The astronomical papyrus Ryland 27

Waerden, B.L. van der + The Astronomical Papyrus Ryland 27 In: Centaurus, 5 RE(i | , 1958, po 177 - 191 we RIEN, BLL VA. e BL. Va ola Wat lan Latas Volume 5. (grb lso f137- 191

1. Introduction and summary The Greek papyrus Ryl. 27 was first published in 1911 in Vol. I of the Catalogue of the John Rylands Library. In 1947, Knudtzon and Neugebauer! published a closely related Greek papyrus Lund 35 a, which shed a new light upon Ryl. 27 and which enabled Neugebauer to publish a commentary on both2. This commentary made clear the main lines of the computation of the lunar apogee and longitude, but the computation of the latitude remained obscure. The importance of Ryl. 27 lies in the fact that this treatise represents a phase of Hellenistic astronomy before Ptolemy. This phase may be described as an elementary astronomy, based upon Babylonian methods but adapted to the Egyptian calendar. “Elementary” means: without or with very little trigonometry. From other sources little is known about this elementary astronomy. The demotic papyrus Carlsberg 9 teaches a rough method for finding the date of new moon, based upon mean motion and a 25-year cycle. The demotic papyrus Berlin P. 8279, the demotic Stobart tablets} and the Greek papyrus fragments Teptunis 274 and Lund 35 b4 give dates of entrance of the five planets into zodiacal signs, but they do not tell us how the dates were computed. I have shown that Babylonian elementary (i. e. non-trigonometric) methods were used to compute these “eternal tables”. Quite recently I have found (Bibliotheca Orientalia 13, p. 108) that the linear part of the Venus section in the Stobart tablets can be explained by rules described in Chapter XVIII of Varäha Mihira’s Pajichasiddhantik4®, but a complete explanation of the planetary texts is still lacking. Centaurus 1958: vol. 5: no. 3-4: pp. 177-191 CENTAURUS, VOL, Y

The only texts giving a complete set of rules for computing one phenomenon, viz. the Moon’s apogee, are the two related papyri mentioned before, viz. L (= Lund 35 a) and R (= Ryl. 27). These rules are based upon the use of two periods C and D, of 248 and 3031 days. They contain approximately 9 and 110 anomalistic periods of the Moon. The first period is also known from cuneiform texts7. Both periods are mentioned by the 6th century Indian astronomer Varäha Mihira (Pafchasiddhäntikä U, 1-4 and VII, 5) and used by Tamil calendar makers in the region of Pondicherry’. Both Varäha Mihira and the Tamil computers give rules for computing the moon’s apogee by means of the periods C and D, and quite analogous rules are given in our papyri L and R. These facts show once more the importance of our papyri. They supply the missing link between Babylonian and Indian astronomy. Between the cuneiform texts, written in the last 3 centuries B.C., and Varäha Mihira, who lived after 505 A.D., there is a gap of 5 centuries. Now a large part of this gap is bridged by our papyri L and R, written about 100 and 300 A.D. respectively. In this paper the rules for computing lunar longitudes and latitudes used in RX will be completely explained. It will be shown that zieéroc means: argument of latitude, reckoned from the point of greatest latitude, divided by 15. The division by 15 appears reasonable, because near the nodes a change of 15° in the argument of latitude implies a change of approximately 1° in the latitude. The elements upon which the calculation of R is based are shown to be (i) the daily motion of the moon’s nodes: 0;3,10,49,15, i. e. 0%3'10”49"""15"""" (ii) the mean daily motion of the moon: 13;10,34,52 (iii) the anomalistic month 27;33,16,21,4 + 0;0,0,0,4 days (iv) the difference between the mean and the minimum daily motion 1917 + 1’ The epoch, from which the calculations start, is the midnight at the end of —31 June 30 (= Cleopatra 20, Epiphi 4). The midnight epoch explains an ambiguity in the counting of days: the midnight at the end of Epiphi 4 may be dated Epiphi 4 or 5, at will.

The longitudes were measured with respect to the fixed stars, just as in Babylonian and Indian astronomy. The zero-point of the sign Aries used in R coincides exactly with the zero-point of the Babylonian sidereal zodiac: it has longitude —4° on the tropical zodiac of the year —100. With the aid of the periods of 248 and 3031 days the date and position of the apogee can be found, but this is not sufficient to calculate the moon’s position at any given date. This position may be computed either by elementary or by trigonometric methods. The elementary methods, used by Babylonian and Indian sources, are based upon the use of arithmetical series of order 2 or, what amounts to the same, quadratic functions of time. However, it seems that the methods taught in papyrus R were meant to be combined with a trigonometric table for the day-byday motion of the moon during 248 days, of the same kind as the tables used in Tamil astronomy. The following table summarizes the methods used in lunar theory in the Babylonian, Greek and Indian sources here considered: Method for day-by-da Source Periods used | Cuneiform texts Papyrus R 248 248 and 3031 elementary probably trigonometric Pañchasiddhántiká 248 and 3031 elementary*) Tamil astronomy 248 , 3031 and 12 372 moti on ya trigonometric ® The method explained in Chapter II, stanza 2-6 is based upon the use of a quadratic function at + bt2. I intend to give a detailed explanation of the method on a later occasion. 2. The date of the apogee Section 1 (lines 1-14) deals with the computation of the apogee. Besides the periods C and D, of 248 and 3031 days, two derived periods B= D— 11C = 303° and A = 3C = 9093° = 25 Egyptian years — 32° are used. The starting point of the calculation is an apogee of the moon, 61 days before the New Moon of Thoth 1 of the year Cleopatra 21

(= Augustus —1). From this starting point, periods D (as many as possible) and C (as few as necessary) are counted off until one finds an apogee in the desired year. The whole computation is described in a few lines in the following style: “Moon. To the total of years add 2, divide by 25, (multiply) the rest by 365, the cycles of 25 by 32, then add 61. The total number divide, if possible, by 3031, and the remainder by 248 and the remainder subtract in the case of nodes from 303, in the case of no nodes from 248 ...”. For a detailed explanation, see Neugebauer’s paper?. 3. Longitudes and latitudes According to section 2 (lines 15-31) the motion in longitude and latitude for the periods A, D, C and B is Period A = 9093 -D = 3031° C= 248 B= 303° Longitude Latitude a = 292;33,57,21 d= 337;31,19,7 c = 27;43,24,56 5b = 32;[33,4]4,5[1] a’ = d' = c' = b' = 6;38,11,24,45 9;12,43,48,15 2;43,28,34,0 [13;14,2[9,34,15] The figures for longitudes are certainly correct, for a is equal to 3d but for a multiple of 360°, and b is equal to d — 11c. Moreover, the figures have the right order of magnitude. If the mean daily motion of the moon is assumed to be 13;10,34,52 (Ptolemy’s value is 13;10,34, 58, ...) we obtain for the mean motion during C = 248 and D = 3031 days _ c = 27;44,6,56 and d= 337;31,20,52 _ The values are slightly larger than the text values. The agreement is better in the case of d, which was to be expected, because the period D is more accurate than C. For latitudes, the results are less satisfactory. The expected relations d'-1ilc =P

are fulfilled for the sexagesimal parts, but not for the integers. Neugebauer has already given the correct explanation, viz. that the numbers were reduced by subtracting multiples of an unknown integer m. We shall see presently that the ‘“‘modulus” m is 24, and that we need only multiply the numbers a’, d’, c’, b' by 15 in order to obtain the “argument of latitude” in the sense of Ptolemy, i. e. the distance from the North point of the moon’s orbit to the actual position of the moon. 4. Explanation of the latitudes The draconic month does not differ much from the anomalistic month. This implies that the change in latitude of the moon during one anomalistic month is only small, and hence the change in latitude in 11 anomalistic months cannot be much larger than the change in latitude in 9 such months, i.e. 5’ cannot be much larger than c’. Therefore let’s try the possibility9 b’ = 3;14,29,34,15 From the relation b’ =d'— 1lc' + km where km is an integer multiple of the “modulus” m we now find km = 24 hence m is a divisor of 24. Since d’ cannot exceed m, the only possibilities are m=12 and m= 24 We shall see that m = 24 makes sense. If 12 or 24 is the correct modulus, the integer part of a’ must be 27 — 24 = 3 (not 6). As Neugebauer told me, this correction is graphically possible. Assuming m = 24, we may change the scale, multiplying the numbers c’ and d’ by 15. Thus we obtain c” = 15 c' = 40;52,8,30 d' = 15 d' = 138;10,57,3,45 The modulus now becomes 15m = 360, which is the usual reduction modulus for the argument of latitude. So we may conjecture that c” and d” represent the motion in latitude in the usual sense during the periods C and D.

If this is correct, we may calculate the motion of the nodes, by subtracting from c’’ and d” the motion in longitude c and d respectively. Thus we obtain for the nodes c* =c" — c = 13;8,43,34 d* = da” — d= 160;39,37,56,45 Dividing c* by 248, and d* by 3031, we obtain in both cases exactly the same figure for the daily motion of the nodes, viz. e* = 0;3,10,49,15 (i) The order of magnitude is right, for, according to section 5, line 54 of the same text, the daily motion of the nodes is 0;3,10. Ptolemy’s value is 0;3,10,41, .... So (i) is very good. | This explains the formation of the “latitudes” completely. They are found simply by adding to the figures for “longitudes” the motion of the nodes during the periods A, D, B, C and dividing by 15. 5. The motion in longitude The fact that the motion during the periods C and Dis not proportional to the number of days might be explained in two ways, viz. 1) the figures might represent the motion during 9 and 110 exact anomalistic periods; 2) the figures might represent the true (not the mean) motion during 248 and 3031 days. In the first case the ratio of the motions ought to be as 9 to 110. This is not the case, so only the second possibility remains. Let's follow this path. Let the mean daily motion be 13;10,34,52 + x Let the anomalistic period be 27;33,16,22 + y y is small. If the period of 3031 days is assumed to be exact, we should have y = 0. If the Tamil period R = 12 372° is assumed to be exact, y becomes + 4””. If the Babylonian period 27;33,16,27 is adopted, y is + 5°”. So we may expect y to lie between (say) —2"” and +6”.

Let the difference between the mean daily motion and the slowest daily motion be z. We may expect z to be less than 2°. 9 anomalistic periods are + y) = 248 — (0;0,32,42 — 9y) = 48 — r 9 (27;33,16,22 days. So 248 days are 9 periods plus a small remainder of r days. The motion during this time is equal to the mean motion minus rz. The mean motion is 248 (13;10,34,52 + x) = 9 rotations + 27;44,6,56 + 248 x The motion according to the text is, disregarding complete rotations, 27;43,24,56. So we must have 27;44,6,56 + 248x — rz = 27;43,24,56 or, substituting r = 0;0,32,42 — 9 y, (1) + 248x + 9yz — 0;0,32,42z = — 0;0,42,0 In the same way, 110 anomalistic periods are 110 (27;33,16,22 + y) = 3031 + (0:0,0,20 + 110 y) = 3031 + s days. So 3031 days are 110 periods minus s days, where s is very small. The motion during this time is equal to the mean motion plus sz. The mean motion according to the text is 337; 31, 19, 7. So we must have 337;31,20,52 + 3031 x + sz = 337;31,19,7 or, substituting s = 0;0,0,20 + 110y, (2) + 3031x + 110yz + 0;0,0,20z = — 0;0,1,45 Thus we have 2 equations with 3 unknown quantities x, y, z. Of course, we cannot solve these without additional assumptions. However, it so happens that z can be solved without ambiguity. 248 . We eliminate x by multiplying (2) by 3031 and subtracting (1). The coefficient of yz in the resulting equation is 27280

Now yz is small (less than 12”), and yz 3031 is quite neglegible. Hence yz cancels out and we obtain an equation for z: (3) 248 248 (1962 + 3037 + 20) z = 2520 — 2037 - 105 The solution of (3) is (4) z = 1;16,44 The 44 minutes in (4) cannot be trusted, because the figures —42” and —1”45"” on the right of (1) and (2) may have been rounded off to seconds and to quarter-seconds respectively. If the computation of z is repeated with 0;0,41,31 or 0;0,42,59 instead of 0;0,42,0, we find z = 1;16 and z = 1;18 respectively. So it seems safer to write instead of (4) (5) z=1°17 +1 Having determined z, we may substitute into (1) or (2) and thus obtain an equation for x and y. We begin by considering the figures on the right of (1) and (2) as exact; then we have to substitute z from (4) and obtain two equations for x and y. Of course, the two equations are dependent: we have in reality only one equation for x and y. This equation is most readily obtained by multiplying (2) by 3 and substituting (4): (6) 9093x = — 422y — 0;0,5,32 We have already seen that y lies between —2’” and +6’. It follows that x lies between +4 and — 20’. Now the latitudes in our text are given with 4 sexagesimals, not rounded off, whereas the longitudes have only 3 sexagesimals. If we adopt the most probable assumption that the longitudes were not rounded off either, it follows that the mean daily motion underlying the calculation had only 3 sexagesimals. On this assumption, x = Cis the only possibility. Moreover, x = 0 is the only assumption which leads to the round figures 0;0,42,0 and 0;0,1,45 for the corrections from mean motion to true motion. So there is every reason to suppose x = 0. Hence the mean daily motion is

The equations (1) and (2) now simplify to (7) (— 9y + 0;0,32,42)z = 0;0,42,0 (8) (110y + 0;0,0,20)z = — 0;0,1,45 We may eliminate z by division and solve for y. This gives y = — 0;,0,0,56 (9) The maximal error in y resulting from the rounding off of the figures 42,0 and 1,45 on the right of (7) and (8) is 4”*”. So the anomalistic month is (iii) 27;33,16,21,4 + 0;0,0,4 6. Sidereal or tropical longitudes ? As we have seen, Papyrus R assumed a mean daily motion of the moon (10) v = 13;10,34,52 and a retrograde daily motion of the nodes (11) n = 0;3,10,49,15 Were these motions reckoned with respect to the fixed stars or with respect to the equinoxes ? The Babylonian Systems I and II both used a sidereal zodiac. This means: longitudes were measured from the fixed stars. It does not imply that the authors of systems I and II were aware of the precession of the equinoxes. From the constants of system I, Kugleri0 computed the following mean sidereal motion y, = 13;10,34,51, ... Hipparchus and Ptolemy used a tropical zodiac. This means: they measured longitudes from the spring equinox. Their figures for tropical motion are v,, = 13;10,34,58, ... n,, = 0; 3,10,41,... If Ptolemy’s precession (6’” a day) is subtracted from v, and added to n,,, we obtain the sidereal motions according to Hipparchus and Ptolemy

y, = 13;10,34,52, ... n,= 0; 3,10,47, ... These figures, which agree well with modern observations, are nearly equal to (10) and (11). This is a strong argument in favour of the assumption that R is based upon sidereal motion. Once more, this does not imply that the author of the system of R knew about precession. The epoch of R is Cleopatra 20 (= Nabonassar 716) XI 5 (midnight, morning or noon). To decide between these three possibilities, we compare with Ptolemy’s Almagest. Neugebauer has already made the comparison for longitudes at noon, but here we are hampered by the fact that we do not know whether the sidereal initial point of Aries coincides with Ptolemy’s. We therefore compare not only the longitudes, but also the arguments of latitude. According to Ptolemy, we have for the noon epoch longitude ................ anomaly................. 354;28 73;20 argument of latitude ...... 185;51 longitude ................ 70;2,16,10 anomaly................. 360 (apogee) The argument of latitude would be, according to line 31 of the text, (12) 24 — 12;12,39,19,15 However, Neugebauer has shown that (12) has to be corrected, most probably, into (13) 24 — 12;10,39,19,15 We shall adopt the latter figure (13). Our conclusions would be the same if we had adopted the former one. To compare (13) with Ptolemy’s value, we have to multiply (13) by 15. This gives, rounded off to minutes: argument of latitude 360 — 182;40 = 177;20 The difference from Ptolemy’s argument of latitude is 8°31’. This would imply a difference of 43’ or nearly 14 lunar diameters in the latitude. This is certainly too much if the latitudes are derived from

eclipse observations. Therefore, I am inclined to believe that the epoch is neither morning nor noon, but midnight. The assumption of midnight epoch also explains why the papyrus Lund 35a associates the same longitudes with a date 1 day earlier. Midnight may be reckoned to the following or to the preceding day. Papyrus R apparently uses in section 1-2 the first convention and in section 3-4, where a “‘shorter method” is taught, the second convention. Since it is practically impossible to attach the same lunar longitudes to two different dates, the assumption of a midnight epoch seems to be the only possible explanation. For midnight, calculation according to Ptolemy and modern tables would give the following results year notenie longitude anomaly Ptolemy (mean position)... 66;43 347;56 Modern (mean position)... 67;5 348 Modern (true position) .... 68;45 Papyrus R............... 70;2,16,10 arg. of latitude 179;14 179,20 181;0 360 177;20,10,11,15 The error in the argument of latitude would be 3°40’ if the figure of R were compared with the true position. However, since the argument of latitude cannot be measured with any accuracy near the south point of the orbit, we must consider the text value as computed and therefore compare it with the mean value according to modern computation. Hence, the error is only 2°, which implies an error of only 10’ or 4 lunar diameter in the latitude. The error in the anomaly is 12°, but this is not serious, because the exact determination of the apogee is a very difficult matter. We now consider the longitudes. It seems impossible that the longitude 70;2,16,10 is a result of direct observation. It has seconds and thirds and hence it must have been computed from other observations. If so, it must have been computed as a mean position, because the moon was supposed to be at its apogee. Therefore, it must be compared with the modern mean position. The difference may be partly due to observational errors, but mainly to the difference between the sidereal and the modern tropical zero point of the zodiac. In previous investigations of Kugler and myself about the zero point

of the sidereal zodiac!!, all modern longitudes were reduced to the vernal point of —100. To obtain the reduced “Kugler longitudes”, we have to subtract 58’ from the modern longitudes for the year —31, i. e. we have to add 58’ to the nearly 3° previously found. This gives us 3.9 as an estimate of the difference between R longitudes and Kugler longitudes. From Babylonian tables of ecliptical stars and from planetary tables I have found a mean difference of 4.1 degrees between Babylonian longitudes and Kugler longitudes (1. c.11), p. 223). Hence, the zero point of the zodiac of R coincides exactly with the zero-point of the Babylonian sidereal zodiac. 7. The day-by-day motion of the moon in cuneiform texts Our papyrus R gives rules for computing the position of the moon at apogee only. Babylonian and Indian sources give fuller information: they also teach us how to compute the motion of the moon during a period of 248 days following the apogee or perigee. We shall explain the principle of the Babylonian method, and return to the Indian sources on another occasion. The Babylonians first compute the daily motion of the moon as a “linear zigzag function”, i. e. as a function increasing and decreasing by constant differences. By adding the daily motions to the initial longitude, the daily positions are found. The following example from Neugebauer’s text No 191 (Astronomical Cuneiform Texts II, p. 132) will illustrate this. Date Position IX I 2 3 7;9,30 20;32,40 4;13,50 4 18;13 5 2;30,10 6 17;5,20 7 1;58,30 8 17;9,40 9 2;9,40 10 16;51,40 Difference* Capricorn 13;23,10 13;41,10 13;59,10 Aquarius 14;17,10 Pisces 14;35,10 14;53,10 Aries . 15;11,10 15 Taurus * The differences are not given in the text.

Each difference exceeds the preceeding one by 18’. In line 8, the addition of 18’ to the preceding difference would give 15;29,10, which is 0;14,35 more than the “ideal maximum” 15;14,35. By subtracting the excess from the ideal maximum the next difference 15;0,0 is obtained. From now on 18’ is subtracted for every day, until the difference would become less than the “ideal minimum” 11;6,35, and so on. The mean daily motion is 13;10,35. The difference between the mean and the minimum is 2°4', much larger than in R. The anomalistic month would be, according to this text, 27;33,40 = = days but the Babylonians and Hipparchus also knew the more accurate value 251 27: = 29; 2 525° + 29:31,50,8,20 = 27;33,16,26,57 which is slightly larger than the value of R. 8. The day-by-day motion of the moon in elementary Hellenistic astronomy As we have seen papyrus R teaches the computation of the lunar apogee and node and of the longitude and (argument of) latitude of the moon at its apogee. It does not teach the computation of the longitude and latitude for an arbitrary day. This is very strange, for the whole treatise is eminently practical. It is not a theoretical investigation like the Almagest, but it gives practical computation rules for astrologers just like Ptolemy’s Handy Tables. Now what is the practical use for the moon’s apogee? The apogee may be a good starting point for the computation of longitudes and latitudes, but in itself it has no importance for the astrologer. For this reason it seems probable that R was only an incomplete copy or an excerpt from a more complete treatise which also contained rules for computing longitudes and latitudes at any given moment. We do not know whether these rules were elementary or trigonometrical. From the relation of R to Babylonian methods one might be inclined to conclude that in R too the daily motion was assumed to be a linear

increasing and decreasing function of time. However, this assumption leads to very improbable consequences. We have seen that in R the difference between minimum and mean daily motion is 77’. Now, if the moon’s velocity were assumed to be a linear increasing function of time during the first half of the anomalistic period, the maximum correction from mean to true longitude would be T 37 62 7 = 4°25 joe mean — Ymin) = 45°7 This is an extremely low figure, much below the true value 6° and the Babylonian 7°. On the other hand, if we adopt a trigonometric formula for the correction from mean to true motion, e. g. w = bsin at and if we determine b in such a way that the difference vus — mia iS 77', we obtain for the maximum correction the very reasonable value The combination of the periods of 248 and 3031 days with a trigonometric table for the day-by-day motion during 248 days would yield a quick and accurate method for computing the moon's position. The same combination is also used in Tamil astronomy!2. NOTES 1. E. J. Knudtzon und O. Neugebauer: Zwei astronomische Texte, Bull. soc. roy. lettres Lund 1946-47, H. 2. O. Neugebauer: The astronomical treatise P. Ryl. 27, Kgl. Danske Vid. Selsk. hist.fil. Meddelelser 32, Nr. 2 (1949). 3. O. Neugebauer: Egyptian planetary texts, Trans. Amer. philos. soc. N.S. 32: 2, p. 209 (1942). 4. See footnote 1. 5. B. L. van der Waerden: Egyptian “eternal tables”, Proc. Kon. Ak. Wet. Amsterdam. G. Thibaut and M. S. Dvivedi: The PanchasiddhântikA of Varáha Mihira, Benares

F. Thureau-Dangin: Tablettes d'Uruk No 25 (Paris 1922). P. Schnabel: Z. f. Assyriol. 37, p. 35 (1927). O. Neugebauer, Astronomical Cuneiform Texts (London 1955) I, p. 77 and 178-183. 8. O. Neugebauer: Tamil Astronomy, Osiris 10, p. 252 (1952). 9. I am indebted to O. Neugebauer for communicating this conjecture to me. 10. F. X. Kugler: Babylonische Mondrechnung (Freiburg 1900) p. 110. 11. B. L. van der Waerden: History of the Zodiac, Archiv für Orientforschung 16, p. 216 B. L. van der Waerden: Tamil Astronomy, Centaurus 4, p. 221 (1956). (1953). I. M. V. Krishna Rav: The Motion of the Moon in Tamil Astronomy, Centaurus 4, p. 198 (1956).