The History of Ptolemy’s Star Catalogue

1

Robust Control of Linear Systems and Nonlinear Control: Proceedings of the International Symposium MTNS-89, Volume II Birkhäuser Basel Joseph A. Ball, J. William Helton (auth.), M. A. Kaashoek, J. H. van Schuppen, A. C. M. Ran (eds.) Tahun: 1990 Bahasa: english File: PDF, 13.89 MB

2

Perspectives on Nonviolence Springer-Verlag New York Kenneth E. Boulding (auth.), Dr. V. K. Kool (eds.) Tahun: 1990 Bahasa: english File: PDF, 28.41 MB

Studies
in the History of Mathematics and
Physical Sciences

14

Editor

G. J. Toomer
Advisory Board

R. P. Boas P. J. Davis T. Hawkins
A. E. Shapiro D. Whiteside

Studies in the History of
Mathematics and Physical Sciences
Vol.

1: O. Neugebauer
A History or Ancient Mathematical Astronomy

Vol.

2: H.H. Goldstine

Vol.

3: C.C. Heyde/E. Seneta
I..J. Bienayme: Statistical Theory Anticipated

Vol.

4: C. Truesdell
The Tragicomical History or Thermodynamics, 1822·1854

Vol.

5: H.H. Goldstine

Vol.

6: J. Cannon/So Dostrovsky
The Evolution or Dynamics: Vibration Theory rrom 1687 to 1742

Vol.

7: J. Liitzen
The Prehistory or the Theory or Distributions

Vol.

8: G.H.Moore
Zermelo's Axium or Choice

Vol.

9: B. Chandler/W. Magnus
The History or Combinatorial Group Theory

A History or Numerical Analysis rrom the 16th through the 19th
Century

A History or the Calculus or Variations rrom the 17th through
the 19th Century

Vol. 10: N.M. Swerdlow/O. Neugebauer
Mathematical Astronomy in Copernicus's De Revolutlonlbus
Vol. 11: B.R. Goldstein
The Astronomy or Levi ben Gerson (1288·1344)
Vol. 12: BA. Rosenfeld
The History or Non·Euclidean Geometry
Vol. 13: B. Stephenson
Kepler's Physical Astronomy
Vol. 14: G. Grasshoff
The History or Ptolemy's Star Catalogue
Vol. 15: J. Liitzen
Joseph Liouville 1809·1882: Master or Pure and Applied Mathematics
continued after index

Gerd GraBhoff

The History of
Ptolemy's Star Catalogue

With 113 Illustrations

Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo Hong Kong

Gerd GraD/loff
Universitat Hamburg
Philosophisches Seminar
D-200l Hamburg 13

FRG

Suits Edit'"

Gerald Toomo:r
History of Mathemat ics Department
Brown University
Providence, RI 02912, USA

Library of COIlgress Cataloging·in·Publication Dala
O~5hoff.

Oerd.

"The hi.wry of PtoIemy's sue ulaloguc5 I Oerd Orasshoff.
p. cm ...-{Studies in !he history of malhcmal;cs and plty5ical

scienoe1 ; 14)
".dudes bibtiographical rererencc5 .
ISBN.13; 978· 1-4612·8788·9
l. Slan-Catalogs--Hislory. 2. P; tolemy fl. 2nd cent Almagest.
\. Title .

OB65.0685 1990
523.S·OOI2--<lc20

89·21~7

Printed on acid·frtt paper.
C 1990 Sprinsu- ~dag New VOlt. 11K:.
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DOl: 10.1007197&-1-4612-4468-4

To

B.-S. and the Elephant

Table of Contents
Introduction
1 Tbe Stars of tbe Almagest
1.1 The Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Persons................................
1.1.2 Methodological Background. . . . . . . . . . . . . . . . . . . .
1.1.3 The Almagest on Fixed Stars . . . . . . . . . . . . . . . . . . .
1.2 The Arabic Revision of the Almagest . . . . . . . . . . . . . . . . . .,

6
6
6
7
9
17

2 Accusations
2.1 Tycho Brahe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.2 Laplace and Lalande. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.3 Delambre's Investigations . . . . . . . . . . . . . . . . . . . . . . . . ..

23
23
25
27

3 Tbe Rebabilitation of Ptolemy
3.1 The Number of Hipparchan Stars . . . . . . . . . . . . . . . . . . . ..
3.2 Supplementary Catalogues . . . . . . . . . . . . . . . . . . . . . . . ..
3.2.1 Bjornbo's New Catalogue . . . . . . . . . . . . . . . . . . . ..
3.2.2 Dreyer's 1/4 Degree Stars . . . . . . . . . . . . . . . . . . . ..
3.2.3 Dreyer II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
3.2.4 Fotheringham . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Reconstruction of the Hipparchan Catalogue . . . . . . . . . ..
3.3.1 The Determination of the Precession . . . . . . . . . . . . . ..
3.3.2 Dreyer's 1/4 degree stars. . . . . . . . . . . . . . . . . . . . ..
3.3.3 Peters' Hypotheses of two Observation Instruments . . . . ..
3.3.4 Graduation of the Astrolabe . . . . . . . . . . . . . . . . . . .
3.3.5 The Epoch of Observation for the Hipparchan Coordinates.
3.4 Gundel's List of Hipparchan Stars . . . . . . . . . . . . . . . . . . . .
3.5 Precession and Solar Theory . . . . . . . . . . . . . . . . . . . . . . ..
3.5.1 Pannekoek's Calculation of Precession . . . . . . . . . . . . ..
3.5.2 The Hipparchan Solar Theory. . . . . . . . . . . . . . . . . ..
3.6 Accusations..................................
3.6.1 The Observation of Regulus and Spica . . . . . . . . . . . . .
3.6.2 The Measurements of Declination . . . . . . . . . . . . . . ..
3.6.3 Stellar Positions from Occultations by the Moon. . . . . . ..
3.6.4 Fraction of the Degrees . . . . . . . . . . . . . . . . . . . . . .

34
34
43
43
44
47
49
52
59
61
63
63
64
67
73
73
76
79
80
81
83
84

viii
4 The Analysis of the Star Catalogue
4.1 The Catalogue in the Alma,gest . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Critical Edition of the Catalogue . . . . . . . . . . . . . . . ..
4.1.2 Recalculation of the Coordinates for the time of Hipparchus
4.1.3 Identification of Stars. . . . . . . . . . . . . . . . . . . . . . ..
4.1.4 Errors in the Almagest . . . . . . . . . . . . . . . . . . . . . ..
4.2 Criticism of Vogt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4.2.1 Vogt's Interpretation of Delambre's Precession Table . . . ..
4.2.2 Reconstruction of Coordinates .. . . . . . . . . . . . . . . ..
4.2.3 The Accuracy of the Reconstructed Coordinates . . . . . . ..
4.2.4 Vogt's Proof of Independent Observations . . . . . . . . . . .
4.2.5 Statistical Test for Independent Data . . . . . . . . . . . . . .
4.2.6 Dating.................................
4.3 Gundel's Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92
92
94
95
95
97
99
99
104
106
107
110
117
122

5 Structures in Ptolemy's Star Catalogue
5.1 Star Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
5.2 Multiple Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
5.2.1 Dreyer's Paradigm . . . . . . . . . . . . . . . . . . . . . . . . ..
5.3 Method of Selective Error Distribution. . . . . . . . . . . . . . . . ..
5.3.1 Cluster-Analysis...........................
5.4 Errors of the Solar Theory . . . . . . . . . . . . . . . . . . . . . . . ..
5.5 Fractions of Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
5.5.1 Fractions of Degrees in Latitude . . . . . . . . . . . . . . . ..
5.5.2 Fractions of Degree in Longitude. . . . . . . . . . . . . . . ..
5.6 Hipparchus' Commentary on Aratus . . . . . . . . . . . . . . . . . ..
5.6.1 Sources................................
5.6.2 Numerical Values . . . . . . . . . . . . . . . . . . . . . . . . ..
5.7 Calculation of Phenomena . . . . . . . . . . . . . . . . . . . . . . . ..
5.7.1 Local Sidereal Time. . . . . . . . . . . . . . . . . . . . . . . ..
5.7.2 Simultaneous Rising and Setting . . . . . . . . . . . . . . . ..
5.7.3 Culmination.............................
5.8 Deviations from Reality . . . . . . . . . . . . . . . . . . . . . . . . . ..
5.8.1 Comparing two Catalogues . . . . . . . . . . . . . . . . . . . .
5.8.2 From Observation to Phenomena. . . . . . . . . . . . . . . ..
5.8.3 New Ways of Comparison . . . . . . . . . . . . . . . . . . . ..
5.8.4 The Globe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.5 More Details. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
5.9 Reconstruction................................

129
129
135
136
140
142
148
156
158
164
174
174
175
177
177
178
178
178
178
182
184
190
191
192

6 Theory and Observation
6.1 The Aristotelian Heritage . . . . . . . . . . . . . . . . . . . . . . . . ..
6.2 The Uncertainty of Empirical Data. . . . . . . . . . . . . . . . . . ..
6.3 Radical Empiricism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Holistic Rationalism . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

198
198
199
204
209

ix

7 Appendix A
217
7.1 Stars and Constellations. . . . . . . . . . . . . . . . . . . . . . . . . .. 217
7.2 Identifications................................. 218
8 Appendix B
270
8.1 Transformation Formulae. . . . . . . . . . . . . . . . . . . . . . . . .. 270
8.2 Column Headings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9 Appendix C
9.1 Column Headings

317
317

10 Literature

335

11 Index

344

List of Figures
1.1

Spherical astrolabe.. . .

3.1 Error of the mean sun. .
3.2 Distribution of precession values.
3.3 Distribution of fractions of a degree.
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

11
78
82
88
97
102
104
105
110
111
111
112

4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19

Identification Chart for the Constellation Canis Minor
Precession H-P. . . . . . . .
Precession T-H. . . . . . . . . . . . . . . . . . . .
Precession, all declinations. . . . . . . . . . . . .
Errors of coordinates, uncorrelated (simulation).
Uncorrelated coordinate errors with different standard deviations. .
Strong correlation. . . . . . . . . . . . . . . . . . . . . . . . . . ..
Correlating coordinate errors (Simulation). . . . . . . . . . . . . .
Correlating coordinate errors, with high standard deviation (simulation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Errors in longitude Almagest versus Vogt's reconstruction. .
Errors in latitude, Almagest versus Vogt's reconstruction.
Errors in latitude, for 1/4 degree stars. .
Errors in latitude, for 1/6 degree stars. . . . .
Errors in longitude, for 1/6 degree stars. . . .
Epoch of coordinates in Aratus Commentary.
Histogram of (AGundel - A_l2S)' . . . . . . . .
Distribution (AGundel - L l2s )/ (AGundel - L l2s).
Simulation of independent errors.
Simulation of dependent errors.

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

Ecliptical coordinates. .
Ecliptical coordinates. .
Equatorial coordinates. .
Equatorial coordinates. .
Error in latitude, average of pairs.
Distribution of errors in longitude, all stars.
Stars with 1/4 degree in latitude. . . . . . .
Distribution of errors in longitude, 1/4 degree stars.
Distribution of magnitudes, all stars. . . . . .
Distribution of magnitudes, 1/4 degree stars. . . . .

131
132
133
134
135
137
138
138
139
139

113
113
114
116
116
117
120
123
126
127
128

5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27
5.28
5.29
5.30
5.31
5.32
5.33
5.34
5.35
5.36
5.37
5.38

Differences in longitude for southern stars. . . . . . . . .
Differences in longitude, constellations Lepus, Eridanus.
Differences in longitude, stars of the zodiac. . . . . . . .
Distribution of differences in longitude for one source. .
Distribution of longitudinal differences for two sources. .
Best fit with three normal distributions. . . . . . . . . .
Error in longitude as a function of the longitude. . . . .
Error of the solar theory with the reference epoch = -128; a) Hipparchus +2;40°; b) Ptolemy. . . . . . . . . . . . . . . . . . . . . . . .
Error of the solar theory (-128), with phase shift 180° and errors of
the star catalogue. . . . . . . . . .
Error and interpolation functions. . .
Fractions of the degree in latitude. .
Fractions of the degree in longitude.
Error in latitude for 1.81 < 20°. ...
Error in latitude for 0° > ~ > 20°. .
Distribution of errors in latitude for full degrees.
Histogram of magnitudes, 1/6 degree stars. ...
1/4 degree latitude in combination with 1/2 degree in longitude.
1/4 degree latitude in combination with 50' in longitude. .
Distribution of the degrees of the phenomena.
Correlation of errors for all phemonena. ... .
Correlation of errors for culminating stars. . . .
Correlation of errors without culminating stars.
Correlation of errors for rising stars. . . . . . .
Correlation of errors of reconstructed longitudes.
Correlation of errors of the reconstructed latitudes.
Correlation of well approximated longitudes. .
Correlation of well approximates latitudes. .
Correlation of fraction numbers. . . . . . . .

141
142
142
143
144
145
149
151
152
153
156
157
160
161
162
163
172
173
177
189
192
193
194
195
195
196
196
197

List of Tables
2.1

Precession constants as function of declination variation ..

31

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17

Additional constellations. . ...
Estimation of Hipparchan stars ..
Ratio of star totals.. . . . . . .
Estimated size of star registers.
Equinox observations. ..
Error classes in latitude ...
Error classes in longitude. .
Possibly copied longitudes ..
Possibly copied longitudes ..
Vogt's precession table. ..
Fractions of a degree in the longitudes ..
Longitudes of Hermes. . . . . . . . . . .
Precession and declination measurements.
Derivation of solar apogee . . . . . . . . . .
Newton's standard deviations of measurements ..
Newton's fractions of the degree. . . . . . . . . .
Theoretical distribution of degree fractions (Newton).

37
37
37
38
50
55
56
57
58
61
64

4.1 Possible copying errors. . . . . . . . . . .
4.2 Determination of the precession constant.
4.3 Common error intervals, latitude. .
4.4 Limiting correlation coefficients ..... .
4.5 Dating of the star data. . . . . . . . . .
4.6 Longitudes of Hermes "De XV Stellis" ..
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

Probabilities of normal distributions ...
Time coefficients of the errors in the solar theory..
Distribution of fractions of the degree in latitude ..
Distribution of fractions of the degree in longitude.
Stars with 1/4 degree in longitude. . . . . . . . . .
Theoretical models for the distributions of degree fractions.
Differences ll.d for the fractions of the degree . . . . . . . .
Newton's theoretical distribution of fractions in longitude.
Combinations of fraction in latitude and longitude.
Types of stellar phenomena. . . . . . . . . . . . . . . . . .

69

74

77
80
85

87
94

102
108
114

120
124
146
151
156
157
157
159
162
168
172
175

5.11
5.12
5.13
5.14
5.15
5.16
5.17

Distribution of phenomena in the Commentary.
Additional specifications in the phenomena..
Number of stars associated with phenomena.
Vogt's table of error classes in latitude .... .
Stars with large common error. . . . . . . . .
Examples of varying phenomena for the same stars ..
Phenomena oh' Hya. . . . . . . . . . . . . . . . . .

176
176
177
181
189
191
193

Introduction
Ptolemy's Almagest shares with Euclid's Elements the glory of being the scientific text
longest in use. From its conception in the second century up to the late Renaissance,
this work determined astronomy as a science. During this time the Almagest was
not only a work on astronomy; the subject was defined as what is described in the
Almagest. The cautious emancipation of the late middle ages and the revolutionary
creation of the new science in the 16th century are not conceivable without reference
to the Almagest. This text lifted European astronomy to the high standard of
knowledge on which the new science flourished. Before, the Ptolemaic models of the
orbits of the sun, the moon, and the planets had been refined by Arabic astronomers.
They provided the structural elements with which Copernicus and Kepler ushered in
the era of modern astronomy. The Almagest survived the destruction of its epicyclic
representation of the planetary orbits in the conceptual traces left behind in the
theories of its successors. The clear separation of the sidereal from the tropical year,
the celestial coordinate systems, the concepts of time, the forms of the constellations,
and brightness classifications of celestial objects are, among many other things, still
part of the astronomical canon even today.
The scientific interest of the star catalogue in the seventh and eighth books
of the Almagest lasted longer than any other part. As late as the beginning of
the 18th century the Royal Astronomer Edmund Halley used the catalogue and
discovered through a comparison with his own observations the proper motion
of the fixed stars. Three centuries before Tycho Brahe had been the first European to revise the star catalogue and replace the Ptolemaic coordinates with his
own. In the longitudes of the stars of the Almagest Tycho recognized a large systematic error of one degree. Tycho was one of the first to suspect that the star
catalogue of the Almagest is not the product of Ptolemy's own accomplishments
as observer, as the text would have us believe, but had been obtained through
a simple conversion of measurements made by the most illustrious of Ptolemy's
predecessors: Hipparchus. This speculation could have been promptly confirmed
or discredited if the presumed Hipparchan catalogue had existed, permitting a direct comparison with the coordinates of the Almagest. As in the case of Euclid's
Elements, however, the comprehensive material of the Almagest had the effect of
rendering older works obsolete for scientific use. There was no real necessity to
laboriously copy out older, scientifically outdated texts, which had served as sources
for the Almagest, and to preserve them for the generations to come. For this reason the only work of Hipparchus that has been handed down in its entirety is
one early commentary on the astronomy of Aratus and Eudoxus. Whether a Hip-

2

Introduction

parchan catalogue must have existed or not is one of the central questions of this
book.
Without even the smallest documentary fragment available it can be misleading to
conceive of a Hipparchan compilation of stellar coordinates in the form of a modern
star catalogue. Even if it could be proven that Hipparchus had star coordinates of a
reasonable accuracy documented in an unknown form and left them for Ptolemy's
exploitation, this would in no way prove the existence of a catalogue as it appears in
the Almagest. Thus two distinct historical questions arise: (i) did Hipparchus record
the stellar positions in the form of a catalogue at all and, if so, (ii) in what type
of coordinate system were the positions given? Instead of a catalogue one could
imagine a celestial globe as the documentary medium, and instead of an ecliptical
coordinate system Hipparchus might have used e.g. polar distances and expressions
equivalent to right ascensions. The actual Hipparchan observations of the stellar
positions could be carried out as declination measurements in conjunction with the
times of the meridian transit of a star. In what follows we will therefore speak of
a star register when we refer to the more general types of documentation of the
stellar positions. Star catalogues contain only tables or lists of star names with their
positions and brightnesses.
After Tycho's allegations, the problem arose of deciding between the historical
links of two catalogues, of which only one is preserved. In the large arena for
possible interpretations thereby created, a centuries-long dispute developed: in one
camp Ptolemy was labelled a plagiarist and a forger, in the other he was considered
to be the greatest astronomer of antiquity, with an irreproachable integrity.
"There are no secrets as such, there are only uninformed people of all degrees",
writes Christian Morgenstern. But what about the historian of science who was
not informed about the compilation of the Ptolemaic star catalogue by eye-witness
reports? If the only one who is fully informed is he who experienced or shared
the secret at first hand, or who heard about it from someone involved and became
informed in this way, then it follows that historians constantly struggle without
any real hope against the mechanisms of forgetting and suppression and that they
have to restrict themselves to be archivists of contemporary reports. The secret
of the falling apple which inspired Newton - who should know the secret better
than the boy next door who was searching at the time for the first ripe fruit on
the tree-limbs? Or Galileo's balls falling from the Leaning Tower of Pisa - who
could more truly attest to it than the beggar at the entrance to the church who
had been driven from his place by the experimental mania of the new era in
physics? And who should better be able to testify to the secret of the source of the
Ptolemaic star catalogue than the assistant who carefully copied the manuscripts?
It appears as though the solution of riddles should be easiest for those who lived
closest to the past events and who had access to the widest variety of contemporary
reports.
The history of the interpretation of the Ptolemaic star catalogue shows that this
picture is deceiving. It was not those who used the star catalogue right after Ptolemy
who understood the riddle best. The efforts at understanding, as we intend to show
through the thematic succession of the chapters, are the product of a historical
process of growing insights, the result of an expanding series of arguments, sources
and interpretive strategies which implant their solutions in the increasingly clearer

Introduction

3

and more complex picture of the historical epoch in question. And they are dependent
on our conception of the type of events that can occur in history.
The first half of this work, therefore, covers the main theses of those authors
who are significant for the current discussion, in their historical order within the
framework of the subject. Although this way of presentation leads to repetitions
of the theme discussed in particular cases (for example, the calculation of the
precession constant by Ptolemy), one should always notice how the context of
discussion changes the perspective of interpretation. As a rule the presentations
will not be accompanied by a critical discussion depending on the argumentation
in the second half of the book, in order that the sequence of argument in which
the contributions were made public shall not be disturbed. The main purpose of
this type of presentation is, for instance, to separate the opinion of Laplace or
Tycho regarding the origin of the Ptolemaic star catalogue from interpretations
of a later period with other historical and methodological backgrounds, as, for
instance, the analysis of Vogt. Only in the second half of this study are the historical
interpretations integrated into the current debate about the origin of the Ptolemaic
star catalogue.
The first chapter describes the time in which the Almagest defined the standards
of astronomy. The first section of it summarizes without commentary Ptolemy's
explication of the astronomy of the fixed stars in the form in which it served as
the foundation for the following generations of astronomers and as it appears to
the reader who faces the text for the first time. The second section highlights the
difficulties that the Arabic astronomers had with the star catalogue and especially
with the determination of the precession constant. It explains why the systematic
errors of the longitudes could not have attracted their attention and that as a
consequence the historical problems of the star catalogue had necessarily to remain
outside their ken.
The second chapter examines the period in which the accusation of forgery was
raised and the historical evaluation of the Almagest was undertaken solely against
the backdrop of the modern concept of science with its high estimation of empirical
data. The chief figures here are Tycho and Delambre. The third chapter summarizes
the reaction of the historians since the beginning of this century in which, inspired
particularly by Vogt's contributions, an attempt was made to rehabilitate Ptolemy
as an accomplished observer.
The fourth and fifth chapters evaluate the previous arguments. The investigation
uses the new critical revisions of the catalogue edited by Toomer and Kunitzsch with
newly recalculated positions of the identified stars. A catalogue with the Ptolemaic
data along with the accurate positions of the stars is printed in Appendix A.
The critique of Vogt's previously uncontested thesis illustrates quite clearly
that the errors of a large group of stars from the Almagest correlate significantly
with his reconstructed Hipparchan coordinates. With that result, Vogt's claim to
have proven the independent observation of the two star registers is discarded.
Even more, strong correlations of errors in the two star registers provide evidence
for originally common observations. Still, if one relies solely on an analysis of
Vogt's reconstructed coordinates, it cannot be excluded that common errors in
the observation and evaluation methods could generate correlating errors in two
independent catalogues.

4

Introduction

If other possible causes for the correlating coordinate errors can be excluded, then
it is evident that early Hipparchan coordinates were used in Ptolemy's compilation
of the star catalogue. A list of stellar longitudes first published by Gundel can
go to show that as far as the coordinates are concerned, ecliptical longitudes of a
Hipparchan origin are at work. It exhibits the existence of truly Hipparchan ecliptical
longitudes. However, the question remains open whether Hipparchus recorded his
star register in an ecliptical coordinate system.
Chapter five extends the critical evaluation of the previous arguments. The
detailed analysis of the coordinate errors reveals interesting structures in the star
catalogue. The stellar positions were not determined independently of one another,
but were rather observed in groups relative to a number of reference stars so that
the positional errors of the reference stars were carried over to the positions of the
other related stars. Furthermore, one can find a periodic error in longitude that was
caused by the inaccuracies of the Ptolemaic/Hipparchan solar theory. It proves that
the measurement procedures involve theoretical calculations of the solar longitude,
and that the number of reference stars with a longitude directly related to the
position of the sun must be reasonably large. The mean error in longitude points
up the fact that the epoch of observation for the coordinates given in the Almagest
and measured by Hipparchus coincides with the epoch of the Aratus Commentary
- provided that Ptolemy used Hipparchan coordinates with an additional 2°40' on
the longitudes.
The clearest evidence for the impact of Hipparchan observations on Ptolemy'S
coordinates can be garnered by a new type of error analysis. The new method avoids
reconstructing the Hipparchan coordinates and then comparing the positional errors
of the two registers. When instead the Hipparchan data of the Aratus Commentary
are compared directly with the values of the phenomena as calculated from the
coordinates of the Almagest, the common observational basis of the two sources
becomes obvious.
Consequently one has to assume that a substantial proportion of the Ptolemaic
star catalogue is grounded on those Hipparchan observations which Hipparchus
already used for the compilation of the second part of his Commentary on Aratus.
Although it cannot be ruled out that coordinates resulting from genuine Ptolemaic
observations are included in the catalogue, they could not amount to more than half
of the catalogue.
Finally, the last chapter argues that the assimilation of Hipparchan observations
can no longer be discussed under the aspect of plagiarism. Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access
to the methods of data evaluation using arithmetical means with which modern
astronomers can derive from a set of varying measurement results the one representative value needed to test a hypothesis. For methodological reasons, then, Ptolemy
was forced to choose from a set of measurements the one value corresponding best
to what he had to consider as the most reliable data. When an intuitive selection
among the data was no longer possible - which can occur quite often even with
careful measurements - Ptolemy had to consider those values as "observed" which
could be confirmed by theoretical predictions. Scientific theories are refuted when
no measurement confirms the prediction. For this reason many observations in the
Almagest appear as if they are constructed from the theory alone: in other words,

Introduction

5

they look like fabrications. This misinterpretation ignores the fact that the selection
of observation values is a very legitimate and even necessary step for the construction
of complex theories. The ancient understanding of "observation" does not include
data evaluation of the modern type. Rather, it expresses the particular property of a
certain type of theoretical statement that its truth value can be confirmed or refuted
by the result of measurement procedures.
Seen in this context it can no longer be surprising that the Hipparchan stellar
coordinates, interpreted by Ptolemy as theoretical statements, were accorded more
credibility compared with the positions which he had himself observed, and that
Ptolemy, even if he observed all the positions with the astrolabe, had to compile the
star catalogue of the Almagest from those coordinates that could be derived from a
Hipparchan star register. The dispute about the scientific respectability of Ptolemy
is nothing more than an argumentative dead-end arising from a misinterpretation
of the concept of observation in ancient astronomy.
The history of the Ptolemaic star catalogue, conceived as the history of the
interpretations by its readers, changes into a history about the complex genesis of
the star catalogue in the Almagest.
All quotations are translated by me if not noted otherwise. L. Schafer, Ch. Scriba,
and A. Kleinert made it possible to work on the book at Hamburg University. More
than anybody else I am indebted to Otto Neugebauer: his HAMA initiated my work,
and his enthusiasm for the subject was a permanent source of inspiration over years.
For numerous corrections and comments I thank G. Toomer. P. Kunitzsch patiently
discussed with me all my philological questions concerning the star catalogue. I
had endless discussions about astronomical subjects with Ch. Miinkel, L. Wisotzki,
and 1. Jahn. S. Pramesa helped me translate my German manuscript. For valuable
comments and other assistance I thank J. Dobrzycki, W. Duerbeck, B. Goldstein,
B. Idlavas, H. Schwan, W. Seitter, Th. Spitzley, the Rechenzentrum of Hamburg
University and the Institute for Advanced Study in Princeton.

1. The Stars of the Almagest
1.1

The Documents

1.1.1

Persons

The most fruitful period of ancient Greek astronomy was the time of Hipparchus
and Ptolemy. Up to then Babylonian astronomy succeeded in predicting solar and
lunar eclipses with great precision. Its major concern, the calculation of the visibility
conditions of the moon and planets, could be achieved by comprehensive algebraic
schemes to a high degree of accuracy.
Hipparchus was probably the first to combine the numerical precision of Babylonian astronomy with Greek geometrical models. In his person the two different
astronomical traditions merged to form the powerful astronomical theories that
followed. It has been shown that many of the Hipparchan basic parameters are
of Babylonian origin.' His solar theory was taken over by Ptolemy as well as his
determination of the length of the year and the essential parameters in the lunar theory. Ptolemy reports that Hipparchus did not succeed in formulating a satisfactory
planetary theory, although he did refute the planetary theories of his predecessors. Hipparchus provided the empirical basis and the methodological standards for
Ptolemy'S construction of the astronomical theories.
Several documents mention Nicaea in Asia Minor as the birth place of Hipparch us. The main source of our knowledge about the astronomer Hipparchus is
the major work of his successor Claudius Ptolemy, the Almagest. Hipparchan observations and theoretical considerations are frequently quoted by Ptolemy and they
provide a general framework for Hipparchus' period of scientific activity. The earliest
mentioned Hipparchan observations are determinations of the equinoxes in book III
of the Almagest. Ptolemy assigns an observation of an autumnal equinox on 26/27
September -146 to Hipparchus himself. 2 He also cites a list of observations of autumnal equinoxes, the earliest of 27 September -161, which Hipparchus "considers
to have been very accurately observed".3 Since Ptolemy does not unambiguously
state that Hipparchus actually had observed these himself, one has to consider the
'Toomer, G. J. (1978), Hipparchus, in: Dictionary of Scientific Biography, ed. C. C. Gillispie, New
York, vol. XV, pp. 211ff.
2Ptolemy, C. (1984), Ptolemy's Almagest, trans. and annot. by G. J. Toomer, London, p. 138.
3Ptolemy, c. (1984), pp. 133f.

1.1. The Documents

7

year -146 as the earliest documented reference. The latest Hipparchan observation
quoted in the Almagest is an observation of the moon on 7 July -126.4
Of a similar Hipparchan observation on 2 May -126 Ptolemy says: "Now
Hipparchus records that he observed the sun and the moon with his instruments in
Rhodes ...".5 All the other Hipparchan observations in the Almagest refer to Rhodes
with a geographical latitude of cp = 36°, too. It is only in Ptolemy's partly preserved
treatise "On the Phases of the Fixed Stars and Related Weather Prognostications"
that the Hipparchan observations are attributed to a place called "Bithynia", which
is the kingdom in which Nicaea was located. 6 From this evidence it is plausible
that Hipparchus lived most of his scientific career on Rhodes. 7 For the most part
we have only indirect access to the scientific contributions of Hipparchus. The only
preserved Hipparchan text, titled "Commentary on the Phenomena of Aratus and
Eudoxus",8 contains a detailed criticism of the older Greek texts on fixed stars
by Aratus, Eudoxus and Attalus. The second book with Hipparchus' own account
of the phenomena related to fixed stars reveals his extensive study of the stellar
positions. Almost all other references to the scientific contributions of Hipparchus
are found either in Ptolemy'S quotations or must be reconstructed from calculation
schemes and observations.
The biography of Ptolemy is as fragmentary as that of Hipparchus. 9 The observations in the Almagest, Ptolemy'S main work, cover a time between +127 and +141.
Since the Almagest is quoted in the other major Ptolemaic texts, the "Tetrabiblos",
the "Handy Tables", the "Planetary Hypotheses" and the "Geography", it has to predate them. Ptolemy attributes several observations dating between + 127 and + 132
to the "mathematician Theon", who could be either his colleague or his teacher in
Alexandria. Also, all the other Ptolemaic observations refer to Alexandria in Lower
Egypt and there is no evidence that Ptolemy ever worked at other places. Ptolemy
formulated his astronomical theories as they endured in their main features for the
next millennium. He developed the planetary theory and refined the lunar theory.
Thus, using the Hipparchan solar theory, he was able to predict eclipses accurately.
Furthermore, the central importance of the Almagest as a systematic mathematical
formulation of the astronomical knowledge cannot be underestimated. Easy tabulations for the major mathematical procedures made the Almagest the comprehensive
and practical astronomical handbook for following generations of astronomers.

1.1.2

Methodological Background

The Almagest opens its astronomical exposition with two introductory chapters
where Ptolemy discusses the rank of astronomy among the sciences and the method4Ptolemy, e. (1984), p. 230.
5Ptolemy, e. (1984), pp. 227.
6Ptolemy, C. (1898-52), Claudii Ptolemaei opera quae exstant omnia, ed. J. L. Heiberg et. aI., Opera
Astronomica Minora, vol. II, pp. 3--fJ7.
7For a summary of Hipparchus' biography and scientific work cf. Toomer, G. J. (1978), Hipparchus,
in: Gillispie, e. e. (ed.) (1970--80), Dictionary of Scientific Biography, New York, vol. XV, pp. 207-224.
8Hipparchus (1894), Hipparchi in Arati et Eudoxi Phenomena Commentarium, ed. and German trans.
e. Manitius, Leipzig.
9Toomer, G. J. (1975), Ptolemy, in: Gillispie, e. e. (ed.) (1970--80), Dictionary of Scientific Biography,
New York, vol. XI, pp. 186-208.
.

1. The Stars of the Almagest

8

ological structure of his book. In the tradition of Aristotelian metaphysics, Ptolemy
counts astronomy as part of mathematics whose methods provide "sure and unshakable knowledge", thereby being distinguished from physics, whose investigation
of the material world, due to the "unstable and unclear nature of matter" can offer
no real hope "... that the philosophers will ever be agreed about them."10
The wish to deduce astronomical laws with mathematical rigor imposes a methodological order on astronomy for Ptolemy that is reflected in the thematic
structuring of the Almagest. 11
"We shall try to note down everything which we think we have
discovered up to the present time; we shall do this as concisely as
possible and in a manner which can be followed by those who have
already made some progress in the field. For the sake of completeness
in our treatment we shall set out everything useful for the theory of the
heavens in the proper order, but to avoid undue length we shall merely
recount what has been adequately established by the ancients. However,
those topics which have not been dealt with [by our predecessors] at all,
or not as usefully as they might have been, will be discussed at length,
to the best of our ability."
With the certainty of the deductive form of argumentation Ptolemy first of
all develops the auxiliary mathematical and astronomical hypotheses in order to
formulate the astronomical theories in their logical order. The solar theory is the
fundamental astronomical hypothesis for all others in the Almagest. No measurement of the position of the other celestial objects is possible without it. All positional
data, whether obtained by use of the astrolabe, a meridian instrument, an eclipse or
the times of rising and setting, are based upon the position of the sun. Consequently,
Ptolemy first outlines in the Almagest a theory of the motion of the sun and proceeds
with the closely related lunar theory before continuing with the fixed stars and the
planets. The order of subjects relates to a tree of definitions with the most general
and fundamental definition at the top and all other subsequently defined concepts
below. 12
"Secondly, we have to go through the motion of the sun and of the
moon, and the phenomena accompanying these [motions]; for it would
be impossible to examine the theory of the stars thoroughly without first
having a grasp of these matters. Our final task in this way of approach
is the theory of the stars. Here too it would be appropriate to deal first
with the sphere of the so-called 'fixed stars', and follow that by treating
the five 'planets', as they are called."
Ptolemy's systematic astronomy hereby resembles the Aristotelian methodology
as it is developed in the "Analytica Posteriora".13
IOPtolemy, C. (1984), p. 36.
C. (1984), p. 37.
12Ptolemy, C. (1984), p. 37.
llcr. Aristotle (1984).
11 Ptolemy,

1.1. The Documents

9

The chapters in the Almagest concerning the fixed stars maintained their unquestioned validity until the beginning of modern astronomy. They will be the subject
of the following pages. Ptolemy's crucial statements will be mostly quoted without
further interpretation, which could anticipate later discussion.

1.1.3

The Almagest on Fixed Stars

The celestial phenomena of the fixed stars are discussed in the seventh and eighth
books of the Almagest. In the eleven chapters of these two books Ptolemy develops
a unified theory of the fixed star phenomena which allows the calculation of all the
important configurations and apparent motions of the stars for any given time. The
thematic sequence and the subjects emphasized by Ptolemy provide a glimpse into
the ancient astronomy of the stellar motions. For example, the detailed discussion of
the question whether the celestial sphere rotates uniformly, especially in connection
with the precession motion, clearly illustrates that these views were not yet a part
of the canonical knowledge of astronomy at the time the Almagest was written (ca.
+150).
Rigorously adhering to the principles of a deductive mode of argumentation, the
chapter on the fixed stars begins with the demonstration that the motion of the stars
can be treated as the motion of a sphere with constant distances between the stars.

VII.l (Seventh book, chapter 1): That the fixed stars always maintain the same
position relative to each other.
Ptolemy compares the alignments of the stars in the constellations as they are
reported by Hipparchus with his own observations without finding any difference.
This proves immediately that there is no relative motion of the stars; hence all
motions of the stars can be represented by a superposition of rotations of a sphere.
This conclusion places Ptolemy even by his own testimony in opposition to the
early Hipparchus who initially promoted the hypothesis that "only the stars in the
vicinity of the zodiac effect had a rearward motion, as Hipparchus proposes in the
first hypothesis he puts forward".14 Ptolemy's reveals the procedure of comparison :15
"If one were to match the above alignments too against the diagrams
forming the constellations on Hipparchus' celestial globe, he would find
that the positions of the [relevant stars] on the globe resulting from the
observations made at the time [of Hipparchus], according to what he
recorded, are very nearly the same as at present."
The quoted passage does not unambiguously state whether Ptolemy actually
resorted to a Hipparchan celestial globe or whether he drew Hipparchus' observational data on a globe for a direct comparison with his observations. 16 In another
passage Ptolemy tells the reader that "observations recorded by Hipparchus, which
are our chief source for comparison. have been handed down to us in a thoroughly
satisfactory form."17 In contrast to the excellent recordings of Hipparchus the older
14Ptolemy, C.
15 Ptolemy, C.
16Ptolemy, C.
17Ptolemy, C.

(1984),
(1984),
(1984),
(1984),

p. 322.
p. 327.
cf. Toomer's footnote p. 327.
p. 321.

10

1. The Stars of the Almagest

observations done by Aristyllos and Timocharis are neither well observed nor carefully "worked out".18 All Hipparchan sources mentioned by Ptolemy are lost today.
A copper globe allegedly belonging to Ptolemy, which reportedly had been found
as late as 1043 in a library in Cairo, was never seen again. 19
VII.2: That the sphere of the fixed stars, too, performs a rearward motion along the
ecliptic.
In the first chapter Ptolemy presents his proof that the fixed stars move on a rigid
sphere. With the available observational accuracy of about 10 minutes of arc and
records of astronomical data over a period of several centuries, the motion of the
sphere of fixed stars could no longer be described solely through the daily rotation.
A second slower rotation around the pole of the ecliptic, later called the "precession
motion", must be added.
Hipparchus was the first who realized the necessity of a second motion of the
celestial sphere. He did not discover the additional motion by analysis of star observations, but through the determination of the equinoxes. The dates of the equinoxes
are one of the most fundamental astronomical parameters in ancient science. The parameters of the solar theory, itself fundamental to all other theories, are derived from
equinox observations, as is the definition of the coordinate systems. The equinox is
defined by the moment when the sun's path on the zodiac intersects the celestial
equator. Only at that moment are the lengths of day and night equal. Because of
the precession motion the eclipticallongitude of the equinox was increasing by I ?38
per century in the time of Hipparchus and Ptolemy. Ptolemy recounts Hipparchus'
discovery of precession :20
"For Hipparchus too, in his work 'On the displacement of the solstitial and equinoctial points', adducing lunar eclipses from among those
accurately observed by himself, and from those observed earlier by Timocharis, computes that the distance by which Spica is in advance of the
autumnal [equinoctial] point is about 6° in his own time, but was about
8° in Timocharis' time. For his final conclusion is expressed as follows:
'If, then, Spica, for example, was formerly 8°, in zodiacal longitude, in
advance of the autumnal [equinoctial] point, but is now 6° in advance',
and so forth. Furthermore he shows that in the case of almost all the
other fixed stars for which he carried out the comparison, the rearward
motion was the same amount."
Ptolemy demonstrates the precession motion by observations with a spherical
astrolabe whose construction he describes in full detail in the first chapter of the
fifth book. The spherical astrolabe (fig. 1.1) consists of a ring system which rotates
freely on two axes pointing to the ecliptical and equatorial poles. 21 The ecliptical
coordinates of a celestial object can be read off directly from the graduation on the
ecliptic ring and the inner ring.
18Ptolemy, C. (1984), p. 321.
19Sezgin, F. (1978), Geschichte des arabischen Schrifttums, Leiden, vol. VI, p. 84.
2OPtolemy, C. (1984), p. 327.
21Adapted from Ptolemy, C. (1963), Handbuch der Astronomie, German trans. and annot. by K.

1.1. The Documents

11

zenith

Figure 1.1: Spherical astrolabe.
In relation to other methods of position measurement the spherical astrolabe
is a complicated instrument. It would be much easier to determine the position
of a celestial object by measuring either the horizontal coordinates, i.e. the height
above the horizon and the azimuth at a given time, or the declination and right
ascension. All these coordinates could easily be measured by observations of the
meridian transit, when an object culminates on the north-south meridian. Of course,
an observer has to wait until a star culminates during the night for such an
observation, but that would be no serious objection to an astronomical program
devoted to compiling a star catalogue of the entire visible sky. That Ptolemy prefers
the astrolabe for the measurement of stellar positions of his catalogue could be
motivated by two reasons: The positions in the star catalogue are given in the
ecliptical coordinate system. With the astrolabe one can read off the ecliptical
coordinates directly from the graduation rings. This, firstly, avoids complicated and
laborious transformations to the ecliptical coordinate system, as in the case of
declination measurements, and secondly allows direct observational control of any
catalogued pair of ecliptical coordinates. It is only with the help of an astrolabe that
the coordinates of the star catalogue can be considered empirical data which could
be directly "observed".
To take a measurement the astrolabe was set up in-such a way that the meridian
ring lay in the plane of the north-south meridian with an axial inclination corresponding to the geographical latitude of the observational site. Before one can read
off the ecliptical coordinates from the astrolabe, the system of rings must be adjusted
in such a way that the ecliptic ring is parallel to the plane of ecliptic at the moment
of observation.
For daylight observations, one turns the whole ring system until the outer
Manitius, introduction and corr. by O. Neugebauer, 2 vols., Leipzig, vol. I, p. 255.

12

1. The Stars of the Almagest

ring marks the solar longitude on the graduation of the ecliptic ring. As Ptolemy
describes it, one has to calculate the longitude of the sun and sets the instrument
accordingly. This setting can be controlled by adjusting the ring system so that the
sun casts a shadow exactly on the other side of the ecliptic and the outer ring.
At this moment the ring system is adjusted exactly to the position of the ecliptical
coordinate system in the sky. It is noteworthy that the control measurement of
the solar position is independent of the solar theory. This means that Ptolemy
could measure ecliptical coordinates without making use of the theory and its
possible errors. However, Ptolemy's description clearly requires the adjustment of
the instrument to the calculated position of the sun.
After the initial adjustments one can observe the moon or another object with
a rotation of the inner astrolabe ring by looking through the diopter. Its ecliptical
longitude can be read from the ecliptic ring and the eclipticallatitude from the inner
ring.
At night the cage of the ecliptic ring must be adjusted to either the known
ecliptical longitude of the moon or a reference star. First, the inner ring is turned
until it intersects the outer ring at the known longitude of the moon or the reference
star. Then the cage of the ecliptic ring is rotated until the object of reference is
visible in the plane of the inner ring. At that moment the astrolabe is adjusted,
interestingly, without making use of the latitude of the reference objects. The
alignment of the ecliptic ring is not without its difficulties and has to be continuously
corrected following the daily motion of the sky. The speed of the latter requires an
extraordinary observational talent and constant correction of the set-up, especially
when the position of several stars in a row is to be measured.
For the determination of the precession motion Ptolemy evaluates an observation
of Regulus, the brightest star of the constellation Leo, on 23 February +139.
According to the Almagest, Ptolemy measured the position of the moon at sunset
and then, half an hour later, the position of Regulus relative to the moon. 22 From
this result Ptolemy was able to obtain the ecliptical longitude of Regulus and to
establish an increase of the value by 2°40' since the time of Hipparchus. Hence,
Ptolemy confirms the Hipparchan value of one degree per century.
At the end of the second chapter Ptolemy mentions that he checked the motion
of the fixed star sphere in the direction of the zodiac signs with observations of the
star Spica. 23
"In the same way we took sightings of Spica and the brightest among
those stars near the ecliptic, from the moon, and then [having done that],
were in a better position to use those stars to take sightings of the rest.
We [thus] find that their distances relative to each other are, again, very
nearly the same as those observed by Hipparchus, but their individual
distances from the solstitial or equinoctial points are in each case about
22Ptolemy, C. (1984), p. 328. There are many difficulties in the numerical details of Ptolemy's evaluation,
as analyzed later. Manitius repeats the calculation and finds that Ptolemy did not consider the change of
the parallax, in spite of his own considerations in the chapters on the theory of the moon. With a correct
calculation Ptolemy would have found a precession of 2°30' instead of the reported 2°40'. However,
this small correction cannot account for the difference from the accurate precession value of 3°40'. Cf.
Ptolemy, C. (1963), vol. II, pp. 397ff.
23Ptolemy, C. (1984), p. 328.

1.1. The Documents

13

2f

farther to the rear than those derivable from what Hipparchus
recorded."

Ptolemy cites the lost Hipparchan text "On the length of the year" in which
Hipparchus gives an estimation of the precession constant: 24
"For if the solstices and equinoxes were moving, from that cause, not
less than 1~ th of a degree in advance [i.e. in the reverse order] of the
signs, in the 300 years they should have moved not less than 3°."
VII.3: That the rearward motion of the sphere of the fixed stars, too, takes place
about the poles of the ecliptic.
Until now Ptolemy has only demonstrated that the longitude of Regulus and Spica
increased by 2°40' over the period of 265 years between his observations and those
of Hipparchus. The axis of the rotation is not yet unambiguously fixed.
In the third chapter the orientation of the motion is confirmed by a comparison
of the ecliptical latitudes of Spica for the time of Hipparchus and Ptolemy. In the
case that the precession motion rotates around the pole of the ecliptic, the ecliptical
latitudes of the stars should not show any measurable changes during the time for
which historical records are available.
Ptolemy reports that Hipparchus had already recognized the orientation of the
precession motion around the pole of the ecliptic in another text, entitled "On the
displacement of the solstitial and equinoctial points", though Hipparchus seemed
uncertain of the result since he could base his calculations only on the unreliable
observations of the astronomers of the school of Timocharis. Ptolemy can be
more certain of his findings, because he can rely on the accurate measurements of
Hipparchus. He finds no significant change in the eclipticallatitudes at all.
However, the exact value of the precession motion is confirmed by evaluating the
declinations of bright stars in comparison with older observations. The declinations
of the stars are easily obtained through the observation of the meridian transit:
as soon as the star passes the north-south meridian at the site of observation, the
altitude he of the star over the horizon is measured and, through an uncomplicated
arithmetical operation with the geographical latitude qJ, one obtains the declination
J of the star as

(1.1)
The simple way of measuring declinations might be the reason that these measurements were not only recorded by Hipparchus, but also in the older astronomical
school of Timocharis.
Ptolemy, for his part, records the declinations of two sets of nine stars in both
the northern and southern part of the sky for the time of Timocharis, Hipparchus
and himself, and he calculates from a subset of six stars the precession motion. From
all the quoted calculations Ptolemy obtains results confirming a precession motion
24Ptolemy, C. (1984), p. 329. As Toomer remarks, the mentioned time difference of 300 years refers to
the solstice observation of Meton (-431), reported in book III (solar theory) of the Almagest; Ptolemy,
C. (1984), p. 138.

14

1. The Stars of the Almagest

of one degree per century which, although identical with the minimal Hipparchan
value, deviates substantially from the accurate value of 1~38.25
VII.4: On the method used to record [the positions of] the fixed stars.

The fourth chapter introduces the fixed star catalogue. An appropriate coordinate
system must be chosen so that the positions of the stars can be calculated without
too great mathematical complications for any given time. Interesting phenomena
like rising and setting times, meridian transits, or the relative positions of two
celestial objects to each other should be derivable with little effort. If only the daily
revolution had to be considered for the celestial motions, the equatorial coordinate
system would be most convenient for a catalogue of stars. From the declination of a
star one can calculate the maximal altitude over the horizon and the circumstances
of rising and setting. Together with the right ascension, all positions of a star in the
sky could be derived with only limited inconvenience.
The discovery of the precession motion then made it clear that the motions of
the stars are not so easy to represent. The accuracy of 10 minutes of arc in the
position entries would require the correction of the star catalogue after only a short
period of time. With a precession of 1~38 per century, an adjustment is necessary in
extreme cases only after little more that 10 years. With these more complex motions
one needs a coordinate system with which the precession motion could be integrated
in a particularly simple way.
It has been demonstrated in the second chapter of the seventh book that
the precession motion is a rotation around the pole of the ecliptic. 26 When the
coordinate system is arranged in such a way that its poles coincides with the poles
of the precession motion, an uncomplicated conversion of the star coordinates to the
respective epoch is possible: the eclipticallatitude of a star remains unaffected by the
precession motion, and the eclipticallongitude increases at a constant value with time
by the precession constant. This enormous advantage requires that Ptolemy'S star
catalogue be compiled in ecliptical coordinates. For example, from a star catalogue
for a given epoch, the ecliptical coordinates can be calculated by a simple addition
of the precession to the ecliptical longitudes. The accurate value was 1~38 during
Ptolemy's time. The Almagest, though, takes over the Hipparchan minimal value of
one degree per century. This had to cause problems for the following generation of
astronomers who used the star catalogue of the Almagest reduced to their epoch.27
In the following we quote at full length Ptolemy's important statements on the
method by which the data of the catalogue were obtained :28
"So we thought it appropriate, in making our observations and
records of each of the above fixed stars, and of the others too, to
give their positions, as observed in our time, in terms of longitude
25 Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical
Almanac, Her Majesty's Stationary Office (1961), London, pp. 28ff.
26This is valid for the limited period between Hipparchus and Ptolemy with changes of the latitude of
less than I'. Cf. Explanatory Supplement, pp. 28ff.
27 If not problems for the accuracy of the stellar position, then problems for a sound theory of precession
motion.
28Ptolemy, C. (1984), pp. 339f.

1.1. The Documents

15

and latitude ... Hence, again using the same instrument (because the
astrolabe rings in it are constructed to rotate about the poles of the
ecliptic), we observed as many stars as we could sight down to the sixth
magnitude. [We proceeded as follows.] We always arranged the first of
the above-mentioned astrolabe rings [to sight] one of the bright stars
whose position we had previously determined by means of the moon,
setting the ring to the proper graduation on the ecliptic [ring for that
star], then set the other ring, which was graduated along its entire length
and could also be rotated in latitude toward the poles of the ecliptic, to
the required star, so that at the same time as the control star was sighted
[in its proper position], this star too was sighted through the hole on its
own ring. For when these conditions were met, we could readily obtain
both coordinates of the required star at the same time by means of its
astrolabe ring: the position in longitude was defined by the intersection
of that ring and the ecliptic [ring], and the position in latitude by the
arc of the astrolabe ring cut off between the same intersection and the
upper sighting-hole.
In order to display the arrangement of stars on the solid globe
according to the above method, we have set it out below in the form of
a table in four sections. For each star (taken by constellation), we give,
in the first section, its description as a part of the constellation; in the
second section, its position in longitude, as derived from observation, for
the beginning of the reign of Antoninus ([the position is given] within a
sign of the zodiac, the beginning of each quadrant of the zodiac being,
as before, established at [one of] the solstitial or equinoctial points); in
the third section we give its distance from the ecliptic in latitude, to the
north or south as the case may be for the particular star; and in the
fourth, the class to which it belongs in magnitude."
Ptolemy claims very explicitly to have observed the stars of the catalogue with a
spherical astrolabe in the year +137. It is precisely this statement whose truth or
falsity has been granted or contested for more than one thousand years.
The position of a number of brighter reference stars then - fundamental stars were determined with the help of the position of the moon and sun, and the positions
of the remaining stars were measured relative to them. Through this method any error
in the positions of the fundamental stars would have carried over to the catalogued
positions of the relatively measured stars. Because of their importance, the positions
of the fundamental stars must be measured with particular care, whereby the desired
precession is dependent on the accuracy of the actual measuring with the astrolabe
as well as on the accuracy of the measurements or calculations of the sun.
Ptolemy is aware of possible errors in observation. In the first chapter of the
third book the concept of the length of the year is discussed. There he criticizes
an incorrectly evaluated measurement of Hipparchus and deems possible an inexact
observation or calculation of the lunar position. 29 In addition the influence of
29"lt is more plausible to suppose, either that the distances of the moon from the nearest stars at
the eclipses have been too crudely estimated, or that there has been an error or inaccuracy in the
determinations of the moon's parallax with respect to its apparent position, or of the motion of the sun

16

1. The Stars of the Almagest

optical illusions on the accuracy of the estimates of position is well known.30
Ptolemy considers measurements made with the astrolabe to be reliable. 3!
Ptolemy gives no indication of an earlier star catalogue comparable to that in the
Almagest, though he must have had access to extensive records of stellar data in the
constellations. As for the grouping and configuration of the constellations Ptolemy
admits openly to deviating from the traditional terminology of his predecessors: 32
"Furthermore, the descriptions which we have applied to the individual stars as parts of the constellation are not in every case the same
as those of our predecessors (just as their descriptions differ from their
predecessors') : in many cases our descriptions are different because they
seemed to be more natural and to give a better proportioned outline to
the figures described."
The historical question later emerged whether Ptolemy also catalogued the
positions and magnitudes of the stars independently of his predecessor.

from the equinox of the time of mid-eclipse." Ptolemy, C. (1984), p. 136.
JOPtolemy, C. (1984), p. 421.
Jl Ptolemy, C. (1984), pp. 453f: "We cannot derive this from the ancient observations [of Mercury],
but we can do so from our own observations made with the astrolabe. For it is in this situation that
one can best appreciate the usefulness of this way of making observations, since, even if those stars
with previously determined positions which are visible are not near the planet being observed (which is
generally the case with Mercury, since, for the majority of the fixed stars, it is rare that they are visible
when they are [only] as far from the sun as Mercury is), one can still determine positions of the planet
in question accurately in latitude and longitude, by sighting stars which are at a considerable distance."
J2Ptolemy, C. (1984), p. 340.

1.2. The Arabic Revision of the Almagest

1.2

17

The Arabic Revision of the Almagest

During the time between its composition and the beginning of the 16th century
the Almagest strengthened its unique position as the standard work of astronomy,
especially through scientific activity in the Orient.
Until its decline in the fifth century, Alexandria was the center of influence for the
Ptolemaic texts. Particularly in the fourth century Alexandrian scholars produced a
series of commentaries some of which are at least partially preserved. The so-called
"small astronomy", an allusion to the "great astronomy" of Ptolemy, consists of
a collection of mathematical and astronomical treatises supposed to serve as an
introduction to the more complex parts of the Almagest. 33
Towards the beginning of the fourth century Pappus wrote a commentary to the
Almagest from which only the parts on the fifth and sixth book are still preserved.
His commentary had more the character of an elucidation and added nothing to the
astronomical knowledge contained in the Almagest. 34 It is still unknown whether
the commentary of Pappus covered all the books of the Almagest or whether it
restricted itself to a discussion of the motion of the sun and moon. A century later
Theon of Alexandria included these expositions of Pappus in his comprehensive
commentary.35
Despite the extensive commentaries on the Almagest, no critical inspection of
its contents, above all of the star catalogue, is known to us from antiquity. The
theoretical and practical advances of the Almagest in comparison to the alternatives
of its predecessors must have been so remarkable that small numerical inaccuracies
in the Ptolemaic theories could not force an astronomer to severe revisions. Theon
tells of a number of astrologers before Ptolemy who did not assume a constantly
increasing longitude of the spring equinox due to the precession motion, but rather
a periodical oscillation over an arc of 8 degrees. 36
It is possible that, shortly after the discovery of the precession motion by
Hipparchus, inexact observation or sheer astrological speculations were the source
of such theories. Besides the astronomical difficulties in developing a satisfactory
33In the introduction to his translation Manitius sketches the transmission of the Almagest: Ptolemy,
C. (1963), vol. I, p. V. See also Dreyer, J. L. E. (1953), A History of Astronomy from Thales to Kepler, 2""
edition, New York; Suter, H. (1900), Die Mathematiker und Astronomen der Araber und ihre Werke, Leipzig;
Sezgin, F. (1978), Geschichte des arabischen Schrifttums, vol. VI; The introduction of Kunitzsch, P. (1975),
Zur Kritik der Koordinatenilberlieferung im Sternkatalog des Almagest, Gottingen; Kunitzsch, P. (1974),

Der Almagest. Die Syntaxis Mathematica des Claudius Ptolem1Jus in arabisch-Iateinischer Uberlieferung,

Wiesbaden.
34Rome, A. (1936/43/31), Commentaires de Pappus et de Theon d'Alexandrie sur l'Almageste, 3 vols.,
Biblioteca Apostolica Vaticana, Studi e Testi 72, 106, 54. Roma, vol. I.
35Rome, A. (1936/43/31), vol. II und vol. III. See also Theonis Alexandrini in Claudii Ptolemaei Magnum
Constructionem Commentariorum Lib. XI, Basel, 1538.
36Dreyer, J. L. E. (1953), p. 204: "According to certain opinions ancient astrologers believe that from
a certain epoch the solstitial signs have a motion of 8° in the order of the signs, after which they go
back the same amount; but Ptolemy is not of this opinion, for without letting this motion enter into the
calculations, these when made by the tables are always in accord with the observed places. Therefore we
also advise not to use this correction; still we shall explain it. Assuming that 128 years before the reign
of Augustus the greatest movement, which is 8°, having taken place forward, the stars began to move
back; to the 128 years elapsed before Augustus we add 313 years to Diocletian and 77 years since his
time, and of the sum (518) we take the eightieth part, because in 80 years the motion amounts to 1°. The
quotient (6°28'30") subtracted from 8° will give the quantity by which the solstitial points will be more
advanced than by the tables". See also Neugebauer, O. (1975), pp. 6311[

1. The Stars of the Almagest

18

theory of the precession motion, the strong desire to formulate the motions of the
celestial sphere into a theory preserving traditional astrological interpretations and
the validity of the ancient observations, led to the construction of models with a
non-linear precession motion, even after the composition of the Almagest. 37
However, without an exact knowledge of the precession motion, the Ptolemaic
coordinates of the stars cannot be adequately checked later, not even with the
most accurate method of measuring. Though the stellar coordinates are a result
of observations, they cannot be confirmed or corrected just by a repetition of the
observations some centuries later. Obviously only an exact recalculation of the
stellar positions for the time of the Almagest allows one to check the data contained
therein. Even if the coordinates of a later epoch are accurately measured, the correct
ecliptical longitudes of an earlier period can only be calculated when the actual
motion of the spring equinox due to precession is subtracted. Therefore a test of
Ptolemy's star catalogue requires an adequate theory of the precession motion.
The possibility of critically checking the Ptolemaic star catalogue arose for the
first time after solid knowledge of the motion of the stellar sphere had been gained.
After the decline of Alexandria as the scientific center of the ancient world in the
fifth century, the religious centers of the Orient took over the tradition of the Greek
sciences and with them the astronomical theories of the Almagest. 38
At the end of the eighth century, with the flourishing of Islamic civilization, an
astronomical science was developed which absorbed and revised first the Indian,
then, somewhat hesitantly, the Greek tradition, and later was transmitted through
Spain to medieval Europe. With the decline of the kingdoms in the Orient and
their breakup into a plethora of small dynasties, astronomy received an impulse to
improved formulations on a level of complexity exceeding those of the Almagest. 39
The Abbasid Caliph al-Ma'mun initiated the heyday of the sciences as, first in
Damascus and then in Baghdad (from 829), he built observatories for the testing
and revision of the traditional astronomical knowledge on the basis of independent
observations. 40 A small list of 24 stars with coordinates independent of the Almagest
bears testimony to the observations of the astronomical school of al-Ma'mun. The
earliest translations of the Almagest known today date to that period.41 The activity
of the Islamic astronomers focused on improving the parameters in the astronomical
theories without calling into question the theoretical edifice itself, namely the views
offered in the Almagest. One accomplishment of this time was the accurate measuring
of the meridian with 56~ miles for 1 of the meridian or 20400 miles for the
circumference;42 another the improvement of the astronomical measuring methods
themselves.43
The Ptolemaic catalogue was converted to the epoch of that time in that the
0

37Mercier, R. (1976/77), Studies in the Medieval Conception of Precession, 2 parts, Archives Internationales d'Histoire des Sciences 26 (I), 27 (II), part I, p. 209.
38Cf. Kunit:=h, P. (1974), pp. Iff.
39Dreyer, J. 1. E. (1953), p. 245.
4OPtolemy, C. (1963), vol. I, p. VI.
41 Kunitzsch, P. (1974), Der Almagest. Die Syntaxis Mathematica des Claudius Ptolemaus in arabischlateinischer Uberliejerung, Wiesbaden, pp. 6ff. Kennedy, E. S. (1956), A Survey of Islamic Astronomical
Tables, Trans. Amer. Philos. Soc., N. S. 46.2, pp. 132ff.
42Nallino, C. A. (1944), Raccolta di scritti, vol. V, p. 421.
43Sezgin, F. (1978), Geschichte des Arabischen Schrifttums, Leiden, vol. VI, p. 20.

1.2. The Arabic Revision of the Almagest

19

eclipticallongitudes of certain reference stars were observed and the difference from
the longitudes given in the Almagest was added to the longitudes of the other stars.
After this had been carried out, Islamic astronomers possessed a comprehensive star
catalogue devoid of any significant errors in its coordinates. Each systematic error in
longitude of the Almagest necessarily remained unnoticed in such a procedure. This
explains why Tycho Brahe's later discovery - that the longitudes of the Almagest stars
are systematically one degree too small- could not be detected by these astronomers.
The critical transmission of the Ptolemaic star catalogue during this time is known to
us through the work of al-BattanI (d. 929), a~-~iifi (903-986), al-BIriinI (d. 1048), Ibn
a~-~alal;1 (d. 1154) and Ulug Beg (1394-1449).44 The comprehensive astronomical
treatise of al-BattanI contain,45 besides longer expositions on the lunar and solar
theory, a number of tables among which two star catalogues can be found. One
of these lists 75 stars whose equatorial coordinates were measured as fundamental
coordinates for the other stars. The catalogue contains all bright stars in the
same sequence as they are catalogued in the Almagest. 46 The second, even more
comprehensive register includes 533 Ptolemaic stars whose eclipticallongitudes were
calculated by adding 11 degrees 10' for the epoch 1 March +880 using a precession
constant of 1 degree for 66 years.47 We know from a~-~iifi that al-BattanI had
considered for his register only the Ptolemaic stars whose coordinates show no
variations in the different versions of the Almagest. 48
The extensive philological activity practised by Islamic astronomers shows that
already 700 years after the writing of the Almagest a large quantity of numerical
values had been corrupted through copying errors. Before a critical appraisal of
the genesis of the Ptolemaic catalogue and, the sole matter of importance for the
Arabic astronomers, a scientific use of the coordinates could be made, these errors
had to be eliminated. In the time following a number of astronomers concentrated
their work on removing improbable interpretations by a critical comparison of the
existing copies with'their own exact observations of the stellar positions.
The value of the precession constant itself provides clues to the procedure of its
derivation.
The precession constant of 1 degree every 66 years (54.5" IY ) is larger than the
accurate value of 1 degree every 72 years (50"IY). It was used in the ninth and tenth
century and was borrowed from the star register of ZIj al-mumtahan,49 which was
composed in the school of al-Ma'miin at the new observatory in Baghdad around
the year 830. 50 In response to a decree of the Caliph the astronomers performed
observations in order to check and possibly correct the values coming down through
tradition. The far too small Hipparchan/Ptolemaic value of precession of 1 degree
44Kunitzsch, P. (1974), p. 47.
45Nallino, C. A. (1899-1907), Al-Battani sive Albatenii Opus astronomicum, ed. Carolo Alphonso Nallino,
3 vols., Milano.
46Kunitzsch, P. (1974), p. 50.
47Kunitzsch, P. (1974), p. 50. As G. Toomer pointed out to me, it seems likely that al-Battam used the
constant I! degrees per century.
4SKunitzsch, P. (1974), p. 47.
49Kunitzsch, P. (1974), p. 51.
5OSuter, H. (1900), Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur
Geschichte der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Heft X. Reprint
New York, 1972, pp. 8&10.

1. The Stars of the Almagest

20

per century had been recognized very early as false and for that reason it had to be
newly determined by the Islamic astronomers. To keep the error as small as possible,
the astronomers were forced to select the longest period of time between their own
position measurements and the epoch of the older coordinates whose longitudes had
increased due to the precession motion. Nothing, therefore, seemed more reasonable
to them than to call upon the old Ptolemaic star register and to make a comparison
between the catalogued ecliptical longitudes of the epoch +137 and the longitudes
of the epoch +830 they had measured themselves. The difference should amount to
exactly 10 degrees when the real precession value of 5<Y'/Y is taken as the basis. Now
the longitudes of the Ptolemaic catalogue, however, are on the average 1 degree too
small. Consequently the Islamic astronomers obtained a difference of 11 instead
of 100. If the Almagest is used as the source of the observations, the precession
constant that is calculated is too large by about 10 per cent. Instead of 50/l/y the
astronomers of the Caliph al-Ma'mun obtained a precession constant of 55/1/Y or,
expressed in other terms, one degree in 66 years.
Later, the value of the precession constant was improved still further. The
astronomer Na~Tr ad-DIn at-TusT computed in 1274 a precession constant of 1
degree for 70 years (51.4/1/Y) which was still larger than the accurate value and
possibly included the coordinates of the Almagest in the calculation as welJ.5!
Two pivotal conclusions can be drawn concerning the observational practice of the
Islamic astronomers and the status of the Ptolemaic star catalogue at this time.
0

(i) The Islamic astronomers of the ninth century carried out observations of the
positions of the fixed stars whose accuracy vis- a-vis that of Hipparchus was
improved and which lay in the neighborhood of 10'. The fixed star register of
Zij al-mumtahan, from which one hundred years later the astronomer a~-~ufi
borrowed the precession constant, enjoyed a very good reputation. 52 However,
these star catalogues included only the most important stars and could not,
therefore, replace the Almagest in any way.
(ii) Even though the Islamic astronomers were able to perform their own exact
observations, they nevertheless placed full trust in the essential formulations of
Ptolemy's Almagest. They could improve the value of the precession motion by
using the coordinates of the Almagest for its calculation, but this very process
prevented them from any critical inspection beyond the purely philological
testing of their authenticity. The scope of al-BattiinTs star register and the
remarks of a~-~ufi and Ibn a~-~alii~ tell us that they were well aware of the
deficiencies of the copies. At first there was no reason to doubt the correctness
of the original Ptolemaic catalogue data before the coordinates of the Almagest
had been reconstructed from the ever increasing number of manuscript copies
and before a proper theory of the precession motion was established.

In the history of Arabic astronomy

a~-~ufi

(903-983) wrote one of the most

51 If one amlumes accurate position measurements in the year 1274, one should derive a precession of
lOin 68 years on the basis of the longitudes in the Almagest. It is also possible that the astronomers of
that period neglected the Ptolemaic longitudes and based their calculations of the precession constant
entirely on observations from early Islamic astronomy.
52Kunitzsch, P. (1974), p. 51. Suter, H. (1900), p. 8.

1.2. The Arabic Revision of the Almagest

21

important texts on the fixed stars since Ptolemy.53 A~-~ufi checked how all of the
Ptolemaic constellations and stars had been handed down through tradition and
also, at least partially, their agreement with the positions he had determined himself.
He was the first to maintain that Ptolemy had not observed the stars of the catalogue
himself, but had taken them from an older manuscript and increased the ecliptical
longitudes by the value of precession in accordance with the Hipparchan value of 1
degree per one hundred years.
A~-~Ufi presumes that Ptolemy made use of the data of Menelaus which had
been obtained 41 years before the epoch of the Almagest (+137) as foundation for
his catalogue, and then added 25' to the Menelaic longitudes. It is not clear why
a~-~ufi makes this claim. As he tells us, although several copies of the Almagest were
available to him, he had no Greek source of Menelaus' writing. 54 Disregarding for the
moment a~-~Uf'j's motives for making this claim, it cannot be the longitudinal errors
of the Ptolemaic stars of one degree that had prompted him to his interpretation.
In 41 years the hypothetical longitudes of Menelaus increase by 34'. According to
a~-~ufi, Ptolemy would have added 25', and with that obtained longitudes only 9'
too small, all of which makes up a negligible error. In spite of his allegation that the
positions of the stars of his catalogue were not observed by Ptolemy himself, a~-~ufi
had obviously not yet discovered the systematic errors in longitude of the Almagest.
For a long period following this remains the unique instance of a doubt about
authenticity of the Ptolemaic star catalogue. It is highly speculative whether a~-~ufi
supplied an interpretive model for Tycho Brahe's later examinations of the Ptolemaic
catalogue. From the late middle ages to the 16th century the astronomer a~-~ufi was
indeed known: it is evident from two wooden engravings of the northern and southern hemispheres by DUrer on which a~-~ufi is depicted as one of the four greatest
proponents of astronomy. 55 His texts were not translated into Latin, though.56 In
several medieval manuscripts with star lists attributed to a~-~ufi there are illustrations influenced by the Arabic tradition. The coordinates and the description of the
star positions are taken from the version of the Almagest as translated by Gerhard
von Cremona. 57
For its own part Islamic astronomy formulated no critique on the accuracy
of the original coordinate measurements of Ptolemy, choosing rather to restrict
itself to a philological purification of the copying errors and making new and
independent measurements. As late as ca. +1150 Ibn a~-~alii~ examined with great
meticulousness the transmission of the Ptolemaic star catalogue and restored a
number of coordinates that are extremely helpful today for the reconstruction of
the original Ptolemaic star catalogue. 58 Despite the extensive discussion of the
possible star positions at the time of Ptolemy, the text contains no remarks on the
systematic errors in longitude. The author's only interest was to solve the problem
53Sezgin, F. (1978), p. 212. In French translation: Schjellerup, H. C. F. C. (1874), Description des etoiles
fixes, St. Petersburg. Cf. Kunitzsch, P. ([974), p. 51.
54Bjornbo, A. A. ([90[), p. 202. Cf. section m.2.
55Sezgin, F. (1978), p. 212. The engraving is printed in Strohmaier, G. (1984), Die Sterne des Abd
ar-Rahman as-Sufi, Hanau.
56Kunitzsch, personal communication.
57 Kunitzsch, P. (1965), Sufi Latinus, Zeitschrift der Morgenlandischen Gesellschaft 115, pp. 65-74.
Strohmaier, G. (1984), p. 12.
58Kunitzsch, P. (1975), Zur Kritik der Koordinatenuberlieferung im Sternkatalog des Almagest, Gottingen.

22

1. The Stars of the Almagest

of restoring the original numbers by checking them against his own observations.
Without knowledge of the appropriate model for the precession motion it was not
possible to estimate systematic errors in the Almagest.
With the increasing number of independent observations Islamic astronomy
could emancipate itself from the Ptolemaic star catalogue. Huge instruments were
built to refine the precision of the measurements. Al-BIriinI, for example, owned a
quadrant with a radius of7.5m.59 The high point of independent Islamic observations
was reached with the fixed star observations of Ulug Beg who revised a fraction of
the traditional star catalogue and replaced the coordinates by more accurate ones
in the observatory in Samarkand. 6O
With Spain as a conduit the astronomical knowledge of the Orient quickly spread
to scholarly circles in medieval Europe and was sufficiently comprehensive to permit
a critical evaluation of the accuracy of the Ptolemaic star catalogue.61

59 Wiedemann,

559.

E. (1970), AufsiUze zur Arabischen Wissenschaftsgeschichte, 2 vols., Hildesheim, vol. I, p.

6OSezgin, F. (1978), p. 30. Knobel, E. B. (1917), Ulughbeg's Catalogue of Stars, Washington.
61Sezgin describes the influence of Islamic astronomy in Sezgin, F. (1978), pp. 37-59. See also Mercier,
R. (1976/77); Dobrzycki, J. (1963), Katalog gwiazd w de Revolutionibus, Studia i Materialy z Dziejow
Nauki Polskiej, Seria C, Z. 7.; Swerdlow, N. M., Neugebauer, O. (1984), Mathematical Astronomy in
Copernicus's De Revolutionibus, New York.

2. Accusations
2.1

Tycho Brahe

The Almagest became known in Europe through the Latin translation of Gerard of
Cremona in 1175. Astronomy began to assimilate Ptolemaic theory and its Arabic
revisions into the emerging physical sciences and thereby laid the ground for the
following rapid scientific development. In the 16th century Copernicus succeeded
in overcoming the geocentric construction of Ptolemy's planetary orbits, but the
methodological structure of his "De revolutionibus" was still oriented on the book
that was written a millennium before.
Copernicus' star catalogue is based exclusively on the data of the Almagest.!
Copernicus complained about the inaccuracies of the catalogue as he also complained about the lack of a viable alternative to it, but it was Tycho Brahe, the last
and the most meticulous observer before the introduction of optical instruments,
who was the first to lay the groundwork for a systematic appraisal of the Ptolemaic
coordinate errors through his own highly precise star coordinates.
Tycho was indeed the first European to replace the Ptolemaic star catalogue with
his own, far more exact positional measurements. The appearance and identification
of a new star in the year 1572 inspired him, as, reportedly, a similar event had inspired
Hipparchus, to assemble a new star catalogue. 2 Tycho also calculated the precession
motion anew without the use of the stellar coordinates of the Almagest. This was
the first step to a historical interpretation of the accomplishments of Ptolemy. The
early Arabic astronomers, who were still forced to base their calculations of the
precession motion on the coordinates and the times recorded in the Almagest, could
in principle not discover any systematic errors in Ptolemy's longitudes.
Tycho Brahe had access to their observational material, with which he was able
to justify a simple linear precession motion independently of the Almagest. After
that, he was in a position to compare the Ptolemaic star catalogue with the positions
recalculated from his accurate measurements.
A correct theory of the precession motion is an irreplaceable precondition
for the checking of the Ptolemaic coordinates. As long as medieval astronomy
still formulated and computed the spring equinox with a theory of trepidation
incorporating the observations of the Almagest in its basic parameters, the systematic
ICc. Dobrzycki, J. (1963) and Swerdlow, N. M., Neugebauer, O. (1984).
2Dreyer, J. L. E. (1953), p. 365.

24

2. Accusations

errors of the stellar longitudes of Ptolemy could not be detected.
In the introductory comments to the chapters on the sphere of fixed stars in
"Astronomiae Instauratae Progymnasmata" (1602) as well as in the introduction to
his star catalogue "Stellarum Inerrantium Restitutio" (1598), Tycho sketches out
the historical development of the star catalogues.3 In the "Progymnasmata" a brief
remark can be found that the star catalogue of the Almagest had been compiled
through the conversion of the Hipparchan stellar coordinates. 4 In "Stellarum Inerrantium Restitutio" Tycho came to the conclusion that the lower limit of the
Hipparchan precession constant used by Ptolemy for the conversion of the stellar
longitudes to his epoch could in fact account for the errors in longitudes of the
stars in the Almagest,S although Ptolemy himself was prevented from discovering
these by certain systematic errors of his own methods of observation. As possible
causes for the error in longitude Tycho considers an inadequate solar and lunar
theory6, the reduction of the solar longitude through the effect of refraction at
sunset, and the neglect of the lunar parallax. 7 Tycho studied the Arabic astronomers
and suggested historical reasons for their errors. He discovered that the error in
longitude of Ptolemy'S star catalogue is responsible for the large precession constant
of al-BattanI. 8 Although Tycho committed himself to the thesis of a Hipparchan
origin of the Ptolemaic star catalogue, his astronomical research opens the way for
two different possibilities of historical interpretation.
(i) The errors in longitude of the star catalogue result from a transformation of
the Hipparchan coordinates with a precession constant that is too small. A
series of systematic errors, like the deficiencies in the solar and lunar theory and
the disregard of the effects of refraction and parallax, must have led Ptolemy
to confirm the conversions he made from the Hipparchan star register.
(ii) Because the systematic errors in the solar theory confirm the longitudes in
the catalogue by later observation, these errors could also be the original
cause for the inaccuracies of the fixed star catalogue. If Ptolemy had used
an observational method which assumes the erroneous solar theory, it follows
that the systematic errors of the star catalogue would be generated thereby.
3Brahe, T. (1913-29), Tychonis Brahe Dani Opera omnia, ed. J. L. E. Dreyer, 15 vols., Copenhagen,
vols. II and III.
4Brahe, T. (1913-29), vol. II, p. 151: "Post hos Claudius etiam Ptolemaeus, circa Annum a nato Christo
140, Alexandriae quoque Aeqypti nonnulla in harum progreBione animaduertere, atque Iiteris mandare,
aggreBus est; Hipparchico tamen, circa earum adinuicem, quoad longum & latum collocationem, totaliter
retento Abaco."
5Brahe, T. (1913-29), vol. III, pp. 335f.
6The maximal error of the latter Tycho estimates as 1/4 degree.
7Brahe, T. (1913-29), vol. III, p. 336: "lncedens enim lubrica ilIa uia & ad fallendum prona, quae a
Sole per Lunam Stellarum loca monstraret, facile quartae partis unius gradus, si non dimidiae, errorem
incaute admittere potuit: ueluti alibi a nobis expressius pandetur. lmo cum refractiones Solis iuxta
Horizontem (circa quem, cum hanc pragmatiam exercebat, constituebatur) positi, ut de Parallaxibus non
dicam, neglexerit, praecisionem ipsissimam non attigit, uti et saepius his a1ijsque de causis tam in Sole
quam reliquis Planetis & Stellis fixis deuiationem aliqualem commisisse uidetur. Verum hoc non ob id
refero, quod tanti artificis & de tota re Astronomica adeo praeclare meriti Viri, sine cuius operibus uix
pateret ad hanc Artem accessus, traditiones eleuare praesumam: sed solummodo ut negotij subtilitatem et
labyrinthos, ubi summa requiritur praecisio, maximis etiam artificibus obrepentes, aliquatenus indicem."
sBrahe, T. (1913-29), vol. III, p. 336.

2.2. LapJace and Lalande

25

Therefore, it is not possible to make a decision between the two interpretative
alternatives based solely on the systematic errors in the stellar longitudes. For
the following generations of astronomers, the specific judgment about the origin
of the star catalogue was more and more based on the possibility of a coherent
interpretation of the totality of astronomical claims in the Almagest.

2.2

Laplace and Lalande

Laplace doubted the Hipparchan origin of the star catalogue in his "Exposition du
Systeme du Monde."9 In chapter two of the fifth book he states that Hipparchus'
length of the year was too large and that Ptolemy's assimilation of this theory
explains why the position of the mean sun was too small by one degree at the time
of the Almagest. Since the star positions are determined relative to the position of
the sun using the astrolabe as described in the Almagest, the solar theory alone is
capable of explaining the stellar longitudes in the catalogue. 10
"This remark moves us to examine whether, as generally believed,
Ptolemy'S star catalogue is merely the one prepared by Hipparchus
adjusted to the time of the former through a yearly precession of 111".
This opinion is grounded on the fact that the systematic error of the
longitudes of the stars in this catalogue disappears when one reduces it
to the time of Hipparchus. However, the explanation offered by us for
this error vindicates Ptolemy against the accusation that he had simply
assimilated the work of Hipparchus and it appears justified to believe
him when he says that he himself had observed the stars of his catalogue,
even the ones belonging to the sixth magnitude."
Laplace's statements go beyond those of Tycho in that it clearly offers a coherent
interpretation of the Almagest according to the principle of the greatest possible
credibility.
A significantly richer historical interpretation of the Almagest, which Laplace
refers to in the chapter just mentioned, had already been articulated by Lalande.H
The texts document a newly awakened interest in the Almagest particularly
among the astronomers of the 18th century. In 1712 the Royal Astronomer Edmund
Halley edited the Greek text of Ptolemy'S star catalogue. 12 The vast span of time
from the epoch of the Almagest turned the star catalogue into an historical witness
of ancient star data which promised interesting evaluations in spite of its recognized
inadequacies.
Halley compared the latitudes of the bright stars with the ecliptical latitudes of
his time and so he was the first who successfully demonstrated the proper motion
9Laplace, P. S. (1796), Exposition du Systeme du Monde, Paris. Cited after Laplace, P. S. (1797),
Darstellung des Weltsystems, Frankfurt, vol. II, pp. 253 If.
I°Laplace, P. S. (1797), vol. II, pp. 254f.
II Lalande, J. D. (1757), Memoire sur les equations seculaires, et sur les moyens mouvemens du Soleil,
de la Lune, de Saturne, de Jupiter et Mars, avec les observations de Tycho-Brahe, faites sur Mars en
1593, tilies des manuscrits de cet Auteur. Memoires de mathematique et de physique, tirees des registres
de I'Acadenne Royal des Sciences, de I'Annee 1757. pp. 411-470.
12Halley, E. (1712), Geographiae Veteris Scriptores Graeci Minores. Oxford, vol. III.

26

2. Accusations

of the stars for Sirius, Arcturus and Aldebaran. 13 In 1786 a French translation of
the Almagest star catalogue was published by the Abbe Montignot, followed by the
German translation of the astronomer Bode in 1795. 14 The advanced mathematical
treatment of celestial mechanics in the 18th century led to a highly precise theory of
celestial motions. An accurate approximation to the motions of three celestial bodies
attracting each other was accomplished and the dimensions of the solar system were
successfully determined through the observations of the Venus transits of 1761 and
1769. At that time the data of the Almagest were considered the testing instance by
which the precision of a theory for long periods of time could be controlled.
One of the most famous astronomers of the 18th century was Joseph-Jerome
Lalande, whose textbook "Traite d'astronomie" of 1764 became, with new editions
in 1771 and 1792, a standard work in the field. 15 Lalande, who stood firmly in the
tradition of the encyclopedists, examines the major theories of the Almagest, the
conclusion of which he draws in "Memoires de l'Acad6mt'e Royale des Sciences."16
Lalande investigates here the Ptolemaic measurements of the equinoxes and finds
there an awesome deviation from the accurate values, very different from the high
precision earlier obtained by Hipparchus. All of the inaccuracies of the Ptolemaic
observations are, as Lalande sees it, explicable in a most natural way if all of them
are interpreted as mere theoretical constructions. I? Lalande uses five arguments
to support the claim that Ptolemy had not himself made the observations in the
Almagest, but had only calculated the results from the theory and then claimed
them as the fruit of actual observations:
(i) Ptolemy reports the lunar eclipses of 19-20 March -199 and 12 September -199
which he evaluated for the calculation of the parameters for the lunar theory.
Ptolemy criticizes the Hipparchan analysis whose calculations assume a time
difference between the eclipses of 176 days, one hour and 20 minutes, and he
replaces it with a time difference of 176 days and 24 minutes. 18 For Lalande,
this proves that Ptolemy had undertaken certain changes in the observational
data as reported by Hipparchus in order to bring the values in agreement with
his theory.19
(ii) The Ptolemaic measurements of the equinoxes are highly distorted. According
to Lalande, the measurement of 26 September +139 as well as of 22 March
is incorrect by 11 hours, and it is peculiar that this error tallies with the
theoretical values.
13Halley, E. (1718), Considerations on the Change of the Latitudes of some of the principal fixt Stars.
Phil. Trans. 30, No. 355, pp. 736-738.
14Ptolemy, C. (1963), vol. I, p. XXII.
15 Hankins, T. L. (1973), Lalande, in: Dictionary of Scientific Biography, ed. C. C. Gillispie, New York,
vol. VII, p. 580.
16Lalande, J. J. (1757), Memoires de l'Academie Royale des Sciences, de l' Annee 1757, Paris. Cf.
Wilson, C. (1984), The Sources of Ptolemy's Parameters, Journal for the History of Astronomy 15, pp.
37ff.
17Lalande, J. J. (1757), pp. 420f.
18Ptolemy, C. (1963), p. 214.
19Lalande, J. 1. (1757), p. 420; Toomer shows that the differences in time are caused by inaccuracies in
the calculation of Hipparchus. Cf. Toomer, G. J. (1973), The Chord Table of HipparChus and the Early
History of Greek Trigonometry, Centaurus 18, pp. 6-28.

2.3. Delambre's Investigations

27

(iii) The third indication concerns the star catalogue. From the data of Hipparchus,
al-BattanI, Tycho and his own measurements, Lalande derives a constant
motion of precession of 50.5" per year. This motion would be 2" per year
larger if the coordinates of the star catalogue are included in the reckoning.
Lalande cites in his evaluation Monnier from "Institutions astronomiques"
(1746), for whom it is true beyond any doubt that Ptolemy was not in a
position to determine even one single fixed star position.
(iv) The Ptolemaic measurement of the lunar parallax is plagued by a substantial
error of 42', drastically exceeding the Hipparchan error of 13'.
(v) According to Lalande, Ptolemy had claimed that in early antiquity the obliquity of the ecliptic should have amounted to 24°, and that only for later
periods had Ptolemy used the value 23°51'20" taken over from Eratosthenes
and Hipparchus. For Lalande this absurd thesis proves the incompetence of
Ptolemy as an observer, for the investigations of Kepler showed very clearly
that such a large variation could never have actually happened. 2o
Lalande did not examine the passages of the Almagest under consideration
himself: rather, he refers to others who checked the accuracy of the statements in
the Almagest and who discovered grave errors. After his long list of grievances about
Ptolemy's observational accuracy, Lalande asks the suggestive question whether or
not all this provides a sufficient reason to condemn the authenticity of the rest of
Ptolemy's observations as well. 21
In the standard astronomical text of the time, Lalande's "Astronomie", this
passage of the Memoires is referred to, and Ptolemy is depicted as a very poor
observer. In the second and third edition of the work, Lalande sharpens his judgment
once again and transforms "Ptolemy, the poor observer" into a Ptolemy "who was
actually no observer at all".22 It remained for the most illustrious student of Lalande
to historically undermine these brief and superficial comments on Ptolemy which
also appeared to be the result of a reading of only secondary texts.
In 1780, while Lalande was holding lectures in Paris at the College de France
and mentioned during one of them the Greek poet Aratus, whose didactic poem
was later discussed critically by Hipparchus, the student Jean-Baptiste Delambre
attracted much attention when he stood up and recited the entire passage in question
from memory and was even able to comment on it in great detail. Delambre, who
criticized Lalande's "Astronomie" by fastidiously writing remarks in the margins,
became his assistant and later his colleague. 23

2.3

Delambre's Investigations

After the first historical interpretations of the star catalogue by Tycho, Lalande
and Laplace had been made, 1. B. Delambre dedicated himself at the beginning
20There is no evidence in the Almagest supporting Lalande's assertion.
21 Lalande, J. J. (1757), p. 421.
22Lalande, J. J. (1764), Trait!: d'astronomie, Paris. 2. ed. 1771, 3. ed. 1792. Wilson, C. (1984), p. 38.
23Cohen, I. B. (1971), Delambre, Dictionary of Scientific Biography, ed. C. C. Gillispie, New York, vol.
IV, p. 14.

28

2. Accusations

of the 19th century to a comprehensive investigation of the history of astronomy,
concentrating especially on the astronomy of the Almagest. 24 His work gave legitimacy to the allegations that Ptolemy, in contradiction to his own claims, did
not really observe the stars listed in the Almagest at all, but had assimilated
them from Hipparchus. The second volume of "Histoire de l'astronomie ancienne" after an introductory chapter on Greek mathematics, treats exclusively of
the astronomy of Ptolemy and comments on each book of the Almagest. The
commentary on the seventh book provides the point of departure for all subsequent historical investigations of the origin of the Ptolemaic fixed star catalogue.
The interpretation of the errors in longitude by Laplace could not be accepted
by Delambre. In his commentary to the third book of the Almagest in which the
solar theory is developed and Delambre refers to the error of the mean sun, we
read: 25
" ... but the error in the mean motions which makes the epoch [of the
era Nabonassar] a bit too large, would not produce any inconvenience
for the epoch at which he [Ptolemy] lived. The errors that might obtain in
the solar longitudes, from which the stellar positions were to be deduced,
resulted from the error in his equinox and the error in the [solar] motion
over [only] a small number of years."
The question as to how the errors of the mean sun position can be reconciled with
the error in equinox remains unanswered and is later labeled a "strange statement"
by Dreyer. 26
With the exception of this more favorable interpretation of Ptolemy's observations, Delambre adheres to the inductive argumentative figure of Lalande: the
numerical error of individual parameters, e.g. for the position of the farthest points
of the solar orbit from the earth (apogee) as well as for