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Historical astronomy Ptolemy: on trial for fraud Ptolemy’s reputation as the greatest astronomer of antiquity has come into question since the 10th century. Norriss S Hetherington reviews the search for fraud in the Almagest and highlights other possible explanations for the anomalies that it contains. F stars” (Dreyer 1917, rom AD 150 until 1918). Strengthening the Copernicus, Ptolemy’s argument for actual meaAlmagest dominated astsurements by Ptolemy was an ronomical science. His reported analysis in 1925 comparing Ptoleobservations, however, match corremy’s and Hipparchus’ data. Discrepsponding theory so closely that the ancies between the Ptolemaic and HipFrench astronomer Delambre asked: “Did parchian longitudes were judged to be “too Ptolemy observe? Are the observations which great to be explained on the assumption that he claims to have made not calculations by Ptolemy simply copied Hipparchus’ catalogue means of tables, and examples intended for a with an additional constant” (Vogt 1925; Pedbetter understanding of his theories?” (Delamersen 1974). This conclusion would much later of stars bre 1817). Furthermore, a systematic error in The title be disputed (Graßhoff 1990), but it still stood in HipparPtolemy’s star catalogue led Arab astronomers page of unchallenged in the 1970s. Then a geophysichus’ lost catas early as the 10th century to suspect that Ptolemy’s cist, Robert R Newton, charged that Ptolemy, alogue came to Ptolemy had taken over the results of a pregreat star catala hero to many as the greatest astronomer of about 850. Ptoldecessor, a suspicion shared by Tycho Brahe in ogue, the Almagest. the ancient world, had committed a crime emy’s catalogue, howthe 16th century, by Lalande in the 18th cenagainst his fellow scientists and scholars, ever, contains 1022 stars. Dreyer and others tury, and also by Delambre. Was Ptolemy the betraying the ethics and integrity of his profesconcluded that more stars must have been greatest astronomer of antiquity, or the greatsion. The Almagest had “done more damage to added after Hipparchus, presumably by Ptolest fraud in the history of science? astronomy than any other work ever written, emy. The possibility that 850 stars was only a Ptolemy’s reported longitudes for each star’s and astronomy would be better off if it had lower limit did not arise in scholarly arguments position are too small by about 1°, but his latinever existed. Thus Ptolemy is not the greatest until much later (Goldstein 1982) and thus did tudes are basically correct. Precession – a slow astronomer of antiquity, but he is something not slow Dreyer’s speculative momentum. change in the orientation of the Earth’s axis of still more unusual: he is the most successful Dreyer next tackled the systematic error in rotation – changes the longitude of stars over fraud in the history of science” (Newton stellar longitude. Ptolemy had measured his time. Believing that the change in longitude 1977). The crime was all the worse because longitudes relative to a fundamental star, was 1° per century, Ptolemy might have added Ptolemy’s encyclopaedic summary had which in turn was measured relative to the 22⁄3° to each of Hipparchus’ longitudes for the replaced much of previous astronomy, thereby Sun. Since Ptolemy’s solar theory was incorrect 22⁄3 centuries between 129 BC and AD 137, forever depriving science of fundamental inforin a manner that resulted in too small a solar the epoch for which Ptolemy’s catalogue was mation about an important area of astronomy longitude by about 1°, Dreyer showed, all of prepared. The actual amount of precession, and history. The charge of fraud, though not the star longitudes would be off by the same however, was 32⁄3°. Hence Ptolemy’s longitudes new, was worded more emphatically than hithamount. Dreyer now “found no reason to diswould be 1° too small, as in fact they are, on erto and commanded new attention. believe his [Ptolemy’s] positive statement, that average (Evans 1987a,b). J L E Dreyer initially Newton had been investigating ancient he had made extensive observations of fixed accepted this explanation, in 1906 writing that observations of the Ptolemy’s star cataMoon’s position. logue was “nothing he astronomical data in Ptolemy’s explanation, careful examination of the There is a small accelbut the catalogue of eration in the motion Hipparchus brought great work, the Almagest, has data suggests that a combination of of the Moon in its down to his own time raised suspicions in the minds of observational, calculation and orbit around the with an erroneous scholars for some 900 years. Did rounding errors, plagiarism of existing Earth and also a slowvalue of the constant Ptolemy make all or any of the data and selection of the best ing down in the of precession”. ManuEarth’s spin about its scripts attributing to observations, did he take them from examples to support Ptolemy’s new axis separate from Hipparchus a Greek existing catalogues, or are they solely theories could account for the that strictly predicted list of constellations the results of calculations? Scientific appearance of fraud. But in the from gravitational soon caused Dreyer fraud is the charge levelled at absence of categorical evidence, the theory. The cause is to re-examine the case tidal friction between against Ptolemy. EstiPtolemy. Although this is a possible verdict must remain “not proven”. the seas and the solid mates of the number T 24 April/May 1997 Vol 38 Issue 2 Historical astronomy Earth, slowing the Earth’s axial spin; conservation of angular momentum in the Earth–Moon system is conserved as the Moon speeds up. A comparison of ancient observations with extrapolations from modern theory offers, potentially, a powerful means of studying the acceleration of the Moon and the retardation of the Earth’s rotation over time (Newton 1970). Newton might have saved himself much effort and aggravation, however, had he been aware of earlier studies finding suspiciously close agreement between theory and observation in the Almagest. Subsequently, Owen Gingerich found with regard to Mars that “although Ptolemy’s stated observations are not particularly good…they do match his tables in an uncanny way. In fact, this situation prevails throughout the entire Almagest, which leads to the suspicion that something is going on that does not meet the eye, something craftily concealed in the writing of the Almagest” (Gingerich 1983a). Also with Mercury, a close examination finds agreement between theory and observation too good to be true (Gingerich 1983b). Ptolemy seemingly fabricated his stated observations, but the agreement between his numerical parameters and modern values is too close to be fortuitous. the 18 easily recognizable stars that he listed in the Almagest (VII, 3) with positions recorded by Timocharis, by Hipparchus and by himself, he would have found approximately the modern value. He reported, however, precession of 1° in about 100 years, as Hipparchus had determined earlier. Next Ptolemy showed how 6 of the stars supported this conclusion, without mentioning the other 12. Seemingly, the fact that a few cases fitted well made them appear more important, and was sufficient to provide confidence that the theory was right cations” were “understood by him as little fibs allowable in the neatening up of his pedagogy”, not “as lies intended to mislead his readers about matters he regarded as crucial” (Wilson 1984). “Here we must remember, above all, that the Almagest is not like a modern research paper. It is a well-honed textbook, modelled as much as applicable on Euclid’s Elements. It was no doubt Ptolemy’s intention to present the procedures with sample data so that future astronomers could see how it was done and could introduce their own A new type of science There are several possible explanations. He likely had a large number of observations – either his own or borrowed without acknowledgement, from Hipparchus or from someone else – and the probable errors largely cancelled each other out in calculations of parameters (Britton 1992). Next, Ptolemy might have selected from among his observations a few in best quantitative agreement with the theory and then presented these examples to illustrate the theory. To the ancient Greeks it may not have appeared as paradoxical as it now does to test a theory empirically with a measurement chosen for its best fit with the theory. Ptolemy and his contemporaries lacked our modern understanding of error ranges, standard deviations and the use of mean values from repeated observations – concepts that would have enabled Ptolemy to propound a theory not necessarily in total and absolute agreement with every data point obtained, but in less strict agreement with all of the data points to within a statistically defined interval around a mean value. Instead, without any tolerable fluctuation in the agreement between theory and observation, every measurement would be understood as an exact result. Consequently, judicious selection from among many measurements was required (Graßhoff 1990). Selective use of data by Ptolemy is especially evident with regard to precession: the movement of the equinoxes relative to the stars. Had Ptolemy calculated the precession from all of April/May 1997 Vol 38 Issue 2 This illustration from Gregor Reisch’s “Margarita Philosophica...”, printed in Basel in 1508, shows Ptolemy crowned and in contemporary dress, using a quadrant to measure the elevation of an astronomical object, accompanied and encouraged by a woman labelled “Astronomia”. Note the armillary sphere in the corner. (Pannekoek 1955). In attempting to demonstrate a new type of science, converting observational data into numerical parameters of geometrical models, Ptolemy may have been led to present “an ‘idealized’ account of the way an astronomer operates starting from observations” (Toomer 1980). He was “fibbing about what he observed in order to make the agreement between theory and phenomena appear perfect when it was not”. His “fudging and fabri- observations over a longer temporal baseline in order to derive even better parameters. For such a purpose he surely wanted to exhibit mutually consistent observations, the best he could find.” Also, “anxious to show how the model might in principle be most easily justified, and keen not to perpetuate data with observational errors, he must have also tried in some way to clean up his reported positions” (Gingerich 1983a). Appreciations of Ptolemy’s possible didactic 25 Historical astronomy Expected distribution of fractional values in the Almagest intent and the difficulty of reconciling theory and observation in ancient times were stimulated by Newton’s angry damnation of the Almagest. Another important contribution from Newton is the attention he focused on the distribution of fractional values in the measurements of stellar positions (Newton 1979), discussed in the box above. Nothing in the distribution of fractions in latitude has raised any suspicion among historians or astronomers in search of criminal activity. The distribution in longitude is another matter. Ptolemy reported 226 stars at 0 fractional degree, 182 at 1⁄6°, only 4 at 1⁄4°, 179 at 1 ⁄3°, 88 at 1⁄2°, 246 at 2⁄3° (40 min), no stars at all at 3⁄4° and 102 stars with a fractional 5⁄6° measurement. How else can we account for the peak at 2⁄3°, other than to call it evidence of a Hipparchian peak at 0 min to which Ptolemy, to update the catalogue for precession, added 22⁄3°? The other peak, originally at 1⁄2°, would be shifted to 1⁄6°. Furthermore, if we shift by 2 /3° (40 min) the original 1⁄4° and 3⁄4° categories (15 and 45 min), they become 55 and 25 min. However, these slots are not represented by any of the sixth or quarter fractions. Hence these occurrences now might be shifted into a neighbouring slot. Adding the 55 s (original 15 s) to the zero category and the 25 s (original 45 s) to the 20 slot (1⁄3°) produces numbers nearer to what Ptolemy reported. (Of course, there is no compelling logic that would have prevented Ptolemy from rounding the values down and up instead, or half up and half down 26 250 no. of stars In the Almagest, Ptolemy reported the latitudes and longitudes of stars as so many degrees plus fractional parts of a degree, the fractions being 0°, 1⁄6°, 1 ⁄4°, 1⁄3°, 1⁄2°, 2⁄3°, 3⁄4° or 5⁄6° (corresponding to 0, 10, 15, 20, 30, 40, 45 and 50 min). This reportorial format suggests that the measurements might have been made with two different instruments, one calibrated in sixths of a degree and the other in quarters (perhaps Ptolemy’s, since there is only one bright star, Pollux, among these presumably later additions, and also Ptolemy’s reputation as an accurate observer falls below that of his predecessor, primarily because Ptolemy had a less accurate knowledge of the constant of precession than did Hipparchus). Alternatively, using a single instrument calibrated only in 1⁄2° increments, an observer could have reported as 1⁄6° those stars falling near the zero mark, as 1⁄3° those stars near the 1⁄2° mark, and as 1⁄4° those stars nearly in the centre of the space between 0 and 1⁄2°, and similarly extrapolated for stars between 1⁄2° and the next higher degree. The following speculation allows for one instrument or two. It is equally likely, with a random distribution of stars, for stellar coordinates to fall at each equal interval, say of 5 min. However, with an instrument or instruments calibrated to measure only the random 1 /6 ° divisions Ptolemy 300 200 150 100 50 0 0 10 20 30 40 minutes 50 60 The expected effect of using an instrument or instruments calibrated in sixths or half degrees. above fractions (0°, 1⁄6°, 1⁄4°, 1⁄3°, 1⁄2°, 2⁄3°, 3⁄4°, and ⁄6°), stars at 5, 25, 35 and 55 min (the 5 min intervals not represented by one of these fractions) would not be measured as such. Instead, the observer(s) might round such occurrences up or down into an adjoining category. Thus any star with a true coordinate of 5 or 55 min might have a 50:50 chance of finding itself assigned a value of zero degrees (either 0 or 10 for the non-reported 5 min mark and either 0 or 50 for the non-reported 55 min mark), with the result that we would expect twice as many stars with a longitude of 5 for each.) The new 1⁄4° and 3⁄4° categories were produced, so the speculation goes, by shifting by 40 min the original 35 and 5 min slots, which were “forbidden levels” with no members. Ptolemy reports only 4 stars with a fractional longitude of 1⁄4° and none at 3⁄4°. The other original forbidden levels, 25 and 55 min, are shifted to 5 and 35 min, also forbidden categories, hence also unreported, and not in need of explanation. Shifting reference stars The distribution of fractional longitudes in Ptolemy’s star catalogue is suggestive of plagiarism. However, Ptolemy might actually have made his reported measurements against a reference star, later found that his longitude for the reference star was short by an amount including a fractional part of 2⁄3° (or too much by 1⁄3°), and then added this 2⁄3° to (or subtracted 1⁄3° from) all of his genuine measurements. Alternatively, perhaps he only determined the absolute longitude of the reference star after making the relative measurements, and it had a fractional part of 2⁄3°, which he then added to each star’s relative longitude. Indeed, three (Spica, Aldebaran and Antares) of the four bright stars (Regulus is the other) near the ecliptic that Ptolemy might conveniently have used as reference stars have fractional 2⁄3° longitudes in the Almagest. The addition of the fractional longitude(s) of one or more reference star, Evans notes, can account for the observed some number of degrees and no fractional part to be reported, as otherwise would have been. Similarly, the non-reported 25 and 35 min coordinates, with a 50:50 chance of being rounded up to 1⁄2° or down to 1⁄2°, would double the 1⁄2° category. The 10, 20, 40 and 50 min categories (1⁄6°, 1⁄3°, 2⁄3° and 5 /6°) would receive rounded up or down measurements from one category each (from 5, 25, 35 and 55 min), increasing their membership by half. Hence, if the instrument(s) gave readings only in sixths and quarters of a degree, we could expect peaks in the distribution of the reported measurements at 0 and 1⁄2°. Furthermore, we would expect minimums at 1⁄4° and 3⁄4° (these latter two categories, corresponding to 15 and 45 min, receiving no rounded up or down additions, because both adjoining 10 and 20 min categories and 40 and 50 min categories are represented by fractional measurements). Quantitatively, starting with, say, 100 stars in each 5 min interval, we would expect to see reported 100 each at 1⁄4° and 3⁄4°, 150 each at 1⁄6°, 1⁄3°, 2 ⁄3° and 5⁄6°, and 200 each at 0 and 1⁄2°. This expectation is realized, in a general way, in the reported latitudes in Ptolemy’s star catalogue in the Almagest. There are 236 stars with measurements of 0 fractional degrees and 198 stars with measurements including 1⁄2°. There are only 88 stars at 1⁄4° and 50 at 3⁄4°. At 1⁄6°, 1⁄3°, 2⁄3° and 5⁄6° there are 106, 112, 129 and 107 occurrences respectively. distribution of fractions in longitude without necessarily denying the genuineness of the reported observations (Evans 1987b). More evidence in favour of Ptolemy comes from further analyses subsequent to those of Evans, which suggest that the stars were observed by constellation in distinct observational sequences connected with different seasons of the year. A constant longitudinal shift in constellations from Cancer to Sagittarius could be the result of using a specific reference star (probably Spica), while larger differences in the values of systematic errors for the spring hemisphere hints at the employment of more reference stars. The fraction 40 min predominates overall but not for the five constellations from Cancer to Scorpio, and for northern but not southern constellations, arguably ruling out the transformation of data en bloc from Hipparchus to Ptolemy (Shevchenko 1990; Wlodarcyzk 1990). Ptolemy’s data were transformed en bloc. The star catalogue of the Arab astronomer al-Sûfi, composed in around 964, reports parts of a degree in latitude in the minute marks corresponding to sixths and quarters of a degree, with peaks at 0 and 1⁄2. However, longitudes are reported in minutes as 2, 12, etc (10n + 2), with the main peak at 22 min. The catalogue is simply the Almagest with 12°, 42 min added to all of the longitudes (Evans 1987b). There is also reason to suspect that the star catalogue of Ulugh Beg, the grandson of Tamerlane and ruler of an empire from Samarkand between April/May 1997 Vol 38 Issue 2 Historical astronomy 1409 and 1449, which reported longitudes with peaks at 25 and 55 min, was plagiarized from an earlier one, now lost. However, considerable and detailed information is available regarding Ulugh Beg’s observatory, not only from written records but from archaeological excavation as well, concerning when it was constructed, who observed there and even some of the instruments. Furthermore, no earlier star catalogue that could have been the source for Ulugh Beg’s is known. Suppose that instead of being borrowed from an earlier source, most of the reported longitudes actually were observed in around 1443, and then 5 min was subtracted for 6 years of precession to set the catalogue at the epoch of 1437. Evans concludes that, “when plausible explanations less drastic than imputations of deception and plagiarism are available, some temperance of judgement is called for” (1987a). Alexandria or Rhodes? Another set of data requiring judgement is the distance of stars from the horizon. Not a single star listed in Ptolemy’s star catalogue was invisible from Rhodes, the home of Hipparchus and some 5° north of Alexandria, the purported observational homebase for the Almagest. An astronomer in Alexandria might have been expected to observe stars closer to his extended southern horizon. A detailed statistical analysis, assuming that ancient astronomers attempted to observe right down to the horizon, estimates the probability of any individual star being included in the catalogue as a function of its apparent magnitude, carefully corrected for atmospheric refraction and atmospheric extinction. Dennis Rawlins found a 90% probability that Hipparchus could have compiled the catalogue, but only 1 chance in 1013 that Ptolemy could have (Rawlins 1982). The probability number is impressive, but the premises on which it is based are not beyond dispute. Other astronomers, including Tycho Brahe, did not observe all the way down to the horizon. Also, the frequency function probably should be higher for stars in constellations and lower for stars not so singled out for the observer’s attention, as indeed is the case in Tycho’s work. Furthermore, the conclusion is drawn from data on only 11 stars (Evans 1987a). Another facet of the debate over the relative likelihood of Rhodes or Alexandria having been the observing site for Ptolemy’s star catalogue involves atmospheric extinction. This increases greatly near the horizon, causing a star near the horizon to appear much dimmer to an observer at Rhodes than at Alexandria (with its horizon extended 5° farther south). Estimating amounts of atmospheric extinction and comparing the predicted result with modern measurements of magnitude, Evans has April/May 1997 Vol 38 Issue 2 found that the magnitudes reported by Ptolemy of the six southernmost stars are consistent with having been observed in Alexandria in Ptolemy’s time, and five of the six stars have magnitudes inconsistent with having been observed from Rhodes in Hipparchus’ time. “On the basis of the magnitudes assigned to the southernmost stars of the catalogue [Ptolemy’s], it appears more likely that the ancient observer was located at Alexandria than at Rhodes” (Evans 1987b). In yet another recent and ingenious addition to the debate, Rawlins has analysed the manner of observing with an armillary sphere and the consequences if the ecliptic of the armillary sphere is set slightly out of the plane of the true ecliptic. Given the mean error of about 1° in longitude in Ptolemy’s observations, whatever its cause, Ptolemy would have misaligned his armillary sphere accordingly. This misalignment would be reflected in periodic errors in the measurements. Indeed, Rawlins insists that had Ptolemy actually made his observations as stated, with an armillary, there would be systematic error-waves of half a degree in his reported latitudes. Yet there are not (Rawlins 1982). We cannot be sure, however, that our hypothetical measuring procedure corresponds completely to that of the actual observer. Indeed, the small periodic error present in Ptolemy’s latitudes, while not consistent with observations referred to the equinoctial point, is not inconsistent with observations referred to a particular reference star, especially if we further fiddle with the supposed procedure for orienting and reorienting the instrument during the process of observing. Even if the process of observation was the very pure and simple one described by Ptolemy in the Almagest, a near infinitude of possible sources of periodic errors, both in the construction of the instrument and in its use, forestalls any certain conclusion that the reported body of data was not observed but calculated (Evans 1987b). Indeed, another scholar concludes from his study of observing techniques with an armillary astrolabe that the description of his observations given by Ptolemy is intrinsically consistent and could correspond to the real situation (Wlodarcyzk 1987). Ptolemy did have an incorrect value for the latitude of Alexandria, which could have contributed to a periodic error and also suggests plagiarism. He was off by 15 min of arc, an error several times as large as one would expect had Ptolemy actually measured the latitude by the method he described in the Almagest. Newton concludes that “there is little doubt that he [Ptolemy] simply adopted the obliquity used by Eratosthenes, with no attempt to verify it independently” (Newton 1980). An alternative explanation is that Ptole- my, in calculating the latitude of Alexandria, observed the zenith distance of the Sun at a time other than noon and then attempted to calculate the corresponding correction for the Sun’s change of position from that time until noon, but mistakenly added rather than subtracted the “correction” in his overall calculation of the latitude based on the supposed zenith distance of the Sun at noon (Britton 1969). Ptolemy’s reported observation of an equinox, in precise agreement with that extrapolated from an observation by Hipparchus and 30 h later than its actual occurrence, cannot be passed off as an innocent error (Gingerich 1980). In recent years, much has been learned regarding Ptolemy, with new and ingenious suggestions fitting the known facts frequently enlivening the debate. For every ostensible conclusion put forward, however, one or more possible, and too often plausible, alternative explanations has risen in response. None has yet proved decisive. Success has been measured in the indecisive widening and deepening of our understanding. ● Norriss S Hetherington is at the Office for the History of Science and Technology, University of California at Berkeley, 543 Stephens Hall #2350, Berkeley, California 94720-2350. References Britton J P 1969 Centaurus 14 29–41. Britton J P 1992 Models and Precision: the Quality of Ptolemy’s Observations and Parameters (New York and London). Delambre J-B J 1817 Histoire de l’Astronomie Ancienne, vol. I (Paris; New York, 1965), quoted, in English translation, in Pedersen (1974). Dreyer J L E 1906 History of the Planetary Systems from Thales to Kepler (Cambridge); 2nd edn, 1953, revised by William Stahl, with a supplementary bibliography, and retitled A History of Astronomy from Thales to Kepler (New York) p202. Dreyer J L E 1917 MNRAS 77 528–539. Dreyer J L E 1918 MNRAS 78 343–349. Evans J 1987a J. Hist. Astr. 18 155–172. Evans J 1987b J. Hist. Astr. 18 233–278. Gingerich O 1980 QJRAS 21 253–266. Gingerich O 1983a Ptolemy, Copernicus, and Kepler The Great Ideas Today M J Adler and J Van Doren (eds) (Chicago) pp137–180; reprinted in Gingerich O 1993 The Eye of Heaven: Ptolemy, Copernicus, and Kepler (New York) pp3–51. Gingerich O 1983b Sky and Telescope 66 11–13. Goldstein S J 1982 J. Hist. Astr. 13 195–205. Graßhoff G 1990 The History of Ptolemy’s Star Catalogue (New York and Berlin). Newton R R 1970 Ancient Astronomical Observations and the Accelerations of the Earth and Moon (Baltimore). Newton R R 1977 The Crime of Claudius Ptolemy (Baltimore). Newton R R 1979 QJRAS 20 383–394. Newton R R 1980, QJRAS 21 379–387. Pannekoek A 1955 Vistas in Astronomy 1 60–66. Pedersen O 1974 A Survey of the Almagest (Denmark) Acta Historica Scientiarum Naturalium et Medicinalium (Edidit Bibliotheca Universitatis Haunienis) 30 23. Rawlins D 1982 Publ. Astr. Soc. Pacific 94 359–373. Shevchenko M 1990 J. Hist. Astr. 21 187–201. Toomer G J 1980 Centaurus 24 97–109. Vogt H 1925 Astronomische Nachrichten 224 17–62. Wilson C 1984 J. Hist. Astr. 15 37–47. Wlodarcyzk J 1987 J. Hist. Astr. 18 173–195. Wlodarcyzk J 1990 J. Hist. Astr. 21 283–295. 27