Table Of ContentStudies
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
A History or Numerical Analysis rrom the 16th through the 19th
Century
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
A History or the Calculus or Variations rrom the 17th through
the 19th Century
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
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 Mathematics 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. Ptolemy 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:.
Softcm-er reprint of !he hard",;wcr lSI edition 1990
All rig:ltts reser'oled. This won; may no( be rranslated or copied in whole or in part withoutlhe wrinen
~rmiS$ion of !he publisher (Springer·~rla8 New Von. Inc .. 175 Fifth Avenue, New 'tbrl<.. NY 10010.
USA). except ~ brief e~s in connection with rcoiews or scholarly analysis. Usc in wnnection
with any form of infOrmation Slorage and retrieval. eloctronic adaptation. computer software. or by
Similar 01' di.S$imillr methodology now known or lIen:.o.ftcr dcYelopcd is fotbidden.
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the Trade Marl;, and Merchandise Mar1ts Act. may accordin&ly be used freely by anyone.
Photooompo<Sed copy prepared fn,JlIl dIC author·s u.')eX file.
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987654321
ISBN-I3: 978,1-4612-8788,9 e-ISBN-I3: 978-1-4612-4468-4
DOl: 10.1007197&-1-4612-4468-4
To
B.-S. and the Elephant
Table of Contents
Introduction
1 Tbe Stars of tbe Almagest 6
1.1 The Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Persons................................ 6
1.1.2 Methodological Background. . . . . . . . . . . . . . . . . . . . 7
1.1.3 The Almagest on Fixed Stars . . . . . . . . . . . . . . . . . . . 9
1.2 The Arabic Revision of the Almagest . . . . . . . . . . . . . . . . . ., 17
2 Accusations 23
2.1 Tycho Brahe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2.2 Laplace and Lalande. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25
2.3 Delambre's Investigations . . . . . . . . . . . . . . . . . . . . . . . . .. 27
3 Tbe Rebabilitation of Ptolemy 34
3.1 The Number of Hipparchan Stars . . . . . . . . . . . . . . . . . . . .. 34
3.2 Supplementary Catalogues . . . . . . . . . . . . . . . . . . . . . . . .. 43
3.2.1 Bjornbo's New Catalogue . . . . . . . . . . . . . . . . . . . .. 43
3.2.2 Dreyer's 1/4 Degree Stars . . . . . . . . . . . . . . . . . . . .. 44
3.2.3 Dreyer II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
3.2.4 Fotheringham ............................ 49
3.3 The Reconstruction of the Hipparchan Catalogue . . . . . . . . . .. 52
3.3.1 The Determination of the Precession . . . . . . . . . . . . . .. 59
3.3.2 Dreyer's 1/4 degree stars. . . . . . . . . . . . . . . . . . . . .. 61
3.3.3 Peters' Hypotheses of two Observation Instruments . . . . .. 63
3.3.4 Graduation of the Astrolabe ................... 63
3.3.5 The Epoch of Observation for the Hipparchan Coordinates. 64
3.4 Gundel's List of Hipparchan Stars .................... 67
3.5 Precession and Solar Theory . . . . . . . . . . . . . . . . . . . . . . .. 73
3.5.1 Pannekoek's Calculation of Precession . . . . . . . . . . . . .. 73
3.5.2 The Hipparchan Solar Theory. . . . . . . . . . . . . . . . . .. 76
3.6 Accusations.................................. 79
3.6.1 The Observation of Regulus and Spica ............. 80
3.6.2 The Measurements of Declination . . . . . . . . . . . . . . .. 81
3.6.3 Stellar Positions from Occultations by the Moon. . . . . . .. 83
3.6.4 Fraction of the Degrees ...................... 84
viii
4 The Analysis of the Star Catalogue 92
4.1 The Catalogue in the Alma,gest ...................... 92
4.1.1 Critical Edition of the Catalogue . . . . . . . . . . . . . . . .. 94
4.1.2 Recalculation of the Coordinates for the time of Hipparchus 95
4.1.3 Identification of Stars. . . . . . . . . . . . . . . . . . . . . . .. 95
4.1.4 Errors in the Almagest . . . . . . . . . . . . . . . . . . . . . .. 97
4.2 Criticism of Vogt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
4.2.1 Vogt's Interpretation of Delambre's Precession Table . . . .. 99
4.2.2 Reconstruction of Coordinates .. . . . . . . . . . . . . . . .. 104
4.2.3 The Accuracy of the Reconstructed Coordinates . . . . . . .. 106
4.2.4 Vogt's Proof of Independent Observations ........... 107
4.2.5 Statistical Test for Independent Data .............. 110
4.2.6 Dating................................. 117
4.3 Gundel's Stars ................................ 122
5 Structures in Ptolemy's Star Catalogue 129
5.1 Star Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 129
5.2 Multiple Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
5.2.1 Dreyer's Paradigm . . . . . . . . . . . . . . . . . . . . . . . . .. 136
5.3 Method of Selective Error Distribution. . . . . . . . . . . . . . . . .. 140
5.3.1 Cluster-Analysis........................... 142
5.4 Errors of the Solar Theory . . . . . . . . . . . . . . . . . . . . . . . .. 148
5.5 Fractions of Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156
5.5.1 Fractions of Degrees in Latitude . . . . . . . . . . . . . . . .. 158
5.5.2 Fractions of Degree in Longitude. . . . . . . . . . . . . . . .. 164
5.6 Hipparchus' Commentary on Aratus . . . . . . . . . . . . . . . . . .. 174
5.6.1 Sources................................ 174
5.6.2 Numerical Values . . . . . . . . . . . . . . . . . . . . . . . . .. 175
5.7 Calculation of Phenomena . . . . . . . . . . . . . . . . . . . . . . . .. 177
5.7.1 Local Sidereal Time. . . . . . . . . . . . . . . . . . . . . . . .. 177
5.7.2 Simultaneous Rising and Setting . . . . . . . . . . . . . . . .. 178
5.7.3 Culmination............................. 178
5.8 Deviations from Reality . . . . . . . . . . . . . . . . . . . . . . . . . .. 178
5.8.1 Comparing two Catalogues .................... 178
5.8.2 From Observation to Phenomena. . . . . . . . . . . . . . . .. 182
5.8.3 New Ways of Comparison . . . . . . . . . . . . . . . . . . . .. 184
5.8.4 The Globe .............................. 190
5.8.5 More Details. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191
5.9 Reconstruction................................ 192
6 Theory and Observation 198
6.1 The Aristotelian Heritage . . . . . . . . . . . . . . . . . . . . . . . . .. 198
6.2 The Uncertainty of Empirical Data. . . . . . . . . . . . . . . . . . .. 199
6.3 Radical Empiricism ............................. 204
6.4 Holistic Rationalism . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 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 317
9.1 Column Headings 317
10 Literature 335
11 Index 344
List of Figures
1.1 Spherical astrolabe.. . . 11
3.1 Error of the mean sun. . 78
3.2 Distribution of precession values. 82
3.3 Distribution of fractions of a degree. 88
4.1 Identification Chart for the Constellation Canis Minor 97
4.2 Precession H-P. . . . . . . . 102
4.3 Precession T-H. . . . . . . . . . . . . . . . . . . . 104
4.4 Precession, all declinations. . . . . . . . . . . . . 105
4.5 Errors of coordinates, uncorrelated (simulation). 110
4.6 Uncorrelated coordinate errors with different standard deviations. . 111
4.7 Strong correlation. . . . . . . . . . . . . . . . . . . . . . . . . . .. 111
4.8 Correlating coordinate errors (Simulation). ............. 112
4.9 Correlating coordinate errors, with high standard deviation (simula-
tion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.10 Errors in longitude Almagest versus Vogt's reconstruction. . 113
4.11 Errors in latitude, Almagest versus Vogt's reconstruction. 114
4.12 Errors in latitude, for 1/4 degree stars. . 116
4.13 Errors in latitude, for 1/6 degree stars. . . . . 116
4.14 Errors in longitude, for 1/6 degree stars. . . . 117
4.15 Epoch of coordinates in Aratus Commentary. 120
4.16 Histogram of (AGundel - A_l2S)' . . . . . . . . 123
4.17 Distribution (AGundel - L l2s)/ (AGundel - L l2s). 126
4.18 Simulation of independent errors. 127
4.19 Simulation of dependent errors. 128
5.1 Ecliptical coordinates. . 131
5.2 Ecliptical coordinates. . 132
5.3 Equatorial coordinates. . 133
5.4 Equatorial coordinates. . 134
5.5 Error in latitude, average of pairs. 135
5.6 Distribution of errors in longitude, all stars. 137
5.7 Stars with 1/4 degree in latitude. . . . . . . 138
5.8 Distribution of errors in longitude, 1/4 degree stars. 138
5.9 Distribution of magnitudes, all stars. . . . . . 139
5.10 Distribution of magnitudes, 1/4 degree stars. .... 139
5.11 Differences in longitude for southern stars. . . . . . . . . 141
5.12 Differences in longitude, constellations Lepus, Eridanus. 142
5.13 Differences in longitude, stars of the zodiac. . . . . . . . 142
5.14 Distribution of differences in longitude for one source. . 143
5.15 Distribution of longitudinal differences for two sources. . 144
5.16 Best fit with three normal distributions. . . . . . . . . . 145
5.17 Error in longitude as a function of the longitude. . . . . 149
=
5.18 Error of the solar theory with the reference epoch -128; a) Hip-
parchus +2;40°; b) Ptolemy. ....................... 151
5.19 Error of the solar theory (-128), with phase shift 180° and errors of
the star catalogue. . . . . . . . . . 152
5.20 Error and interpolation functions. . . 153
5.21 Fractions of the degree in latitude. . 156
5.22 Fractions of the degree in longitude. 157
5.23 Error in latitude for 1.81 < 20°. ... 160
5.24 Error in latitude for 0° > ~ > 20°. . 161
5.25 Distribution of errors in latitude for full degrees. 162
5.26 Histogram of magnitudes, 1/6 degree stars. ... 163
5.27 1/4 degree latitude in combination with 1/2 degree in longitude. 172
5.28 1/4 degree latitude in combination with 50' in longitude. . 173
5.29 Distribution of the degrees of the phenomena. 177
5.30 Correlation of errors for all phemonena. ... . 189
5.31 Correlation of errors for culminating stars. . . . 192
5.32 Correlation of errors without culminating stars. 193
5.33 Correlation of errors for rising stars. . . . . . . 194
5.34 Correlation of errors of reconstructed longitudes. 195
5.35 Correlation of errors of the reconstructed latitudes. 195
5.36 Correlation of well approximated longitudes. . 196
5.37 Correlation of well approximates latitudes. . 196
5.38 Correlation of fraction numbers. ....... 197
