude (Introductio astronomiae 1.18–41), for example,though he supposes that it is only apparent because the Sun moves at aconstant speed on a circle eccentric to the Earth, Geminus does not use hisvalues for the lengths of the seasons (cf. 1.13–17) to specify the eccentricityof this circle and so on (cf. Neugebauer 1975, 581–4).Second, and more striking, is that Geminus’ account of lunar motion inchapter 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 fiveplanets move at a constant speed [isotachôs], in a circle, and in adirection opposite to [the daily rotation of] the cosmos. For thePythagoreans, who first came to investigations of this sort, positedthat the motions of the Sun, Moon, and five planets were circular andsmooth [homalas]. Regarding things that are divine and eternal theydid not admit disorder of the sort that sometimes [these things] movemore quickly, sometimes more slowly, and sometimes they stand still(which they call stations in the case of the five planets). One wouldnot even admit this sort of unsmoothness [anômalian] of motionregarding a man who is ordered and fixed in his movements. For theneeds of life are often causes of slowness and speed for men; but as forthe imperishable nature of the celestial bodies, it is impossible thatany cause of speed and slowness be introduced. For which reasonthey have proposed [the question] thus: How can one explain thephenomena by means of circular, smooth motions?Accordingly, we will give an explanation concerning the othercelestial bodies elsewhere; but just now we will show concerning theSun why, though it moves at a constant speed, it traverses equal arcsin unequal times.(Introductio astronomiae 1.19–22)This means that the arithmetical scheme presented in chapter 18 does notdescribe the Moon’s real motion in longitude; at best, it can represent theMoon’s apparent motion—assuming, for the moment, with Geminus thatthe daily variations in the Moon’s longitudinal displacement are indeedobservable. But, if so, Geminus has yet to supply the account of theMoon’s real motion in longitude that he has promised. Such an explanationwould, of course, have to overcome a serious problem; namely, that thereis no way, using resources presented in the treatise thus far, to construct acoherent argument that begins with the qualitative geometry of the Moon’sreal motions and concludes with the arithmetical detail of his scheme forthe Moon’s apparent motion. In short, by introducing the sort ofarithmetical detail that he does in his account of the Moon’s ‘observed’variable motion in longitude, Geminus undermines his ostensible project ofexplaining this motion in terms of the smooth circular motion(s) that itsupposedly makes in reality. In effect, chapter 18 exposes a problem at theheart of Greco-Latin astronomy of the time that becomes evident once itattempts to incorporate in its explanatory structure arithmetical proceduresand results from Babylonian astronomy.Geminus’ mean daily displacement can only be an apparent lunar motionin longitude and not one the Moon really makes, if the mean in question isarithmetic. If the mean is periodic, however, the Moon’s mean dailydisplacement can become a basis for specifying its true or real motion. Butit would take Ptolemy to straighten out Geminus’ conflated notion of meanmotion and its relation to real and apparent planetary motion. Indeed, partof Ptolemy’s genius lay in seeing that texts such as Geminus’ Introductioastronomiae were typical of what was wrong with the astronomy of his time;that, in assimilating Babylonian astronomy, earlier and contemporaryGreco-Latin writers betrayed a confused, inconsistent, and insufficientlysophisticated grasp of the proper role of arithmetic, geometry, andobservation 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 toPtolemy is a question that involves understanding not only his literary andscientific context but also how he appropriates and transforms this contextin his own highly technical work.<sup>23</sup> Granted, there are scatteredthroughout Ptolemy’s treatises tantalizing passages in which he talks ofmethod and indicates a conceptual framework in which the sciencesdiscussed somehow fit. There is even a treatise, the De iudicandi facultate,in which Ptolemy sets out an epistemology that is intended to explain andjustify 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 becausescholars have yet to interpret this treatise and the related passages found inPtolemy’s other works in the light of the technical, scientific matters whichgive 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 thequestion of Ptolemy’s philosophical views by examining the first twochapters of the first book of his Harmonica (Düring 1930, 3.1–6.13) withoccasional 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 onthe related matter of the goal of the harmonic theorist, though he doesmention astronomy and astronomers as well.By Ptolemy’s time, argument about the criteria of truth was prominent inintellectual circles: in fact, by then, the problem was to explain thecontributions of reason and the senses to knowledge of external objects,and to determine what infallible means there are for distinguishingparticular truths about these objects from falsehoods (cf. Long 1988, 180,192). But Ptolemy recasts the problem. To begin, he decides to ignore thetechnical vocabulary current among philosophers of his time in favor of asimpler vocabulary that suffices to aid non-experts and to clarify reflectionon the realities signified (cf. De iudicandi facultate 4.2–6.3). Accordingly,he proposes to use ‘criterion’ (kritêrion) to designate (a) the object aboutwhich one makes judgments, (b) the means through which and the meansby which judgments about such objects are made, (c) the agent ofjudgment, (d) the goal of the judgments made, as well as the more usualsense, (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 criterionqua goal of judgment (cf. De iudicandi facultate 2.1–2), Ptolemy representsthe general problem as one of discovering the criterion of what there is (cf.1.1). In the context of harmonic science (harmonikê), this becomes theproblem of determining the criteria of what there is in the domain ofmusic, that is, the criteria of harmonia, where harmonia is ultimatelytunefulness or the way pitches should or do fit together properly.Chapter 1 of the first book of the Harmonica opens with the assertionthatHarmonic science is a capacity for apprehending intervals of high andlow 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—andhearing and reason are criteria of tunefulness [harmonia] though notin the same way.(Düring 1930, 3.1–4)I take this to mean that harmonic science is a branch of knowledge bywhich one is able to account systematically for intervals among pitches andto determine their harmonia. Obtaining and exercising this knowledge,however, is to draw on two faculties, hearing and reason, that serve ascriteria in different ways:<sup>27</sup> as Ptolemy says, ‘hearing is [a criterion] inrelation to matter and experience (pathos); but reason [is a criterion] inrelation to form and cause’.<sup>28</sup> To explain why hearing and reason areunited in this way in developing or using harmonic science, Ptolemy firstpoints out thateven in general that which can discover what is similar [to suneggus]and admit precise detail [to akribes] from elsewhere is characteristic ofthe senses; while that which can admit from elsewhere what is similarand 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 whereasexperiences [are defined and delimited] by the causes of motions, andsince [matter and experience] are proper to sense-perception but [formand cause] are proper to reason, it follows fittingly that our sensoryapprehensions are defined and delimited by our rationalapprehensions, in that, at least in the case of things known throughsense-perception,<sup>29</sup> the sensory apprehensions first submit their rathercrudely [holoscheresteron] grasped distinctions to the rationalapprehensions and are guided by them to distinctions that areprecisely detailed and coherent.(Düring 1930, 3.8–14)The key to understanding Ptolemy’s account thus far of the roles of reasonand the sense of hearing in harmonic science is the distinction between tosuneggus and to akribes. One possibility is that Ptolemy is concerned withtruth, that he means to affirm the approximate character of perception andthe accuracy of reason. Thus, as Barker translates the critical lines:…it is in general characteristic of the senses to discover what isapproximate and to adopt from elsewhere what is accurate, and ofreason to adopt from elsewhere what is approximate, and to discoverwhat is accurate.(Barker 1991, 276)The problem here, in the first place, is that Ptolemy does not actually saythat the senses characteristically discover to suneggus and so forth. Whathe maintains instead is that some thing, which is capable of discovering tosuneggus and admitting to akribes from elsewhere, is characteristic of thesenses. Likewise for reason, he does not say that it characteristicallydiscovers to akribes and so forth, but that some thing, which is capable ofdiscovering to akribes and admitting to suneggus from elsewhere, ischaracteristic of it. Now these items are, I submit, the perceptum andthought, respectively. Second, it is important to realize that Ptolemy is nothere directly concerned with truth but with how information istransformed into knowledge. In short, as his talk of matter and formsuggests, the contrast he has in mind is one between sensory informationbefore and after it has been articulated as knowledge, and not one betweenthe approximate and the accurate.Thus, I take Ptolemy’s point to be that the scientific analysis of pitchrequires hearing to discern similarity and difference among pitches andintervals, and reason to articulate this similarity or difference byquantifying it numerically according to a theoretical system. This processof informing or articulating and thereby appropriating what hearingdiscerns into a system of knowledge involves introducing precision ornumerical detail. Thus, for Ptolemy, what hearing grasps is rather crude(holoscheresteron), either because it is not numerically quantified at all orbecause it is quantified in a way not involving theory (as when someonehears an interval and simply says that it is a fifth, for instance). Thus, whatis holoscheresteron is rather crude because it lacks the sort of precise detailit must have to be scientific knowledge, which does not mean of itself thatit 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 initself, fixed, and always the same in relation to the same things; butthat sense-perception is always involved with matter which isconfused and in flux. Consequently, because of matter’s instability,neither the sense-perception of all people nor even that of the samepeople is ever observed to be the same in relation to objects similarlydisposed, 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 theirobjects, reason alone is fit for articulating consistently what is grasped bythe senses: the senses themselves cannot do this.In saying this, Ptolemy has raised the related issues of disagreementamong listeners and error. If hearing does not on every occasion discern thesame distinction in what is heard though the circumstances are such that itshould, it is important to discover whether reason may ever rely on hearingand in what way, if one is to account fully for the roles of reason andhearing in harmonic science. The question is, then: does hearing everdiscern similarities and differences among pitches correctly and, if so,whose hearing is it?Ptolemy answers by pointing out first that hearing may be brought torecognize its errors by reason.<sup>31</sup> So, under some circumstances at least, it ispossible for hearing to discern things accurately. Then, Ptolemy affirms thestronger thesis that sometimes what hearing presents to reason does notneed any correction at all. Let us consider these claims in turn.Ptolemy maintains that reason can bring hearing to a knowledge oferrors in its apprehensions, by way of an analogy:So, just as the circle drawn by the eye alone often seems to beaccurate until the circle made by reason brings [the eye] to therecognition of one that is in reality accurate [akribôs echein], thuswhen some definite interval between sounds is taken by hearing alone,it will initially seem sometimes neither to fall short nor to exceed whatis appropriate, but is often exposed as not being so when the intervalselected according to proper ratio is compared, since hearingrecognizes by the juxtaposition the more accurate [one] as somethinggenuine, 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 properratio) so that hearing may apprehend it and thereby come to discernerror.<sup>32</sup> Obviously, reason will be obliged to be equip itself with aninstrument that it can employ in a way consistent with theory in order toproduce the correct sounds—a point Ptolemy makes explicit later. In anycase, Ptolemy clearly holds that both reason and hearing can detect errorsin the apprehensions of hearing. But, though this entails that theapprehensions of hearing are sometimes accurate, it does not yet followthat reason may rely on hearing for information about differences amongsounds.The analogy illustrating how reason can bring the eye and hearing todiscern error when none was recognized previously also suggests that thesenses are better as judges than as producers of percepta. But to establishthat hearing may apprehend distinctions correctly unaided by reason,Ptolemy must evaluate the capacity of the senses to make distinctions ontheir own.He begins by affirming that it is in general easier to judge something thanit is to do it. His elaboration of this premiss makes it clear that hearing willhave better results in recognizing that an interval or melody is out of tunethan when it guides the production of the interval or melody by means ofsome instrument such as the voice or aulos. Indeed, he explains,this sort of deficiency of our sense-perceptions does not miss the truthby much in the case of [our] recognizing whether there is a simpledifference between them [sc. our sense-perceptions] nor, again, in thecase of [our] observing the excesses of things that differ, at least when[the excesses] are taken in greater parts of the things to which theybelong.<sup>33</sup>(Düring 1930, 4.10–13)The locution here may strike the modern reader as odd. The deficiency inquestion is, I think, the deficiency of the senses in apprehending what is inreality accurate, which he has just described. Now, as I understand it, theclaim that this deficiency ‘does not miss the truth by much’ is a figure ofspeech: Ptolemy actually means that, when the senses discern a meredifference or report the amount of this difference (providing that theamount is suitably large), they do not under these circumstances miss thetruth at all.<sup>34</sup> In other words, I maintain that, for Ptolemy, theapprehensions of hearing are in fact correct and accurate, when hearingattends to the mere occurrence of an interval between sounds or when itreports the amount of this interval (if the amount is large enough).It is important that the amounts of the differences between the sounds belarge in comparison to the sounds, that is, for example, that the differencebetween two intervals be large in comparison to the two intervals. AsPtolemy says of the senses in general, if the amounts of the differences theyapprehend are a relatively small part of the things exhibiting them, thesenses may not discern any difference at all; yet, when such apprehensionsare iterated, the error or difference accumulates and eventually becomesperceptible.The upshot is that Ptolemy assigns the senses a well-circumscribedreliability: sensory apprehensions of sameness and difference are for themost part deemed unreliable. Only in apprehending the fact of differenceor the amount of this difference (when the amount is suitably large) do thesenses such as hearing provide a reliable and fitting empirical basis forscience (cf. De iudicandi facultate 12.4),<sup>35</sup>The question of whose hearing it is still remains, and Ptolemyapproaches it by considering the class of those instances when hearing bynature goes astray. After all, what hearing apprehends or reports canbecome scientific only when integrated by reason in an explanatorysystem;<sup>36</sup> and this means that reason will often have to deal with error inwhat hearing reports. As he says,just as for the eyes there is a need for some rational criterion throughappropriate instruments—for example, for the ruler in relation tostraightness and for the pair of compasses in relation to the circle andthe measurements of parts—in the same way as well there must be forthe ears, which are with the eyes especially servants of the theoreticalor reason bearing part of the soul, some procedure [ephodos] fromreason for things which [the ears] do not by nature judge accurately,a procedure against which they will not testify but will agree that it iscorrect.(Düring 1930, 5.3–10)Ptolemy begins chapter 2 by identifying the instrument for correcting auralapprehensions as the harmonic canon or ruler (kanôn), adding that thename 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 thirdcriterion that Ptolemy has designated as such thus far in the openingchapters of the Harmonica?According to Ptolemy,it should be the goal of the harmonic theorist to preserve in every waythe rational hypotheses [hupotheseis]<sup>37</sup> of the canonas neverconflicting in any way with the senses in the judgment of most people,just as it should be the goal of the astronomer to preserve thehypotheses of the celestial motions as in agreement with theirobserved periods, hypotheses that while they have themselves beentaken 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 todisplay the works of nature as crafted with a certain reason and fixedcause, and [to display] nothing as produced [by nature] without apurpose 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 truthfrom falsehood in musical sound and on which it may rely, turns out, infact, to be the consensus of the majority about what is heard when thecanon is properly set up according to theory and actually struck.<sup>39</sup> For thiscriterion entails that, on such occasions, the standard of accuracy indetermining not only the fact of differences among sounds but also, undercertain circumstances, how great these differences are, is what most peoplehear. In sum, the hearing that stands as the reliable counterpart of, andstandard 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 topursue 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 nowbriefly address the question ‘What does the harmonic theorist actuallyknow?’In the first place, the harmonic theorist understands harmonia, that is, theorganization of differences in pitch. To say more than this, however, it isnecessary to discover just what it is that the majority reports about suchdifferences. In particular, one should at least ask whether the consensus ofthe majority concerns a subjective experience or the objects underlying thisexperience. Now, Ptolemy’s answer to this question comes in the nextchapters (Harmonica i 3–4), which discuss (a) the causes of high and lowpitch in sound (psophos), and (b) musical notes (phthoggoi) and theirdifferences. But this very distinction between sounds and musical notessuggests another feature of harmonia that one must not neglect, namely,that harmonia is fundamentally an aesthetic phenomenon, that thedifferences in pitch have an intrinsically aesthetic character. That is,implicit in Ptolemy’s account of the third criterion is the view thatharmonia is ultimately defined by the musical sensibilities or tastes of acommunity—no matter whether one assumes (as I do) that the phrase katatên tôn pleistôn hupolêpsin means ‘in the opinion of most people’ or that itmeans ‘in the opinion of most experts’:<sup>41</sup> in either case, the harmonictheorist is to appeal to, and to rely on, a shared sense of differences in pitchand their melodic propriety.It would seem, then, that in answer to the question ‘What does theharmonic theorist know?’, one might point out that harmonic sciencearticulates systematically by means of number a communal sense ofmusical propriety. But, if so, does this science change over time? There is,after all, a tension in Ptolemy’s account between reason and what mostpeople hear, and I suspect that it is essential to his understanding ofharmonic science itself: such tension is certainly built into his third criterionto the extent that agreement or consent is an issue.One way to cast the problem is to ask, does hearing ever correct or bearwitness against theory? Obviously, it must as the theoretical account of themusic characteristic of a culture becomes more scientific and accurate, apossibility implicit in Ptolemy’s criticism of contemporary and earliertheorists at the close of chapter 2, for instance, and elsewhere. But doestheory ever have to adapt to changes in what most people hear when thecanon is set up according to theory and struck? This is a question to bringto a careful reading of the Harmonica. For if Ptolemy denies that musicalsensibility changes over time, harmonic science has a perfection it can reachin articulating the sense of musical propriety shared by most people. But, ifhe allows that it does change, then harmonic science must too and so itcannot have a final form. In this case, then, what most people hear whenthe canon is set up according to theory and then struck will serve not onlyto confirm theory and to correct practice, it will on occasion serve also toconfirm practice and to correct theory.<sup>42</sup> And if observation may take onsuch a role in harmonic science, may it do the same in astronomy?<sup>43</sup>CONCLUSIONIt is perhaps appropriate to finish with a question, since a series of casestudieswill hardly generate global results. To philosophers it is often giventhat one may grasp the universal in the particular; but rarely is this grantedto historians of ancient science. Thus, for now, I content myself with themore mundane hope that the preceding studies of particulars in detail willat least raise questions leading to other particulars in a fruitful way.NOTES1 I take the exact sciences to include arithmetic, geometry and all those sciencesinvolving arithmetic and geometry in a significant way (for example,astronomy, astrology, harmonics, mechanics, and optics). Isolating thesesciences as a class is not a uniform characteristic of Greco-Latin thought.Still, it is a useful starting-point, particularly if one considers the variousexact sciences throughout their histories and inquires of each whether it wasin fact (always) viewed as scientific by the ancients, and to what extent the roleof mathematics affected this decision.2 When writing of Greek and Latin science, philosophy, and so on, I refer onlyto 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 thislemma 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 wholenumber n such that n•a>b), it is essential to Archimedes’ indirect calculationsof 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 amagnitude of the same kind, it excludes the possibility of infinitesimals suchas 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 arcsmay be equal or similar: they are similar if they lie on parallel circles and arecut 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 twelveequal arcs of the ecliptic in the Phaenomena would seem to place it after thefourth 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 theannual variations in the length of daytime, a concern characteristic of thesecond and first centuries BC. (Hypsicles’ Anaphoricus, a treatise presenting aBabylonian arithmetical scheme for determining the length of daytimethroughout the year, is commonly thought to belong to the second centuryBC.)8 It is difficult to determine the relative dates of Autolycus and Euclid. Theargument 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 Autolycusand 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 hewas active in the first century BC; but Neugebauer (1975, 579–81) hasargued 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 textwhich organizes information according to a (modified) Babylonian scheme ofthe 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 theexeligmos: see pp. 300–1 below, on the Saros.14 Geminus represents numbers in two ways, either as whole numbers plus asequence of unit-fractions (in decreasing order of size) or as sexagesimals. I willfollow convention by writing unit-fractions by means of numerals with barsover 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 thesixtieths, and commas to separate sexagesimal places to the right of thesemicolon. See Introductio astronomiae 18.8, for an explanation—Manitiusviews this as a marginal gloss that has been moved into the text—ofGeminus’ nomenclature for sexagesimal fractions of a degree: first sixtiethsare units of ; second sixtieths, units of •; and so on.15 Since (4) rules out and , it follows that and16 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 orpoint where its orbit crosses the plane of the ecliptic in the same direction.Determining the length of the draconitic month is useful in understandingeclipses, 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 Suncompletes 18 zodiacal revolutions (sidereal years) and 10; 30° in the sameperiod.19 Not all the manuscripts of Pliny, Historia naturalis ii 56 have 223 as thenumber of synodic months: cf. Mayhoff 1906, 144; Neugebauer 1969, 142.20 In Geminus’ version of the exeligmos—as in Ptolemy’s (Almagest iv 2)—theMoon makes 723 zodiacal revolutions and then travels 32° farther, whereas,if one triples the Babylonian Saros-cycle, the Moon circles the zodiac 723 timesbut then travels only 31; 30° farther.21 Ptolemy’s accounts of the shorter cycle (the Babylonian Saros) and of theexeligmos are consistent: he posits that in one Saros the Moon makes 241zodiacal revolutions and 10; 40°,and that the Sun makes 18 such revolutionsand 10; 40°.22 I take the opportunity in what follows to revise and develop the analysisgiven in Bowen and Bowen 1997, 104–12.23 See Grasshoff 1990, 198–216 and Taub 1993 for two recent attempts todiscover 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, albeitmisleading in critical matters of philosophical and technical detail.26 Cf. Gersh 1992, 149. See Bowen and Bowen 1997, 137 1122 for criticism ofSolomon’s analysis (1990, 71–2) of Ptolemy’s definition of harmonic science.27 Hearing and reason are both instrumental, though the mode of theirinstrumentality differs, as Ptolemy’s use of different instrumentalconstructions at De iudicandi facultate 1.5, 2.2–4 indicates: hearing, like anyother 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 andpathos in relation to explanation. Barker renders pathos by ‘modification’;but it makes little sense to compare hearing to a modification (that is, to achange or enmattered form) in the present context. So, I propose instead torender 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 thingsthat can be detected through sensation’. See Ptolemy, De iudicandi facultate10.5 which allows that there are things known by reason without the aid ofthe 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). Inmost cases, the astronomical observations criticized by Ptolemy involvemeasurement; 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 notknown. 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 ofakribes changes from ‘detailed’ to ‘accurate’ or ‘true’.32 The force of this analogy is not, as Barker (1989, 277 n9) supposes, thathearing 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 todetermine (again on the basis of theory) that the sound produced is incorrect.In De iudicandi facultate 10.4–5, Ptolemy writes that on certain occasionsreason may choose to correct sense-perception through the means of senseperceptions.Thus, if a sense is affected in a way inappropriate to the objectsensed, reason may determine the error either through similar, unaffected oruncorrupted sense-perceptions when the cause of error involves the senseperceptions,or through dissimilar sense-perceptions of the same object whenthe cause does not involve them but something external.33 On Solomon’s version (1990, 73–4) of these lines, see Bowen and Bowen1997, 140 n33 and Barker 1989, 277.34 Barker (1991, 118) does not recognize a rhetorical figure here, though thefact that hearing does sometimes disclose the truth is assumed in the nextlines, when Ptolemy considers how imperceptible error in sense-perceptionmay accumulate and eventually become perceptible. Cf., for example, Düring1930, 23.19–24.8.35 Long’s claim (1988, 193) that, for Ptolemy, ‘sense-perception is limited to theimmediate experiences it undergoes and it cannot pass judgment on anyexternal objects as such’, though it neglects kai epi poson apallagentôn at Deiudicandi facultate 8.5, does draw attention to an important puzzle.According to 8.4, sense-perception judges only its experiences (pathê) andnot the underlying objects. The same is said at 10.1–3 (cf. 11.1), exceptPtolemy here remarks that sense-perception sometimes reports falsely aboutthe 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 theHarmonica becomes, ‘when hearing reports veridically about the occurrenceof sounds and certain of their differences, does it simply report truly itsexperiences or does it somehow disclose true information about the physicalstate of affairs producing these experiences?’ At issue is Ptolemy’s idea ofwhat sound and, in particular, musical sound, is (see p. 310).36 In De iudicandi facultate 2.4–5, Ptolemy distinguishes phantasia, which is theimpression and transmission to the intellect of information reached bycontact through the sense-organs, and ennoia (conception), which is thepossession and retention of these transmissions in memory. The conceptionsare what may become scientific if integrated into theory (cf. 10.2–6).37 In the Almagest, Ptolemy uses hupotheseis in reference to his planetarymodels: 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 morerational of the senses, sight and hearing’. It is not clear just what theseconstructions belonging to sight and hearing are supposed to be. They aremost likely not the objects of these senses: Ptolemy’s meaning here is that thetheoretician 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 apparentlymistaken.If one recalls the various meanings of ‘criterion’ that Ptolemy lists at Deiudicandi facultate 2.1–2, the argument of Harmonica i 1–2 would seemto indicate that: harmonia is a criterion qua object of judgment; hearing andreason are criteria qua means through which and means by which,respectively; the harmonic theorist is a criterion qua agent of judgment; whatmost people hear when the canon is set up according to theory and struck is acriterion qua standard; and preserving truth, that is, the concordance of thehypotheses of harmonic science with this standard, is a criterion qua goal.40 Thus Ptolemy counters scepticism that harmonic science does in factconstitute knowledge, because of the acknowledged variations in hearingfrom 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 limitsthis criterion to experts or cognoscenti.42 Until such matters are settled, I hesitate to follow Long (1988, 194) ininferring that the De iudicandi facultate presents science as ‘a stable andincontrovertible state of the intellect, consisting in self-evident and expertdiscrimination’, especially since there is, so far as I can tell, nothing in thistreatise that favors this view and excludes the role of third criterion I havejust indicated.43 On progress in astronomy, see Almagest i 1 with Toomer 1984, 37 n11. Atits most general, the question is, How does Ptolemy understand the Almagestand its place in the history of astronomy?BIBLIOGRAPHYPRIMARY LITERATUREAncient documentsIn 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 Frenchtrans.).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.), Aujac1975 (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, Deinstitutione musica v 2 with Bowen and Bowen 1997.Saros Texts. See Aaboe, Britton, et al. 1991 (with trans.).Modern editions and translationsAaboe, A., Britton, J.P., Henderson, J.A., Neugebauer, O., and Sachs, A.J. (1991)Saros Cycle Dates and Related Babylonian Astronomical Texts, Transactionsof the American Philosophical Society 81.6, Philadelphia: AmericanPhilosophical Society.Aujac, G. (1975) ed. and trans. Géminos, Introduction aux phénomènes, Paris: LesBelles Lettres.——(1979) ed. and trans. Autolycos de Pitane: La sphere en mouvement, Levers etcouchers, testimonia, Paris: Les Belles Lettres.Barker, A.D. (1989) trans. Greek Musical Writings: II. Harmonic and AcousticTheory, Cambridge/New York: Cambridge University Press.Bluck, R.S. (1964) ed. Plato’s Meno Edited with an Introduction, Commentary andan Appendix, Cambridge: Cambridge University Press.Blumenthal, H., Long, A.A., et al. (1989) ed. and trans. ‘On the Kriterion andHegemonikon: 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 Aufgangszeitender Gestirne mit einer Einführung von O.Neugebauer, Abhandlungen derAkademie der Wissenschaften in Göttingen, philologisch-historische Klasse,dritte Folge, Nr. 62. Göttingen: Vandenhoeck and Ruprecht.Düring, I. (1930) ed. Die Harmonielehre des Klaudios Ptolemaios, GöteborgHögskolas Ärsskrift 36, Göteborg: Elanders Boktryckeri Aktiebolag.(Reprinted, New York and London: Garland 1980).——(1934) Ptolemaios und Porphyrios über die Musik, Göteborg HögskolasÄrsskrift 40, Göteborg: Elanders Boktryckeri Aktiebolag. (Reprinted, NewYork and London: Garland 1980).Friedlein, G. (1867) ed. and trans. Anicii Manlii Torquati Severini Boetii deinstitutione arithmetica libri duo, de institutione musica libri quinque, Leipzig:Teubner.——(1873) ed. Procli Diadochi in primum Euclidis elementorum librumcommentarii, 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 exstantomnia, 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 AstronomicalCompendium 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'éditioncritique 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: BabylonianEphemerides of the Seleucid Period for the Motion of the Sun, the Moon, andthe Planets, London: Lund Humphries. Repr. Sources in the History ofMathematics and Physical Sciences 5, New York/Heidelberg/Berlin: Springer-Verlag.Ross, D. (1955) Aristotle’s Physics: A Revised Text with Introduction andCommentary, Oxford: Clarendon Press.Toomer, G.J. (1984) trans. Ptolemy’s Almagest, New York/Berlin: Springer-Verlag.SECONDARY LITERATUREBooksAaboe, A. (1964) Episodes from the Early History of Mathematics, NewMathematical Library 13, Washington, DC: Mathematical Association ofAmerica.Barbera, A. (1990) ed. Music Theory and its Sources: Antiquity and the MiddleAges, Notre Dame, IN: University of Notre Dame Press.Bowen, A.C. (1991) ed. Science and Philosophy in Classical Greece, Institute forResearch in Classical Philosophy and Science: Sources and Studies in theHistory and Philosophy of Classical Science 2, New York/London: GarlandPublishing.Dijksterhuis, E.J. (1987) Archimedes, trans. by C.Dikshoorn with a newbibliographic essay by W.R.Knorr, Princeton: Princeton University Press.Dillon, J.M. and Long, A.A. (1988) eds. The Question of ‘Eclecticism’: Studies inLater 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 NotreDame 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 theHistory of Mathematics and Physical Sciences 14, New York: Springer-Verlag.Heath, T.L. (1921) A History of Greek Mathematics, 2 vols, Oxford: ClarendonPress.Huby, P. and Neal, G. (1989) eds. The Criterion of Truth: Essays in Honour ofGeorge 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 mathematischenWissenschaften. 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 theHistory of Mathematics and Physical Sciences 1, Berlin/Heidelberg/New York:Springer-Verlag.Taub, L.C. (1993) Ptolemy’s Universe: The Natural Philosophical and EthicalFoundations of Ptolemy’s Astronomy, Chicago/La Salle: Open Court.Walbank, F.W. et al. (1984) eds. The Cambridge Ancient History. VII.1: TheHellenistic World, 2nd edn, Cambridge/London/New York: CambridgeUniversity Press.Wallace, R.W. and MacLachlan, B. (1991) eds. Harmonia mundi: Musica efilosofia 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: BritishMuseum Press/St Martin’s Press.Papers and reviewsBarker, A.D. (1991) ‘Reason and Perception in Ptolemy’s Harmonics’. See Wallaceand 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 CenturyBC as Found in Euclid’s Phaenomena’, Physis 39:7–33.Blumenthal, H. (1989) ‘Plotinus and Proclus on the Criterion of Truth’. See Hubyand 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 Sectiocanonis 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 theAmerican Philosophical Society 135:233–54.Britton, J.P. and Walker, C.B. F. (1996) ‘Astronomy and Astrology inMesopotamia’. See Walker 1996, 42–67.Gersh, S. (1992) ‘Porphyry’s Commentary on the “Harmonics” of Ptolemy andNeoplatonic 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 PracticingScientist’. 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’sHarmonics’. 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.