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Hipparchus' Celestial Globe

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Bianca
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« Reply #15 on: December 08, 2007, 09:04:21 am »








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.
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« Reply #16 on: December 08, 2007, 09:07:18 am »








                        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.
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« Reply #17 on: December 08, 2007, 09:10:20 am »








                                           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.
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« Reply #18 on: December 08, 2007, 09:12:07 am »








                                                         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.

(Cool
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.
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« Reply #19 on: December 08, 2007, 09:14:13 am »








                                               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.
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« Reply #20 on: December 08, 2007, 09:15:50 am »








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 
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« Reply #21 on: December 08, 2007, 09:17:20 am »








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.
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« Reply #22 on: December 08, 2007, 09:18:36 am »








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

α Ari  45.6  8.0  –4.2  19.2  17.6 

β Per  63.3  25.8  29.0  30.6  27.9 

α Tau  60.9  46.2  47.5  13.5  12.3 

ε Ori  64.3  67.0  62.6  –2.7  –2.5
 
α And  47.4  20.7  340.0  17.0  15.5
 
α Cas  72.2  27.6  331.9  41.1  37.2
 
ε Peg  66.4  55.7  303.2  10.6  9.7 

γ Psc  47.2  51.9  316.4  –4.0  –3.7 
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« Reply #23 on: December 08, 2007, 09:42:50 am »








The angular distances from the equator, along the colure, for the points on the Tropic of Capricorn, the Tropic of Cancer, and the Arctic Circle are then 24.8°, 25.9°, and 57.8° respectively. These numbers are averaged together with similar numbers from other pictures and along other colures so as to get the values reported in Section A.1.2.

I have adopted two points to define the globe’s coordinate system. These reference points are where the equinoctial colure crosses the two tropics, and these points were chosen as they give a long baseline which does not get near to the edge. For each of the eight constellation positions, I then calculate the angular distances from the star to both reference points (Γ*1 and Γ*2) in the (Φ, Ψ) coordinate system using the formula near the end of Section A.1.1.

By trial and error (which can be done fast within EXCEL), I then find a position on the sky whose right ascension (α) and declination (δdist), for the distorted coordinates of the globe has identical distances from the two reference points (assumed to be at α = 0° and δ = ±26.2° in the globe coordinate system). These distances and globe positions are presented in Table 2. The final step is to correct the distorted declination to the correct declination for the sky by means of a linear interpolation. This linear interpolation has a subtractive correction of 0° on the equator, 2.25° at δdist = ±26.2°, and 5.8° at δdist = ±57.5°.

The final declination is presented in the last column of Table 2.

The end result is measured α and δ for all eight positions on this one photograph. These values will be averaged with corresponding measures for the same point as made on separate photographs, with the final coordinates presented in Table 5.
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« Reply #24 on: December 08, 2007, 09:47:08 am »







APPENDIX 2: RESULTS FROM POSITIONAL ANALYSIS

The positions of the constellations on the globe carry information about the date of the observations ultimately used by the sculptor. In addition, the declinations of the tropic and Ant/Arctic Circles will give us the information about the adopted obliquity






TABLE 3. Positions of stars on circles.



#  Circle  Position Description  Star  α–125 (°)  δ–125 (°)  Dev (°) 

 α=0°  Westernmost Aries’s horn  γ Ari  0.5  7.9  0.5 

 α=0°  Westernmost star in Perseus  θ Per  9.1  38.5  9.1 

 α=0°  Andromeda left foot  γ And  1.4  31.0  1.4 

 α=90°  Tail of Dog  η CMa  90.1  −27.1  0.1 

 α=90°  Westernmost shield on stern  p Pup  92.6  −25.7  2.6
 
 α=90°  Just east of Pollux  φ Gem  85.1  29.2  −4.9 

 α=180°  Centaur’s chest front  θ Cen  182.5  −25.1  2.5
 
 α=180°  West edge of Libra  λ Vir  187.0  −2.4  7.0 

 α=180°  Bootes’s western toes  τ Boo  180.9  28.9  0.9
 
 α=270°  Just west of Capricorn’s head  ξ Cap  273.1  −16.3  3.1
 
 α=270°  Just east of Lyra’s edge  η Lyr  270.5  37.2  0.5
 
 α=270°  Just west of Cygnus’s beak  2 Cyg  270.2  27.5  0.2 

 Equator  Aries hoof, above Cetus’s head  σ Ari  14.7  4.8  4.8
 
 Equator  Taurus’s right hoof  ν Tau  33.3  −2.1  −2.1
 
 Equator  Middle of Orion’s waist  δ Ori  56.3  −4.4  −4.4
 
 Equator  Top of Cup  θ Crt  147.3  1.3  1.3 

 Equator  Between Crow and Virgin  21 Vir  161.4  2.3  2.3
 
 Equator  Ophiucus’s right hand  ν Oph  240.9  −6.8  −6.8
 
 Equator  Top of Aquarius’s head  25 Aqr  297.5  −5.5  −5.5
 
 Equator  Top of Pegasus’s head  35 Peg  309.8  −4.8  −4.8
 
 Equator  Between Pegasus and Fish  55 Peg  320.0  −1.1  −1.1
 
 Cancer  Perseus’s foot  ζ Per  27.2  23.2  −0.6
 
 Cancer  Taurus’s shoulder  ψ Tau  30.6  20.8  −3.0
 
 Cancer  Bottom of Pollux’s head  φ Gem  85.1  29.2  5.3
 
 Cancer  Middle of Crab  35 Cnc  108.4  26.4  2.5 

 Cancer  Lion’s chest  α Leo  122.6  20.6  −3.2 

 Cancer  Bootes’s left foot bottom  υ Boo  181.4  27.2  3.4
 
 Cancer  Bootes’s right foot bottom  ζ Boo  195.0  24.2  0.3
 
 Cancer  Hercules’s head  α Her  234.8  19.1  −4.8
 
 Cancer  Beak tip of Cygnus  β Cyg  271.4  25.5  1.6
 
 Capricorn  Bottom of Lepus  ε Lep  54.2  −27.3  −3.5
 
 Capricorn  Dog’s front forefoot  β CMa  72.5  −19.2  4.7
 
 Capricorn  Argo’s shield middle  1 Pup  94.5  −25.3  −1.5
 
 Capricorn  Centaur’s shoulder  θ Cen  182.5  −25.1  −1.3
 
 Capricorn  Bottom of Scorpion’s body  τ Sco  217.5  −21.1  2.7
 
 Capricorn  Top of Sagittarius’s bow  λ Sgr  244.5  −23.5  0.3
 
 Capricorn  Sagittarius’s shoulder  σ Sgr  250.9  −25.7  −1.9
 
 Arctic  Top of Perseus  χ Per  2.5  46.0  −5.7 

 Arctic  Auriga’s head top  ξ Aur  46.8  51.2  −0.5
 
 Arctic  Bootes’s head top  β Boo  205.2  50.0  −1.7 

 Arctic  Hercules’s western knee  τ Her  229.4  52.7  1.0 

 Arctic  Hercules’s eastern foot  82 Her  250.7  51.2  −0.5 

 Arctic  Cepheus’s chest  ξ Cep  316.0  55.1  3.4 

 Arctic  Cassiopeia’s foot  ι Cas  2.6  56.3  4.6 

 Antarctic  Lower rudder end  α Car  84.4  −52.7  −1.0
 
 Antarctic  Centaur’s rear hoof  α Cru  161.4  −51.4  0.3
 
 Antarctic  Centaur’s front hoof  α Cen  186.1  −50.0  1.7 




and latitude. To obtain these results, I have used four procedures based on the photographs.

The first procedure is to identify singular points of the constellation fi gures that fall exactly on the various circles inscribed on the globe.

The second procedure is to use photogrammetry to measure the declinations of the tropics and Ant/Arctic Circles. This work reveals a small distortion in declination by the sculptor that must be corrected to derive positions on the sky.

The third procedure is to use photogrammetry to measure the positions of many individual points within the constellation figures.

The fourth procedure is to take all the positions from the globe and fi t them to the real sky as a function of the year by means of a standard chi-square analysis, with the result being a best fit year and a quantitative estimate of the uncertainty.
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« Reply #25 on: December 08, 2007, 09:51:39 am »







A.2.1. Stars Along Circles Direct examination of the photographs of the Farnese Atlas can show what specifi c parts of various constellations are depicted as being exactly on the globe’s circles. For example, the westernmost edge of Aries’s horn is drawn as being on the equinoctial colure, and this is unambiguously identified with the star γ Ari. The chest of Leo is on the Tropic of Cancer, and this is identified with the star α Leo (Regulus). Lists of these stars can be used to determine the epoch of the Farnese Atlas’s constellations as being the year in which these stars-on-circles most closely match the circles. When calculated for this best epoch, the average positions of these stars-on-circles will provide an accurate measure of the positions of the circles.



I have compiled a list of specific stars that correspond to specific positions within constellations depicted as being exactly on one of the celestial circles. This list is presented in Table 3. The individual columns are

(1) a running number for counting and reference,

(2) the identification of the circle,

(3) the position description of the part of the constellation that is exactly on the circle,

(4) the modern name for the star that matches the position description,

(5) the right ascension (α) in degrees of the indicated star for the epoch 125 B.C.,

(6) the declination (δ) in degrees of the indicated star for the epoch 125 B.C., and

(7) the deviation in degrees between the target and the observed value. This deviation will depend on the circle; for example item number 1 is the colure for which α = 0° is the target while the observed value is α = 0.5° for a deviation in observed-minus-predicted equalling 0.5°.

Another example is item number 31, where the target is δ = –23.7°19 and the star had a declination of –27.3° for a deviation of (–27.3°) – (–23.7°) = –3.6°. The precession was calculated with the exact formula given by J. Meeus.

20 The choice of the date (125 B.C.) is justified in Sections 3.1 and A.2.4 as being the best fit date.
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« Reply #26 on: December 08, 2007, 09:55:50 am »








                            TABLE 4. Average positions of stars on circles for 125 B.C.





Circle Average Position

 

α = 0° α = 3.7° ± 2.7°

α = 90° α = 89.3° ± 2.2°

α = 180° α = 183.5° ± 1.8°

α = 270° α = 271.3° ± 1.6°

Equator δ = –1.8° ± 1.3°

Tropic of Cancer δ = 24.0° ± 1.1°

Tropic of Capricorn δ = –23.9° ± 1.1°

Arctic Circle δ = 51.8° ± 1.3°

Antarctic Circle δ = –51.3° ± 0.8°
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« Reply #27 on: December 08, 2007, 09:56:42 am »








The RMS scatter of the deviations will provide a measure of the accuracy of the original observations plus whatever later errors accumulate. For each target circle, the RMS scatter varies somewhat about an average of 3.5°. For the observations along the colure, the deviations must be corrected by a factor of cos (δ) to account for the convergence of meridians towards the poles. The deviations in Table 3 are scattered with a Gaussian distribution, as 34/47 = 72% are within one-sigma and 2/47 = 4% are farther than two-sigma. These average deviations (3.5° for targets of declination circles and 3.5°/cos (δ) for targets of colures) will arise from many causes in addition to the usual uncertainties in the original data. There will be scatter added in when the original data were transferred to a format (likely a working astronomical globe) useable by the original Greek sculptor. And the sculptor will, for purely artistic reasons, have had to shift the figures slightly so that their parts do not appear distorted on the globe (even though the figure on the sky might have some distortions). Then the Roman sculptor will not have made a perfect copy of the original Greek statue. More scatter might have been introduced because the points that I chose on the constellation figures might not have been the positions that corresponded to the modern stars.

Table 4 presents the average coordinates (either right ascension or declination, depending on the circle) for all stars along the given circle as derived from Table

3. The uncertainties quoted on the average equal the RMS scatter divided by the square root of the number of measures. The average positions for the colures will vary systematically with date, increasing roughly by 1.3° per century. The average positions for the circles of constant declination will vary by only a small amount, one that is substantially smaller than the quoted error bars. For example, the average declination of the stars on the equator and those on the Tropic of Cancer changes respectively from –1.0° ± 1.5° to –1.7° ± 1.3° and from 23.4° ± 1.3° to 23.9° ± 1.1°, in the period from 500 B.C. to A.D. 1. So, not surprisingly, the circle declinations are not sensitive to the date and hence it is sufficient to use the average declinations for the best fi t date (125 B.C., see Sections 3.1 and A.2.4). We are now in a position to get preliminary answers concerning the Farnese Atlas.

An estimated date for these stars-on-circles can come from looking for the minimum average deviations for the stars along the colures. The average deviation is zero for the year 280 B.C. However, such a criterion is crude since it does not allow for compression of meridian lines far from the equator, nor does it use the optimal chi-square weighting. In addition, there are substantial amounts of further data (see Section A.2.3) which should be combined to derive a final answer. In all, the conclusion about the date of the Farnese Atlas is discussed in Sections 3.1 and A.2.4.

The obliquity of the ecliptic is not sensitive to the epoch, so we can derive it here. The declinations of the two tropics are essentially identical (as expected) and thus we can average together the results of both tropics. From this, I infer that the Farnese Atlas was made for an obliquity of 23.95° ± 0.8°. This is to be compared to the obliquity adopted by both Ptolemy and Hipparchus of 23.85°, as well as with the true value of 23.71°. Note that this derived obliquity is obtained by comparing positions of the tropic circles with respect to the constellations, and this is independent of any distortions that may have been built into the coordinate grid of the globe by the sculptor.

The declinations of the Arctic and Antarctic Circles are identical to within the error bars. The combined average of the equatorial distance (51.7° ± 0.9°), and this appears to be the placement intended by the sculptor.
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« Reply #28 on: December 08, 2007, 10:00:16 am »








                                                A.2.2. Obliquity and Latitude





The obliquity of the ecliptic can be measured by photogrammetry as the average declination of the tropic circles. The latitude of the observer will be related to the average declinations of the Arctic and Antarctic Circles as measured by photogrammetry. Appendix A.1.2 presents full details on my procedures for measuring the declinations of the circles on the Farnese Atlas.

I find that the tropics and Ant/Arctic circles are parallel to the equator to within an accuracy of one degree. By averaging together all my photogrammetric measures of the circles’ declinations I get the best measures of their declinations. Thus, the Arctic Circle is at a declination of +57.8° ± 0.5°, the Tropic of Cancer is at a declination of +26.3° ± 0.2°, the Tropic of Capricorn is at a declination of –26.0° ± 0.4°, and the Antarctic Circle is at a declination of –57.0° ± 0.5°.

The equatorial distances of the tropics should equal the obliquity adopted by the Farnese Atlas. The two tropics are at virtually identical distances from the equator, so it is reasonable to form a weighted average and as a result we get an obliquity of 26.2° ± 0.2°. Similarly, we can combine the Arctic and Antarctic Circles to get an equatorial distance of 57.5° ± 0.4°.

These photogrammetry values are substantially different from those based on the circle positions with respect to the background constellations. In fact, the photogrammetry values are larger by 2.25° ± 0.8° for the tropics (26.2° versus 23.95°) and larger by 5.8° ± 1.0° for the Ant/Arctic Circles (57.5° versus 51.7°). These differences are too large to be by random chance or measurement error. An offset of one degree in declination corresponds to a misplacement of 0.57 cm on the surface of the globe, so the tropics are off by 1.3 cm while the circles of invisibility are off by 3.3 cm.

How can we reconcile these differences between the declinations from the stars-on-circles method and the declinations produced by photogrammetry? It is implausible that the sculptor correctly placed the circles on the globes and then systematically misplaced the constellations with respect to the primary coordinate grid on the globe. Also, there is no precedent for there ever being an intentional placement of the tropics for an obliquity of 26.2° ± 0.2°. However, there is excellent precedent for an obliquity of 23.95° ± 0.8°, and this argues that the sculptor was intending to place the tropics and the constellations simultaneously. So the obvious interpretation is that the sculptor placed the declination circles onto the globe with a small distortion that increases with distance from the equator, and then placed the constellations accurately with respect to the coordinate grid. Such a distortion in the placement of the grid circles could arise either from an unintentional error on the part of the sculptor or from an intentional decision on his part to improve the display of the constellations for artistic reasons.

The distortion of the globe’s coordinates is apparently independent of right ascension (because the tropics and Ant/Arctic Circles are parallel to the equator to an accuracy better than it would be if, for example, the distortion were in ecliptic latitude). The distortion is symmetric north/south. As for the magnitude of the distortion, we only have two values: 2.25° for the tropics and 5.8° for the Ant/Arctic Circles. Some model of the distortion is required for photogrammetry of positions away from the circles. While the distortion might well be smooth, many correction functions are plausible, given that we know the curve at only two points. I have therefore adopted the simple piecewise linear function, and this surely produces an additional uncertainty that is small compared to other sources of error.

These two ways of measuring the declination of the circles also accounts for a large disagreement between authorities on the derived obliquity and latitude. Gialanella and Valerio place the tropics at 25.5° and the Arctic Circle at 58°, so it is clear that they used photogrammetry. Fiorini gives an obliquity of 23° and a declination of the Ant/Arctic Circles of 50°, so it is clear that he was deriving the declinations of the circles from their positions with respect to the constellation fi gures.

In all, I conclude that the obliquity used in constructing the Farnese Atlas was 23.95° ± 0.8° while the Ant/Arctic Circles were 51.7° ± 0.9° from the equator. The globe has a small distortion in the declination which increases with equatorial distance, and this must be accounted for by the photogrammetry.
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« Reply #29 on: December 08, 2007, 10:01:49 am »








A.2.3. Stars Off Circles


The placement of constellation figures away from the various circles also contains information on the epoch of the globe. That is, a greater or lesser ecliptic longitude on the globe will correspond to a later or an earlier date for the observer. The previous analyses in this paper have not used this source of information, and this task is reported in this section. The idea is to use photogrammetry to derive the right ascension and declination of points within constellations that are readily identifi ed with specific stars, and to convert these positions to ecliptic coordinates. The resultant ecliptic longitudes can then be part of a fi nal fit to derive the approximate year of the observations incorporated into the Farnese Atlas by the sculptor.

Complete details of my photogrammetry are presented in Appendix A.1, along with a worked example for one of my pictures. This provides a mechanism to go from my pictures of the Atlas to positions on the globe.

I have determined 23 specific points within constellation figures that can be unambiguously identified with single stars in the sky. For example, Perseus is depicted as holding a head on his western side near the south (i.e., the Medusa) and this is certainly the star β Per (Algol). And the beak of the bird (Cygnus) corresponds to the star β Cyg (Albireo).

These identifications are listed in Table 5.
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