THE EPOCH OF THE CONSTELLATIONS ON THE FARNESE ATLAS AND THEIR ORIGIN IN HIPPARCHUS’S LOST CATALOGUE

BRADLEY E. SCHAEFER, Louisiana State University, Baton Rouge

1. BACKGROUND

The Farnese Atlas is a Roman statue depicting the Titan Atlas holding up a celestial globe that displays an accurate representation of the ancient Greek constellations (see Figures 1 and 2). This is the oldest surviving depiction of this set of the original Western constellations, and as such can be a valuable resource for studying their early development. The globe places the celestial figures against a grid of circles (including the celestial equator, the tropics, the colures, the ecliptic, the Arctic Circle, and the Antarctic Circle) that allows for the accurate positioning of the constellations. The positions shift with time due to precession, so the observed positions on the Farnese Atlas correspond to some particular date. Also, the declination of the Arctic and Antarctic Circles will correspond to a particular latitude for the observer whose observations were adopted by the sculptor. Thus, a detailed analysis of the globe will reveal the latitude and epoch for the observations incorporated in the Atlas; and indeed these will specify enough information that we can identify the observer. Independently, a detailed comparison of the constellation symbols on the Atlas with those from the other surviving ancient material also uniquely points to the same origin for the fi gures.

The Farnese Atlas¹ first came to modern attention in the early sixteenth century when it became part of the collection of antiquities in the Farnese Palace in Rome, hence its name. The statue was later transferred to the museum in Naples now called the Museo Archeologico Nazionale di Napoli. It is carved in white marble and depicts the bearded Atlas crouched down on one knee with a cloak over his shoulder and holding the celestial globe on his shoulder with both hands. The globe is 65cm in diameter. Its top has a substantial hole knocked into it and this has obliterated the constellations of Ursa Major and Ursa Minor. A total of 41 constellations² are depicted, each drawn in positive relief as the classical figure, with no individual stars shown. Art historians conclude that the statue is a Roman copy from the second century A.D. of a Greek original dating to before the birth of Christ.³

What is the date of the observations used for depicting the constellation positions on the Farnese Atlas? A very wide range of plausible answers is possible. First, the Roman sculptor could have updated the constellation positions with his own observations (or those of a contemporary), hence suggesting a date of c. 150 A.D. Second, the Roman sculptor could have used the latest star catalogue to place the constellations accurately onto the coordinate grid of the sky, and this would be from the Almagest of Ptolemy,

0021-8286/05/3602-0167/$10.00 © 2005 Science History Publications Ltd

FiG. 1. The Atlas holds the celestial globe on his shoulders. In this fi gure, we see the colure dividing the sky between Canis Major and Argo. This one observation immediately tells us that the constellations were placed to represent the sky as it was around the time of Hipparchus. Photograph by Dr Gerry Picus, Griffi th Observatory.

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FIG. 2. The rear side of the globe. We see the equator, the ecliptic, the equinoctial colure, and the two tropics. The edge of the horn of Aries is exactly on the colure, and this immediately tells us that the constellations were placed for around the time of Hipparchus. Photograph by Dr Gerry Picus, Griffi th Observatory.

suggesting a date of c. 128 A.D. Third, art historians all point to the original Greek sculptor as using the constellations based in Aratus’s poem Phaenomena, which has a date of c. 275 B.C. An origin with Aratus was the dominant opinion amongst scholarly publications in the last century. Fourth, we know that Aratus’s work was substantially a copy of an earlier book of the same name by Eudoxus with a date of c. 366 B.C. Fifth, a precessional dating of 172 lore items derived from Eudoxus’s book proves that all of the lore actually dates from 1130 ± 80 B.C.⁴ So we are left with many candidates, all reasonable, for the date of the observations used to place the constellations: from 1130 B.C. to A.D. 150.

The possibility of deriving a date and latitude from the Farnese Atlas has not been lost on earlier researchers. E. L. Stevenson claims (purely on the basis of the positions of the solstices) that the constellations date “at least three hundred years before the Christian era”, while C. Gialanella and V. Valerio as well as M. Fiorini agree that the constellation positions suggest a date in the fourth century B.C., although Valerio later changed the date to A.D. 150. Thiele points to an origin by Hipparchus and Eudoxus based on stylistic considerations, and he also points to a latitude of 23° which he specifies as being greatly different from that of Rhodes. For the latitude of the observer, Fiorini gives 40° and points to Macedonia, Gialanella and Valerio give 32° and point to Alexandria, while Valerio later gives 33.5° and points to Middle Phoenicia. For the obliquity of the ecliptic, we hear values of 23° from Fiorini, 25.5° from Gialanella and Valerio, 25° from Valerio, and 24° from G. Aujac. Throughout all the few sentences of discussions by the various authors, no evidence is ever presented, nor are methods of measurement ever discussed, nor is any indication of the accuracy of the claims ever made. With the total lack of these crucial details, we cannot take these off-hand claims seriously. In her appraisal of this situation, Aujac concludes: “A critical review of existing studies of the globe, together with detailed reproduction and careful analysis, is urgently needed to resolve these questions.”

This paper will provide the requested analysis. I start with a detailed analysis of the constellation symbols and descriptions on the Farnese Atlas as compared to all relevant surviving sources from Antiquity. In addition, I have taken detailed photographs of the Farnese Atlas under conditions for which photogrammetry can be performed. From these photographs, I have measured the positions of the constellations in the coordinate system of the globe. I have then performed a chi-square analysis to determine the best date (as well as the uncertainties in this value) for the constellation positions. In all, my results will point to the source of the observations with high confi dence.

The plan of this paper is to start in Section 2 by making a detailed comparison of the symbols on the Farnese Atlas with those from all other ancient sources. This analysis will include detailed comparison of descriptions of each constellation as compared with the works of Aratus (from the Phaenomena, and hence also Eudoxus), Hipparchus (from his Commentary on the Phaenomena of Aratus and Eudoxus), Ptolemy (from the Almagest), and Psuedo-Eratosthenes (from the Catasterismi and hence also Eratosthenes). In Section 3, I will present the results of my extensive photogrammetry, such that I will derive a very confident date for the original observations used to place the constellations onto the globe. This section will also give the declination of the Arctic and Antarctic Circles on the globe and discuss the implications for the latitude of the observer. A third part of Section 3 will discuss the accuracy of the placement of the constellations and the implications for the source of the original observations. Section 4 will put all the results together and a strong case will be made for the identification of the original observer. Section 5 will discuss some implications and applications of this conclusion. Section 6 will summarize the conclusion. This paper has an extensive Appendix, into which I have placed all the technical details relating to the photogrammetry. The goal is to provide complete details so that readers can test my results or perform their own analyses.

2. DETAILED COMPARISON OF CONSTELLATION FIGURES WITH ANCIENT SOURCES

The Atlas’s globe contains a wealth of information in the form of constellation fi gures. From a detailed comparison of these symbols with all known sources describing the ancient constellations, we might be able to select and/or eliminate possible sources for the sculptor. Here, I will make this comparison with the works of Aratus (and hence Eudoxus), Hipparchus, Pseudo-Eratosthenes (and presumably Eratosthenes), and Ptolemy.

As a preview, I will highlight several of the more distinct differences. First, the Farnese Atlas is completely missing the later Greek constellations of Equuleus, Coma Berenices, and Antinous. Second, Hercules is depicted as a kneeling man with no clothes and no objects in his hand instead of as the Greek hero. These two facts suggest that the source of the constellations was not in later Greek times. Third, the modern constellation of Libra is depicted as a separate balance, even though the claws of Scorpius extend up to the balance. Fourth, the summer solstice is depicted as being at the start of Cancer near the head of Pollux, in stark contrast to the tradition from Aratus and Eudoxus that the solstice is near the start of Leo. These last two items suggest that the source is after the time of Aratus.

Before I perform the detailed comparison with the individual books, I will give a list of the details that are different from all ancient sources. These details then cannot be useful evidence against any one source, but rather point to changes incorporated after the data left their source (e.g., by the sculptor). Here are the universal differences: (1–2) The horn of the Bull does not touch the foot of the Charioteer, and the head of Andromeda does not overlay the navel of Pegasus, at odds with all ancient descriptions of these constellations. Likely, the sculptor avoided the overlap simply for artistic reasons and clarity. (3) The curious rectangular feature above Cancer corresponds to nothing recorded in any of the ancient sources, and is undoubtedly a later addition by either the Greek or Roman sculptor. (4–5) The globe does not depict Sagitta or Triangulum, whereas every ancient source explicitly discusses both. These are inconspicuous constellations in crowded areas, so perhaps their absence is just an artistic decision by the sculptor. (6) The globe does not depict Ursa Minor, although this is attested by all ancient sources. But the Little Bear should be on the very edge of the hole in the top of the globe, so there is a small chance that the fi gure was in the hole and that there is some other error relating to the positioning near the north pole (cf. Section A.2.2). (7) The ecliptic crosses the equator 5° west of the colure lines. This arrangement is wrong by definition, as precession moves the sky along the ecliptic, suggesting that the sculptor made the change because of his lack of astronomical knowledge. (8) The string attached to the northern fi sh of Pisces is missing, although the string is present in all ancient sources. (9) Sagittarius appears to have a bare back, even though the Almagest and Commentaries talk about the cloak-strap. Since all nine differences are universal, they cannot be used to point towards or away from any one source. Instead, these differences indicate changes made after the information had left the astronomer, likely by the sculptor. Similarly, the universal similarities between the globe and all ancient sources (e.g., that only the stern of Argo is depicted) cannot be used to distinguish the origin of the fi gures. Thus, the only data relevant for determining the origin of the figures are the differences between the globe and sources, which vary with the source.

The Phaenomena of Aratus was a popular description of the constellations; it dates to around 275 B.C, and is the earliest surviving discussion of the ancient Greek constellations.⁵ Its popularity served to freeze the development of the constellations as well as to define the basic properties of the group. The text is largely a version of an earlier (now lost) book of the same name by Eudoxus from around 366 B.C., with substantial further fragments from the work by Eudoxus appearing in the only surviving work of Hipparchus (his Commentary). The Phaenomena gives descriptions of the constellation figures and also tells how these figures relate to the various circles on the sky. A detailed comparison with the Farnese Atlas shows many differences: (1) Aratus stated the summer solstice to be at the start of Leo, whereas the statue shows it to be near the head of Pollux with the entire constellation of Cancer between. (2) The human part of the Centaur is said to be under Scorpius, whereas the statue places it under Virgo, with the entire constellation of Libra between. (3) Eridanus is said to stop at the neck of Cetus while the stars under Lepus are said to be nameless, whereas the Farnese Atlas shows Eridanus as extending all the way through Columba over to the feet of Canis Major. (4) Aratus says that Auriga has the Goat and Kids, whereas the globe in Naples shows the Charioteer instead to be holding a whip. (5) Serpens is said to encircle the waist of Ophiucus, contrary to what is seen on the statue. (6) The constellation of Libra is depicted as a balance on the globe, but Aratus simply calls it “The Claws”. (7) The knees of the Charioteer are said to be on the Tropic of Cancer, whereas the globe shows even his feet to be north of the tropic. (8) The left shoulder and shin of Perseus are said to be on the Tropic of Cancer, whereas on the statue all of Perseus is north of the tropic. (9) The head and neck of Cygnus are said to be on the Tropic of Cancer, whereas the sculptor depicts the constellation with the beak tip touching the tropic. (10) Ophiucus’s shoulders are said to be along the tropic, whereas the globe shows the top of his head to be there. (11) The knees of Ophiucus are not on the equator as in Aratus, but are depicted as being substantially south of the equator. (12) The belt of Cepheus is said by Aratus to be on the Arctic Circle, whereas the Atlas has the neck of Cepheus so drawn. There are many further discrepancies for which the case is less clear, for example Aratus says that the head of Draco is on the Arctic Circle and that Crater and Corvus are on the equator, while the globe shows the head only as being near the Arctic Circle and the raven and cup as only tangent to the equator.

Hipparchus was perhaps the greatest astronomer of Antiquity, his outstanding discovery being the precession of the equinoxes. He is said to have compiled a large star catalogue with at least hundreds of stars, although this catalogue has not survived. Indeed, only one of Hipparchus’s works has survived, the Commentary, which gives extensive quotes from both Eudoxus and Aratus.⁶ The thrust of the Commentary is to correct perceived errors in the Phaenomena by quoting the author’s own data and interpretation of the constellation figures. As a result, we can make a detailed comparison of Hipparchus’s personal descriptions with the Farnese Atlas. First the differences between Hipparchus and the globe: (1) The head of Pollux is close to the Tropic of Cancer (and might even lie upon it), whereas Hipparchus says that the heads of both Twins are north of the Tropic and he even says that Pollux is north of the Tropic by 6°. But that is all: I can find only one non-universal difference between the globe and the Commentary. This difference is only quantitative, and it is only 1.7-σ in error (cf. Section A.2.1), which is not adequate grounds to claim a significant discrepancy. That is, the small shift of Gemini is typical of the scatter in the placement of the constellations, and so this is not grounds for considering the misplacement to be indicative. Another potential difference is that the globe depicts the modern Libra as a balance with the scorpion’s claws extending into it, while Hipparchus usually calls the constellation “The Claws”; however, in one place Hipparchus does recognize the constellation as “The Balance” (Commentary 3.1.5), so this cannot be regarded as being a true difference.

Now let us examine items in which the globe matches details particular to Hipparchus: (A) Hipparchus explicitly corrects Aratus’s claim about the position of the Centaur (see item 2 above), and Hipparchus’s claim is matched by the depiction on the globe. (B–G) Hipparchus also explicitly corrects Aratus’s items 7–12 above, and these corrections are entered onto the globe. (H–J) Hipparchus explicitly corrects the Phaenomena by saying that the head of Draco is only close to the Arctic Circle, and that Crater and Corvus are south of the equator; all these items are as represented on the globe. (K) Hipparchus states that the Arctic Circle is 37° from the north pole (i.e., at a declination of 53°), and this is reasonably consistent with my measured declination of 51.7° ± 0.9° for the Farnese Atlas. (L) Item 3 (from the penultimate paragraph) is apparently corrected to agreement with the Atlas, as Hipparchus states that Eridanus has a second westward segment. In all, the one difference is statistically insignificant while the dozen agreements between Hipparchus’s personal observations and the globe are close, many, and detailed. With this, we see that the comparison between the Atlas and Hipparchus is arguably perfect (other than the 9 itemized differences that are true for all ancient sources).

The Catasterismi that survives to today is an epitome from c. A.D. 100 (by an author known as Pseudo-Eratosthenes) of an original work of the same name by the famous Eratosthenes from c. 245 B.C.⁷ It is unclear what fraction of the surviving text is from Eratosthenes’s composition. The Catasterismi contains a verbal description of the myths and constellations, these being for the most part simply the usual descriptions such as are from Aratus. The Catasterismi does give some non-traditional details that match with the statue, in particular that (A) the Arctic Circle is along the neck of Cepheus, (B) the upheld wrist of Bootes is also along the Arctic Circle, and (C) Eridanus appears above Canopus (hence it must extend through Columba). The Catasterismi differs in many details from those visible on the Farnese Atlas:

(1) Auriga is not shown on the Atlas with either the Goat or the Kids. (2) The Asses are not shown on the west side of Cancer. (3) Hercules is not shown on the Atlas as standing. (4) Hercules is not shown as holding a lion skin. (5) Hercules is not shown as holding a club. (6) Pegasus is shown as having wings, despite the explicit denial in the Catasterismi. (7) The constellation Corona Australis is depicted on the globe but never mentioned by the Catasterismi. (9) The modern constellation of Libra is not separated out and is described as simply the claws of Scorpius, unlike what we find on the statue.

Ptolemy’s Almagest dates from c. A.D. 128 and contains a long catalogue of stars individually labelled by their positions in the constellations.⁸ These labels allow us to visualize the constellation figure for comparison with the Farnese Atlas. There are many differences between the Almagest and the Atlas: (1) Auriga is not shown on the statue as carrying the Kids. (2–4) The constellations of Equuleus, Coma Berenices, and Antinous are not depicted on the globe. (5) The Almagest identifi es the Asses near the middle of Cancer, while these are not displayed on the globe. (6) The modern constellation of Libra is called “The Claws” by the Almagest but is drawn as a balance on the Farnese Atlas. (7) Sagittarius does not have a cloak over the shoulders as stated in the Almagest. (8) In the Almagest Aquarius in not said to have a water jar, although this is clearly depicted on the statue. (9) Canis Major does not have a crown, as shown on the globe, although the crown may simply be a depiction of light rays from Sirius. (10) In the Almagest Eridanus turns south along the modern track near υ Eri, rather than extending to near the feet of Canis Major as shown on the Farnese Atlas. (11) Ptolemy explicitly assigns two legs to Cygnus, whereas the Atlas shows only one.

From this detailed analysis, we see that the Farnese Atlas is virtually identical to the constellation description by Hipparchus, yet is greatly different from the descriptions from all other ancient sources. This obviously strongly suggests that the ultimate source of the position information used by the original Greek sculptor was Hipparchus’s data, which must be closely related to his (now lost) star catalogue.

3. EPOCH AND LATITUDE

The constellations in the sky move slowly with respect to the declination circles and the colures, as a result of precession. The epoch for the observations that were incorporated into the Farnese Atlas is near that year for which the constellation positions on the globe most closely match those in the sky. The latitude of the observer is related to the declination of the Ant/Arctic Circles as presented on the globe. Both of these calculations require that we get the positions of the constellations in the reference frame of the globe’s coordinate system. In principle this could be performed by taking a tape measure to the globe, but in practice an equivalent method is to take pictures of the globe and then carefully measure the pictures.

I took 49 pictures of the Farnese Atlas in Naples on 1 June 2004. I used a digital camera that allowed for good recording of the details without special lighting. For photogrammetry, it is important to know the distance between the camera and the near surface of the globe, as this is required to transform positions on the photograph to spherical coordinates on the globe. (All previous published photographs were taken at unknown distances, and that is why a new set of photographs was required.) My photographs were all taken with the camera at a distance of either 6 or 20 feet from the surface of the globe. There was substantial duplication and some pictures were not useable for various reasons, so I ended up doing photogrammetry on twelve pictures. Detailed explanations and examples for my photogrammetry techniques are presented in Appendix 1. Detailed results and analysis of the constellation positions and the declinations of the tropics and Ant/Arctic Circles are presented in Appendix 2.

3.2. Latitude

Appendix 2 presents an analysis that derives the declination of the Ant/Arctic Circles to be ±51.7° ± 0.9°. This datum must be related in some sense to the latitude of the observer. There are three reasonable interpretations.

The obvious interpretation is that this value is 90° minus the observer’s latitude. This case is where a mathematically-inclined observer measured his own latitude and derived the position of the Ant/Arctic Circles as being that angle from both poles. Thus, the observer (whose report was used ultimately by the sculptor) was at a latitude of 38.3° ± 0.9°. This parallel cuts through the Straits of Messina, Athens, and central Turkey. This would likely rule out that the observer was in Mesopotamia (30°–36°), Alexandria (31.2°), or near Rome (41.9°). This latitude is consistent with an origin in classical Greece (36°–40°) as well as being not greatly inconsistent with Hipparchus in Rhodes (36.4°).

The second interpretation is that the depicted circles might be intended to match actual observations of the lowest declination where the stars never set and of the most southerly limits of visibility. For the Arctic Circle, the observer might have a northern horizon that is higher up than an ideal horizon or he might have adopted a visibility definition such that a star is circumpolar only if it is actually visible at its lower meridian passage. In the latter case, the adopted declination would be closer to the pole and lead to our deriving a latitude that is too far north, and hence the latitude of the observer might be closer to 34°. For the Antarctic Circle, the effects of normal extinction in the atmosphere results in a significant difference between the ideal southern horizon and the actual southernmost visible star depicted. This difference is roughly 4° for the bright southern stars of relevance.¹⁰ Hence, the visibility conditions would suggest an observer farther south, perhaps at a latitude of 34°. Other interpretations are possible as an intermediary between the ‘obvious’ value (38.3°) and the visibility value (~34°). For example, perhaps the placement of the circles on the globes was made by the mathematical calculation based on the known latitude while the actual visible constellation figures were stretched to reach these circles (causing the distortion in declination noted in the Appendix). With this interpretation, the latitude of the observer could be from roughly 34° to 38°. With this extension, the latitude of Hipparchus in Rhodes (36.4°) becomes easily acceptable.

James Evans has suggested a reasonable third interpretation. He points out that Geminus implies that there was something like a standard latitude for the manufacture of celestial globes when he says “all the spheres are inscribed for the horizon of Greece”, and he explicitly remarks in this context that the Arctic Circle is 6/60 of a full circle (i.e., 36°) from the pole.¹¹ That is, apparently the Arctic Circle on globes are standardized to be at 54° in declination, although the universality of this convention is not complete. For example, Hipparchus’s Commentary gives a value of 53°. The existence of such conventions is common, for example in the placement of α = 0° at the vernal equinox for some standard epoch even for mapmakers at other epochs, as well as the placement of the prime meridian at Greenwich even for mapmakers far from England. The Farnese Atlas might either have its Arctic Circle slightly misplaced as a standard or have adopted some alternative standard. In such a case, the observer could have been anywhere in the Greco-Roman world.

In all, the declinations of the Ant/Arctic Circles (±51.7° ± 0.9°) has an unknown relation to the latitude of the observer who provided the constellation positions. Any observer in the Greco-Roman world is consistent with this constraint.

3.3. Accuracy of the Constellation Placement

The constellations are placed onto the Farnese Atlas with remarkable accuracy. From Appendix 2, I find that the constellation positions have an accuracy of 3.5° along the various celestial circles and of 5° away from those circles. (The difference between on-circle and off-circle accuracy is likely due simply to the sculptor’s being less well able to interpolate the positions between the marked grid lines.) Given the many and various factors contributing to this observed accuracy, the original data source must have been substantially more accurate than 3.5°. An estimate of the sizes of the other sources of scatter suggests that the original source must provide the positions at least as accurately as ~2° or better. This fact can give us an indication of the nature of the data source.

The constellation positions in Aratus (and Eudoxus) are simply verbal descriptions. The accuracy at which they place points along the various declination circles and colures is 4°.¹² This is substantially worse than what is required to place the constellations onto the Farnese Atlas. As such, the known verbal descriptions of the constellations are not likely to be the source for the sculptor.

A star catalogue (with measured positions for stars identified as particular parts of the figures) allows for accurate placement of the constellations onto a celestial globe. The typical positional error for stars in the Almagest is rather better than 1°, and the star catalogue of Hipparchus undoubtedly had comparable accuracy. This is fully consistent with the observed accuracy for the Farnese Atlas. The Farnese Atlas will have additional errors added to the star catalogue errors, due to the sculptor (both in his not placing the constellations correctly according to the catalogue in hand and in his drawing the figures in natural poses) and to my measurement errors (resulting from both the usual uncertainties in photogrammetry of 1° – 2° and my choice of the exact place in the figure to identify with the star position). In all, the total error in my derived positions for stars on the Farnese Atlas should be ~3° or worse, if the sculptor based the figures on a star catalogue. In practice, the original Greek sculptor might well have been working from a functional globe made by some astronomer and based on a star catalogue.

Thus, the fine placement of the constellations implies that the original source of astronomical data was a star catalogue. Only two star catalogues are known from the ancient Western world, those of Hipparchus and Ptolemy.

4. THE FARNESE ATLAS AND HIPPARCHUS’S LOST STAR CATALOGUE

Let me summarize the main results of what we know about the source for the constellation positions on the Farnese Atlas. First, the constellation symbols and relations are identical with those of Hipparchus and are greatly different from all other known ancient sources. Second, the date of the original observations is 125 ± 55 B.C., a range that includes the date of Hipparchus’s star catalogue (c. 129 B.C.) but excludes the dates of all other known plausible sources. Third, the accuracy of the original data source must be ~2° or better, which implies that the source was a star catalogue, and the only known star catalogues are those of Hipparchus and Ptolemy. These three strong results all compel us to the conclusion that Hipparchus’s lost star catalogue is the source of the constellations on the Farnese Atlas.

Nevertheless, it is prudent to take a further step, to check in every way possible that the conclusion is consistent with everything else we know about ancient Greek astronomy. Many aspects of the claim can be checked for consistency:

(1)

Is it plausible to date celestial globes back to the time of Hipparchus? The Almagest (Book VIII, chap. 3) gives a detailed discussion on the construction of solid globes for showing stars. The concept of star globes was common in Greek times, as evidenced by remarks of Geminus (fi rst century A.D.) that assumed widespread familiarity with the concept, by remarks by Cicero that Eudoxus (c. 366 B.C.) and Archimedes (c. 287–212 B.C.) possessed globes, and by the existence of many Greek and Roman coins and engraved gems that show such globes.¹³ In particular, a small bronze coin from Roman Bithynia depicts Hipparchus seated in front of a globe resting on a table. But the primary evidence that star globes date back at least to Hipparchus is that Ptolemy specifically states that Hipparchus had a celestial globe (Almagest, Book VII, chap. 1).

(2)

Is the obliquity of the Farnese Atlas consistent with the value used by Hipparchus? From the Almagest (Book I, chap. 12), we are told that Hipparchus adopted an obliquity of 23.85°. As we shall see in the Appendix, I found that the obliquity adopted for the Farnese Atlas was 23.95° ± 0.8°. These two values are consistent.

(3)

The latitude of Hipparchus in Rhodes was 36.4°, and this is consistent with all three interpretations for the position of the Ant/Arctic Circles.

(4)

The art-historical view of the Farnese Atlas is that it is a copy of a Greek original statue made sometime before around 1 B.C. Presumably, the sculptor made use of some Greek astronomer’s observations that were known at the time. Again, this is fully consistent with the source’s being Hipparchus.

(5)

Likely the original Greek sculptor was not knowledgeable in astronomy, perhaps even to the point of his not being able to use a star catalogue. In this plausible case, the sculptor would need some visual aid, and maybe that aid came as a working celestial globe with the constellations already laid out with respect to the grid of circles. We know that Hipparchus made such globes, so it is quite possible that the Greek sculptor got hold of one of Hipparchus’s globes and based the Atlas’s globe on this model.

For every point on which we can check, therefore, the Farnese Atlas is found to be consistent with what is known about Hipparchus’s lost star catalogue, which strongly supports our conclusion that the Farnese Atlas is indeed based on Hipparchus’s catalogue.

The globe on the Farnese Atlas is not a perfect rendition of the Hipparchus star catalogue, as there are small random errors in position introduced by the scupltors as well as a variety of universal differences that must have been made after the figures left Hipparchus. There may be substantial uncertainties in taking a fi gure’s position on the globe to be identical to that in Hipparchus’s catalogue for purposes of comparison with the Almagest.

5. IMPLICATIONS AND APPLICATIONS

As a result of this investigation we can see the skies as observed by the greatest ancient astronomer, and recorded by him in the earliest Western star catalogue. This discovery also sheds light on several major questions that have been debated among historians.

One concerns the type of coordinate system used by Hipparchus. This question has been widely discussed, even in recent years. The conventional view is that “it is quite obvious that at Hipparchus’s time a definite system of spherical coordinates for stellar positions did not yet exist”.¹⁴ Nevertheless, some particularly large errors for three partial star positions given in Hipparchus’s Commentary can simply be explained as errors that could occur only if Hipparchus was using ecliptic coordinates.¹⁵ Alternatively, a variety of arguments can be presented in support of the view that Hipparchus used equatorial coordinates, the simplest being that the Commentary reports most of the fragmentary star positions in the equivalent of right ascension and declination.¹⁶ Duke goes further and points out that the possession of a celestial globe by Hipparchus is possible only if he employed “some sort of ‘defi nite system of spherical coordinates’, which Neugebauer assured us ‘did not yet exist’ at the time of Hipparchus”. I believe that the Farnese Atlas will be the key to the continuation of such debates, but I do not know how the arguments will play out. My fi rst reaction is that the globe shows clear circles of constant declination and the colures, and hence is manifestly an equatorial coordinate system. But it could be that the various circles are included merely as part of a tradition for demarcating the sky with the circles mentioned by Aratus and Eudoxus, with no implications for what (if any) coordinate system was used by Hipparchus. (A terrestrial analogy would be that my old hometown has a grid of main streets that are cardinally oriented, but this does not prove that the townfolk use latitude and longitude. A celestial analogy is that modern constellation boundaries are orthogonal for the equinox of 1875, whereas all working astronomers now use J2000 coordinates.) And Duke’s prior argument now has more force, as the existence of accurately placed constellations on a globe (as well as the underlying star catalogue) virtually demands the existence of a coherent spherical coordinate system by the later years of Hipparchus, even if Hipparchus had no single system in his early Commentary. We will have to wait to see what the implications of the Farnese Atlas are for this question.

A second question concerns the relation between Hipparchus’s star catalogue and that of the Almagest. This debate has been long and bitter over the centuries, and it has only gotten harsher in the last few decades.¹⁷ A primary approach has been through efforts to make partial reconstructions of Hipparchus’s catalogue, based on fragmentary measures discussed in the text of his Commentary.¹⁸ Now, with the full sky coverage of the Farnese Atlas, we at last have access to all of Hipparchus’s star catalogue (and not only to partial positions for a small fraction of the stars). As such, I foresee that the Farnese Atlas will take centre stage in the dispute, as it is the only new source of information for over a century. With this, someone should make a very complete catalogue of all constellation positions on the globe, perhaps involving all positions that correlate with the thousand stars in the Almagest and not merely the 70 positions reported in the appendices of this paper. A substantial disadvantage of this approach will be that the globe positions will be less accurate than the original star catalogue. Nevertheless, I predict that there will be a ‘cottage industry’ of comparing the Farnese Atlas with the Almagest.

6. CONCLUSIONS

This paper provides the first effective examination of the positions of the constellations on the Farnese Atlas. Here are my conclusions:

(1)

The epoch for the observations that were used ultimately by the sculptor to place the constellations onto the coordinate grid is 125 ± 55 B.C. This is a very strong conclusion, with no real likelihood that this date could simply be the result of historical vagaries or errors (random or systematic).

(2)

The declinations of the Arctic and Antarctic Circles are ±51.7° ± 0.9°. There are three reasonable explanations of this value. The obvious one is that the observer was at a latitude of 38.3° ± 0.9°, which is a circle that runs through the Straits of Messina to Athens and to the middle of Turkey. A second explanation is that the intention of the observer or sculptor was to follow the real visibility of the stars, and this allows the observer to be up to ~4° farther south, i.e., from roughly 34° to 38°. A third possible explanation is that the sculptor placed the Ant/Arctic Circles to correspond to some ‘standard’ latitude.

(3)

The obliquity of the ecliptic on the globe is 23.95° ± 0.8°. This is easily consistent with the value adopted by both Hipparchus and Ptolemy (23.85°) as well as with the actual obliquity of the time (23.71°).

(4)

The positional accuracy for the placement of constellation figures shows that the original source of the data had a positional accuracy of ~2° or better. This makes it likely that the original observations were recorded as a star catalogue and not as a verbal description.

(5)

All previously published proposals for the origin of the observations are easily ruled out with high confidence as a result of the above results.

(6)

A detailed comparison of the Farnese Atlas with all surviving ancient sources

shows a virtually perfect match with the constellation descriptions of Hipparchus. In contrast, all other ancient sources differ profoundly from the Atlas.

(7)

The constellations on the Farnese Atlas are based on the now-lost star catalogue of Hipparchus. This is proved by the perfect match with the constellation symbols used by Hipparchus and only for these, by the perfect match with the date of Hipparchus (with the exclusion of all other known candidate sources), by the requirement that the source be a star catalogue such as that compiled by Hipparchus, and by the many points of consistency with what we know about ancient Greek astronomy.

(8)

The obvious scenario is that Hipparchus constructed a small working globe based on his (now lost) star catalogue, that this globe was then used by the original Greek sculptor as a model for the constellation placement on a statue, and that the later Roman sculptor used the (now lost) Greek statue to create the globe that is now in Naples.

(9)

The existence of this ‘new’ source for Hipparchus’s catalogue is likely to be valuable for our understanding of Hipparchus’s astronomical methods and for investigations of the origin of the star catalogue in the Almagest.

APPENDIX 1: PHOTOGRAMMETRY

Photogrammetry is the process of deriving quantities by the detailed measurement and analysis of photographs. In the case of the Farnese Atlas, I want to be able to measure the declination of the tropics, the declination of the Ant/Arctic circles, and the right ascensions and declinations for many points within the constellation fi gures. This appendix will present the detailed procedure that I used for my photogrammetry, as well as one worked example.

A.1.1. Angular Distances Between Two Points on the Globe When each photograph was taken, I noted the distance between the camera and the surface of the globe (Dcamera). The physical radius of the globe (Rglobe) is 32.5 cm. Each picture was printed onto a sheet of paper with a zoom such that the globe fi lled the page. On the printed picture, the radius of the globe was then measured (ρglobe), typical values being 100 mm. The centre of the globe’s image was then found either by use of construction techniques straight from simple geometry or by trial and error with a compass. The accuracy of the centre determination was typically ~1% of the globe radius. The globe is spherical in shape to within ~1% of the radius, the dominant scatter being caused by the relief depictions of the constellations. (The only exception to this spherical shape is related to the hole gouged in the northern skies which has obliterated Ursa Major and Ursa Minor.) Onto this printed picture, I then drew an orthogonal coordinate system with the origin at the centre of the globe. With this system, every point on the visible surface will have coordinates X and Y, as measured with a ruler in millimetres from the appropriate axis. The precision of my measures is one millimetre.

The first transformation is from this rectangular coordinate system on the photograph (X, Y) to polar coordinates on the photograph (ρ, θ). The polar coordinates are the distance from the centre, ρ = (X² + Y²)^0.5, and the angle from the positive X axis, θ = tan^–1(Y/X).

The second transformation is from the polar coordinate system on the photograph (ρ, θ) to a spherical coordinate system centred on the camera (ζ, η). The angle η is the azimuth angle from the direction of the positive X axis, so that η = θ. The angle ζ is the angle between the sub-camera point on the globe to the point of interest on the globe as viewed from the camera. In this coordinate system, the edge of the globe will satisfy the equation

sin ζedge = Rglobe/(Rglobe + Dcamera). The angle ζ can be found from tan ζ = tan ζedge × (ρ/ρglobe). The third transformation is from this spherical coordinate system (ζ, η) centred on the camera to a spherical coordinate system (Φ, Ψ) centred on the middle of the globe. The azimuth of the point, Φ, will simply be the same value as η. As viewed from the centre, the angle between the zenith (sub-camera) point and the point of interest will be Ψ. By applying the Law of Sines to the triangle defined by the camera, the centre of the globe, and the point of interest, we fi nd sin(A)/(Rglobe + Dcamera) = sin(ζ)/Rglobe, where A is the angle subtended between the camera and the globe centre as viewed from the point of interest. In this same triangle, the angle Ψ is simply 180° – ζ – A. We can now convert all the positions measured on the photograph into spherical coordinates for the globe. The next task is to calculate the angular distances Γ (within the spherical coordinates) between any two points on the globe. Let the two points have coordinates (Φ1, Ψ1) and (Φ2, Ψ2). We can define a spherical triangle from the sub-camera point and the two points labelled with subscript ‘1’ and ‘2’. From the Law of Cosines for spherical triangles, we fi nd that cos Γ = cos Ψ1 × cos Ψ2 + sin Ψ1 × sin Ψ2 × cos (Φ1 – Φ2). Thus, we can determine the angle between any two visible points on the globe.

A.1.2. Declinations for Tropics and Ant/Arctic Circles from Photogrammetry With this framework, we can now calculate the angular distances between the equator and the other circles along the great circles formed by the colures. For each picture, along each of the colures visible, I placed a dot of coloured ink at the exact crossing point with each of the equator, tropic, and Ant/Arctic circles. I then measured the X and Y rectangular coordinates of each dot. With an EXCEL spreadsheet, the conversion to Φ and Ψ coordinates was easy. Then, for a given colure, I calculated the angle between the equator and the circles. This procedure yields the declination for each circle as based on that one photograph. For each intersection, I have an average of 3.5 measures of declination. The RMS scatter of these separate measures has a typical value of 0.5°, and this represents my measurement error. These values are averaged together to get the best value for the declination of the circle along that colure. The declinations of each circle are all constant to within the rather small uncertainties, and this demonstrates that the sculptor made good parallel circles to within an accuracy of 0.2° – 0.5°. The RMS scatter in all the measured values is divided by the square root of the number of independent measures to determine the one-sigma uncertainty in the measured declinations.

A.1.3. Right Ascensions and Declinations for Any Position The primary task for my photogrammetry is to go from the measured position on the photograph to the right ascension and declination of the star in the reference frame as defined by the grid of circles on the globe. Section A.1.1 of this Appendix tells how to go from measured positions on the photograph to spherical coordinates on the globe with the sub-camera point being the ‘pole’. In principle, a suitable triple of rotations in the spherical coordinates will transform to the equatorial coordinate system. Instead, I have adopted an easier method: I (a) choose two widely-spaced cross points of grid circles, (b) calculate the angular distance between the point of interest and both of the reference points using the formula from the first section, (c) adopt some approximate right ascension and declination for the point of interest,

(d) calculate the distance between the currently adopted position on the sky and the right ascension and declination of the reference points, (e) compare the observed angular distances from steps (b) and (d), and (f) repeat steps (c)–(e) with successive refinements in the adopted position until the agreement is satisfactorily close. This iterative numerical procedure is fast and accurate.

The reference points are usually taken to be where a colure intersects the two tropic circles. The adopted declinations for these points of intersection must be those of the photogrammetric coordinate grid, so the tropics are taken to be at ±26.2° while the Ant/Arctic circles are taken to be at ±57.5°. In principle, there will always be two points on the sky that have the same angular distances from the two reference points as on the globe, but this ambiguity is always easy to resolve with certainty on the basis of the visible position on the globe. In this iterative process, “satisfactorily close” is to better than 0.1° for my calculations. The result will be a position in the photogrammetric coordinate system and must be corrected to the real sky. As discussed in the next appendix, there is a small distortion in declination, such that positions in the photogrammetric coordinate system must have their declination corrected to that of the real sky. This correction is made by subtracting an offset to the magnitude of the declination which is found by a linear interpolation to vary from 0.0° on the equator to 2.25° on the tropics and 5.8° on the Ant/Arctic circles. The result will be the derived right ascension and declination for the object as based on that one photograph. Measures of the position on multiple pictures of the Farnese Atlas will provide largely independent measures of the coordinates, and the averaging together of these positions will help reduce the measurement error. So, fi nally, the end result is an averaged right ascension and declination of the indicated position on the sky as depicted on the globe.

TABLE 1. Measured positions for fi rst photograph.

Point	X (mm)	Y (mm)	ζ (rad)	A (rad)	Ψ (rad)	θ=η=Φ (rad)
α = 0°, δ = 0°	–7	–14	0.023	2.987	0.131	–2.03
α = 0°, δ = –26.2°	5	–60	0.089	2.512	0.541	–1.49
α = 0°, δ = 26.2°	–17	38	0.062	2.721	0.359	1.99
α = 0°, δ = 57.5°	–25	85	0.131	2.101	0.910	1.86
α Ari	–3	30	0.045	2.841	0.256	1.67
β Per	28	60	0.098	2.438	0.606	1.13
α Tau	70	36	0.116	2.266	0.759	0.48
ε Ori	94	10	0.139	1.975	1.027	0.11
α And	–52	15	0.080	2.583	0.478	2.86
α Cas	–57	59	0.121	2.215	0.806	2.34
ε Peg	–96	–5	0.142	1.934	1.066	–3.09
γ Psc	–79	–33	0.126	2.155	0.860	–2.75

Many uncertainties contribute to the error bars. First, there is my measurement errors, which are 1° – 2° as based on the repeatability of positions as measured from picture-to-picture. Second, there is the uncertainty as to my placement of the dot on the constellation figures. For example, does the star α Her correspond to the top or the middle of the head of Hercules? Third, the sculptor will not have placed the constellation figure perfectly with respect to the position of the star, for example because the sculptor has a high priority in not making the constellation fi gures look wrongly elongated. Fourth, the original observations on which the sculptor is working will not be perfectly accurate. The star catalogue in the Almagest has positional accuracies of a little better than a degree, whereas the verbal descriptions in Aratus are accurate only to around 4° for placing parts of constellations onto the celestial circles. Fifth, the Roman sculptor did not make a perfect reproduction of the original Greek statue, and this introduces yet more errors.

A.1.4. A Worked Example I will here present a detailed example, with all intermediate values presented. This will allow researchers to test my procedures and to see typical values, and will provide a known example to check later applications. I will take for my example the fi rst of my photographs, which is a typical case with neither large nor small error bars.

The photograph was taken with the camera 6 feet from the edge of the globe (Dcamera = 183 cm). Recall that Rglobe = 32.5 cm. The angular radius of the globe as viewed from the camera has ζedge = 0.151 rad. The image of the globe was expanded and printed onto paper such that the radius of the image was ρedge = 10.3 cm. The centre of the image was found by repeated trials with a compass until all edges were within

0.1 cm of a circle drawn around this centre (except for a 45° arc to the north caused by the hole in the globe). I constructed a rectangular coordinate with an origin at this centre and with the Y axis roughly towards the north. I next placed red dots where the vernal equinox colure intersected the two tropics and the Arctic Circle. I also placed green dots at 8 positions in constellations that can be identified with distinct stars in the modern sky. Then, with a millimetre ruler, I measured the distance of each dot from the two axes (see Table 1). With these positions, I then made calculations (in EXCEL) as presented in Appendix A.1.1 so as to convert to spherical coordinates on the globe. Intermediary and final values are presented in Table 1.

Point	Γ*1(°)	Γ*2(°)	α(°)	δdist(°)	δ(°)
α Ari	<