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ApPENDIX 1 's Table of Chords

The construction of this table is based on the facts that the chords of 60° and 90° are known, that starting from chd 8 we can calculate chd(180° - 8) as shown by Figure Al.1, and that from chd S we can calculate chd ~8. The calculation of chd is goes as follows; see Figure Al.2. Let the AOB be 8. Place F so that CF = CB, place D so that DOA = i8, and place E so that DE is to AC. Then ACD = iAOD = iBOD = DCB making the BCD and DCF congruent. Therefore DF = BD = DA, and so EA = iAF. But CF = CB = chd(180° - 8), so we can calculate CF, which gives us AF and EA. Triangles AED and ADC are similar; therefore ADIAE = ACIDA, which implies that AD2 = AE·AC and enables us to calculate AD. AD is chd i8. We can now find the chords of 30°, 15°, 7~0, 45°, and 22~0. This gives us the chords of 150°, 165°, etc., and eventually we have the chords of all

R

P 8 = PQ,

C chord (180 - 8) = QR, QR2 = PR2 _ PQ2.

FIGURE A1.l.

235 236 Appendix 1. Hipparchus's Table of Chords

Ci"=:...-----""----...... ~A

FIGURE A1.2. multiples of 71°. The table starts:

2210 10 8 2 30° 45° 522 chd 8 1,341 1,779 2,631 3,041

We find the chords of not listed and angles whose chords are not listed by linear interpolation. For example, the angle whose chord is 2,852 is

( 2,852 - 2,631 1)0 . 45 + 3,041 _ 2,631 x 72 = 49° approxImately. ApPENDIX 2 Calculation of the Eccentric• Quotient for the , and the of its Apogee

This is Hipparchus's method as described by . However, Ptolemy used his own table of chords; I use the figures from Hipparchus's table as reconstructed by Toomer [103]. The basic are that the interval from spring to summer is 94~ days and the interval from then to autumn equinox is 92~ days. In Figure A2.1, T is the , 0 is the center of the sun's , H and L are the , and] and K are the .

U N H Spring

x o Summer P i------"--t<-:---t------i S K Q -~TT------J

V L Autumn

FIGURE A2.1.

237 238 Appendix 2. Calculation of the Eccentric-Quotient for the Sun

/ / " / "y): / /" / " / " / "

FIGURE A2.2.

The sun turns through the angle HOL in 92~ + 94~ days = 187 days. It turns through a whole in 365;14,48 days. Therefore the angle HOL is 184°20' , and so NOH + VOL = 4°20', and so UOH = 4°20'. Therefore, by linear interpolation, HU = 260. The sun turns through the angle HOK in 94~ days. Therefore HOK = 93°9'. But NOH = !UOH and so NOH = 2°10'. and so POK = 59'. Then KOX = 1°58'. Therefore, by linear interpolation, KX = 118, and so OQ = 59. But TQ = !HU = 130, and so, because T02 = TQ2 + OQ2, TO = 143. Thus TOION = 143/3438 = 1124 approximately. This is the eccentric-quotient. As above OQIOT = 59/143 = 2830/6876 and so (see Figure A2.2) OQIOY = 2830/3438. Then by linear interpolation, OYQ = 49°, and so OTQ = 241°. Therefore the apogee is 241° west of the summer solstice, i.e., its longitude is 651°. ApPENDIX 3 Ptolemy's Table of Chords

Ptolemy's table of chords is much more sophisticated than the one that we think Hipparchus used (see page 128). The chords are in a circle of 60 instead of 3,438, which makes calculations much easier. The interval between entries is ~o instead of 7!0; and the smaller the interval the smaller the errors introduced by linear interpolation. Attaining a smaller interval is not merely a question of subdividing more finely. Hipparchus could easily have produced a table with intervals of 3~o or l~o or ~~o by the halving process, but such a table would have been awkward to use. Ptolemy stated and proved a (usually known today, in fact, as Ptolemy's theorem) which enabled him to calculate the chords of x + y and x - y if the chords of x and yare known. He used 's construction of a regular to find the chord of 36°, which, since he knew the chord of 37~0, enabled him to find the chord of 1!0. It is not possible to trisect an angle of 1~0 by Euclidean methods, but it is possible to find a good enough approximation to the chord of ~o by using the result that chd x x if x> y, then --<-. chd y y

Taking x = 1~0 and y = 10 , we have chd 1° > ~ chd l~o > ~ x 1;34,14,41 > 1;2,49,47.

Similaly, taking x = 10 and y = ~o, we can show that chd 1° < 1;2,49,55.

Between them these show that chd 10 = 1 ;2,50 correct to two places. It is now easy to calculate the chord of ~o to two places and to complete the table. (Ptolemy's own explanation of his calculations was a trifle careless. He worked to only two sexagesimal places, and stated that the chord of 10 was both greater than 1 ;2,50 and less than 1 ;2,50.)

239 ApPENDIX 4 Calculating the Radius of the 's Epicycle

On page 133 we saw how Hipparchus (or Ptolemy) could calculate the radius of the moon's epicycle from data obtained by observing three eclipses. Here are the details of one such calculation carried out by Ptolemy using eclipses observed by the Babylonians in the first and second years of the reign of Marduk-apal-iddina, about 720 B.C. The intervals between the eclipses, reduced to mean , were 354 days, 2 hours, 34 minutes from the first to the second, and 176 days, 20 hours, 12 minutes from the second to the third. From the anomalistic period Ptolemy calculated how far round the epicycle the moon traveled in these two intervals. If the positions of the moon on the epicycle at the of the eclipses are , P2, and P3, respectively, then, measured clockwise

(1)

From the times of the eclipses, converted to time, Ptolemy found the of the sun and hence of the moon. From these, as described on page 133, he found (see Figure A4.1, in which T denotes the earth)

(2) (1) and (2) are the numerical data for the calculation. Ptolemy several times used the table of chords to find the proportions of a right-angled . This is how it is done. Let ABC he a triangle with a at B (see Figure A4.2). Suppose that the anglc ACB is ~x and we want to find ABIAC. If 0 is the midpoint of AC, then A 0 B = x. If we look up x in the tahle of chords and find that chd x = y, this means that AB = Y on a scale in which AO = 60. Thus

ABIAC = y/120.

This is the for such items as ~ x 6°48' or ~ x 1°14' in various steps of the calculation.

240 Appendix 4. Calculating the Radius of the Moon's Epicycle 241

P2

./ ./ / / ./ ./ /

T

FIGURE A4.1.

A~------~~------~~~~C

FIGURE A4.2. 242 Appendix 4. Calculating the Radius of the Moon's Epicycle

P2

T

FIGURE A4.3. Appendix 4. Calculating the Radius of the Moon's Epicycle 243

Let PzT cut the epicycle at X (Figure A4.3). Drop XY and XZ to TP3 and TP1 • Drop a perpendicular P3W to PlX. Choose a scale in which XT = 120. Then:

XTZ =! x 6°48' from (2) Therefore XZ = 7;7 from tables (3) arc P2P1 = 360° - arc P1P2 = 53°35' from (1) Therefore P2XPI = t x 53°35' by the angle-at -the- theorem P2 TPI = t x 6°48' from (2) Therefore XPIT =! x 46°47' by subtraction Therefore XZIP1X = 47;38,30/120 from tables Therefore PIX = 17;55,32 from (3) (4) Again, XTY = ! x 1°14' from (2) Therefore XY = 1;17,30 from tables (5) P2XP3 =! x 150°26' from (1) and the angle-at· the-circumference P2 TP3 = ! x 1°14' theorem from (2) Therefore XP3 T = ! x 149°12' by subtraction Therefore XY:P~ = 115;41,21/120 from tables Therefore P3X = 1;20,23 from (5) (6) arc PIP3 = arc P2P3 - arc P2P 1 = 96°51' from (1) Therefore P~ =! x 96°51' from the angle· at-the· circumference theorem and WP~ =! x 83°9' being 90° - P~ Therefore P3WIP3X = 0;44,53,7 and XWIP~ = 0;39,48,57,30 from tables Therefore P3W = 1;0,8 and XW = 0;53,21 from (6) (7) Then PIW = PIX-XW = 17;2,11 from (4) and (7) (8) Then Pl~ = PIW 2 + P3W 2 = 290;14,19 + 1 ;0,7 from (7) and (8) = 291;14,36 Therefore PIP3 = 17;3,57 (9) But arc PIP3 = 96°51' from (1) as above Therefore (see Figure 10.5, in which C is the center of the epicycle) PIP3 /CR = 1;29,46,14 from tables Therefore P~/CR = 1;29,46,14 x 1;20,23/17;3,57 from (6) and (9) = 0;7,2,50 (10) Therefore arcP~ = 6°44'1" from tables Therefore arc P2X = 157°10'1" from (1) Therefore P2XICR = 1 ;57,37,32 from tables But XTiCR = 0;7,2,50 x 120/1;20,23 from (6) and (10) = 10;31,13,48 Therefore P2 TICR = 12;28,51,20 by Then PzT·XTICR2 = 131;18,20,5,32 by multiplication But P2 T·XT = RT·ST by a theorem in and TC 2 = RT·ST + CR 2 by another. Therefore TC 2 /CR 2 = 132;18,20,5,32 and so TCICR = 11 ;30,8,42 giving CRITC = 0;5,13. ApPENDIX 5 The Eccentric-Quotient and Apogee of Mars

As pointed out on page 166, Ptolemy could calculate the eccentric• quotient and the direction of apogee of Mars if he knew the angles marked Z lTZ2, Z2TZ3, ZlEZ2, and Z2EZ3 in Figure 6.32. This is by no means obvious, so let us follow the method in some detail. In Figure AS.l, the points Zl, Z2, Z3, E, and T are as in Figure 6.32, K is the where Z3T cuts the circle ZlZ2Z3 again, and KF, KG, ZlH, and EN are perpendicular to ZIT, Z2 T, Z2K, and Z3K, respectively, A is the apogee. Knowing Z2TZ3, we know the angles KTG and TKG, and KGITK. (i) We know Z2KZ3 (it is ~Z2EZ3)' and so by (i) we know Z2KG and KGIZ2K. (ii)

A

K

P

FIGURE AS.1.

244 Appendix 5. The Eccentric-Quotient and Apogee of Mars 245

Knowing ZI TZ3, we know FTK and KFITK. (iii) We know ZjKT (it is ~ZIEZ3) and Z\ TK (by iii) and therefore TZ\K and KFIZ\K. (iv) We know Z1KH (it is ~Z1EZ2)' so we know Z\HIKH and Z\KIKH. (v) Now in terms of KG we know Z2K (by ii), TK (by i), KF (by iii), Z1K (by iv), and KH (by v). Therefore we know Z2H. We also know Z\H (by v), so we know Z1Z2 (in terms of KG). But we also know Z1Z2 in terms of the radius r of the circle Z1Z2Z3, because we know the angle ZIEZ2. Therefore, we know all these lengths in terms of r. In particular, we know Z1K, therefore Z)EK, therefore Z3EK, therefore KZ3. Knowing Z3K and TK, we know Z3T. Since AT· TP = Z3T· TK, we know AT· TP. But AT· TP + TE2 = r2, so we know TE (in terms of r) -we have found the eccentric-quotient TElr. We know Z3N (it is ~Z3K) and Z3T. Therefore we know NT. We know also TE, so we know the angle NTE. This gives us the direction of TE (the direction of apogee) in term of the observed direction TZ3 . ApPENDIX 6 Reversed Epicycles

In Figure 6.9, let T be the earth, let C be the center of the epicycle of a revolving about T anticlockwise in a circle of radius 60, and let P be the planet revolving clockwise round C in an epicycle of radius r. The minimum of P as seen from T (counting anticlockwise as positive) occurs when P is at the point A beyond C on the TC. Let the sidereal period of the planet be x years and its synodic period y years. Then the (angular) velocity of the line TCA about T is 1/x revolutions per year. The distance Tis 60 + r and so the linear velocity of A is 21t(60 + r)/x. The of the line CP relative to CT is l/y revolutions per year, and so the linear velocity of P, when it is at A, relative to A is 21trly. Thus the condition that the planet should retrogress is 21trly > 2rc(60 + r)lx, i.e., y < rxl(60 + r). Figures from the Almagest are as follows:

r x rxl(60 + r) y

Mercury 2212 1 0.03 0.3 1 433 1 0.4 1.6 Mars 3912 1.9 0.7 2.1 1112 11.9 1.9 1.1 ~2 29.4 2.9 1.0

This shows that the first three will not retrogress. This conclusion would not be reversed if we made the orbit of C eccentric and introduced an . Besides this, it is possible that if Ptolemy went through the detailed calculations to find the parameters of the planets' using clockwise epicycles, his data would not yield coherent results. And, for the outer planets, making it part of his theory that CP points toward the mean sun, coupled with the fact that the synodic periods are greater than a year, requires the epicycle to rotate anticlockwise.

246 Appendix 6. Reversed Epicycles 247

If the motion of the sun is presented as epicyclic motion (see Figure 7.1) then, because the line joining the mean sun to the sun is in a fixed direction in , the sun must move clockwise round its epicycle, like someone walking down an up-escalator at precisely the of the escalator. It is possible that the clockwise epicycle for the moon was copied from the theory for the sun. In spite of all this, there is evidence that some early Greek did use clockwise epicycles [154]. Further Reading

General H. Spencer Jones, General Astronomy, Arnold, London, 1934.

General J.L.E. Dreyer, The History of the Planetary System from Thales to , Dover, New York, 1953 (second edition). The classic text: very readable, though outdated. A. Pannekoek, A History of Astronomy, Allen & Unwin, London, 1961. (Dutch original, 1951.)

Preliterate Astronomy Evan Hadingham, Early Man and the Cosmos, Heinemann, London, 1983. Gerald Hawkins, Beyond Stonehenge, Harper and Row, New York, 1973. In• vestigates, in various localities, alignments of the type believed to exist at Stonehenge. Douglas C. Heggie, Megalithic : Ancient and Astronomy in Northwest , Thames & Hudson, London, 1982. E.W. MacKie, The Megalith Builders, Phaidon, Oxford, 1977. Alexander Thorn, Megalithic Sites in Britain, Oxford University Press, Oxford, 1967; Megalithic Lunar , Oxford, 1971; (with A.S. Thorn) Mega• lithic Remains in Britain and Brittany, Oxford University Press, Oxford, 1978. J.E. Wood, Sun, Moon and Standing Stones, Oxford University Press, Oxford, 1980.

Egyptian Astronomy R.A. Parker, Ancient , Philosophical Transactions of the Royal Society, volume 276, (1974). There are also brief references in A. Pannekoek's, A History of Astronomy, London, 1961; and Otto Neugebauer's, The Exact in Antiquity, Brown University Press, Providence, 1957.

249 250 Further Reading

Babylonian Astronomy Otto Neugebauer, The Exact Sciences in Antiquity, Brown University Press, Providence, 1957. This concise beautifully written text opened up the subject to the general public. Otto Neugebauer, History of Ancient Mathematical Astronomy, Springer-Verlag, New York, 1975. An extensive and detailed compendium with considerable mathematical detail. B. van der Waerden, Science Awakening, volume 2, Noordhoff, Leyden, 1974, (and Oxford University Press, New York). H. Hunger and D. Pingree, Mul'apin, Archiv fur Orientforschung, Beihefte 24, (1989).

Ancient China's Technology and Science [no author named], Foreign Languages Press, Beijing, 1983. Joseph Needham, Science and Civilization in China, Cambridge University Press, Cambridge, 1954. This extensive is the main source of information in English on the history of Chinese science and technology. Volume 3 contains three hundred pages on astronomy, with particularly full coverage of early Chinese sources, , the history and organization of the xiu, maps and armillaries, the last two topics abundantly illustrated. Ho Peng Yoke, Li, Qi and Shu: an Introduction to Science and Civilization in China, Hong Kong University Press, Hong Kong, 1985. Greek Astronomy D.R. Dicks, Early Greek Astronomy to , Cornell University Press, London, 1970. The strong point of this book is the author's careful treatment of Greek texts, rather than his understanding of astronomy. Otto Neugebauer, History of Ancient Mathematical Astronomy, Springer-Verlag, New York, 1975. An extensive compendium with considerable mathematical detail of Ptolemy's work and Greek astronomy immediately preceding and following him. R.R. Newton, The Crime of Claudius Ptolemy, Johns Hopkins University Press, Baltimore, 1977. This author brought doubts on Ptolemy's reliability into the limelight. In the course of denigrating Ptolemy he gives some very clear explanations of parts of the . Olaf Pedersen, A Survey of the Almagest, Odense University Press, Odense, 1974. G.J. Toomer, Ptolemy's Almagest, Springer-Verlag, New York, 1984. Supersedes all previous translations. Its introduction, footnotes, and appendices almost render commentaries (such as Pedersen's Survey) superfluous. Gerd Grasshoff, The History of Ptolemy's , Springer-Verlag, New York,1990. D.A. Somayaji, A Critical Study of the Ancient Hindu Astronomy, Kamatak University Press, Dharwar, 1971. Further Reading 251

David Pingree, History of mathematical astronomy in India, in the Dictionary of Scientific Biography, Scribner, New York, 1978, volume 15, pages 533 to 633. Astronomy We are in sore need of a general study of Arabic astronomy by a special• ist. Meanwhile, the best source of extra information is J.B.J. Delambre's Histoire d'Astronomie du Moyen Age, Paris, 1819; and articles under the names of individual astronomers in the Dictionary of Scientific Biography, New York, 1978. Maya Astronomy Floyd G. Lounsbury, Maya numeration, computation, and calendrical astronomy, in the Dictionary of Scientific Biography, Scribner, New York, 1978, volume 15, pages 759 to 818. John E. Teeple, Mayan Astronomy, Carnegie Institute of Washington, Washing• ton, 1930. J. Eric S. Thompson, Maya Hieroglyphic Writing, University of Oklahoma, Norman, Oklahoma, 1960 (second edition). The writing treated is largely concerned with astronomy. J. Eric S. Thompson, A Commentary on the Dresden Codex, American Philo• sophical Society, Washington, 1972. Later European Astronomy Max Caspar, Kepler, Abelhard-Schuman, London, 1959. J.L.E. Dreyer, , Dover, New York, 1963. Alexander Koyre, The Astronomical Revolution, Cornell University Press, Ithaca, 1973. Thomas S. Kuhn, The Copernican Revolution, Harvard University Press, Cambridge, Massachusetts, 1957. Edward Rosen, Copernicus and the Scientific Revolution, Krieger, Malabar, 1984. Bruce Stephenson, Kepler's Physical Astronomy, Springer-Verlag, New York, 1987. Noel M. Swerdlow and Otto Neugebauer, Mathematical Astronomy in Copernicus' De Revolutionibus, Springer-Verlag, New York, 1984. Victor Thoren, Tycho Brahe, in volume 2A of the General History of Astronomy, edited by Rene Taton and Curtis Wilson, Cambridge University Press, Cambridge, 1989. Sources of Information

1. Tribal from M.P. Nilsson, Primitive Time-Reckoning, Lund, 1920. 2. Babylonian constellations: B.L. Van der Waerden, Science Awakening, Ley• den, 1974, volume 2, pages 63 to 74. 3. E. Walter Maunder, The Astronomy of the Bible, New York, 190~. 4. Stansbury Hagar: The celestial bear, Journal of American Folklore, volume 13 (1990), pages 92 to 98. 5. Change in obliquity: Vistas in Astronomy, volume 10 (196~), page 54, or almost any standard tables. The obliquity in 2800 B.C. was 24.01°. 6. Ant on mill-stone: Jin shu, Chapter 11. (See note 67.) 7. Primitive observers: from M.P. Nilsson, Primitive Time-Reckoning. ~. Temples in Egypt: Joseph Norman Lockyer, The Dawn of Astronomy, London, 1894. Temples in Mexico and Guatemala: Gerald Hawkins, Beyond Stonehenge, New York, 1973. 9. Newgrange: C. O'Kelly, Illustrated Guide to Newgrange, Oxford, 1971. 10. D. Lewis, Voyaging , Philosophical Transactions, volume 276 (1974), pages 133 to 148. Also Kjell Akerblom, Astronomy and in Polynesia and Micronesia, Stockholm, 1968. 11. Anthony F. Aveni, Venus and the Maya, American Scientist, volume 67 (1979), pages 274 to 2~5. For further details on this, including a re• identification of the pyramid as Cehtzuc instead of Nohpat, and a suggestion that the sight-line was from the pyramid to the Casa del Gobernador, see Ivan Sprajc, The Venus-Rain-Maize complex, Journal for the History of Astronomy, volume 24, (1993), pages 18 to 48. 12. Intervals between solstices: Schiaparelli, Le Sfere Omocentriche di Eudosso, di Callipo, e di Aristotele, Milan, 1875, page 46. 13. Van der Waerden, Science Awakening, volume 2, page 103. 14. Almagest, Book 1, Chapter 12. 15. Yuan shi, Chapters 48 and 52. 16. E.C. Krupp, Shadows cast for the sun of heaven, Griffith Observer, volume 46, 8 (19~2), pages 12 to 17. Wen wu (1976), pages 92 to 95. 16a. AydIn SayIll, The in Islam, New York, 1981. 16b. E.W. Piini, A giant astronomical instrument of stone: the observatory, Griffith Observer, volume 48, number 9 (1984), pages 3 to 19.

252 Sources of Information 253

17. Joseph Needham, Science and Civilization in China, Cambridge, 1954, vol- ume 3, pages 339 to 343. 18. Almagest, Book 5, Chapter 1. 19. Ancient China's Technology and Science, Beijing, 1983, page 28. 20. Opere Storiche del P Matteo Ricci, Macerata, 1911, volume 1, page 135. 21. Stonehenge and sunrise: William Stukely, Stonehenge, a Temple Restored to the British Druids, London, 1740. 22. Car-park post-holes: C.A. Newham, Supplement to "The Enigma of Stone• henge," 1970. 23. Aubrey holes for counting: Gerald Hawkins, Nature, volume 202 (1964), page 1258. Also Fred Hoyle, On Stonehenge, San Francisco, 1977. 24. Distance between centres: Thorn, Journal for the History of Astronomy, volume 5 (1974), page 84. 25. height: National Geographic Survey Research Reports for 1965, pages 101 to 108. 26. Directions of sunrise: My own calculations. The most uncertain factor is the correction for refraction. Hawkins, Hoyle, and Thorn all used different figures (in Vistas in Astronomy, volume 10 (1968), page 54, On Stonehenge, page 141, and Journal for the History of Astronomy, volume 5, page 84, respectively). I have followed Thorn. The effect on the final result is a variation of just under 0.1°. 27. Directions from heel-stone to center, etc.: National Geographic Survery Research Reports for 1965, pages 101 to 108. To realize how measurements vary, note that J.F.C. Atkinson (Journal for the History of Astronomy, volume 7 (1976) page 144), got 49.4° and 50.6° for the short sides of the station . That the short sides of the station rectangle point to midsummer sunrise and midwinter sunset was first noticed by Edward Duke in 1846 (according to Peter Lancaster Brown, Megaliths, Myths and Men, Poole, 1976, page 107). 28. Stone 93 cutting the horizon: Fred Hoyle, On Stonehenge, page 76. 29. Alignments of the long sides of the station rectangle: discovered by G. Charriere (Societe Prehistorique Fram;aise, Bulletin, volume 58 (1961), pages 276 to 279); rediscovered by C.A. Newham and written up (together with stone-hole G alignments) in the Yorkshire Post, 16 March, 1963. 30. Alignments in diagram 1.6: Gerald Hawkins, Nature, volume 200 (1963), pages 306 to 308. 31. Central Stonehenge alignments: as note 30. 32. William Stukely, The History of the Temples and Religion of the Antient Celts, 1723 (quoted in Aubrey Burl, The Stonehenge People, London, 1987.) 33. Dacia, volume 4 (1960), pages 231 to 254. 34. Stuart Piggott and D.D.A. Simpson, Excavations of a stone circle at Croft Moraig, Perthshire, Scotland, Proceedings of the Prehistoric Society, volume 37 (1971), pages 1 to 15. 35. Use of post-holes as fine graduations: C.A. Newham, Nature, volume 211 (1966), page 456. 36. Stone-holes, F, G, H: R.J.C. Atkinson, Stonehenge, page 70. 37. G.S. Hawkins, Stonehenge Decoded, New York, 1966, pages 135 to 136. 254 Sources of Information

38. Atkinson: Moonshine on Stonehenge, Antiquity, volume XL (1966), pages 212 to 216. 39. Fred Hoyle, Speculations on Stonehenge, Antiquity, volume XL (1966), page 270. 40. Alexander Thorn, Megalithic Sites in Britain, Oxford, 1967; and Megalithic Lunar Observatories, Oxford, 1971. With A.S. Thorn, Megalithic Remains in Britain and Brittany, Oxford, 1978. 41. Callanish. First suggestion of astronomical alignments: Henry Callendar, Proceedings of the Society of Antiquaries of Scotland, volume 2 (1857), pages 380 to 384. The moon alignments were first suggested by Boyle Somerville, Journal of the Royal Anthropological Institute, volume 42 (1912), page 23 onward. The latest investigation is by 1.A. Cooke and three colleagues in the Journal for the History of Astronomy, volume 8 (1977), pages 113 to 133. 42. Thorn, 1967, page 151. 43. Notch and observers: Thorn, 1971. 44. C.L.N. Ruggles, Megalithic Astronomy, B.A.R. British 123, 1984. 45. Vincent H. Malmstrom and lames T. Carter, Stenalderskalendrar i Sverige? Forskning och Framsteg, volume 5 (1979), pages 1 to 5; and Curt Roslund, Aleforntidsmatematiker (the next article, on pages 6 to 11). 46. Neugebauer, The Exact Sciences in Antiquity, Providence, 1957, pages 58 to 66 and 110 to 121. 47. Peter I. Huber, Astronomical dating of I and Ur III Occasional Papers on the Near East, volume 1, issue 4 (1986). 48. Successful eclipse prophecy: report 272C in R.C. Thompson's Reports of the Magicians and Astronomers of Nineveh and Babylon, London, 1900, foretells the eclipse; report 274F confirms that it occurred. 49. Earliest reference to 8° placement: Manilius, , III, 257, III, 680 to 681 (A.D. 15). 8° placement in A.D. 1396: F. Kaltenbrunner, Die Vorgeschichte der Gregorianischer Kalenderreform, 1876, page 294. Hip• parchus, In Arati et Eudoxi Phaenomena Commentarium, page 132 of the Manitius edition, stated that most of the ancient astronomers used the 0° placement. Some modern writers state that Meton (about 450 B.C.) used the 8° placement, but the only evidence is from Columella (De re Rustica, IX, XIV, 12). In fact, Meton is unlikely to have used degrees at all. His close collaborator placed the solstices at the beginnings of the signs (A. Rehm, Das Parapegma des Euktemon, Sitzungsberichte der Heidelberger Akademie der Wissenschaft, 1913). 50. Nearly all the known tablets have been reproduced, transcribed, translated, and annotated by Otto Neugebauer in his Astronomical Cuneiform Texts, London, 1955, usually abbreviated to ACT. 5!. ACT, table 13, reverse side (second half). The names of the as transcribed in ACT differ from the names given here because each cunei• form symbol can be pronounced in more than one way. 52. Otto Neugebauer, History of Ancient Mathematical Astronomy, New York, 1975, volume 1, page 368. 53. ACT tablet 80/1. (See note 50.) 54. ACT 200. 55. Ephemerides for new crescent moon: ACT 5 and 18. Instruction-tablets; ACT 200 and 201. Sources of Information 255

56. A. Aaboe, Scientific astronomy in antiquity (Philosophical Transactions of the Royal Society) volume 276A (1974), pages 21 to 42. 57. ACT, 812, §1O and 813, §20. 58. A.H. Gardiner, Ancient Egyptian Onamastica, Oxford, 1947. 59. O. Neugebauer and R.A. Parker, Egyptian Astronomical Texts, Providence, 1969. 60. R.A. Parker, Ancient Egyptian astronomy, Philosophical Transactions of the Royal Society, volume 276 (1974), pages 51 to 65. 61. F.R. Stephenson, Astronomy in the monasteries, New Scientist, 1984 April 19, pages 27 to 31. 62. Si feng almanac: Hou Han shu, Chapter 13. 63. Hsiieh Chung-san, A Sino-Western for 2000 years, 1-2000 A.D., Beijing, 1956. 64. Hou Han shu, Chapter 13. 65. H. Maspero, Les instruments astronomiques des Chinois au temps de Han, Melanges Chinois et Bouddhiques, volume 6 (1939), page 235. 66. Yuan shi, Chapter 53. 67. Ho Peng Yoke, The Astronomical Chapters of the Chin Shu, Paris, 1966. 68. F.R. Stephenson, Quarterly Journal of the Royal Astronomical Society, volume 17 (1976), page 121. 69. A. Beer et al., An 8th-century line, Vistas in Astronomy, volume 4 (1960), pages 3 to 28. 70. Shigeru Nakayama, Accuracy of pre-modern determinations of length, Japanese Studies in the , volume 2 (1960), page 102. 71. Yuan shi, Chapter 52. 72. Laplace: Exposition du Systeme du Monde, fifth edition (1876), page 458. 73. Shigeru Nakayama: Accuracy of pre-modern determinations of tropical year-length, Japanese Studies in the History of Science, volume 2 (1963), pages 101 to 118. 74. Pan Nai, Guo Shoujing, Shanghai, 1980; and Li Ti: Guo Shoujing, Shanghai, 1966. 75. Yuan shi, Chapter 55. 76. Yuan shi, Chapter 54. 77. Yuan shi, Chapter 55. 78. As note 20, pages 175, 184 to 5, and 207. 79. i, 74, 2. 80. Eclipse Periods and Thales' Prediction of a : Historic Truth and Modern Myth, , 1969, page 60. 81. Parapegmata: details in Pauly's Real-Encyclopiidie der Classischen Alter- tumswissenschaften. 82. Meton and Euctemon's observations: Ptolemy, Phaseis, 67.2. 83. 19-year period: , Isagoge, Chapter VIII. 84. Eudoxus's mathematics: anonymous comment in Euclid, Book V (page 275, volume 5 in Heiberg's edition) and , introduction to On and . 85. Geminus: Isagoge, Chapter I. 86. Aristotle on Eudoxus: Metaphysics, A8, 1073, b17. Simplicius on Eudoxus: In de Caelo (page 493 of Heiberg's edition). 256 Sources of Information

87. Otto Neugebauer, On the "hippopede" of Eudoxus, Scripta Mathematica. volume 19 (1953), page 225. 88. Schiaparelli: as note 11. 89. Callipus: from Aristotle. Metaphysics, A8, 1073b, 32 and Simplicius In de Caelo (page 497 of Heiberg's edition). 90. Phaenomena, lines 147 and 148. 91. , De Motu Circulari Caelestium, i 10, edited by Ziegler. 92. Later writers' 252,000 stades: , Geographia, II, 5, 7. 93. Pliny on the length of the stade. Historia Naturalis, II: universum autem circuitum CCLII milium stadiorum prodidit, quae mensurae Romana computatione efficit trecentiens quindeciens centena milia pas• suum. Ihid. XII, xxx: Schocnus patct Eratosthenis ratione stadia XL, hoc est p. v, aliqui XXII stadia singulis schoen is dedere. Both these passages make Eratosthenes's stade equal to one-eighth of a Roman mile, the first one equating 252,000 stades to 32,500 miles, the second one saying that Eratosthenes took a schoenus to be 40 stades, i.e., 5 miles. The second passage notes that other people took a schoen us to be 32 stades. 94. Dennis Rawlins: The Eratosthenes-Strabo Nile map. Is it the earliest surviv• ing instance of spherical cartography? Did it supply the 500-stade arc for Eratosthenes' experiment? Archive for the History of Exact Sciences, volume 26 (1982), pages 211 to 220. 95. Strabo on /Alexandria distance: Geographia, L 4, 6. 96. 's 180,000 stades: Strabo, Geographia, n. 2, 2. 97. Columbus's mistake: Irene Fischer. Quarterly .Tournai of the Royal Astro• nomical Society, volumc 16 (1975), page 164. 98. Simplicius on Heraclides: In de Caelo (page 519 of Heiberg's edition). 98a. Dennis Rawlins, Ancient heliocentrists, Ptolemy, and the equant. American .Tournai of Physics, volume 55 (19S7), pages 235 to 239; and B.L. van der W

107. Eratosthenes's star-map is reprinted in the Loeb edition of Aratus's Phae• nomena. 108. Aratus on stars between Argo and : lines 366 onward. Hipparchus thereon: i 8, 2. 109. Phaenomena, line 518. 110. Steven C. Haack, Astronomical of the Egyptian pyramids, , no. 7 (1984), pages S119 to S125. 111. D.H. Fowler, The Mathematics of 's Academy, a New Reconstruction, Oxford, 1987. 112. Severin: Non tantum erasse ilium dixit observando sed finxisse obser• vatum quod ex Hipparcho computaverit, Introductio in Theatrum Astro• nomicum, Copenhagen, 1639, L i f 33. 113. The most detailed investigation of Ptolemy's calculation of the obliquity of the is by John P. Britton, in Centauru.~, volume 14 (1969), pages 29-41. 114. B. L. van der Waerden, Greek astronomical and their relation to the Athenian civil calendar, Journal of Hellenic Studies, volume 80 (1960), pages 168-180. 115. A. Rehm, Das Parapegma des Euktemon, Sitzungsberichte der Heidelberger Akademie der Wissenschaft, 1913. 116. Foreshortening error: Olaf Pedersen, A Survey of the Almagest, Odense, 1974, page 200. 117. Newton on the epicycle-radius: The Crime of Claudius Ptolemy, Baltimore, 1977. This title will be abbreviated to Crime. 118. Simplicius: In de caelo. Geminus: Isagoge, Chapter I. 119. : the temple of Canopus is at Abu Qir, latitude 31°19'. The itself is at 31°13'. 120. Error in moon's longitude: Viggo M. Petersen, The three lunar models of Ptolemy, Centaurus, 14, page 169. 121. Newton: Crime, pages 218 to 237. 122. James Evans, On the origin of the Ptolemaic star catalogue, Journal for the History of Astronomy, volume 18 (1987), pages 155 to 172 and 233 to 278. 123. Jaroslaw Wlodarczyk, Notes on the compilation of Ptolemy's catalogue of stars, Journal for the History of Astronomy, volume 21 (1990), pages 283 to 295. 123a. The latest summary of the controversy is N.M. Swerdlow's The enigma of Ptolemy's catalogue of stars, Journal for the History of Astronomy, volume 23 (1992), pages 173 to 184. 124. "Ptolemy assumed that the converse is true." It isn't: Comments in Crime, page 289. 125. Accuracy of Venus theory: Crime, page 211. 126. Crime, page 322. 127. Bernard R. Goldstein, The Arabic version of Ptolemy's planetary hypoth• eses, Transactions of the American Philosophical Society, volume 57 (1967), pages 3 to 13. 128. Both are available in English: Aryabhatlya of Aryabha(a, edited by K.S. Shukla, New Delhi, 1976; and The KhafJtJakhiidyaka (an Astronomical ) of Brahmagupta, edited by Bina Chatterjee, Calcutta, 1970. 129. Almagest [9, 2]. 130. Part 2, stanza 23 (page 109 in Shukla's edition). 258 Sources of Information

131. Quoted in Chatterjee's edition, pages 146 to 147. 132. D.A. Somayaji, A Critical Study of the Ancient Hindu Astronomy, Dharwar, 1971, page 97. 133. Otto Neugebauer, The transmission of planetary theories in ancient and medieval astronomy, Scripta Mathematica, volume 22 (1956), pages 165 to 192. Hugh Thurston, Greek and Indian planetary longitudes, Archive for History of Exact Sciences, volume 44 (1992), pages 191 to 195. 134. Otto Neugebauer, Tamil astronomy, Osiris, volume 10 (1972), pages 252 to 276. 135. Dictionary of Scientific Biography [DSBj, New York, 1978, volume 7, page 360. 136. G.J. Toomer, Ptolemy's Almagest, New York, 1984, page 2. 137. B. R. Goldstein, The Arabic version of Ptolemy's planetary hypotheses, Transactions of the American Philosophical Society, volume 57 (1967), page 3. 138. N .M. Swerdlow and O. Neugebauer, Mathematical Astronomy in Copernicus's De Revolutionibus, New York, 1984, page 44. 139. Swerdlow and Neugebauer, pages 45 to 47. 140. Swerdlow and Neugebauer, pages 47 and 196. 141. DSB (see note 135), volume 1. page 510. 142. J.B.J. Delambre, Histoire d'Astronomie du Moyen Age, Paris, 1819, page 209. 143. DSB, volume 1, page 51l. 144. Delambre, page 211. 145. Mayan data from John E. Teeple, Mayan Astronomy, Washington, 1930; and J. Eric S. Thomson, Maya Hieroglyphic Writing, Norman, 1950. 146. Venus almanac: extracted from pages 46 to 50 of the Dresden Codex. 147. Aveni: Archaeoastronomy in the Maya region, Archaeoastronomy, volume 3 (1981), pages SI to S8. 148. Teeple, pages 71 to 74. 148a. Dialogo di Galileo Galilei Zinceo . .. sopra i due massimi sistemi del mondo, Tolemaico e Copernicano, Firenze,1632. (Third .) 149. Walter G. Wesley, The accuracy of Tycho Brahe's instruments, Journal for the History of Astronomy, volume 9 (1978), page 42. 150. Tychonis Brahe Dani Epistolarum Astronomicarum, , 1596, page 167. 151. J.L.E. Dreyer, History of the Planetary System from Thales to Kepler, second edition, New York, 1953, page 356. 152. The earth's does cause bodies to fall nonvertically-by about 1 cm in a fall of 70 meters. See Alexander A. Mikhailov, on the quest of direct proofs of the earth's motion, Vistas in astronomy, volume 19 (1975), page 169. 153. , edited by Caspar, Munich, 1929, Chapter VII. 154. A Aabce, On a Greek qualitative planetary model of the epicyclic variety, Centaurus, volume 9 (1969), pages I to 10. 155. The Selected Works of Pierre Gassendi, Johnson Reprint Corporation, New York, 1972, page 121.

Answer to the question on page 197: The next day is 2 Ik 5 Pop. Index

a 67 Ara 136 Aaboe, A. 255,258 Arabic astronomy 32, 190-195 absin 67 Arabic miles 121 Abu Oir 257 Aratus 117,124,137-138 acrostic 117 Archimedes 122 adaru 20 Arcturus 24, 136 Akerblom, K. 252 -rule 226 Ale 62 Ares 21 Alexandria 119, 148, 150, 152 Argo 2,137 Algonkin 4 67, 112 allul 67 Aristarchus 122-123, 126, 128 Almagest 138-171. Also 129,192, Aristotle 117-119,124,206,212 205 on Eudoxus 113 almanac 69,92, 199 Ptolemy's criticism of 173 Alphonsine tables 194 Aristylius 152 11 armillary 35-41,94, 214 Amenhope 82 148,178-188 Ammasalik eskimos 11 Aryabha{fya 178-188, 207 Ammisaduqa 64 astrolabon 35-38, 153-154 110 2, 177, 213 110 -priests 53, 109 Andromeda 3 Astronomiae Instaurata Mechanica annular eclipse 149 215 anomalistic period 75,78, 106, 127, Astronomiae Instaurata 131-133 Progymnasmata 215 anomaly 75 Atkinson, R.J.C. 54,253 epicyclic 133, 162 Aubrey holes 46, 55 of the moon 133, 162 Aveni, A.F. 252,258 of a planet 162, 169 6, 23 zodiacal 164 of moonrise 13 -15, 49 ant on a millstone 9 of the rise of Venus 23 antikhthon 111 at Stonehenge 50 Anyang 84 of sunrise 9-11, 49 aphelion 223 Aztecs 96 apogee 129,160,208,214,244-245 Apollonius 170 apparent direction 133 7,64-81 67, 112 constellations 2

259 260 Index

Babylonian astronomy (continued) celestial 6, 32 coordinates 66, 74 latitude 7,32,36 eclipse intervals 19,201 longitude 32, 36, 38 kings 141 pole 5, 7, 26, 36, 97 length of the year 128 1, 109 months 20 model of 5 numerals 124 rotation of 5,24, 188 periods, synodic and sidereal 80, size of 1, 90, 94, 172, 179 127 Cetus 137 67 Charles's wain 4 backsight 10, 215 Charriere, G. 51 Baghdad 193 chi 87 Baikari 2 Chinese astronomy 84-109 baktun 197 calendar 87-89 Ballochroy 56 recorded by 124 barley-corn 74 constellations 4 al-Battan! 194 coordinates 4, 85 Batu Salu 10 instruments 38-41 bear 4 names of planets 21-22 Bei hai 101 vertical rod (gui biao) 96-97,101 bei tau 4 chords 128,138,237-238 Beijing 101 Chukchi 2 ~ Scorpii 26 circumpolar template 94 big dipper 4 Cleomides 121 Brahe, Tycho (Tyge) 41, 152, 174, clockwise epicycles 170-171, 246- 210-217 247 Brahma 178, 187 Codex, Dresden 23, 199-201 Brahmagupta 178 coffin lids 83 branches, celestial 87 Columba 138 brightness, apparent 152 Columbus 121 Britton, J.P. 257 Columella 254 Brown, P. Lancaster 253 comet 84,85,124, 174,207,211 bushmen 2 conjunction 69,95,212 constellations 2,85, 155 calculations (contrasted with constancy of 125, 150 observations) 64 on the moon's path 65 calendar 20,87-89, 141, 196,202, origin of 4, 135 203 continued fractions 199 and see under Babylonian, Chinese, Cooke, l.A. 254 Egyptian, Greek, Mayan Copan 203 "calendar," diagonal 83 Copernicus 1, 113, 146,205-211,222 calendar-maker 189 Coriolis 207 calendar-round 197 cosmology 90 Callanish 55 crank 160, 163; see also 145 Callender, H. 254 Croft Moraig 51 Callipic cycle 126 cross-staff 212 Calli pus 112, 117 crystalline spheres 173 67 65 Canopus 120, 138 cun 87 temple of 257 Cygnus 2 67 Cas a del Gobernador 23 Cassini 113 Da ming 93, 100 Cassiopeia 12, 213 Da tong 107 Index 261

Dadu 101 trios of 133, 143,240-243 date, Ptolemy's method of citation in the Aryabhatrya 187 141 in the Babylonian tablets 69, 74-75 see also calendar in the Dresden Codex 201-202 day 9 used for lunar parameters 133, 143, daylight, length of 124 240-243 De caelo 117 ecliptic 9, 13,68 De nova stella 213 see also obliquity De revolutionibus 205 Egyptian months 141 decan 82 Egyptians 7,8,24-26,82-83 7,32,57, 151 elongation 22, 79, 160 of the moon 15, 56 Enoptron 117 of the sun 11, 44, 57 -58 Eosphoros 21, 110 deferent 133, 143, 145, 148 epagomenal (= extra) days 20, 196 71 ephemerides 71, 78 Delambre, J.B.S. 258 epicycle 132-133,143,156-173, diagonal calendar 82 181-184,208,246-247 Dilbat 21 epicyclet 208, 218 aI-Din, Na~ir 191 epicyclic anomaly 132, 146, 162 ai-Din aI-Shirazi, Qutb 192 Epitome Astronomiae Copernicanae dioptra 124 233 distances of heavenly bodies 91, 113, (= zero-date) 14 L 172 123,147 equant 164,217,222,225 dodecametorion 66 142 Doig, P. 256 equinox 10. 112 draconitic 18 in Arabic astronomy 94 Dresden Codex 23, 199-201 in Babylonian astronomy 73 Dreyer,l.L.E. 210 in Chinese astronomy 94 du 84 in Greek astronomy 112, 125, 130, Duke, Edward 253 140 Dun Ruadh 51 Eratosthenes 119-120,137,139 Eridanus 137 errors (inaccuracies) 100, 124, 128, earth, movement of 4, 111, 121-122, 141 188 in longitudes 169, 170 shadow of 18,19,111,149 in the moon's latitudinal parameters shape of 90, 111, 118, 138 144, 148 size of 118,119-121,179 in the winter solstice 104-105 eccentric 121,131,158,173,222 see also length of the year, eccentric-quotient (or -distance) 129, foreshortening, distance of the 164, 168, 183, 208-209, 218, sun and moon, obliquity 224,244-245 eskimos 11 eclipses 17 -19 Euclid 112 annular 149 Euctemon 66, 128, 254 causes of 13,85,95,111 Eudemus 117 consequences of 95 Eudoxus 112-118 intervals between 18-19, 150, 202 Evans, James 257 of 75,85, 143, 150 exeligmos 127 prediction of 19,212 by the Babylonians 19,66 by the Chinese 19, 85 falling bodies 210-211 by the Indians 189 finger 75 by Thales 110 first gleam sunrise 49 at Stonehenge 55 Fischer, Irene 256 262 Index foreshortening 143 Hakemite tables 174 foresight 10, 53, 215 half-moon 122, 145 Foucault 10, 207 Harkhebi 82 Fowler, D.H. 139 Harmonice Mundi 233 Frederick II 214 Hartner, Willy 110 full moon 13, 15, 71, 145 Hawkins, Gerald 51,55 al-Haytham, Ibn 192 heel-stone 45-49 gaitian 5,90,94 Heggie, Douglas 213 Galileo 206, 233 (= first visibility) 24, gan 88 69, 199 Gardiner, A.H. 255 in Babylonian tablets 79-80 Gassendi, P. 211 in Egyptian astronomy 82 Ge Hong 94 in the Almagest 171 2,67,117 heliacal setting (= last visibility) 69, Geminus 113, 146,255 79,199 Genna 21 heliocentric 206 (Strabo) 119 heliostatic 206 geographical treatise (Ptolemy) 177 Heraclides 122 geometry 122 110 geostatic 206 Hermes 21 gir 3,67 Herodotus 110 gir-tab 67 24 glyph 23 Hesperus 21, llO (= vertical rod) 26-31, 66, Hipparchus 123-135; also 8,78,208 119 in the Almagest 140,143-146 Chinese (gui biao) 27,91,96-99 on Aratus 124 goal-year 69 Hipparchus's ring 112, 126 Goldstein, B.R. 257 hippopede 115 graha 1112 hippos 3 Great Bear 2,82, l17 Ho Peng Yoke 255 2 Hopi 10 Greek astronomy 1,24,110-177 horizon 50, 137 and Babylonian astronomy 123. Hottentots 2 128 Hou Han shu 93 constellations 3 Hoyle, Fred 54, 55 coordinates 32 Hsiieh Chung-san 255 and Indian astronomy 178 hull down 118, 138 months 20, 141 hun 67 names of the planets 21 huntian 90, 94 numerals 123 Huo xing 21 203 Hveen = Hven 214 gu 67 3, 136, 138 gu-ad 21 Hypotheseis ton planomenon 1,172- gu-utu 21 173, 179, 207 guest stars 84, 95 gui hiao 27 gu[a 67 iku 3 Guo Shoujing 29, 102 inclination (of the moon's orbit) 13, Gustav Adolf 213 16, 148 "Indians" (American) 4, 121 infinite sky 1, 206 Hadingham, Evan 249 instruction-tablets 71 Ilagar, Stansbury 252 intercalation 20,89,92 Index 263

Iroquois 4 Li Ti 255 irregularity 7 67 of the moon 17,75,114 Lui Chuo 86 of the sun 72, 78, 114 Loanhead of Daviot 51 Islamic ( = Arabic) astronomy 32 Lockyer, Norman 252 long-count 197 longitude 32, 69 Jesuits 109 longitudinal period 166 jian 89 see also sidereal period lin shu (older spelling Chin shu) 94- Lounsbury, Floyd G. 251 95 Loyang 100 jin xing 21 lu 67 Jones, H. Spencer 248 luhunga 67 Jupiter 21 luni- 19 in Babylonian astronomy 69, 79-81 in Chinese astronomy 94, 108-109 in Greek astronomy 159,169,170 MacKie, E.W. 249 in Indian astronomy 179, 184 madhya 182 Jupiter stations 94 magnitude of eclipses 75, 85, 143, 150 of stars 152 Malmstrom, V.H. 254 kak$yamandala 180, 183 al-Ma'mun 193 kak$yavrtta 180 manda 182,187 Kaltenbrunner, F. 254 mandakendra 182, 187 Katasterismoi 137 mandocca 182, 184 katun 197 Manilius 254 ke 90 Maoris 2 ke xing 95 maps, Chinese 85 Kepler 13,217-233 Maragha 193 al-Khwarizmi 190 Mars 21 Kronos 21 in Babylonian astronomy 69, 79 Krupp, E.C. 252 and photographs in Chinese astronomy 86 Krupp, R.R. photographs in Greek astronomy 116, 159, 166- kushu 67,68 170 Kusumapura 178 in Indian astronomy 179, 184 Kepler on 218-232 mash 67,68 LaPlace 99, 104 mash 67 latitude 32 mashtabba 67 of the devisers of the constellations Maspero, H. 255 135 Maunder, E.W. 8,135 of Mars 221 Mayas 14,23, 196-204 of the moon 74, 143 mean elongation 160 of planets 114, 170 mean longitude 133 of stars 153 mean moon 133 latitudinal period 18,75, 78, 127 mean solar day 141 laws mean solar time 141 Kepler's 222-223 mean sun 130, 223 Newton's 233 Mecca 195 2,67,117,135,160 Melanesia 10 Lepus 137 21 Lewis (Scotland) 55 in Babylonian astronomy 69, 80 Lewis, D. 252 in Chinese astronomy 86 Ii 87 in Indian astronomy 179, 184 264 Index

Mercury (continued) moonshine 110 Copernicus on 208 motion of the earth 4, 111, 121-122, Ptolemy on 159-163 188 Meru 187 muxing 21 Metaphysics 117 mul 66,67 Meton 66, 111, 128 mul-apin 64 20 mul-babbar 21 mi 95 music of the spheres 111 Micmac 4 Micronesia 11 Nabu-nasir 141,172,179 midday (= noon) 26 Nakayama, S. 255 midnight 9 nangar 67 midsummer 8, 30 Nanhai 101 interval to midwinter 11 Nanjing (f'ianking) 99 at Stonehenge 48-50 Na~ir ai-Din 191 see also solstice navigation 11 midwinter 11,54 Needham, Joseph 250 see also solstice Neugebauer, Otto 64,71,78,115, Mikhailov, A.A. 258 187,204,255 miles, Arabic 121 Newgrange 12 141 new moon 15,71 94 new year (Chinese) 87 Ming shi 98, 107 Newham, C.A. 253 moxian 107 Newton, Isaac 207 month 13,20,82,87,88 Newton, R.R. 144,171 length of 68,75-78,89, 127 Nibiru-maruk 21 names of 20 Nile 8,119 moon 12-21 Nilsson, M.P. 252 in Babylonian astronomy 65-78 Nindaranna 21 in Chinese astronomy 85, 106 Nine maidens 57 in Indian astronomy 181 nisannu 20 in Mayan astronomy 198-199 node 13-14,74,143,179 Copernicus on 208-209 Nohpat 23 Eudoxus on 113-114 noon 26-27,30,96-99,102-103 Hipparchus on 131-133,240-241 26 Ptolemy on 143-148,240-241 Norway 10 anomalistic period of 75, 106, 127, notch (in horizon) 57-58,60-61 131-133 nova 95 declination of 15, 16 numerals full 13, 15, 71, 145 Arabic 190 half 122, 145 Babylonian 69, 124 inclination of (to the ecliptic) 13, Greek 123 16, 148 Indian 178 irregularity of 17, 75, 114 Mayan 198 latitude of 74-75, 143 latitudinal period of 127, 143 megaliths and 49,56 obliquity 9,25 new 15,71 estimates of 9, 44, 96, 105-106, orbit (path) of 13, 64 148, 194 of 146-148 in armillaries 35 size and distance of 113, 122, in Chinese astronomy 105 147-148 in Indian astronomy 179 velocity of 76-77 Ptolemy on 139-140 moonrise 13, 15-16,49,68 observations (contrasted with Index 265

calculations) 68 in Indian astronomy 181-187 O'Kelly (0 Ceallaigh) 252 Plato 146 omens 94-95 Pleiades 2, 66 13,21,168-169,219-224 plinth 27, Figure 1.9 oracle bones 84 Pliny 119, 124 orbit, shape of 227-233 plough 4 orientation (of edifices) 11, 14,23, plumb-line 2, 31 25-26,45-63,82 pole (gnomon) 27-31,66,91,96-99, 2,82 119 Osiander 205 pole, celestial 5,7, 24, 41, 97, 136 Ovenden, Michael 4, 135 pole of a great circle 113 pole-star 5,24, 141 Polynesia 11 pa 67 Pondicherry 188 pabilsag 67 Posidonius 120 Palenque 199,202 -drive 41 Pan Nai 255 218 Pannekoek, A. 249 24,34, 79, 135, 150-153, Pappus 124 170 57 prediction of eclipses 19,212 parallax 6, 133 priestesses 11 of Mars 220 Prodromos dissertation em of the moon 133, 146-149 cosmographicarum 218 of stars 149, 201 8 parameter 129, 161 prosthaphaeresis 133, 145 parapegma 111,141 212 parhelion 95 Psammites 122 Parker, R.A. 83 Ptolemaic system 173 110 Ptolemy 138-177,206,239,244-245 Pedersen, Olaf 250 his geographical treatise 121 3 on Hipparchus 124, 134 Peking = Beijing his instruments 27, 35-38 10 on the size of the earth 121 Peri megathon kai apostematon heliou on the size of the universe 1, 172 kai selenas 122 pyramids 25-26 Peri takhon 117 111 perigee 160, 163 Pythagoreans 111, 146 periodicity 110 Persia 86 Petersen, Y.M. 257 Qian Han shu 93 Phaenomena 117, 136 qi-lin 92 Phaseis aplanon asteron 177 quadrant 28, 32, 212 Philip of Macedon 172 It 187 Piggot, s. 253 radius-rule 226 Piini, E.W. 258 Rahu 179 Pingree, David 250 Raqqa 194 2, 67, 112 129, 161 planetarium, mini- 5 Rawlins, Dennis 119, 120, 128 planets 8, 21 rectangle 45, 50, 53 in Babylonian astronomy 79-81 reference-object 38 in Chinese astronomy 86 reference-stars 155 in Greek astronomy 115-117, 155- refraction 6, 24, 50, 112, 214 173 regression of the nodes 14, 74 266 Index regular (uniform) 146 Seleucid era 72 Rehm, A. 254, 257 Severin, Christian 140,214,218 reign-period 87 sexagesimals 69, 123 Relativity 207 212, Figure 10.6 retrograde motion shadow, earth's 18, 111, 118 in the Almagest 157, 164, 170 of a rod 27-31,96-103 in Babylonian tables 79 Shandu 101 in Chinese astronomy 86, 109 al-Shatir, Ibn 192 of 207 she 74 in Eudoxus's theory 114 shi 90 retrogression 21, 164 ShiShen 92 of Venus 22 Shi ji 93 Rhodes 120,124,137,152 Shou-shi 102-108 rifa 93,105 Shu jing 84 Ricci Matteo 41,253 Shui xing 22 32, 142 sidereal day 9, 179 rin 67 period 80-81,116,127,159,179 rising, direction of 9-10,13-15,23, see also longitudinal period 166 136 year 25 rising-times of ecliptic arcs 74 si fen almanac 86 of constellations 136 sfghra 182, 186 river of heaven 94 sfghrakendra 185, 186 rod, vertical (= gnomon) 26, 66, 91, sfghrocca 182, 184, 188 119 of the zodiac 66-68, 112 Roman months 20 Simplicius 113, 116, 146 Roslund, C. 254 179 Rothman, Christopher 210 Sirius 20, 82, 138 Rudolph II 215 Skorpios 67 236 solar day 9, 142 Ruggles, Clive 58 solar time 142 Russia (reports from) 85 solstice 10 Chinese time of 90 determination of 10,57,97-99 67 observations at 44,91,96-99, 103 sahurmash 67 on ecliptic 68 Salbatani 21 time-intervals from 11, 25, 111, 130 Samarkand 32, 193 in tribal astronomy 10-11 San guo zhi 93 used by Hipparchus 125 San tong 92-93 used by Ptolemy 140-141 Sanskrit numerals 178 see also midsummer Sarawak 10 Somayaji, D.A. 187 Sarmizegetusa 51-52 Somerville, Boyle 56 Saros 19, 75-78, 127 Sosigenes 146 Saturn 21-22, 159 south 26 in Babylonian astronomy 80 Southern Cross 2 in Chinese astronomy 86 sphuta 182, 185 in Indian astronomy 179, 184 sphuramadhya 185 Saylh, A. 258 135 Schiaparelli 116 stade 119 schoenus 252 stars, Ptolemy on 150-155 Schwartz, C. 256 list (catalogue) of 38,64-65,85, or 3, 67, 135 124 sea-shell 189, 198 station-stone 45, 50 , length of 111,131 statistics 54, 137, 152 Index 267

Stephenson, F.R. 96 Thales 110 Stilbon 222 Thorn, Alexander 51-58,249 Stonehenge 10,45-55 Thompson, J.E.S. 251 Strabo 119,256 Thompson, R.C. 254 Stukely, William 51,256 thoth 20, 141 Sui xing 22 tides 206 Sultan of Turkey 213 135, 152 sun 8-13 Tlingit 2 in Babylonian astronomy 71-73 Toledan tables 194 in Chinese astronomy 91 Tong dian 105 in Greek astronomy 113,128-131, Toomer, G.J. 128,237 140-143 top, spinning 25 in Indian astronomy 179-181 215 declination of 11,44,58 tribal names of stars 2 distance of 91, 123, 149, 179 trigonometrical tables 128, 139, 180 irregularity of 11, 72, 92, 114 trilithon 45, 50 megaliths and 45,48-62 tropical year 25 size of 110-111,122 true direction 132 sunrise 9-11,48-50,68 true (as opposed to mean) 223 sunset 4, 9-11, 57, 68 trunks, celestial 88 sunspots 85, 172 tu xing 21 95,212 tun 197 Swerdlow, N.M. 258 al-Tiisi 192 Syene 119 Tycho (= Tyge) Brahe 175,210-218 synodic month 13 synodic period 21-23,80,86,157, 159,200 Udaltar 21 Syntaxis 138 uinal 196 systems A and B 78 Ulugh Beg 32, 193 ululu 20 umbrella 5 table of chords 128,139,236-237, uniform (regular) circular motion 146 239 ura 67 of eclipses 201-202 Uraniborg 214 of the moon's motion 106-107 ush 71 of parallax 149 Uxmal 23 of planets' motions 86, 108-109 of 180 of the sun's motion 141 Van der Waerden 250 Alphonsine 194 variance 152 Hakemite 194 Varuna 187 Prutenic 212 Vedic 178 Rudolphine 236 Veen 214 Toledan 194 velocity of the moon 76, 77, 106-108 Tai shi ling 101 of the sun 72, 78 Tang dynasty 87, 96 Ven 214 target-practice analogy 152 Venus 21-23 67, 135 in Babylonian astronomy 69,80 Techne Eudoxou 117 in Chinese astronomy 86 Teeple, John E. 202, 251 in Greek astronomy 154, 163-166 telescope (constellation) 2 in Indian astronomy 179, 181-185 template 94 in Mayan astronomy 14, 199-201 temple 11, 53 = Hesperus 110 177 vernal (spring) equinox 38 268 Index

"Viking ship" 62 YIJlng 94 67 ying fu 29 yojana 179 Yuan shi 98, 101-109 Wang Chong 85 Yucatan 23 WangXun lUI yuga 179 Wesley, W.G. 258 Yunus, Tbn 194 whole disc sunrise 49 winter solstice 90, 104 Wlodarczyk 155 137 Wood, J.E. 249 zero 70, lU9, 198 zero date 141,172,197 Zeus 21 Xanadu 10 zib 67 Xia xiao zheng 92 Zieljahrtexte 69 xiu 4,85 zhi 87 xuanye 90, 94 zhi yuan 102 Zhou bei suan jing 90 zibatanu 67 Yangcheng 32,94 zib-me 67 year 9,25,87 zig-zag 76-78 Babylonian estimate 128 zodiac 66-67, III Brahe's estimate 214 and precession 142 Chinese estimates 92-93, 101, 105 zodiacal anomaly 164 Hipparchus's estimate 126 zodion 66 Indian estimate 179 Zu Chonzhi 97 length of 194 Zulu 10 rough estimate 9 Zuiii 10 supposed Mayan estimate 204