The Three Lunar Models of Ptolemy

3665 a f VU. Petersen aba O+TSoPATy, A e SOE 1. Introduction In this paper Ptolemy’s theory for the motion of the moon (Almagest IV, 1 to V,9) is expounded. This has previously been done, e.g. by P. Kempf p. 1-35 and O. Neugebauer p. 192-198, 210-214, but contrary to these I shall try to give an examination of the theory of the moon, based on modern results. On the other side, in this work, no attempt has been made to check the internal calculations made by Ptolemy, e.g. trigonometrical calculations and conversion of true solar time into mean solar time. Making such a check would be interesting, but this exceeds the limits of this paper. The sections 2-7 contain a rather detailed explanation of the theory of the moon, and they may not give much new information to the reader, who is beforehand familiar with the nature of Ptolemy’s three geometrical models for description of the motion of the moon. This explanation has been included for the sake of completeness, it is more systematical and it is written in a more modern mathematical language than usually done. The greater part of the investigations relating to modern results are in the sections 3, 8, and 9, and the main result is a proof of the fact that in reality Ptolemy has been able to describe the motion of the moon by means of a complicated model 3 which is superior to not only the simple model 1 handed over by Hipparchus, but also to his own model 2. The calculations made by a computer in section 9 have been done with skilled assistence of stud. scient. Gert Jacobsen and stud. scient. Jens Thyge Kristensen, both University of Copenhagen. The Three Lunar Models of Ptolemy 149 Remarks about the notations The decimal point is used to separate the integral part of a number from the decimal part, whereas semicolons and commas are used when writing a number in the sexagesimal notation. Whenever an angle is mentioned it is reduced modulo 360°, so that its value is in the principal interval, 0° s a < 360°. Where modern values are to be compared with values from the Almagest, the modern values are marked with a star, *. It is a wellknown fact that any calculation involving triangles can be made by means of the Pythagorean theorem and Ptolemy’s table of chords. Whenever Ptolemy makes such a calculation it will not be expoundedin details here. The models for the motion of the moon will be given in the following coordinate system: The earth is at the origin O, the directions of the x¡-axis and the x-axis are the directions from the earth to the vernal and from the earth to the summer solstice. The angle, which a equinox ai vector AB makes with the x1-axis, is called Arg AB, the length 14B| of the vector is often called AB. 2. The Epicycle Model for the Description of the Motion of the Moon When Ptolemy sets up a model for the description of the motion of the moon, it is beforehand certain that only uniform circular motions will be used. One might think that the motion of the moon can be described by an epicycle model, analogous with the one used for the sun. However, from two fundamental facts it appears that this is not the case. These facts have been discovered by steady observations of the moon, made by “the old ones” (i.e. the astronomers before Hipparchus), and mentioned by Ptolemy at the beginning of Almagest IV,2: 1. The moon moves irregularly in such a way that in the course of time it has assumed its maximal velocity for every value of the longitude À. 2. The moon moves not only in the ecliptic, but has latitudes between approximately +5° and —5°, and in such a way that in the course of time it has assumed its maximal latitude for every value of 4. If the motion of the moon is described by an epicycle model the following facts are valid. The epicycle centre C moves on the deferent and * Amalievej 7, 3330 Garlase, Denmark. Centaurus 1969: vol. 14: no. 1: pp. 142-171 represents the mean position of the moon, so that Arg OC

1. Introduction In this paper Ptolemy’s theory for the motion of the moon (Almagest IV, 1 to V,9) is expounded. This has previously been done, e.g. by P. Kempf p. 1-35 and O. Neugebauer p. 192-198, 210-214, but contrary to these I shall try to give an examination of the theory of the moon, based on modern results. On the other side, in this work, no attempt has been made to check the internal calculations made by Ptolemy, e.g. trigonometrical calculations and conversion of true solar time into mean solar time. Making such a check would be interesting, but this exceeds the limits of this paper. The sections 2-7 contain a rather detailed explanation of the theory of the moon, and they may not give much new information to the reader, who is beforehand familiar with the nature of Ptolemy’s three geometrical models for description of the motion of the moon. This explanation has been included for the sake of completeness, it is more systematical and it is written in a more modern mathematical language than usually done. The greater part of the investigations relating to modern results are in the sections 3, 8, and 9, and the main result is a proof of the fact that in reality Ptolemy has been able to describe the motion of the moon by means of a complicated model 3 which is superior to not only the simple model 1 handed over by Hipparchus, but also to his own model 2. The calculations made by a computer in section 9 have been done with skilled assistence of stud. scient. Gert Jacobsen and stud. scient. Jens Thyge Kristensen, both University of Copenhagen. * Amalievej 7, 3330 Gorlose, Denmark. Centaurus 1969: vol. 14: no. 1: pp. 142-171

Remarks about the notations The decimal point is used to separate the integral part of a number from the decimal part, whereas semicolons and commas are used when writing a number in the sexagesimal notation. Whenever an angle is mentioned it is reduced modulo 360°, so that its value is in the principal interval, 0° < a < 360°. Where modern values are to be compared with values from the Almagest, the modern values are marked with a star, *. It is a wellknown fact that any calculation involving triangles can be made by means of the Pythagorean theorem and Ptolemy’s table of chords. Whenever Ptolemy makes such a calculation it will not be expounded in details here. The models for the motion of the moon will be given in the following coordinate system: The earth is at the origin O, the directions of the x ;-axis and the x2-axis are the directions from the earth to the vernal equinox and from the earth to the summer solstice. The angle, which a vector AB makes with the x¡-axis, is called Arg AB, the length ¡AB! of the vector is often called AB. 2. The Epicycle Model for the Description of the Motion of the Moon When Ptolemy sets up a model for the description of the motion of the moon, it is beforehand certain that only uniform circular motions will be used. One might think that the motion of the moon can be described by an epicycle model, analogous with the one used for the sun. However, from two fundamental facts it appears that this is not the case. These facts have been discovered by steady observations of the moon, made by “the old ones” (i.e. the astronomers before Hipparchus), and mentioned by Ptolemy at the beginning of Almagest IV,2: 1. The moon moves irregularly in such a way that in the course of time it has assumed its maximal velocity for every value of the longitude 1. 2. The moon moves not only in the ecliptic, but has latitudes between approximately +5° and —5°, and in such a way that in the course of time it has assumed its maximal latitude for every value of 1. If the motion of the moon is described by an epicycle model the following facts are valid. The epicycle centre C moves on the deferent and represents the mean position of the moon, so that Arg OC = im = the

Ye 3 a M Cc a A N > O x, Y / Fig. 1 mean longitude. Consequently, the vector OC turns with a constant angular velocity, Am, relating tc to the e x¡-axis. The moon M moves on the epicycle. Ptolemy calls 7 (CM, OC) the anomaly vm, and the vector CM turns with a cc constant angular velocityvm relating to OC, or, since Arg CM = Arg OC + / (OC, CM) = Am — vm, With the angular velocity Am — Um relating to the X1-axis. If as in the solar theory Am = Vm, then vm(t) = Am(t) + c (c = a constant). Consequently, the greatest apparent velocity of the moon, which is assumed in the lower point of the epicycle, is always assumed for the

same longitude (namely Am = 180° — c) in contradiction to 1. Hence From the fact 2 it appears that the plane of the orbit of the moon makes an angle of approximately 5° with the plane of the ecliptic. If the line of intersection of these two planes (the nodal line) is fixed, the moon has always the greatest latitude for the same longitude, namely the longitude of the northern limit point, which is in contradiction with 2., hence, the nodal line turns relating to the x-axis. Based on such rather rough reflections on the observation material Ptolemy assumes: The motion of the moon can be described by the above mentioned epicycle model, and the deferent makes an angle 5° with the plane of the ecliptic. However, to make the theory simpler, Ptolemy ignores this angle, so that in the following it is supposed that the epicycle model is in the plane of the ecliptic. Only a minor inaccuracy is involved by this approximation, the projection factor being cos5° = 0.996 ~ 1. The position vector OM of the moon M is a sum of two vectors OM = OC + CM, where OC = R=a constant, CM = r =aconstant, Arg OC = Am = the mean longitude Arg CM = Adm — Vm (Vm = the anomaly). Ptolemy calls the angle 7 (OC, OM ) = p1, at which the epicycle radius CM is seen from the earth O, “prostaphairesis”. This is the angle which is to be added to the mean longitude À, to give the true longitude 4. So, at any time ¢ the following relation is valid Arg OM = Mt) = Am(t) + pi(t). Note that p, can be regarded as a function of the anomaly vm. In the following this model is called “model 1”. Having made this assumption Ptolemy’s first problem is to find the constant velocities Am = the mean velocity of the moon in longitude Vm = the mean velocity of the moon in anomaly. If the mean longitude of the sun is called Âom, Ptolemy calls the difference Am — Agm = nm the “mean elongation”. CENTAURUS, VOL. XIV

3. The Periods of the Moon To solve this problem Ptolemy has to use the results of his predecessors. First he mentions how “the old ones” have looked for a period L, with the following property: If at the time ¢ there is a lunar eclipse there is again a lunar eclipse at the time £ + L, and the increase of the longitude of the moon, A(t + L) — Alt) is a constant. It is possible to prove that a period L with this property must contain an integral number of returns in anomaly (under the assumption: that the moon can be described by model 1). Ptolemy mentions this fact without a proof. Ptolemy states that Hipparchus has found such a period by comparing his own observations with older chaldean observations: L = 1260073; days in which the moon completes 4267 synodic periods 4573 returns of anomaly and 4612 revolutions in the ecliptic except for 74°. From these results the mean values of the synodic month, and the anomalistic month L d L 2267 °° 4573 and the corresponding daily mean motions in elongation, and anomaly . 4267 o . 4573 7m = —— * 360°, and Vm = T 360 o can be found. Ptolemy finds the daily mean motion in longitude of the moon by means of L and the daily mean motion in longitude of the sun Jom. The mean motion in longitude of the sun in a syn. month = peje"

The mean motion in longitude of the moon in a syn. month = mee” + 360° The mean motion in longitude of the moon in a day = L : L (ole + 360 oés) 5 . . = 4267... 360° + dom = fim + dom. Consequently, at once he could have said: The mean velocity in longitude of the moon = the mean velocity in longitude of the sun + the mean velocity in elongation. The lunar theory is now based on the mean values . 4267 : Am = 360° T + dom = 13;10,34,58,33,30,30 °/d . 7 Vm = 360° = = 13; 3,53,56,29,38,38 °/d . 4267 Nm = 360° T = 12;11,26,41,20,17,59 °/d Hipparchus knew these mean values and they were handed over to Ptolemy. However, he re-examines the results and finds a minor correction for vm. Even if this correction is so little (order of magnitude 12 - 60-4 = 2'/100 years) that in practice it plays no role for the lunar theory, it is to Ptolemy’s credit that he tries to check and correct the results handed over to him. The mean motions in longitude, anomaly, and elongation as functions of time are given in four tables in Almagest IV,4. We are here going to examine how accurate Am and Am are. According to P. V. Neugebauer [1] I, p. 35 the following formulas are valid Am = const; + 1732564394.50”7 + 12.200”72 + 0.0066"T3 23m = const, + 129602764.37’T + 2,600"T2 from which by subtraction nm = const; + 1602961630.13"T + 9.600”72 + 0.0066” T3. Here T means the number of Julian centuries which have elapsed since the year 1800. These are P. V. Neugebauer’s original elements, supplied with the corrections based upon the investigations by C. Schoch.

By differentiation we find the mean velocities 2 Af, = 1732564394.50" + 24.400°T + 0.0198"7 7m = 1602961630.13” + 19.200”7 + 0.019872 Note. Since the remaining terms (T3-term, 74-term, etc.) in 1% and Agm are not known, such a differentiation may be risky. But it is reasonable to assume that the mentioned polynomials approximate so well that in a bounded time interval the real derivatives do not differ essentially from the derivatives of the polynomials. If T = —16.4 is inserted the mean velocities for the year + 160 (the end of Ptolemy’s lifetime) are obtained in the unit seconds/100 years. Division by 3600 - 365.25 - 100 gives the unit degrees/day. Results: 1% = 13.176393640 °/d *m— 0.985646658 °/d 7% = 12.190746981 °/d. Ptolemy has the values Am = 13;10,34,58,33,30,30 = 13.176382215 °/d Jom = 7m 0:59, 8,17,13,12,31 = 0.985635278 °/d = 12;11,26,41,20,17,59 = 12.190746937 °/d. The errors in Ptolemy’s values for the year + 160 are then Alm = 4% — Âm = 0.000011425 °/d = 0.417 °/100 years Alon = ax Io m = 0.000011380 °/d = 0.416 °/100 years Arm = 7% — nm = 0.000000044 °/d = 0.002 °/100 years. By differentiation of 15,1%om, and 7m the accelerations are found. hin = 0.0066°/100 years per 100 years om = 0.0014°/100 years per 100 years = 0.0052°/100 years per 100 years. Note. As above.

Consequently, at the beginning of the time of Hipparchus (— 160) the following formulas are valid: Alm = 0.417 — 0.0066 - 3.2 °/100 years = 0.396 °/100 years Ad m= 0.416 — 0.0014 - 3.2 °/100 years = 0.412 °/100 years Anm = 0.002 — 0.0052 - 3.2 °/100 years = —0.015 °/100 years. During the whole period from the beginning of the time of Hipparchus to the end of the time of Ptolemy, —160 to +160, the followingis valid with good approximation: Alm — Alm = = 0.41°/100 years, while Anm is practically equal to zero. This examination shows that nm is a good basis for the lunar theory, while Am is not good, On the whole, the error in the mean velocity in longitude of the moon, Am, is the same as the error in the mean velocity in longitude of the sun, om. In Almagest III dom was found with a rather great error, originating in an error in the length of the year of 0.0044 days (see Petersen and Schmidt, p. 88-89). This error from the solar theory is undiminished handed over to the lunar theory and causes that the moon according to Ptolemy on an average loses about 0.41 degrees per 100 years. 4. The Computation of the Longitude of the Moon According to Model 1 To compute the longitude of the moon at a moment t from the model 1 it is necessary to know two things, namely A. The dimensions of the model B. The position of the moon at a definite moment. As only the proportion R is Important for the computation of prosthaphairesis pı, Ptolemy can arbitrarily chose R to 60 units. In what follows all other distances will be measured in these units. Then the problem A is reduced to fix r. Now, the problems A and B are solved using the moments 7), 7, and T; for 3 lunar eclipses, observed in Babylon shortly after the beginning of the era Nabonassar. By means of the solar theory Ptolemy computes the longitude, 4,(T:) of the sun at these three moments, and since 7;, T>, and Ty are moments

of lunar eclipses (mid-eclipses), 4(7:) = 4,(7:) + 180°. He finds A(T,) = 174°30’, A(T>) = 163°45’, and A(T3) = 333°15’. Note. The moments T1, T2, and 73 are around the year — 720 (see section 8). On account of errors in the solar theory (Petersen and Schmidt, p. 88-90), Ptolemy’s sun in the year —720 is about 24° ahead of the real sun. This causes a one-sided error in the Jongitude of the moon, A(T;), of about —24°. The moments 7;, 72, and 73 are measured in true solar time. Ptolemy converts the time intervals T> — T,, and 73 — T> into mean solar time. The results are given in the table 1, column I. In column II the differences MT) — MT |), and A(T3) — A(T) are given. The one-sided error of —2}° disappears by this subtraction. The column III gives the mean motion of the moon in longitude in any of the two time intervals, calculated by Ptolemy from the moon tables, Almagest IV,4. The column IV contains the differences between the true motion of the moon in longitude and the mean motion of the moon in longitude (column II-column III), and . finally the column V gives the motion of the moon in anomaly in every of the two intervals, calculated by Ptolemy by means of the moon tables. I II MI IV V T2-T1....... 35442h34m 349°15’ T3-T2....... 176220n12m 169°30° 345°51’ 3°24’ 306°25’ 170°7’ —0°37" 150°26’ Table 1 Since the motion of M on the epicycle (which is the motion in anomaly, column V) determines the difference between the true motion in longitude and the mean motion in longitude, a close connection between the columns IV and V exists. This solves the problems A and B in the following manner. At first the epicycle is regarded alone, without consideration of its real position in the coordinate system at the three moments 71, 7, and 73. The position of the moon at these moments are called M,, Mo, and M; (fig. 2). Now, the results of column V give Z (CM, CM;) = Vm(T2) — Vm(T}) = 306°25' M3M> = Z (CM, CM) = va(T3) — Vn(T2) = 150°26'

from which follows MM, = ¿ (CM, CM») = 360° — 306°25’ = 53°35’ M3M; = M3M> — MM; = 150°26’ — 53°35’ = 96°51’ Since A(T:) = Am(Tı) + pi(T+), the results from column IV give Pı(T2) — pi(T1) = 3°24’ (1) Pi(T3) — pi(T2) = —0°37 (2) and by addition (3) Pi(T3) — pi(T1) = 2°47' Since prostaphairesis is the angle p;(7:;)= £ (OC, OM:) this can be inserted in (1)-(3), OC can be eliminated, and this gives Z (OM, OM:) = / MOM, = 3°24’ 4 (0M:, OM») = Z M30M») = —0°37 LZ. (OM), OM3) = / MOM; =

The problem is now equivalent to the following one: “On a circle of radius r three points M1, M2, and M; are given, which are seen under given angles from a point O. Find r and the position of O with respect to M1, M), and M3.” Neugebauer gives a good and detailed description of the general solution of this problem, see O. Neugebauer, p. 211-214. The results of the calculations are R = 60 and r = 5;13 the anomaly at the moment T> = yn(T>) = 12°24’ the prosthaphairesis at the moment T> = p,(T>) = —0°59' Above was found 4(72) = 163°45’, which with p;(T>) gives the mean longitude at the moment 77 = Am(T>2) = 164°44’ Note. Of course, here it plays a role that A(T;) has an error of about — 24°, In this way the dimensions of the model and the position of the moon at a definite moment (72) are found from the moments 71, T2, 73 of three lunar eclipses. Ptolemy repeats all the calculations from a new set of moments of three lunar eclipses, this time someones, he himself has observed at the moments Ty, Ts, and Tg. The result is r = 5;14, after which he goes on with the approximate value r = 5:15. From the values of Am(T2) and vm(T>) is found Am(Tp) and vm(To) at the beginning of era Nabonassar (= Nabonassar 1, thoth 1, noon) by first converting 72 — Ty into mean solar time and then finding from the moon tables the position of the moon at the moment To: Vm(To) = 268°49' Am(To) = 41°22’ From the solar theory is known 46 m(70) = 330°45’, from which also the mean elongation at the beginning of the era is known: nm(To) = Am(To) — Ao m(To) = 70°37’. Now, from these starting values the mean longitude A„(t) and the anomaly vm(t) can be calculated at an arbitrary moment £ by means of the moon tables. Prostaphairesis p;(t) can be calculated from the triangle OCM (fig. 1), in which OC = Rand CM = rare known, and 7 OCM can be found immediately from the anomaly vm. To make the calculations easyer, Ptolemy has once for ever tabulated p; as a function of the anomaly

vm in Almagest IV,10. Then, in this table p, is read, and the true longitude of the moon is given by A(t) = Am(t) + pi(t). Ptolemy states in Almagest IV,11, that model 1 was already known by Hipparchus, but declares that Hipparchus had certain errors, which are now corrected. Finally, it is noted that prosthaphairesis p, for all vm is given by the formula —r Sin Vin pı = tan R + rcosvm (4) which is easy to find using the triangle OCM (fig. 1). 5. First Change, from Model 1 to Model 2 Ptolemy begins Almagest V by stating that model 1 has shown good agreement with observations made by elongations of 0° or 180° (the syzygies), but with other values of 7 the agreement is sometimes good, sometimes bad, and worst by elongations of 90° or 270° (the quadratures). Ptolemy has masterfully selected the observations on which this discovery is based, partly among the older observations handed over to him by Hipparchus and partly among his own observations. After an analysis of these observations he finds: A. If prostaphairesis p; > 0, the moon is observed to be ahead of the position, determined by model 1, and the error increases with p1. B. If prostaphairesis pi < 0, the moon is observed to be behind the position, determined by model 1, and the error increases with |pı|. Since then model 1 does not correspond to the observations, it must be changed into a model, say “model 2”. In this the mean longitude of the moon, An, is provided with a new prostaphairesis p to give the true longitude À, and compared with prostaphairesis p; the following must be valid. For7 = 0°, 180°: p = pi For y = 90°, 270° : |p| > [pil 2 These demands are satisfied by inserting a mechanism into the model, bringing the epicycle closer to the earth at elongations 7 = 90° or 270°, while the distance remains unchanged at elongations 7 = 0° or 180°.

Ptolemy inserts this mechanism assuming that the motion of the moon can be described by model 2, fig. 3. Now, the epicycle centre does not move on the deferent of radius R and centre O, but on an excentre of radius R-s and centre Z which moves on a small circle of radius s and centre O (the constant s is found below). The epicycle centre steadily represents the mean moon, consequently Arg OC = Am, while the centre Z of the excentre moves on the small circle in such a way that 7 (OZ, OC) = 21m = the double of the mean elongation. The motion on the epicycle of the moon M is with respect to OC unchanged from model 1. Now, the position vector of the moon is a sum of three vectors OM = OZ + ZC + CM, where |OZ| = $ = a constant. Arg OZ = Arg0C— z (OZ, OC) = Am — 29m IZC| = R-s = aconstant. ArgZC = ArgOC + Z(OC,ZC) = Am + à ICM|= r =aconstant. ArgCM = ArgOC — Z (CM, OC) = Am— Vm

The small angle « = / (OC, ZC) is given by sina = — sin 27m, (5) which is easily found from the law of sines applied to the triangle OCZ. Then the position vector OM is a sum of three vectors with constant lengths of which the two OZ and CM turn with a constant angular velocity, while the third turns more irregularly. OM can also be regarded as a sum of two vectors: OM = OC + CM, where OC has a constant angular velocity Am, but variable length OC, given by | OC = ZC: cosa + OZ : cos 27m = (R—s)- VI —sin2« + s-cos2nm (6) Now, (6) shows that for 7m — 0°, 180°: OC=(R—s):l+s=R for nm = 90°, 270°: OC = (R-s) 1-—s=R— 2 This shows that the epicycle has the least possible distance R — 2s from the earth at the quadratures, while at the syzygies it has the same distance R from the earth as in model 1. Thus model 2 agrees with the observations. The remaining problem, to find the constant s, is solved by Ptolemy in Almagest, V 3-4 in the following way. Since the numerical value of prostaphairesis p has maximum when 1. nm = 90°, 270° (the epicycle as close as possible to the earth) 2. vm = 90°, 270° (OM nearly is a tangent to the epicycle), he observes the moon at a moment T6, where this is the case. He finds The longitude of the moon 4716) = 219°40’ The longitude of the sun 42(T16) = 318°50’ The elongation n(716) The time interval T6 — To is converted into mean solar time, and from the moon tables the following mean values are calculated

The mean longitude of the moon 4m(T6) = 227°20' The mean longitude of the sun Agm(7i6) = 316°27 The mean elongation nm(716) = 270°53’ (22707) The anomaly vm(716) = 87°19’ (= 90°) Then the conditions 1 and 2 are satisfied, and Praz can be found: p max = |p(Ti6)| = |AT16) — Am(Ti6)| = 7°40’. Ptolemy ends Almagest V,3 by stating that he has reached the same result from many other observations, and in Almagest V,4 the radius from this result is calculated. Since 1 and 2 are satisfied, with good approximation the following is valid: OC = R — 2sand 7 CMO = 90°. Then, in the rectangular triangle OMC (fig. 4) MC = rand / COM = 7°40’ are known, from which the hypotenuse OC = R — 2s can be found. Finally, this gives (in the units of which R = 60): s = 10;19. Model 2 is now completely determined, and Ptolemy ends his description of this part of the complicated lunar theory ascertaining that model 2 agrees well with observations at 7 ~ 0°, 90°, 180°, and 270°. Briefly, the change from model 1 to model 2 consists in replacing the constant radius R with the variable length OC given by (4). Therefore, the formula (4) can be used for model 2 with OC instead of R, and gives —r sinYm tanp = OC + r cosvm 0) 6. The Second Change, from Model 2 into Model 3 Now, in true scientific way Ptolemy starts examining whether model 2 gives good results compared with Hipparchus’ and his own observations. As mentioned above this is the case for the elongations 7 = 0°, 90°, 180°, and 270° but it appears not to be the case for y = 45°, 135°, 225°, and 315°, i.e. around the octants. After an analysis of a large number of observations Ptolemy has found that the deviations relating to the moon positions determined by model 2 are largest when

Fig. 4 1. y = 45°, 135°, 225°, or 315° and 2. vm z 0°, or 180°. Since model 2 does not agree completely with the observations it must be changed into a new model, say “model 3”. In this the mean longitude Am of the moon is provided with a new prostaphairesis p, which compared with the prostaphairesis p from model 2 satisfies the following conditions A. For n = p =p 0°, 90°, 180°, 270°: o 90°) and m= O':p<p vm © 180°:p > p C. For y = 135°, 315° (i.e. 2n = 270% and l['"= OP HP Vm = 180°:p <p (; 27 4 o 225° o (1.e. B. For y ~ = 45°, = Note. Ptolemy does not formulate these demands so sharply. But in his construction of model 3 in Almagest V,5 he applies two observations which satisfy two of the conditions mentioned above. These conditions are satisfied if model 2 is changed in such a way that Z (CM, OC) is slightly larger than vm for y = 45°, 225°, and slightly smaller than vm for y = 135°, 315°. Ptolemy makes this change by assuming that the motion of the moon can be described by model 3, fig. 5. In model 3 the points Z and C move as in model 2, i.e. the vectors OC and ZC are determined as in model 2. M is still to move on the epicycle of radius r and centre C, but now the anomaly is to be counted

© Y Fig. 5 from the variable direction CN, where N is placed diametrically opposite to Z on the small circle of radius s and centre O. Since the motion of M in model 3 on the epicycle is somewhat different from the motion of M in model 2 on the epicycle it follows that the prostaphairesis of model 3 may differ from the prostaphairesis of model 2. In what follows the angle / (CM, OC) is called the true anomaly, y, and then v= £ (CM, NC) + Z (NC, OC) = vm + B

From fig. 5 it follows that: for nm = 0°, 90°, 180°, 270°: 7 ZOC = 0° or 180° and then B = 0° for nm = 45°, 225°: / ZOC = 90° and then $ positive for nm = 135°, 315°: / ZOC = 270° and then f negative consequently, complete agreement with the conditions A, B, and C. The position vector OM of the moon is a sum of three vectors: OM = OZ + ZC + CM, where OZ and ZC are as in model 2, while |CM| =r = 5;15, Arg CM = Am — (vm + 8) = Am — Y. The small angles « and f are given by . sino = sinf = S R sina 3 Vi ) + (3 a -) cos(27m + a) (9) — $ The last formula is found by at first computing NC by the triangle and then applying the law of sines to triangle NCO. NCZ Ptolemy begins by mentioning how model 3 is constructed and then he shows that the so constructed model agrees with two observations. One of these is made by Hipparchus at the moment 7;g where the moon is around an octant. The following was observed: The longitude of the moon A(T¡g) = 351°27’ 37°45’ The longitude of the sun 4,(T13) = The elongation n(71g) = 31342" The time interval T¡g — Ty is converted into mean solar time, and from the moon tables the following mean values are calculated: The mean longitude of the moon An(T13) = 352°13' The mean longitude of the sun Ag m(T1s3) = 36°4l' = 315°32 The anomaly vm(T¡5) = 185°30’ The mean elongation nm(71g) From these results prostaphairesis p(T¡g) is found: p(Tis) = MT 13) — An(T18)

Note. Since vm(T1g) > 180°, model 2 gives a positive prostaphairesis, P(T;3), while in fact the observation shows a negative prostaphairesis, corresponding to the condition C. By a detailed calculation Ptolemy shows that this negative prostaphairesis is obtained, if the anomaly vm is counted from the direction NC, where N is opposite to Z on the line through O and Z, and ON = 10;18. Now, in model 2 OZ = 10;19 was found, hence, approximately ON = OZ, which agrees with the construction of model 3. Ptolemy makes quite analogous calculations from another observation, made by Hipparchus at the moment T0, and finds ON = 10;20 = OZ. He finishes by saying that many other observations have given the same result and states that the model now set up gives good results in syzygies, quadratures, and octants, and with this he is satisfied. Briefly, the change from model 2 into model 3 consists in replacing the anomaly vm with the true anomaly v. Therefore, the formula (7) can be used in model 3 with v instead of vm, and gives —rsinv t an p = + rcosv OC (10) The next problem is to find simple methods for computing the longitude of the moon according to model 3 at an arbitrary moment. The solution of this problem is shortly expounded in the next section. 7. Computation of the Longitude of the Moon According to Model 3 To compute the longitude of the moon at an arbitrary moment 7, the mean values Am(t), vm(f), and nm(f) can at once be found from the moon tables. Since Mt) = Ant) + p(t) the problem is to calculate p(t), which is given by (10), where v = vm + 8, 8 is given by (5) and (9), and OC is given by (5) and (6), such that p= Pm, Nm). In principle, it is no problem to find A(t) according to model 3, but in Almagest V,7-9 Ptolemy sets himself the problem to set up few and simple tables, by means of which p can be found from the mean values Vm and Nm.

A table for computing p(vm, 7m) could be set up as a table with two entries. If the prostaphairesis is tabulated for 45 values of vm, and 45 values of 7m (and one might expect about 45 values since Ptolemy in model 1 has tabulated prostaphairesis for 45 values of the anomaly) then the table would contain altogether 2025 values. Ptolemy succeeds in avoiding that many values by using (without proving) the fact that approximately Pm, 7m) = Pi) + f{nm) : Lpa(r) — PO] where f(nm) and p2(v) will be defined later. In this way he constructs 4 tables, (namely the values of B, p1, p2 — Di, 4-45 = 180 values. These 4 tables are the columns 3, 4, 5, and 6 in the collection of tables Almagest V,8. and f) with a total of Column 4 contains the function p;, given by —rsinv tan pı = —__ — PI R-+rcosv corresponding to prostaphairesis as a function of v, when the epicycle centre has the maximum distance R from the earth, consequently precisely the prostaphairesis known from model 1. Column 5. The function p» is defined by —r sin y tan p, = (R — 25) + rcosy corresponding to prostaphairesis as a function of v, when the epicycle centre has the minimum distance R — 2s from the earth. Column 5 contains the values of Pv) — pit) which is the angle by which prostaphairesis increases, when OC changes from the maximum to the minimum value. Column 6. If nm has such a value that OC is between its maximum value R (p(ym, v) = pı(v)) and its minimum value R — 2s (p(ym, v) = p;(v)), prostaphairesis has a value between pı(v) and p(y). A factor q exists, so that P(nm, Y) = pi) + 9 (226) — pr), CENTAURUS, VOL. XIV

namely _ Pm, Y) — Pi) P2(v) — pi(v) This factor y is a function of v and 7». But it can be shown that it is approximately independent of v. This implies that for any value of 7m it is sufficient to compute (nm, v) for one suitable value of v, and the values of y can be tabulated in a table with one entry. Obviously, Ptolemy has been aware of these circumstances, and by means of this he finishes the solution of the problem in the following way. In column 6 a function f given by (Am) = max |p(7m, v)| — max |p1(v)| max |p2(v) — py)! (where maximum is made for 0° < v < 360° and 7m fixed) is tabulated. Column 4 shows: max |pı(v)| = 5°1’ Column 5 shows: max | p2(v) — pi(v)| = 2°39' so that fis given by Alam) = 5555 (max | (nm, WI — 51 Finally Ptolemy states that the prostaphairesis is given by p=pi + £-(p2— pi) Perhaps, the connection between Ptolemy’s function f and my function gy is not quite obvious. The connection between f and y appears from the fact that maximum of prostaphairesis is obtained for y = 90° or 270°. This means that the function f is approximately the function (rm, 90°). By a closer examination it can be proved that for all 7m [p(rm, 90°) — f(nm)| S 5 - 1073, Further it can be proved that for all nm, v \P(nm, Y) — P(7m, 90°)| < 2 - 10-2 so that on the whole Ip(nm» v) — f(nm)| < 2.5 - 102.

Since the maximum value of pp — pi is 2°39’ = 159’, the error made in the computation of p by replacing (7m, v) with /(nm) is rather unimportant, namely smaller than 2.5 - 10-2- 159’ ~ 4°. It is a very small error, compared with the accuracy of the model, see section 9. 8. An Examination of the Observations in the Lunar Theory In connection with setting up of model 1, 15 observations of lunar eclipses made at the moments 71, ..., 715 are mentioned. These moments Ptolemy has converted into the Egyptian calendar, Alexandrian time. By means of P. V. Neugebauer’s [2] tables, the moments 7), ..., Tıs can be converted into the civil Julian calendar, Alexandrian time. (Concerning details of this conversion, see Petersen and Schmidt p. 85-87.) Exactly computed moments T}*, ..., 715* for the corresponding eclipses can be found in Oppolzer’s “Canon der Finsternisse”, these moments are given in Greenwich time, mean solar time. Addition of two hours converts 71*, ..., 7 5* into Alexandrian time, mean solar time, and finally the true solar time is obtained by applying the equation of time, also found in Oppolzer’s tables. The absolute errors AT; in the moments 7), ...,71s are given by AT; = Ti* — Ti. The results of these subtractions appear from table 2, where 47; is given in minutes. Ti Ti Date — 720 AT; Ti 14 Tg March 19 72 | —719 — 719 6 T7 +133 —36 Tg +134 Oct. 20 +125 —490 —2 To Ti | —381 — 501 14 Tıo | —382 Dec. 23 Date AT: 22 June 18 31 Ti2 | —381 —3 Dec. 12 —41 T13 | —200 — 8 Sept. 22 50 Tia | —199 Nov. 19 Table 2 11* —30 April 25 May 6 Ts Ti April 5 Sept. 1 T4 +136 AT; March 6 March 8 T3 Date —13 March 20 33 Tıs Sept. 12

All the errors are smaller than 1 hour, half of them even smaller than 4 hour. The average of the numerical values of the errors is 22 min. This shows that just from the beginning of era Nabonassar it has been possible to observe lunar eclipses with great accuracy. It is especially noted that the error in 7>, used for starting the moon in the mean motion (section 4), is only 6 minutes. The setting up of model 2 and model 3 is based on observations of the longitude A of the moon, the longitude 4, of the sun, and the elongation y = À — As. In Almagest V,1-9 altogether 4 such observations made at the moments T6, ..., Tio are mentioned. The first one Ptolemy has made himself, the three others are made by Hipparchus. Now, the moments 716, ..., 719 are converted from Egyptian calendar, Alexandrian (or Rhodesian) time, true solar time, into the civil Julian calendar, Greenwich time, mean solar time, the results are given in table 3, column I. Column II contains Ptolemy’s (or Hipparchus’) values for 4(7;), column V contains their values for A,(7:). In the columns III and VI the corresponding modern values 2*(7:) and A;(T:), computed according to Tuckerman, are given. The columns IV and VII contain the absolute errors AA(T:) = A*(T:) — MT) and 41,(T) = 44(T) — A(T:1). Ti T16 I +139 1 I A(T) A*(Ti) IV 219.7° 221.9° 2.2° 42.3° 41.0° 351.5° 149.0° V VI VII AST) SAS (TM) 318.83° 319.43° 0.60° —1.3° 128.58° 128.73° 0.15° 351.3° —0.2° 37.75° 37.64° —0.11° 149.2° 0.2° 100.90° 100.96° 0.06° AMT:) | Ae(T) Febr. 9.210 Tu — 127 Aug. Tıs — 126 May Tio 5.162 2.176 — 126 July 7.583 “Table 3 The values 4*(7;) are found by linear interpolation in Tuckerman’s tables, which can give an inaccuracy of 1.6° (Tuckerman I, p. 5-6). On account of this rather great error no definite conclusion can be made

about the accuracy of A(7;), i = 17, 18, 19. But as for A(T 16) we are justified in maintaining that it is in error of at least +0.6° (= 2.2° — 1.6°). The values A5(7;) are also found by linear interpolation, but here the accuracy is 0.01°, so that AA, is quite precisely defined. In particular we note that at the time 716 Ptolemy has observed the sun 0.60° behind its real place, and he has observed the moon at least 0.60° behind of its real place. This may indicate that his instruments had a systematical error. Hipparchus’ values for À, are unquestionable better than Ptolemy’s. The average of the numerical values of Hipparchus’ errors is 0.32°/3 = 6.4’, which confirms the generally accepted statement that Hipparchus was a good observer and that the accuracy of observations in antiquity is approximately 5-10 arc minutes. 9. An Examination of the Quality of the Models Having expounded the three models and examined the basic observations an answer is given below to the following question: How accurate can these models describe the motion of the moon at Ptolemy’s time, and are Ptolemy’s complicated models 2 and 3 in reality better than model 1, known by Hipparchus ? From Tuckerman’s tables the longitude A* of the moon during the years +150 to +169 at moments with five days intervals can be found, totally 1461 values. This time interval begins in Ptolemy’s last years of activity and its length, 20 years, is selected to have a fairly large material. By means of the above expounded methods, A can be computed according to the three models at the same 1461 moments and then the absolute errors AA in À can be found. To make these computations quickly and safely, they have been made by means of a computer (GIER, Kobenhavns Universitets Matematiske Institut) in the following way. From Ptolemy’s tables is at first manually calculated Am(t}), vm(¢1), and 7(t,) at the moment 1; = +150, Jan. 5.75, Alexandrian local time. The computer is fed with these mean values and the mean velocities 4m, Vm, 7m. The true longitudes A of the moon, according to the three models, are given by the formulas —r Sin vm Model 1 :A=A m + tan —1 REF COVA

—r SIN Vm OC + room tan Model 2 :AzA m + tan”! Model 3 + tan —r sin y OC Tr cosy where OC and v are given by the formulas (5), (6), (8), and (9). Now, by means of this information the values for À according to the three models at the 1461 moments are computed, and by comparing these with Tuckerman’s values 4*, the absolute errors 44 = 4* — À can be found at once. The computer has also found 44, being fed with Tuckerman’s values before running the program. All results are computed in degrees with one decimal, since Tuckerman calculated with this accuracy. In fig. 6 pillar diagrams of the result of this examination are drawn, the height of the pillars for each value of AA signifies how often this error appears. It is at once noted that in model 1 —2.0° < 44 < 4.1°, in model 2 the corresponding interval is somewhat smaller —1.3° < 44 < 3.4°, and model 3 has the smallest interval —0.3° < 44 s 2.5°. This shows that Ptolemy actually has improved the lunar theory by constructing model 2 and model 3, but it is remarkable that the longitude of the moon computed according to the three models has a tendency to be smaller than the real longitude of the moon. This is especially clear by model 3, where only about 1% of the errors are negative. This distortion of the errors must be a result of a systematical error in the lunar theory, originating in the following facts. The mean motion of the moon in longitude according to Ptolemy is too slow (section 3) and the moon is started with an error of 24° (section 4). Both errors Originate in corresponding errors in the solar theory. At a definite moment fo in history, Ptolemy’s moon coincides in the mean with the real moon. Before this moment Ptolemy’s moon is ahead of the real moon, after this moment behind it. Since we know that the longitude of Ptolemy’s moon diminishes about 0.41°/100 years, the moment £ can be fixed if only the systematical error at one moment (e.g. +160) is known. If there was not a systematical error in the computation of the longitude of the moon according to Ptolemy, but only casual errors, the mean values of the errors would be about 0°. If, on the other hand, the mean

MODEL MODEL 167 1 -1$ “20 -a5s 0.8 es 410 18 20 25 30 3.5 S& #2 Vv 2 Ps -19 -05 0.9 0.5 4.0 15 2.0 25 3.0 [to as oA eas 70 75 20 45 30 Vv MODEL 3 d Fig. 6 value of the errors differ essentially from 0°, the size of the mean value must originate in a systematical error, the size of which is then fixed. A calculation shows the following mean values of the 1461 errors: Model 1: 1.110° Model 2: 1.108°} average 1.108° — 1.11°. Model Now, it is reasonable to suppose that on an average Ptolemy’s moon

Fig. 7 is 1.11° behind the real moon in the year +160. The mentioned moment to is then given by the solution of the equation (160 — t) 0.0041 0.656 — 1.11 {di 1.11 0.0041 : tp 0.454 lo o = — === 0.0041 We — 110 If the mean error in Ptolemy’s moon is called Adm, then 44m(160) = 1.119 and 44m(—110) = 0.00? is now known. This gives rise to drawing a graph (a straight line, fig. 7), from which the mean error 44m is read at any time in the time interval —160 to +160 (where Ahm is regarded as a constant: 44m = 0.0041°/year). Note. It is remarkable that 44, has very near the same size as the corresponding mean error 4A, m in the solar theory, found and described by Petersen and Schmidt p. 88-90. The graph of Al, n is given as a dotted line in fig. 7. So Ptolemy has a fine computation of the mean elongation 7m; during the whole period from —160 to +160 it is true that Anm = Adm — diem = —0.1°. Corresponding to the accuracy, with which the errors are computed (1 dec.) it is assumed that 44m = 1.1° during the years from +150 to

+169. The three pillar diagrams in fig. 6 also show symmetry relating to this value. If a true picture is wanted of the value of Ptolemy’s lunar theory it is necessary to correct for this systematical error. After such a correction the following intervals are obtained: Model 1: —3.1° < AA s 3.0° A D So Model 2: —2.4° s NA 2.3° Model 3: —1.4° < AA < 1.4° which definitely shows the superiority of model 3, and also shows that model 3 is much better than model 2. An other measure for the quality of the models can be obtained by finding the “standard deviation” S, given by i > (amp S= Vaa The result of such a computation is Model 1: S = 1.37°, model 2: S = 0.83°, model 3: S = 0.57°, consequently, again model 3 is the best one. It is interesting to get an answer to another question: Do the causes, on account of which the models are determined, make themselves known in the numerical material, i.e. have the three models approximately the same (small) errors at syzygies, is model 2 better than model 1 at quadratures, and is model 3 better than model 2 in the octants ? The answer is given in fig. 8, made in the following way: When the computer ran the above sketched program, it also gave the mean elongation 7m for each of the 1461 moments. Now, it is possible to go through the material and for each tabulated value of 7m read the corresponding errors AA. For practical reasons the interval of 7m from 0° to 360° is divided into intervals (decades) of the length 10°, and the result of this is that then in any of these decades the smallest and the largest value of 42 is found. These extremes are marked in fig. 8. From this it is at once seen that all three models have the smallest errors around the syzygies, while model 2 is definitely better than model 1 at the quadratures and further that model 3 has smaller errors than model 2 at the octants (even if model 3 has still the largest errors at the octants). In this examination the systematical error of 1.1° is taken into

2ed.__.-ı!-UÜ---A.-y.---|I--etÙIC EL“—+e+-.41ÙtLali4n+.<RXp+--—-4!Ùtl--4+[A+Y$3i ‘a=Yotnotaot1!'u=o-Hd‘0oe“.2l*Ÿ°Li2lN.2i-2=u»i*xyAh"t’'+!o.‘dL"o‘eoyt"oos"> I1to«a!txo Viggo M. Petersen . Ùt aSt,i - a N) -——cn——ww i i I } ‘ t 1 I I I -ES r 0 3 +2 oct. tk vel. 8 Fig. account so that regarding fig. 8 the dotted horizontal line is to be used as starting point. Note. In all three models Ptolemy’s moon has a tendency to be behind the real moon for 7m land 45°, 215°, while it is ahead of the real

moon for nm = 135°, 315°. This is due to the fact that the models do not include the variation which causes that the moon in Ist and 3rd octant can be nearly 40’ ahead of the ecliptical position, while in 2nd and 4th octant it can be nearly 40’ behind of the ecliptical position. Finally, it is noted that all the results in this section are based on Tuckerman’s tables and are therefore as accurate as these tables. However, the main result is firm: Ptolemy’s complicated model 3 is superior to not only Hipparchus’ model 1, but also to his own model 2. BIBLIOGRAPHY 1. Kempf, P. Untersuchungen über die Ptolemäische Theorie der Mondbewegung. Berlin 1878. 2. Manitius, Karl. Des Claudius Ptolemäus Handbuch der Astronomie, Erster Band. Verlag von B. G. Teubner, Leipzig, 1912. New Edition: 1963. 3. O. Neugebauer. The exact Sciences in Antiquity. Second Edition. This book was originally published in 1952 by Princeton University Press, a second edition was published by Brown University Press in 1957, and is reprinted by arrangement. Harper & Brothers, New York. 4. P. V. Neugebauer [1] Astronomische Chronologie, Bd. I und II. Walter de Gruyter und Co., Berlin, Leipzig, 1929. 5. P. V. Neugebauer [2] Hilfstafeln zur technischen Chronologie. Astronomische Nachrichten, Bd. 261, 1936/37. 6. Oppolzer, Th. Ritter von. Canon der Finsternisse. Denkschriften der Kaiserlichen Akademie der Wissenschaften. Matematisch-Naturwissenschaftlicher Classe. Bd. 52, Wien 1887. 7. Petersen and Schmidt. The Determination of the Longitude of the Apogee of the Orbit of the Sun according to Hipparchus and Ptolemy. Centaurus, Vol. XII, 1967. 8. Tuckerman, Bryant. Planetary, lunar and solar positions 601 B.C. to A.D. 1649 at five-day and ten-day intervals. Vol. I and II. The American Philosophical Society, Independence Square, Philadelphia, 1962-1964.