Determination of the Suns Orbit

Arch. Hist. Exact Sci. 53 (1998) 1-49. © Springer-Verlag 1998 Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe Y. MAEYAMA Communicated by J. NORTH Introduction Symbols ... 1. Preliminary TEMAS... 2 roo encens sets a 2. Deviation of Hipparchus’ solar hypothesis from the modem theory and influence on the determination of the two orbital elements .................. 4 3. Observational errors and the determination of the orbital elements: 9 Derivation of the formulae 4. Errors in the underlying observations and the orbital elements of the Sun ..... 13 5. Explanatory remarks on the underlying observations of the Sun . x „26: Summary È Billiger ins SEIT pry » NYEAMA, \qa® G \o MA RT 47 Introduction Much has been said and written about historical determinations of the Sun's orbit; the subject is indeed an essential part of every planetary theory and world-system, both geocentric and heliocentric. The historical records show us that the solar eccentricity — a parameter which, after Hipparchus, no astronomer could avoid discussing — decreased linearly from Hipparchus and Ptolemy (ca. 0.0416) through al-Battani (0.0346...) and many other Islamic astronomers down to Copernicus (0.0323). This decrease, together with other similar phenomena, led Copernicus to his most bizarre theory of the 1717-year cycle. Barely 70 years later, the heavens, as rigorously observed by Tycho, forced him to take the historical way back and to adopt a much greater value (0.03584),! just between Copernicus’ and Hipparchus’ values, This in turn induced him, or made him the more determined to measure the Universe himself, independently of all his predecessors. The problem of all these numerical values for the Sun and their deviations from the modern values has already been studied and explained in terms of the errors in the

underlying observations.2 Our present study aims at finding the cause of those errors. We shall follow as closely as possible the range of the immediate observations, which are assumed to be normally distributed about the true value of each observed object. We shall thus deal mainly with: 1. the theoretical deviation of Hipparchus’ solar hypothesis3 from modern theory (ch. 2), 2. the mathematical derivation of the historical solar elements by means of the underlying observations and the modern values (ch. 3), 3. the analysis of the underlying observational errors (ch. 4). The results obtained are entirely new, and our main achievements are easily explained with the help of few equations and figures. Particularly interesting is the fact that our results could be compared with each other, based on my early study on the length of the seasons (1988), the determination of which was the first step in deriving the solar elements. This paper has a long history. I first noticed the problem presented here in 1972 whilst working on Tycho Brahe and Kepler. I published its partial solution in my paper on Tycho’s solar theory (1974). In 1986–87 I solved the problem, but in order to present it better, I first finished my study on the length of the seasons (1988). Now at last I have been able to finish the task I first undertook 20 years earlier. Symbols A(◦ ) d(◦ ) e h (◦ ) k K (◦ ) l(◦ ) L (◦ ) p p0 P (◦ ) r r0 t v (◦ ) v1 (◦ ) longitude of the apogee of the Sun’s orbit declination solar eccentricity according to the modern theory (Sun’s meridional) altitude solar eccentricity according to Hipparchus’ solar hypothesis, ≈ 2e Sun’s true longitude at t2 , = l2 − l1 = l2 Sun’s true longitude at t, mostly used as l1,2,3 at t1,2,3 Sun’s mean longitude at t, mostly used as L1,2,3 at t1,2,3 mean equatorial horizontal parallax of the Sun, 8.8000 mean equatorial horizontal parallax of the Sun, e.g. 20 5100 (Ptolemy), 30 (al-Battānı̄ , Copernicus, Tycho) longitude of the perigee of the Sun’s orbit coefficient of refraction, ≈ 6000 at h > 10◦ (cf. e.g. Expl. Suppl. 54–56, Woolard/Clemence, 79–96) atmospheric refraction for the Sun after Tycho (Op. om. II, 64), = f (h) time, mostly used as t1,2,3 true anomaly Sun’s true anomaly at t1 , l1 = 0

γ, δ (◦ ) ε (◦ ) θ (◦ ) ϕ (◦ ) ϕ̄ (◦ ) ω (◦ ) 3 Sun’s zenith-distance mean anomaly equation of the center at l1,3 and l2 respectively obliquity of the ecliptic angle between the solar longitude at t2 and the apogee, cf. Fig. 2.1 geographical latitude colatitude, = 90 − ϕ Sun’s mean daily motion (e.g. tropical, 0.9856...◦ /d) Symbols for historical values are marked by (0 ) and the deviations by (1) like 1l = l 0 −l. 1. Preliminary remarks: Determination of two orbital elements of the Sun, the position of the apsidal line and the eccentricity In order to derive two orbital elements of a planet revolving about its central body we need two equations, which involve two sets of two fundamental quantities relating to the transition of the planet in time and position, which are given by three observations. Since the observation of the planet gives us only its celestial direction but not its orbital position, we further need a hypothesis which defines the celestial direction into the orbital point such that the planetary orbit can be defined by the three points. Hence, to determine our two orbital elements we need: 1. a hypothesis on the planetary motion, 2. three longitudinal observations l1 , l2 , l3 at times t1 , t2 , t3 . We shall deal only with Hipparchus’ hypothesis concerning the apparent solar motion around the Earth (Alm. III, 4). This hypothesis was virtually the only one which was constantly employed for the solar determination; it remained relevant at least up to the time of Vieta’s new invention (1600) and Kepler’s Astronomia nova (1609). For all the astronomers we consider, the three solar observations at t1,2,3 can be defined as l1 = 0 (vernal equinox), l2 = l1 + K, K = 90◦ (summer solstice; Hipparchus, Ptolemy, al-Battānı̄) = 45◦ n [n = 1, 3, 5, 7(fus.ūl, sections); Islamic astronomers after c. + 830,1.1 Copernicus, Tycho], l3 = 180◦ (autumnal equinox). Hipparchus’ method differs from other determinational methods of the orbital elements such as those of Ptolemy with the punctum equans, of Vieta (1600) and Kepler

(1609) which are all iterative and approximative. Hipparchus’ method is mathematically accurate and it yields, given the year length and three observational values, one single answer. Since all historical values for the relevant tropical year can be considered sufficiently accurate for our present problem, the errors in the solar elements can only arise from two sources: 1. inaccuracy of the hypothesis, 2. inaccuracy of the underlying observations. The values in the historical records deviate from the modern ones. The deviations are due to a complex agglomerate of many different factors. In order to distinguish between these factors we are obliged to make use of the expansion in series. This is particularly useful, since the deviations are small. Therefore, with only a few terms in each equation, all basic problems become visible to any desired order of accuracy. We shall follow the above two items separately. 2. Deviation of Hipparchus’ solar hypothesis from the modern theory and its influence on the determination of the two orbital elements, the position of the apsidal line and the eccentricity If the mean anomaly α is known, we can roughly compare the solar motion according to Hipparchus and the solar motion according to Kepler’s laws: 2 Hipparchus: vH = α + k sin α + k2 sin 2α + . . . , Kepler: vK = α + 2e sin α + 45 e2 sin 2α + . . . , (2.1) where k is the eccentricity in Hipparchus’ solar model (Fig. 2.1). If we now put vH = vK (in connection with some longitudinal observations of the Sun), k will correspond nearly to 2e, the double of the modern eccentricity, their difference amounting only to the e2 -order: k = 2e − 23 e2 cos α + . . . (2.2) The theoretical error in the position of the apogee is supposed to be correspondingly small. For our quantitative analysis of the two orbital elements below we shall therefore take as our basis the double eccentricity and apogee, 2e and A, both modern parameters. It is therefore convenient to start with the modern parameters at the epoch t. From the geometrical construction given in Fig. 2.1 we have the relations 2e sin θ = sin δ, 2e sin(K + θ) = sin γ, hence we obtain the eccentricity and the position of the apogee A(= K + θ) as functions of γ and δ:

circular orbit about „O“

sin2 γ0 + sin2 δ0 − 2 sin γ0 sin δ0 cos K k= 0 θ = tan −1  sin K sin δ0 sin K sin γ0 − sin δ0 cos K  = f(γ0 , δ0 ) = f(γ0 , δ0 ). (2.5) Putting γ0 = γ + 1γ, δ0 = δ + 1δ, (2.6) we obtain the deviation of the solar eccentricity 1k as a function of the two modern equations of the center γ and δ and the deviations 1γ and 1δ to the first approximation: 1k = k − 2e = 1γ cos γ(sin γ − sin δ cos K) + 1δ cos δ(sin δ − sin γ cos K) 2e sin2 K = f (γ, δ, 1γ, 1δ). (2.7) In the following we shall express the above quantities in modern parameters. 1. γ, δ Simplifying the formulas by putting v1 = 180 − (K + θ), we insert sin γ, cos γ, sin δ and cos δ according to (2.3) into (2.7) and express 1k as a function of 1γ and 1δ: 1k = k − 2e = 1 {1γ[− cos(v1 + K) + . . .] + 1δ[cos v1 − . . .]} sin K = f (1γ, 1δ). (2.8) Likewise from (2.4) and (2.5): 1θ = (θ0 − θ) = A0 − A   sin δ 1 1δ cos δ − 1k = 2e cos θ 2e =− 1γ sin 2(v1 + K) + 21δ[sin K − cos v1 sin(v1 + K)] 4e sin K cos(v1 + K) = f (1γ, 1δ). (2.9) 2. 1γ, 1δ Unlike the lengths of the seasons the historical values for the length of the year can be considered as being mostly accurate and the small errors can be neglected as far as our present problem is concerned.

According to Hipparchus’ method the two equations of the center γ0 and δ0 are given by the observational quantities as γ0 = 21 [(L3 − L1 )] − 90 δ0 = 21 [(L3 + L1 )] − L2 + K − 90. (2.10) Since the mean and true longitude are given by L = P + α, l = P + v, (2.11) and since the mean anomaly α can be expressed as a function of the true anomaly of the Keplerian motion v as2.1 α = v − 2e sin v + 43 e2 sin 2v − 13 e3 sin 3v + . . . = f (v), (2.12) we can express γ0 and δ0 , for the case where no observational errors are involved, by means of the modern values: γ0 = 2e sin v1 + 13 e3 sin 3v1 + . . . , δ0 = 2e sin(v1 + K) − 23 e2 cos(2v1 + K) sin K + 13 e3 sin 3(v1 + K) + . . . , (2.13) where according to our definition above (2.11) P2 + v2 = l2 = l1 + K = P1 + v1 + K P3 + v3 = l3 = l1 + 180 = P1 + v1 + 180. (2.14) Now we obtain from (2.3) γ = 2e sin v1 + 43 e3 sin3 v1 + . . . , δ = 2e sin(v1 + K) + 43 e3 sin3 (v1 + K) + . . . , (2.15) hence, from (2.13) and (2.15)
γ0 − γ = 1γ = e3 − sin v1 + 23 sin 3v1 + . . . , δ0 − δ = 1δ = − 23 e2 sin K cos(2v1 + K) + . . .

These are the theoretical deviations of the two equations of the center γ0 and δ0 according to Hipparchus’ hypothesis from the modern values which yield the accurate solar parameters 2e and A (= K + θ). In our general computation to the e2 -order we therefore put 1γ = 0 and 1δ above into (2.8) and (2.9) and obtain the deviations expressed only by the two modern parameters, e and v1 : 1kh = (k − 2e)h = − 23 e2 cos v1 cos(2v1 + K), = f (e, v1 ), | 1kh |max = 23 e2 (≈ 0.0004), 1θh = (θ0 − θ)h = (A0 − A)h = 3 cos(2v1 + K) e[sin k − cos v1 sin(v1 + K)] 4 cos(v1 + K) (2.17) = f (e, v1 ). These are the theoretical deviations of the two solar elements, as determined by Hipparchus’ hypothesis from the modern parameters if the observations contain no errors. If we roughly put v1 = 90◦ in (2.17) for t = 1250 ± 450 = 800 ∼ 1700 (v1 = 97 ∼ 83◦ ), we have 1kh = f (e3 ) ≈ ±0.000005, k ≈ 2e, 1θh = 43 e cos K, | 1θh |max ≈ 0.7◦ ; (v1 = 90◦ ). (2.18) If we further put K = ±90◦ [Hipparchus, Ptolemy, al-Battānı̄ (v1 = 113.5◦ , 108.9◦ , 96.2◦ ), the apogee is also obtainable accurately: 1θ ≈ 0; (v1 = 90◦ , K = ±90). (2.19) Putting now in (2.17) only K = ±90◦ (Hipparchus, Ptolemy, al-Battānı̄) we obtain 1kh = ± 23 e2 sin 2v1 cos v1 1θh = ± 23 e sin2 v1 cos v1 . (2.20) For accurate observations the deviation of the eccentricity from 2e and the error of the apogee are bounded by an approximately quantity of ±0.0003 and of ±0.6◦ respectively. Thus, for ca. 1800 years between Hipparchus and Kepler the theoretical error of Hipparchus’ solar hypothesis had, as far as it exceeded the values above, nothing whatsoever to do with the well-known numerical errors in the historical records on the solar orbit (Fig. 2.2). The deviations of the two solar elements from the modern values must therefore be attributed to the problem of observations, immediate or corrected.

Fig. 2.2. The solar eccentricity as a function of time, t = −500 ∼ +2000 The theoretical deviation of Hipparchus’ solar hypothesis from the modern theory (cf. Tabs. 3.1., 4.1., Figs. 4.2–4.4) modern value 2e with the maximum theoretical deviations of Hipparchus’ determinational method as a function  of time  k = 2e − 23 e2 cos v1 cos(2v 1 + K) max,min ; K = 0 − 360◦ , cf. (2.17). If the observations and the corrections for solar parallax and refraction are accurate, the determined eccentricities are to be bounded within these two lines. ——— − ◦ ——— maximum theoretical deviations, 2e ± 23 e2 ; cf. (2.2), (2.17). historical records and our approximate reproductions by means of the historical observational data (Tab. 3.1., cols. o, p). Note that the reproduced solar eccentricities agree well with the recorded ones. Note also alBattānı̄’s outstanding accuracy, due to his accurate observations and his neglect of the solar parallax; another reason for his accuracy was that because of the high colatitude of his observing place, the effect of refraction was small. 3. Observations and the determination of the orbital elements: Derivation of the formulae of the elements as a function of observational errors After deducing the theoretical deviations of the solar elements determinable by Hipparchus’ hypothesis from the modern parameters, 1kh and 1θh , we can complete the formulae (2.8), (2.9) by simply retaining 1γ and 1δ as standing for observational errors only. From (2.8), (2.9) and (2.17) we obtain the total deviations from the modern parameters as functions of 1γ and 1δ to an approximate order of e2 for the eccentricity 1k

and, for simplicity, to the first order in e for the apogee 1θ(= 1A) : 1k = k − 2e 1 [−1γ cos(v1 + K) + 1δ cos v1 ] − 23 e2 cos v1 cos(2v1 + K) = sin K = f (1γ, 1δ), 1θ = θ0 − θ = 1A = A0 − A =− 1γ sin 2(v1 + K) + 21δ[sin K − sin(v1 + K) cos v1 ] 4e sin K cos(v1 + K) + 43 e cos(2v1 + K) [sin K − sin(v1 + K) cos v1 ] cos(v1 + K) = f (1γ, 1δ). (3.1) In the final formulae above the two terms 1γ and 1δ are now free from incorrectness of the adopted solar hypothesis and related solely to the quantities of observational errors: 1γ, 1δ = [quantities deduced from historical records according to (2.10)] minus γ0 , δ0 [theoretical quantities according to (2.13)]. (3.2) Thus, the total deviations in (3.1) consist of two parts: 1. errors of observations (1γ-, 1δ-term), 2. errors of the hypothesis[1kh (e2 -term), 1θh (the last e-term)]. If we put the observational errors according to (3.2) into 1γ and 1δ and the modern values for the true anomaly v1 (= 180 − A) at the vernal equinox t1 and the eccentricity e at the epoch, the Eqs. (3.1) give us the total deviations at the epoch, and by adding the modern parameters, we obtain the values corresponding to the attested solar elements. Tab. 3.1. shows the historical parameters and our reproduced values (cols. o/p, r/s). The agreement is reasonable, which means that our reproducing computation, in particular (3.1), is correct to the first approximation. If we put v1 = 90◦ for t = 830 (al-Ma’mūn) ∼ 1630 (Kepler) [v1 (= 180 − A) = 93 ∼ 83◦ ], we obtain a very appropriate orientation for explaining the historical records; the error in the most fundamental observation of the equinoxes is directly transformed into that in the eccentricity: 1k ≈ 1γ, (3.3) hence

According to this, all historical values of the Sun’s eccentricity should deviate from the double modern eccentricity 2e by the quantity of the observational errors of the equinoxes 1γ(>< 0; cf. chs. 4, 5). Likewise we have 1θ ≈ −1γ cos K + 1δ 3 + 4 e cos K, 2e sin K (v1 = 90◦ ). (3.5) Putting further K = ±90◦ we have 1θ ≈ ± 1δ , 2e (v1 ≈ 90◦ , K = ±90◦ ). (3.6) Putting now only K = ±90◦ in (3.1) we have 1k = ±1γ sin v1 + 1δ cos v1 ± 23 e2 sin 2v1 cos v1 1θ = − 1γ cos v1 ∓ 1δ sin v1 ± 23 e sin2 v1 cos v1 2e (K = ±90◦ ). (3.7) As we shall see below, all historical records of the solar elements reveal their very characteristic errors in accordance with the tendencies shown above. Due to Hipparchus and Ptolemy the above Eqs. (3.7) deserve special attention. With K = +90◦ (Hipparchus, Ptolemy, al-Battānı̄) we put (3.7) as consisting of two terms relating to observations (1γ, 1δ) and the hypothesis (e2 , e) respectively: 1k = (1obs)k + (1hyp)k 1θ = 1A = (1obs)A + (1hyp)A (3.8) and for simplicity we put |1γ| = |1δ|. We then obtain √ (sin v1 ± cos v1 )max,min = ± 2 at v1 = ±π/4 + π √ (sin 2v1 cos v1 )max,min = ±1/ 2 at v1 = ±π/4 + π (sin2 v1 cos v1 )max,min = ±0.385 at v1 = ±54.7◦ + π. (3.9) With (3.9) we obtain small maximum errors arising from the hypothesis: 3 (1hyp)k,max,min = ± √ e2 = ±0.0003 2 2 (1hyp)A,max,min ≈ ±0.6◦ Neglecting these small error sources and putting simply |1γ| = |1δ| = 1 we may describe

Table 3.1. Computational deduction of the historical solar elements - eccentricity and apogee - by means of the modern parameters. a b c d e f g h i Length of the seasons recorded t K (◦ ) t3 − t1 dhm d modern t2 − t1 dhm d t3 − t1 d t2 − t1 d 94 1/2 186.3661 94 1/2 recorded – modern 1(t3 − t1 ) d h 1(t2 − t1 ) d h 94.0154 +0.6339 +15.213 +0.4846 +11.630 186.4622 93.9039 +0.5378 +12.908 +0.5961 +14.307 1 Hipparchus Alm.III, 4 c. −130 90 187 2 Ptolemy Alm. III, 4 c.+140 90 187 3 al-Battānı̄ Op. ast. XXVIII 882/883 90 186 14 45 186.6146 93 14 93.5833 186.5899 93.5166 +0.0247 +0.593 +0.0668 +1.603 4 Copernicus De rev. III,16 1515 225 186 8 36 186.3583 231 15 0 231.6250 186.5407 231.7199 −0.1823 −4.376 −0.0949 −2.277 5 Tycho Op. om. II, 19–24 1583 45 1588 45 186 18 41 186.7785 186 18 30 186.7708 46 2 48 46.1167 46 2 55 46.1215 186.5269 46.0727 186.5258 46.0704 +0.2516 +6.038 +0.2450 +5.881 +0.0440 +1.056 +0.0511 +1.226 1588 135 „ 140 8 50 140.3681 „ 140.1665 „ „ +0.2015 +4.837 6 Remarks 3-d, -e: ca. Maeyama (1988), Figs. 1.1–1.7 1−, 2−d, −e: = Geminus, Manitius, 8, cf. also 210ff. col. c: solar longitudes at t2 . 4-d: 365 41 (III, 13) −178;53,30 (III, 16). For the notorious value 186;5 21 cf. e.g. Swerdlow-Neugebauer, 172. 4-e: (4-d) +45;16. cols. j, k : 1(t3 − t1 )ω/2, 1γ − 1(t2 − t1 )ω. Note a generally good agreement between the historical records and our reproduced values (cols. o/p, r /s). √ 1kmax,min ≈ ± 21, 1(rad.) 1 1Amax,min ≈ ± √ ≈ ±401, 2e 1(◦ ). (3.11) Both extreme errors are therefore proportional to the underlying observational errors, 1. As we shall see below, both observational quantities, 1γ and 1δ, roughly correspond to individual observational errors: 1γ ≈ −1t1 ω ≈ 1t3 ω ≈ −1l1 ≈ 1l3 1δ ≈ −1t2 ω ≈ −1l2 [cf. (4.1), (4.2)]. Putting therefore 1 ≈ |1t1,2,3 |ω ≈ |1l1,2,3 | for simplification, we obtain approximate numerical connections between observational errors and the resulting errors of the

Table 3.1. (Continued) j k l m n o p q r modern computed s Apogee (◦ ) Eccentricity recorded computed recorded 1γ (◦ ) 1δ (◦ ) A (◦ ) e 1k k +0.312 −0.165 66.527 0.017558 +0.006284 0.041400 2;30 0.041667 −1.278 65.249 65;30 65.5 +0.265 −0.323 71.124 0.017457 +0.006289 0.041203 2;29,30 0.041528 −6.719 64.405 „ −1.672 82.109 82;17 82.283 96 2/3 96.667 k p 1θ A0 A0 +0.012 −0.054 83.782 0.017169 +0.000323 0.034661 2;4,45 0.034653 −0.090 +0.004 94.615 0.016912 −0.001472 0.032351 0.0323 +2.122 96.737 +0.1240 +0.0806 95.780 0.016883 +0.002174 0.035941 0.0359194 −0.114 95.667 +0.1208 +0.0704 95.866 0.016881 +0.002094 0.035857 0.0358416 −0.351 95.515 95;44 95.733 95;30 95.5 „ −0.0769 „ „ +0.002090 0.035853 0.0358388 −0.414 95.453 95;27 95.45 (3.1) = 2e + 1k 1-p: ca. Conn. d. temps Expl. suppl. (3.1) = A + 1θ dependent parameters. Through a combination of observational errors of ±1/4d and ±1h e.g. the error in the location of the apogee at the time of Hipparchus and Ptolemy can attain extreme values of ±9◦ and ±1.5◦ (Tab. 3.2, 2-h, -i, 1-h, -i). Only the maximum attainable errors are shown above. As we can easily see in (3.7), the actual errors can vary greatly within these limits depending on the combination of and variation in observational errors.3.1 4. Errors in the underlying observations and the orbital elements of the Sun In the preceding section our computation reproduced the historical records of the solar elements to a sufficient order of accuracy by taking into account only the errors in three observations necessary for their deduction. It follows that all other error sources such as year length and the motion of the perigee are of minor magnitudes and the three solar longitudes recorded as observed are solely responsible for those attested orbital

Table 3.2. Observational errors and the resulting maximum attainable errors in the dependent parameters (eccentricity and apogee) a b |(d ) 1 ≈ |1t1,2,3 ≈ |1l1,2,3 |(◦ ) 1 2 3 4 1/24 = 1h 0.04 1/4 0.25 1 1 v1 (= 180◦ − A) in (3.7) with K = +90◦ c d |1k|max e |1A|max (◦ ) 0.001 1.7 0.006 10 0.025 40 ±π/4 + π f |1k|max g at t = h i j |1A|max (◦ ) at t = −130 +140 +880 −130 +140 +880 0.001 0.006 0.023 113.5◦ 0.0009 0.006 0.022 108.9◦ 0.0008 0.005 0.019 96.2◦ 1.6 9 38 113.5◦ 1.5 9 36 108.9◦ 1.3 8 31 96.2◦ elements of the Sun. Our final problem is thus to search for the origins of the true solar longitudes as recorded by Ptolemy, al-Battānı̄ etc. Since we may now consider the two terms 1γ and 1δ in (3.1) as resulting only from the errors in the observed times, 1tn = tn0 − tn (n = 1, 2, 3), we obtain from (2.10) 1γ = 21 (1t3 − 1t1 )ω = 21 (1L3 − 1L1 ) = 21 [1l3 − 2e1v3 cos v3 + . . . − (1l1 − 2e1v1 cos v1 + . . .)] ≈ 21 (1l3 − 1l1 ) ≈ 1l3 ≈ −1l1 , 1δ = 1γ − (1t2 − 1t1 )ω ≈ 21 (1l3 + 1l1 ) − 1l2 ≈ −1l2 . (4.1) (4.2) Here the only error sources 1γ and 1δ are directly connected with the longitudinal errors, 1l1−3 .4.1 We shall analyse our problem by following Tycho’s rigorous computation and clear demonstration. Our main concern below is with the deduction of Tycho’s longitudinal errors–the deviation of the Sun’s positions in the orbit l 0 , which he actually observed, from the longitudes l which he sought to determine: 1ln = ln0 − ln ; n = 1, 2, 3. Tycho determined his three solar longitudes ln first by measuring the Sun’s meridional altitudes, which was virtually the only observation to be made, and then by converting them into longitudes with his parameters (Fig. 4.1). His longitude l is thus given by his parameters as sin dT , (4.3) sin ε0 where he deduced the declination dT from his immediately observed solar altitude at the meridian hob corrected for refraction and solar parallax: sin l = dT = hob + p 0 cos h − r 0 − ϕ̄0 . (4.4) The modern theoretical declination at this instant is given by d 0 = hob + p cos h − r cot h − ϕ̄.

Meridian a(hlt0itu)de Tycho's aim Tycho's aim the Sun T — h Ad] — hob (modern) the apparent Sun, Tycho's immediate observation Ì in the orbit ob d' NK id 9 X A é ¿> 2 e. A x, > a 4 (Tycho) SS a 5 % Right ascension

Putting this into (2) we derive Tycho’s longitudinal error 1l as a function of his erroneous parameters. However, it is convenient to consider the total error 1l as consisting of two terms, the 1dand the 1ε-term [cf. (4.6)]. Note: By our description above Tycho’s solar observations can be followed accurately, an indication that Tycho strived for obtaining his solar position, l, with a high precision (cf. Tab. 4.1., 5-q, -r). The same quantity 1l is also obtainable to the first approximation by simply adding (dT − d 0 ) to the modern declination d as indicated by „- -•- -“ [cf. Maeyama (1974), 39, (15)]. For a general application it is convenient to express longitudinal error 1l approximately as a function of two error sources, 1d and 1ε. Since the former consists of three error sources, Tycho’s errors in the determination of the solar orbit all originate from the following four sources (1ϕ − 1ε):4.2 declination (1d) geographical latitude (1ϕ) solar parallax (1p) refraction (r 0 − r cot h) obliquity of the ecliptic (1ε) 1d cos d − 1ε sin d cot ε 1l = l 0 − l ∼ =− cos l sin ε = f (1d, 1ε), 1d = d 0 − d = 1p cos h − rh0 + r cot h − 1ϕ̄, hob ∼ = 6000 at h > 10◦ , =h∼ = hT ∼ = h0 for cos h, cot h; r 0 at h; r ∼ 1p = 30 − 900 = 0.048◦ , 1ln = ln0 − ln ; n = 1, 2, 3; (cf. Fig. 4.1). (4.6) Putting the longitudinal errors 1l1–3 obtainable from (4.6) into (4.1)–(4.2) we can derive from (3.1) the deviations 1k and 1θ(=1A) and, by means of the modern parameters at the epoch, we finally arrive at the solar orbital elements corresponding to the historical records. Our computed values agree closely with Tycho’s parameters (Tab. 4.1., 5-q, -r). It follows that the above series of Eqs (3.1) can be assumed to be an adequate mathematical expression which transmitts, simply and accurately, Tycho’s actual procedure of deducing the solar orbital elements. Three of the above four error sources, namely the three contributions to the error in declination, are in principle those astronomical quantities which should be, but were not necessarily, determined as independent parameters. This is a long complex problem.4.3 According to the above equations all these three error sources of declination involve similar contributions to the longitudinal error 1l: cos d . 1ld ∼ cos l

Hence, their individual contributions at the equinoxes are of the same order in absolute values: 1l ∼ = ∓2.51d ∼ = ∓2.5×[1p cos h; −(rh0 −r cot h); −1ϕ̄]; l1,3 = 0, 180◦ , d = 0. (4.8) The obliquity of the ecliptic can be determined by observation virtually in three ways. From the two observable, extreme altitudes of the Sun, hmax,min hmax = ϕ̄ + ε − p cos hmax + r cot hmax , hmin = ϕ̄ − ε − p cos hmin + r cot hmin , (4.9) we obtain ε01 = 21 [(hmax + p 0 cos hmax − rh0 max ) − (hmin + p 0 cos hmin − rh0 min )], ε02 = (hmax + p 0 cos hmax − rh0 max ) − ϕ̄, ε03 = ϕ̄ − (hmin + p 0 cos hmin − rh0 min ), hence, the originating errors 1ε1 = 21 [1p(cos hmax − cos hmin ) − rh0 max + rh0 min + r(cot hmax − cot hmin )], 1ε2 = 1p cos hmax − rh0 max + r cot hmax − 1ϕ̄, 1ε3 = 1ϕ̄ − 1p cos hmin + rh0 min − r cot hmin . (4.10) Likewise, if one determines the colatitude ϕ̄ by means of the Sun’s observation as ϕ̄01 = 21 [hmax + hmin + p 0 (cos hmax + cos hmin ) − rh0 max − rh0 min ], ϕ̄02 = hmax − ε0 + p 0 cos hmax − rh0 max , ϕ̄03 = hmin + ε0 + p 0 cos hmin − rh0 min , we obtain its errors, similarly to the derivation above, as dependent on that body: 1ϕ̄1 = 21 1p(cos hmax + cos hmin ) − rh0 max − rh0 min + r(cot hmax + cot hmin )], 1ϕ̄2 = −1ε + 1p cos hmax − rh0 max + r cot hmax , 1ϕ̄3 = +1ε + 1p cos hmin − rh0 min + r cot hmin . (4.11) Although refraction as an atmospheric phenomenon was known in antiquity and Islam and later in the west, it seems that its values for astronomical use, as a function of altitude, were first determined systematically by Tycho.4.4 Another particularly important parameter was the solar parallax. Ptolemy’s original mean value of this parallax,

Table 4.1. Computational deduction of the historically recorded lengths of the seasons by means of the underlying parameters, historically recorded and modern (cf. Tabs. 3.1., 5.1., 5.2., Figs. 4.1, 4.2) a b c d e ◦ K (◦ ) g h ◦ Colatitude ( ) recorded f Solar altitude ( ) modern t1,3 t2 i Refraction ϕ̄0 ϕ̄ 1ϕ̄ h1,3 h2 0 r1,3 r20 1 2 Hipparchus Ptolemy 90 90 c. 54 59;2 59.033 58.800 +0.233 58.800 82.478 0 0 3 al-Battānı̄ 90 53;59 53.983 54.074 −0.090 54.074 77.658 0 0 4 Copernicus 225 35;40,30 35.675 35.641 +0.034 35.641 19.261 0 0 5 Tycho 45 34;5,30 34.092 34.093 −0.001 34.093 50.464 4500 0.013◦ 0 45 „ „ „ „ 135 „ „ „ „ 6 Remarks 3-c: Nallino II, 12, n.4. cols. j, k: mean horizontal parallax, p = 8.8000 , Expl. Suppl., 490. l . 1: Hipparchus, not deducible due to the lack of his data. 2-p, -r, 3-p, -r: Ptolemy/al-Battānı̄ not deducible from (4.6) because l2 = K = 90◦ . ls. 1–4, cols. o−r: p0 = 0 for all but Tycho; If p 0 = 2;51 − 30 applied, then 1(t3 − t1 )(d ) = −1.010 (Ptolemy), +0.5759 (al-Battānı̄), +0.1394 (Copernicus); 1(t2 − t1 ) = +0.2066 (Copernicus), thus the deviations larger than the results given above for p0 = 0; cf. Tycho +2.888 and +2. 210h (5-q, -r) for p0 = 0. Note the historical records [( ),cols. q, r], reproducible roughly (Copernicus), accurately (Tycho with p 0 = 30 ) and not reproducible (Ptolemy and al-Battānı̄). In the latter case their recorded parameters contain problems (see text). Hence, Tycho’s observational behaviour can accurately be known.

Table 4.1. (Continued) j k p 0 (0 ) 2;51 c. 3 3 3 1p (◦ ) m l Solar parallax n o ε0 ε 1ε p Longitude (◦ ) Obliquity of the ecliptic (◦ ) 1l1,3 1l2 q r Lengths of the seasons (recorded - modern) 1(t3 − t1 ) d h (h) 23;51,20 +0.045 23.856 23.678 +0.177 ±0.559 −1.134 −27.2 (+12.9) 23;35 +0.048 23.583 +0.427 +10.251 (+0.593) 23.584 −0.001 ∓0.211 1(t2 − t1 ) d h (h) „ 23;28,30 23.475 23.502 −0.027 ±0.033 −0.026 −0.067 −1.617 (−4.38) „ 23;31,30 23.525 23.492 +0.033 ∓0.132 −0.077 +0.2676 +0.0558 +6.422 +1.339 (+6.038) (+1.056) „ „ „ +0.077 (4.6) „ −0.060 −1.437 (−2.28) „ (+5.881) (+1.226) („ ) +0.2118 +5.084 (+4.837) col. q := 2(−1l1 )/ω col. r := (1l2 − 1l1 )/ω

Fig. 4.2. The time interval between two equinoxes (l3 − l1 = 180◦ ); historical measurements and modern values as functions of time ◦ ——— ----- • Hipparchus/Ptolemy ——— al-Battānı̄ modern values t = −500 ∼ +2000 (Tabs. 3.1., col. h; 4.1, col. q; 5.1; Figs. 4.3–4.5, 5.4–5.6) historical records and our reproducing computations. our different reproductions – 1 = +13.82h (al-Battānı̄), +3.35h (Copernicus), +2.89h (Tycho), cf. Tab. 4.1., 6 Remarks – they do not agree with the records, hence they are not valid. computed with Copernicus’ data, 186;5 21 , not valid (Tab. 3.1., 4-d, 6). the historical record (187d ) with the error-allowance ±0.5d , cf. text. our tentative computation with ϕ̄02 = 54.077◦ [(4.11)], p 0 = 0; 1(t3 −t1 ) = +0.034d = +0.81h , but see the most plausible explanation in ch. 5.2, Fig. 5.3. Maeyama (1988), Fig. 1.4. Note al-Battānı̄’s and Tycho’s high accuracy of the immediate observations and their accurate and inaccurate values due to the neglect and application of the erroneous solar parallax of 30 respectively (cf. ch. 5). 20 5100 (Alm. V, 18; H 442–3), remained virtually unaltered up to the time of Kepler. All these parameters are directly connected with the determination of the lengths of the seasons, as shown in Fig. 4.2. Their great influence on the determination of the solar elements will be treated separately in the next chapter. For a great time span, t = 1250 ± 450 ∼ = 800 ∼ 1700, during which the position of apogee remained at a nearly right angle to the vernal equinox, we approximate the true anomaly of our first longitude (l1 = 0) by v1 ∼ = 90◦ and obtain from (3.1), (4.1)

[1γ(sin K − 43 e cos2 K) + 43 1δe cos K] + 1kh sin K   1l3 − 1l1 1t3 − 1t1 ∼ ∼ ω∼ = −1l1 = 1l3 ∼ = = 1γ = = 2.51d, 2 2 1k ∼ = (v1 = 90◦ ), 1θ ∼ = (4.12) 1 (−1γ cos K + 1δ) + 43 e cos K, 2e sin K (v1 = 90◦ ), 1l2 1δ (4.13) =− (v1 , K = 90◦ ). 2e 2e (4.12) shows a remarkably simple relationship, consisting of proportionalities, between observation [time (t) and position (l)] and dependent parameter (eccentricity, k). This only becomes visible if their quantities are evaluated as deviating from the modern values at the epoch, as shown above. The deviation in time and position (1t, 1l) comes from the one in declination (1d) which ultimately originates from the errors in a few independent parameters (r 0 , p0 , ϕ0 ) [cf. (4.6)–(4.8)]. Figs. 4.3–4.5 illustrate the simple proportionalities just mentioned. If we now put in (4.6) roughly 1ϕ̄ = 0, 1p ∼ = p0 , we obtain = 0 −1r1,3 + p cos ϕ̄ 1l1 = −1l3 ∼ (l1 = 0, l3 = 180◦ ; 1r1 = 1r3 = r 0 − r cot ϕ̄), =− sin ε (4.14) hence, we roughly have from (3.1) −1r1,3 + p 0 cos ϕ̄ ∼ p0 cos ϕ̄ ∼ 0 1k = k − 2e ∼ = 1γ ∼ = −1l1 ∼ = = 2p , = 1l3 ∼ = sin ε sin ε (4.15) (roughly for ϕ̄ ∼ = 35◦ ; e.g. Copernicus and Tycho). Consequently, those who took the solar parallax seriously into account had an inevitable tendency to obtain too great an eccentricity: k∼ = 2e + 2p 0 ∼ = 2e + 0.00175 ∼ = 0.0369(t = −130) ∼ 0.0361(+880) ∼ 0.0355(+1600) > 2e, (p0 ∼ = 30 ). (4.16) or, bisecting the distance between the central body and the reference point for the uniform angular motion and denoting it as e0 e0 ∼ = e + p0 ∼ = e + 0.0009 ∼ = 0.0178 (t ∼

Fig. 4.3. The error of the time interval between two equinoxes as a function of the error of declination (cf. Figs. 4.2, 4.4, 4.5) The error of the time interval between two equinoxes changes proportionally to the error in declination and their relationship can simply be expressed as 1(t3 − t1 ) = 2(−1l1 )/ω ≈ 51d1 /ω(d ), (1) 0 hence roughly, e.g., for 1d( ) ≈ 21d1,3 (h ), cf. (4.8). (2) Therefore each minute of an arc in declination yields an error of 2 hours in the length between the equinoxes. ◦ historical records - modern values relating to our computed values of 1d1 [(4.6)]; +0.400 (al-Battānı̄, our tentative computation, cf. Fig. 4.2, esp. ch. 5.2), −0.790 (Copernicus), +3.160 (Tycho). -◦our reproductions [(4.6)–(4.8) and (1), (2) above]. • Copernicus’ value computed with p0 = 30 ; 1d = +1.640 (not valid). Note the high validity of the simple correspondence, 10 (1d) ∼ 2h [1(t3 − t1 )]. (4.15)–(4.17) show that the eccentricity of the solar orbit to be determined from two equinoctial observations at roughly t = 1250 ± 450 ∼ = 800 ∼ 1700 exceeds the modern value by the value adopted for the solar parallax. Thus, the above eccentricities correspond precisely to e.g. the well-known value of Tycho (0.03584) and to its rounded and bisected one of Kepler (0.018). This has already been shown elsewhere.4.6 Since the values adopted for the solar parallax decreased because of its new determinations as an independent parameter, the determined eccentricity necessarily decreased

Fig. 4.4. Deviation of the solar eccentricities (dependent parameter) from the modern values as a function of the error in the time interval between two equinoxes (observations); cf. Figs. 4.2, 4.3, 4.5 ◦ —— historical records - modern double eccentricity (2e) 1k ≈ −1l1 = 1l3 ≈ 1(t3 − t1 )ω/2 [cf. (4.12), A, v1 = 90◦ ], e.g. here 1k = 0.0003584 · 1(t3 − t1 ). Hence, an error of ±1h in the time interval between two equinoxes (observations) changes the solar eccentricity k (dependent parameter) by ±0.00036. Note: At A, v ≈ 90◦ [cf. (4.12)] the deviation 1k is proportional to the error of the underlying time interval between two equinoxes, hence roughly applicable for t = 1250 ± 450 = 800 ∼ 1700. too [(4.15)–(4.17)]. This actually began to occur at around Kepler’s death (1630) and, as evident from the equations, the solar eccentricity then converged rapidly towards the modern value.4.7 Contrary to this later historical process, those astronomers who neglected the solar parallax in their computation were – independently of the epochs due to the actual small amount p = 8.800 ≈ 0 – close to obtaining an accurate eccentricity corresponding to the modern value. An outstanding example was al- Battānı̄ (cf. Figs. 2.2, 4.2–4.5; ch. 5.2). It may be of interest to present our greatly simplified computation. Once again we put v1 = 90◦ (= 180 − A) in (3.1) for the whole time interval between Hipparchus

cu Pua [= ++ $2 cos S 8 + + u o tua do ao <d 4 sò dv Tycho ner +0.002- o +0.001- ai-Battant . Error in declination -0.02 0.01 A 0.01 4 -0.001- Copernicus q 0.0027 0.02 0.03 0.04 2 0.05 (9) Ad4=Ads

Table 4.2. Approximate solar elements computed from (4.18); cf. Table 3.1. K(◦ ) Hipparchus Ptolemy al-Battānı̄ Copernicus Tycho Remarks 1A = 1θ(◦ ) k0 1k 90 +0.005445 0.04056 −4.70 [+0.006284] (0.04167) [−1.28] 90 +0.004625 0.03954 −9.25 [+0.006289] (0.04153) [−6.72] 90 +0.000209 0.03455 −1.57 [+0.000323] (0.03465) [−1.67] 225 −0.001571 0.03225 +1.98 [−0.001472] (0.0323) [+2.12] 45 +0.002108 0.03587 −0.116 [+0.002094] (0.03584) [−0.351] 135 +0.002108 0.03587 −0.156 [+0.002090] (0.03584) [−0.414] [ ]: accurately computed from (3.1),Table 3.1., cols. n, q ( ): recorded values, Table 3.1., cols. p, s. A00 (◦ ) 61.83 (65.5) 61.87 (65.5) 82.21 (82.28) 96.59 (96.67) 95.75 (95.5) 95.71 (95.45) v1 = 90◦ in (3.1); cf. (3.3) – (3.6), (4.12), (4.13) 1(t3 − t1 ) ω; (4.1), Tab. 3.1., cols. h, j 2   1(t3 − t1 ) − 1(t2 − t1 ) ω ≈ −1t2 ω ≈ −1l2 ; (4.2), 1δ = 2 1γ = Tab. 3.1., cols. h, i, k et , At : modern values at the epoch t.4.8 (4.18) Tab. 4.2. shows the high accuracy of our much simplified Eqs. (4.18) in the case of al-Battānı̄, Copernicus and Tycho; this accuracy is due to the good approximation of v1 ≈ A ≈ 90◦ . If we, once again, simplify our equations above and put v1 ≈ A ≈ 90◦ , K = ±90◦ in (4.12), (4.13), (4.18), we obtain numerically simple expressions: 1(t3 − t1 ) π · 2 180 1t2 ≈ ∓30 · 1t2 (◦ ). 1A ≈ ∓ 2e 1k ≈ In the case of al-Battānı̄ we then derive 1k ≈ 0.0247 π · = +0.00022 2 180 k 0 ≈ 0.034554

1A ≈ −30[0.0668 + (−0.0247/2)] ≈ −1.63◦ A0 ≈ 82.15◦ (82.109◦ ), (−1.672◦ ) (cf. Tab. 3.1., l.3), agreeing with the accurately computed values in parentheses. Consequently, the deviation of the eccentricity from 2e and the error of the apogee are approximately equal to the error in the time-determination of the equinoxes, 1t1 ≈ −1t3 , and to 30 times the error of the solstices respectively. 5. Explanatory remarks on the underlying observations of the Sun Our computation has shown that the solar elements of all astronomers under consideration can be reproduced accurately by means of the modern values if we take the underlying observations, the lengths of the seasons, into account (Tab. 3.1., cols. o/p, r/s). There are, however, inconsistencies between observations and some independent parameters, in particular, in the case of Ptolemy and al-Battānı̄. The outstanding example is the geographic latitude, which is supposed to be one of the most basic parameters in any observation, and yet the recorded geographic latitudes do not agree with the actual observations of the lengths of the seasons. We claim, however, that this problem of apparent inconsistencies can be solved to a certain extent. For this purpose we shall first make our preliminary study using Tycho’s observations. In the years 1584–1588 Tycho rigorously observed the instants of the equinoxes, vernal and autumnal, and obtained a constant interval between the two equinoxes (Op. om. II,15). Our comparison of his observations with Tuckerman’s Tables (Tab. 5.1.) clearly shows, as expected, that the total error of Tycho’s time interval between the two equinoxes consists of two observational errors of equal magnitude (cols. d, g, note), and that therefore his observations at the two equinoxes were made under the same condition. Assuming that this argument is applicable in the case of other astronomers as well as in Tycho’s case, we can deduce approximate errors at the equinoxes in time and position from the lengths of the seasons recorded by these astronomers: 1t1,3 = ∓ 1(t3 − t1 ) , 2 1l1,3 ≈ ∓ 1(t3 − t1 ) ω, 2 and hence: 1t2 = 1(t2 − t1 ) + 1t1 (d ), 1l2 ≈ [1(t2 − t1 ) + 1t1 ]ω(◦ ).

Table 5.1. Tycho’s determination of the vernal and autumnal equinoxes, 1584–1588 (Op. om., II, 15). a b c d e Vernal equinox t1 f modern dhm d Tycho - modern h Autumnal equinox 1t1 Tycho g t3 Tycho 1t3 modern Tycho |1t1 | + 1t3 - modern 1 1584 9 21 30 9.8958 d dhm h −0.1318 12 16 0 10.0276 −3.162 12.6667 2 1585 10 3 19 10.1382 −0.1281 10.2663 −3.074 12 21 49 +0.1275 12.9090 12.7815 +3.060 +0.2556 +6.134 3 1586 10 9 8 10.3806 −0.1329 10.5134 −3.189 13 3 38 13.1514 +0.1212 13.0302 +2.908 +0.2540 +6.096 4 1587 10 14 56 −0.1299 10.6222 10.7521 −3.117 13 9 26 13.3931 +0.1334 13.2597 +3.202 +0.2633 +6.318 5 1588 9 20 45 9.8646 12 15 15 +0.1271 12.6354 12.5083 +3.051 +0.2580 +6.191 6 Remarks 1̄t1 = −0.1307d = −3.136h σ = 0.0016 = 0.039 9.9954 −0.1308 −3.140 d d d h h +0.1262 +0.2580 12.5404 +3.029 +6.192 1̄t3 = +0.1271d = +3.050h σ = 0.0039 = 0.094 |1t1 | + 1t3 = +0.2578d = +6.186h |1t1 | − 1t3 = +0.086h = +5.2m Tycho’s equinoctial determinations (the immediate observations, corrected for parallax, refraction and equation of time and interpolated) are accurately reproducible by means of his astronomical data (ϕ0 , p0 , r 0 ) and the modern parameters [cf. (4.6)]. Unlike his length of the tropical year his time interval from the vernal to the autumnal equinox is thus systematically too long by ca. 6h (Figs. 4.2–4.5, 5.4–5.6): Tycho–modern values: |1t1 | + 1t3 = +6.19h (av., 6 Remarks):Tuckerman 1(t3 − t1 ) = +5.88 (Tab. 3.1., 5-h): Maeyama (1988), Fig. 1.4 2(∓1l1,3 )/ω = +6.42 (Tab. 4.1., 5-q): Computed from (4.6) The small differences in the above values are primarily due to the neglect of perturbations in the apparent solar motion [ca. 30m max(= 15m × 2); cf. Maeyama (1988), 5]. Tycho’s error is therefore mainly caused by his adoption of the traditional solar parallax too great by 1p 0 ≈ 30 [cf. Maeyama (1974), passim, esp. 51]. The corrections to Tuckerman’s tables at Tycho’s time are small and were here omitted [cf. Britton’s study (1992), 17, 153–178]. Note: Tycho’s determination of two equinoxes shows systematic errors of the same amount [|1t1 | ≈ 1t3 ≈ 1(t3 − t1 )/2 ≈ 3.1h (cols. d, g, h)] indicating that his solar observations were equally conditioned at those two instants. Putting 1l into (4.6) we obtain the error of the colatitude 1ϕ̄, on which the whole observational operations are supposed to have been based. With 1l1,3 we may deduce a simple equation giving the relevant value, namely the error of the meridian altitude at which the equinoctial observations of the Sun were actually made:

Table 5.2. The colatitudes (= 90◦ -geographical latitudes), practically used and recorded (cf. Tab. 5.1.) a b c Errors of colatitude (1ϕ̄) d e p0 , r 0 Remarks practically used recorded 1 Hipparchus −0.115◦ −6.900 uncertain 2 Ptolemy −0.098 −5.86 +0.233◦ +140 0 3 al-Battānı̄ 4 +0.006 +0.35 −0.074 −0.090 −4.42 −5.42 0 +0.034 +2.07 5 Copernicus +0.057 6 +0.026 7 av +0.042 +3.43 +1.55 +2.49 8 Tycho 9 1588 10 11 +0.0034 +0.205 +12.300 +0.0026 +0.154 +9.2 −0.0014 −0.084 −5.0 av +0.0015 +0.091 +5.5 12 Remarks Deduced from the recorded lengths 1-c:ϕ, ϕ of the seasons [from (5.2); ls. 6,9, 10 from (4.6) and (5.1)]. 1−, 2−b: A striking agreement with the systematic error in declination, ca. −60 ∼ −70 , as found by Fortheringham, 412 ff. and Britton (1992), 18–24. −0.067 −400 −0.0011 0 ≈ 36◦ 0 ϕ0 = 36◦ ϕ0 = 36◦ 10 ; Op. astr. II, 12 0 K = 225◦ p0 = 30 0 = 4500 K = 45◦ r1,3 r20 = 0 K = 135◦ ls 1–7: p0 = 20 5100 ∼ 30 not applied, cf. text. For r 0 (ls. 8–11) cf. Op. om. II, 64. Note the great discrepancies between the observing conditions (col. b) and the recorded values (col. c) in the case of Ptolemy and al-Battānı̄ (ls. 2–4, cols. b–c) and Tycho’s excellent agreement; −5 ∼ +1200 versus −400 (ls. 8–11, cols. b–c); cf. text. 1ϕ̄ = − 1(t3 − t1 ) ω sin ε + 1p cos ϕ̄ − r 0 + r cot ϕ̄. 2 (5.2) Our computation shows Tycho’s marvellous agreement between the descriptions (−400 ) and the practice of observations ( −5 ∼ +1200 ; Tab. 5.2., ls. 8–11, cols. b-c). Based on the results derived above we shall now analyse the individual observational conditions. 5.1. Hipparchus-Ptolemy (ca.−130/+140) Since we have more general data about Ptolemy than about Hipparchus, we shall first try to clarify the problem in Ptolemy’s case. In principle we presuppose that Ptolemy actually did what he said with regard to his observations. His fundamental parameters for ϕ and ε were deduced from his observations of two 0 extreme zenith-distances of the Sun, zmax,min : 0 ) = 30◦ 580 , 0 ϕ0 = 21 (zmax + zmin (Alm V, 13; H 407, 409), 0 0 − zmin = 47◦ 400 ∼ 47◦ 450 , (Alm I, 12; H 68), 2ε0 = zmax

from which we have 0 = 54◦ 480 ∼ 54◦ 50.50 , zmax 0 = 7◦ 80 ∼ 7◦ 5.50 . zmin Now, from (5.1) and the lengths of the seasons measured by Hipparchus which Ptolemy claimed to have reconfirmed by his own observations,5.1 we may deduce 1t1,3 = ∓0.2689d = ∓6.45h 1t2 = +0.3272 = +7.85, (Tab. 3.1., 2-h,-i), hence, from (5.2) we obtain the colatitude, the meridian altitude of the Sun which he observed at the equinoxes 1ϕ̄ = −5.860 ≈ −60 , (Tab. 5.2., 2-b). Since the zenith-distance of his equator gave him his geographic latitude ϕ0 = 30◦ 580 (ϕ = 31◦ 120 ; 1ϕ = −140 ), and we can put ϕ0 = ϕ − 1z − 1ϕ̄, his zenith must have been inclined to the south by the amount (which we will call Ptolemy’s error of zenith): 1z = +140 (−1ϕ; his error) + 60 (−1ϕ; deduced from his lengths of the seasons) = +200 . (5.3) His extreme zenith-distances of the Sun corrected for this error will then be 00 = 7◦ 80 ∼ 7◦ 5.50 + 200 = 7◦ 280 ∼ 7◦ 25.50 zmin 00 zmax = 54◦ 480 ∼ 54◦ 50.50 + 200 = 55◦ 80 ∼ 55◦ 10.50 . Comparing these values to the modern ones deducible from (4.9) zmin = 7◦ 31.20 zmax = 54◦ 51.40 ,

Fig. 5.1. Ptolemy’s meridional observation of the Sun (simplified and not to scale) An approximative reproduction of Ptolemy’s meridional observation of the sun based upon Hipparchus’ lengths of the seasons as reconfirmed by Ptolemy and his two parameters, ϕ0 and ε0 , 0 arranged by me such that the errors of his two supposed immediate observations, zmin and h0min , 0 are of the same magnitude, ±3 . Note: Assuming Ptolemy’s error of the zenith-position to be ca. 200 and that of the immediate observations ca. 50 max., we can reproduce all his solar determinations: lengths of the seasons (eccentricity and position of the apogee), 2ε0 = 47◦ 400 ∼ 47◦ 450 and ϕ0 = 30◦ 580 (cf. ch. 5.1). we obtain the large deviations 1zmin ≈ −30 ∼ −60 1zmax ≈ +170 ∼ +190 . However, if he measured the minimum meridian altitude instead of the maximum zenithdistance of the Sun, we would obtain h0min = 90 − (54◦ 480 ∼ 54◦ 50.50 ) = 35◦ 120 ∼ 35◦ 9.50 , which implies a small deviation from the modern value 1hmin ≈ +30 ∼ +10 . This would mean that if he measured the solar distance at two solstices from the zenith and the horizon, then we ought to be able – assuming that the observations contain the same amount of errors ±30 – to obtain a series of figures in agreement with all his numerical records (Fig. 5.1):

two observations at the solstices and the resulting parameters: 0 = 7◦ 80 [+200 (error of zenith) = 7◦ 280 ] ; 1zmin ≈ −30 zmin 0 hmin = 35◦ 120 ; 1hmin ≈ +30 0 0 0 ◦ ◦ 0 2ε = (90 − hmin ) − zmin = 47 40 ; 1 ≈ +200 0 ϕ0 = [(90 − h0min ) + zmin ]/2 = 30◦ 580 ; 1 ≈ −140 . (5.4) None of the above values contradicts his allegedly reconfirmed lengths of the seasons – correct to 0.6d (Tab. 3.1., 2-h, -i) – or his observational conditions which we deduced by means of these lengths (the meridian altitude of his equator, 1ϕ̄ = −60 ; Tab. 5.2., 2-b). His error of the zenith as deduced by us, 200 , is large indeed, however, so is the error of the maximum difference of the solar altitudes 2ε (c. 200 ) as well as the error of the geographic latitude (140 ). In fact, our computation shows that if he had related his geographic latitude to the correct zenith he would have obtained entirely different lengths of the seasons, 1(t3 − t1 ) = −1.13d (Tab. 4.1., 2-q) instead of +0.54d (Tab. 3.1., 2-h). In that case he should also have found the solar elements to be entirely different from his records. By means of the modern values and his errors Ptolemy’s obliquity of the ecliptic is approximately given as ε0 = 21 [2ε − 1zmin (−30 ) − 1hmin (+30 ) + error of zenith (200 )] = 23◦ 500 ; 1ε ≈ +100 , which is too large simply by half the error of zenith. Also his geographic latitude of Alexandria, ϕ0 = ϕ(31◦ 120 ) − error of zenith (200 ) − 1ϕ̄(−60 ) = 30◦ 580 , 1ϕ = −140 , (5.5) is too small by his error of zenith and of the meridian altitude of the instrumental equator [−200 − (−60 ) = −140 ]. It is particularly interesting that we find the same phenomenon in al-Battānı̄’s observations (ch. 5.2). Our description above may be offered in addition to several other possibilities hitherto claimed.5.2 The fact that Ptolemy, ca. 300 years later, still found no need to correct Hipparchus’ solar data has frequently been called into question.5.3 Due to the fraction 1/4d in the year’s length, of which no one was unaware in their time, Hipparchus and Ptolemy sought to attain at least that order of accuracy in their solar theories. The time interval, (t3 − t1 ) and (t2 − t1 ), can therefore contain roughly the double order of accuracy, ±0.5d . Within that range Ptolemy seems to have found that Hipparchus’ values agree with his own measurements (0.5, 0.6d resp. at his time; Tab. 3.1., 2-h, -i). Though both show very similar deviations from the modern values (Fig. 5.2, 1), Ptolemy’s error in the equation of the center 1δ is twice as much as Hipparchus’ (2), so that his final error turns out to be much greater than Hipparchus’ (3).

This can also be shown by a simple computation. If we put K = 90◦ and (just for a simple comparison) v1 = 90◦ for 113◦ and 109◦ (Tab. 3.1., 1-, 2-l), we obtain roughly from (3.1) 1θ = 1A ≈ + Putting the difference 1δ ≈ 30 · 1δ 2e 1(1δ) = 1δpt − 1δHi ≈ −0.16◦ , [cf. (3.7)]. (5.6) (Tab. 3.1., 1-, 2-k), we obtain the difference in their errors as 1(1A) = 1Apt − 1AHi ≈ 30 · 1(1δ) ≈ −4.8◦ , (Tab. 3.1., 1-, 2-q), (5.7) which roughly agrees with the actual longitudinal motion of the apogee in 270 years, 4.6◦ , which virtually corresponds to Ptolemy’s error, −5.6◦ (Tab. 3.1., 1-, 2-l, -q, -s; Fig. 5.2). Contrary to the case of the eccentricity, Ptolemy’s solar apogee shows a much greater error than Hipparchus’ because his underlying errors in time determination, though very similar to Hipparchus’, tended to enlarge the error of the equation of the center 1δ, which plays an important role in determining the solar apogee [(4.13), Fig. 5.2]. The seemingly complex problem of the apogee is in fact simple, as appears from the expression (3.7). The accuracy of the determined location of the apogee corresponds precisely to the errors of the underlying observations. If these errors are given within a certain amount, say ±1/4d , the accuracy can vary (neglecting the small amount of ±0.6◦ due to the hypothesis [(3.10)]) within the corresponding maximum range, ±9◦ (Tab. 3.2, 2-h, -i), including, of course, the possibility of becoming „0“. This was the case for Hipparchus and Ptolemy.5.4 Because of the low order of accuracy in general, the question of whether the solar parallax was taken into account cannot be answered, nor is it of importance. 5.2. Al-Battānı̄ As in the case of Ptolemy, al-Battānı̄’s lengths of the seasons show no agreement with his still more basic observational data such as the geographic latitude. The error in his geographic latitude 1ϕ = 0.09◦ = 5.40 (Tab. 4.1., 3-e) would cause an error of more than 10h (Tab. 4.1., 3-q) in the length of the half-year, substantially greater than his actual error of 0.6h (Tab. 3.1., 3-h). Unlike Ptolemy, however, al-Battānı̄ gives us his observations of two extreme zenithdistances of the Sun (Op. astr. IV, Nallino I, 12): 0 = 12◦ 260 zmin 0 zmax

1 9 Ù È o N m + a + a (4) 94.5 187 - dv = o z & o o modern values 94.0 - t3-+1 t2-t] 93.5 186 4 | Y T T T T (9) alt?” 0.54 | A T A6 1110 “en 46 AAA Ay J 0 T T T T =~ TS34(t3-t} dw T A (9) | moder® 70 - 6073 -200 ¡ -100 0 +100; 8 Li 5

This quantity eventually turned out to be the error of Ptolemy at t2 , l2 = 90◦ , which was added to the error of Hipparchus. 3. Hence, the same solar apogee resulting from the same underlying observations shows two entirely different deviations from the actual position at the epochs. Ptolemy’s result therefore shows an additional error of 1(1Apt ) ≈ 1(1δ)/2e ≈ −4.6◦ , corresponding roughly to the motion of the apogee in that time interval, 0.0172 × 270 years = 4.6◦ , connected to the phenomenon that the length of the seasons changes as a function of the tropical longitude of the solar apogee. This can be shown by means of our much simplified equation. The deviation of a time interval from its mean value is given as a function of the initial mean anomaly αo by
(s1S) ≈ − πS 2e cos α0 + M2 sin M2 . . . = f(α0 ), (d ), [Maeyama (1988), eq. (3.7)], where S is the length of the mean tropical year in days. Putting α0 ≈ v1 and v1 + 1v for Hipparchus and Ptolemy respectively at l1 = 0 and further for the mean anomalistic motion of the Sun for the two time intervals M ≈ l2 − l1 = 90◦ and ≈ l3 − l1 = 180◦ , we obtain the error which Ptolemy had to derive additionally to that of Hipparchus according to the equations (4.13), (4.18), (5.1) above as 1Apt − 1AHi ≈   1(1δ) 360 1v sin(v1 + 45◦ ) sin 45◦ − 21 cos v1 . ≈ 2e π Putting v1 = 90◦ for simplification, roughly corresponding to the decrease of the true anomaly v1 , the increase of the longitude of the apogee, in that time lapse becomes ≈ 1v ≈ 109 − 113.5◦ = −4.5◦ . Note: From the same underlying data, satisfying approximately the presupposed condition of both astronomers, one logically obtains the same orbital elements which, however, show entirely different errors at two different epochs, since the lengths of the two seasons concerned decreased and increased respectively in relation to the tropical longitude of the apogee. The resulting additional error of Ptolemy therefore corresponds to the longitudinal advance of the solar apogee since Hipparchus’ time. The case of the eccentricity is entirely different due to its slow change. where the modern values are zmin = 90 − hmax = 12.339◦ = 12◦ 200 2200 zmax = 90 − hmin = 59.484◦ = 59◦ 290 200 ; hence [cf. (4.9)], 1zmin = +50 3800 1zmax = +60 5800 . (5.9) Since the deviations would have been still greater if he had corrected his observed values for the solar parallax (1zmin ≈ +6.30 , 1zmax ≈ +9.50 ), we are certain that al-Battānı̄, like most astronomers up to Tycho’s time, neglected the solar parallax for his solar determination.

Knowing this, we can now determine the colatitude, at which his instrument is supposed to have been fixed: 1ϕ̄ = +0.006◦ = +2100 hence (Tab. 5.2., 3−b), ϕ00 = 90 − (ϕ̄ + 1ϕ̄) = 35◦ 550 1400 = 35.921◦ , which deviates considerably from his own determination: 0 0 + zmax )/2 = 36◦ 10 ϕ0 = (zmin (Nallino I,12, n.4), ϕ0 − ϕ00 = +5.80 . Notably, all three deviations above [(5.9)] are of the same magnitude, ca. + 60 , which would indicate that there is only one systematic error-source, the zenith. Fig. 5.3 shows our approximate picture of how al-Battānı̄ is supposed to have made his solar determinations. Although the error in his assumed zenith-direction is very great (0.1◦ = 60 ), we are able to reproduce all numerical determinations of al-Battānı̄’s solar elements to ±4000 = ±0.01◦ , if we give to his immediate observations this amount as a limit of uncertainty. Due to the remarkable accuracy of his immediate observations we can see no other feasible way of explanation. As to the fact that al-Battānı̄ gives all solar observational values only to a minute of arc, we have to consider a rounding error of ±3000 , which is close to our above allowance of ±4000 . Only in this way, it seems, can we clarify his extremely accurate measurements of the equinoxes and the summer solstice with small errors of 1t1,3 = ∓0.3h and 1t2 = +1.3h [(5.1); Tab. 3.1., 3-h, -i; Fig. 5.3]. With respect to the immediate observations of the Sun, al-Battānı̄’s accuracy is comparable with that of Tycho. 5.3. Copernicus As is well known and as we mentioned above,5.5 Copernicus’ numerical data frequently contain ambiguities and sometimes notorious inconsistencies. Unlike in the case of Tycho and Kepler, this was certainly a consequence of his huge task to propagate a new astronomical system which he hoped would function quantitatively for many centuries – replacing Ptolemy’s traditional world system which was based on observations of which the earliest, Babylonian, ones dated from the year − 720. With regard to the geographical latitude of Frauenburg – 54◦ 19 1/20 given in the “De revolutionibus” (III, 2) – we note that Copernicus seems to have employed this value consistently, because he gives there −8◦ 400 for the declination of Spica which is too small (+1.90 ) precisely by that amount of his error in the colatitude 1ϕ̄ = +2.070 .5.6 This quantity in fact agrees precisely with what we deduced from his observations on the lengths of the seasons, +1.550 and +3.430 (Tab. 5.2., 5 ∼ 7-b, -c).

Fig. 5.3. Al-Battānı̄’s meridional observation of the apparent Sun (approximately reproduced) We put al-Battānı̄’s zenith as deviating from the true one by the quantity deducible from his two immediate observations of the solar zenith-distance [(5.9)]: 1zenith = (1zmin + 1zmax )/2 = 6.30 = 0.105◦ . We then obtain the two extreme zenith-distances as measured from his zenith by means of the modern values: 0 zmin = ϕ − ε − r tan zmin + p cos(90 − zmin ) + 1zenith = 12.444◦ = 12◦ 260 4000 ; 1 = −4000 0 zmax = ϕ + ε − r tan zmax + p cos(90 − zmax ) + 1zenith = 59.589◦ = 59◦ 350 2000 ; 1 = +4000 0 0 ε0 = (zmax − zmin )/2 = 23.572◦ = 23◦ 340 2000 ; 1 = +4000 0 0 ϕ0 = (zmax + zmin )/2 = 36.017◦ = 36◦ 10 ; 1 = 0, where al-Battānı̄’s values are reproduced within a range of ±4000 . With a deviation of only 0.009◦ ≈ 3000 from his geographic latitude [ϕ0 = 36◦ (recorded); = 36◦ 10 (computed); Tab. 5.2., 3-, 4-e], the equator of his instrument is situated as deduced by us (Tab. 5.2., 3-b). Thus he was able to make most accurate observations of the Sun at the equinoxes and summer solstice: 1l1,3 = ∓0.012◦ , 1t1,3 = ∓0.3h 1l2 = +0.054◦ , 1t2 = +1.3h [according to (5.1); Tab. 3.1., 3-h, -i]. Note: The coincidence of the two locations of the equator (1 ≈ 0.009◦ ≈ 3000 ; cf. e.g. Tycho, −5 ∼ +1200 versus −400 , Tab. 5.2., 8 ∼ 11-b, -c) – the one computed from his extreme zenithdistances of the Sun [(5.9)] and the other deduced from his lengths of the seasons (Tab. 5.2., 3-b) – indicates a high plausibility of our reproduction above.

With his colatitude and his other data such as the obliquity of the ecliptic, but discarding his notoriously erroneous year length (186;5 1/2 + 178;53 1/2 = 364;59: III,16), we could reproduce reasonably well his errors in the time intervals, taking also accidental errors into account (Tab. 4.1., 4−q, −r). Copernicus most probably determined the colatitude and the obliquity of the ecliptic by means of the two extreme altitudes of the Sun: hmax = 59.152◦ = 59◦ 9.10 hmin = 12.213◦ = 12◦ 12.80 [(4.9)]; hence ϕ̄01 = 35.682◦ = 35◦ 40.90 [(4.11); p0 , r 0 = 0] ε01 = 23.469◦ = 23◦ 28.10 [(4.10); p0 , r 0 = 0]. These two values are close to Copernicus’, the deviations being only ca. 2000 : Copernicus: ϕ̄ = 35◦ 400 30" (Tab. 4.1., 4-c) ε = 23◦ 280 30" (Tab. 4.1., 4-l). Copernicus’ actual deviations from the modern theoretical values, 1ϕ̄ = +2.10 (Tab. 4.1., 4-e) and 1ε = −1.60 (4-n), therefore correspond to 1ϕ̄01 = +0.042◦ = +2.50 [(4.11); p0 , r 0 = 0] 1ε01 = −0.033◦ = −2.00 [(4.10); p0 , r 0 = 0], (5.10) and are attributable primarily to the neglect of refraction. The above correspondence shows that on determining his solar elements Copernicus also neglected the traditional solar parallax 30 (Tab. 4.1., Remarks), despite having mentioned it (IV, 21). As we have seen above, both al-Battānı̄ and Copernicus followed almost the same procedures based on similar observations in order to determine the solar orbit. The question that arises is why their results differ strongly from one another and why the former was more accurate than the latter. Neglecting all minor factors we can reduce the question to the problem of refraction, particularly because both al-Battānı̄ and Copernicus neglected the solar parallax of 30 . Designating for simplification the deviation between their errors by “1”, we obtain roughly 1(1ϕ̄) ≈ 21 r[(cot hmin )Cop − (cot hmin )al-B ] ≈ +1.40 [p 0 , r 0 = 0 in (4.11)];

hence from (4.8) 1(1l1 ) ≈ +3.50 ≈ +0.06◦ , thus the deviation in the time interval 2[−1(1l1 )] ≈ −0.12d ≈ −2.8h , 1[1(t3 − t1 )] ≈ ω agrees roughly with Copernicus – al-Battānı̄ = −4.4 − (+0.6) = −5.0h (Tab. 3.1., 3-, 4-h), or more precisely with = −1.6 − (+0.6) = −2.2h (Tab. 4.1., 3-, 4-q). (5.11) The above value 1(1l) is introduced into the equation of the center and further into the eccentricity as 1(1k) ≈ 1(1γ) ≈ −1(1l1 ) ≈ −0.06◦ ≈ −0.001 rad, corresponding to (1k)Cop − (1k)al-B ≈ −0.0015 − 0.0003 = −0.0018 or, neglecting the decrease of the value in the time interval of ca. 630 years (1kt ≈ 21et ≈ 0.0005), roughly to kCop − kal-B ≈ 0.0323 − 0.0346 = −0.0023. (5.12) The different orders of accuracy of the solar determinations of two astronomers can therefore be reduced essentially to the effect of refraction at two different minimum solar altitudes, or simply to two different geographical latitudes: Copernicus: hmin ≈ 12.1◦ ; r cot hmin ≈ 4.60 ϕ = 54.359◦ (Frauenburg) al-Battānı̄: hmin ≈ 30.5◦ ; r cot hmin ≈ 1.70 ϕ = 35.926◦ (al-Raqqa) 1 refraction = 30 1ϕ = 18.4◦ ; Tab. 4.1., 3-, 4-d. (5.13) It is strange that in determining the solar orbit Copernicus – unlike many other astronomers, especially Tycho – used the low solar altitude at l2 = 225◦ as one of the three fundamental observations for his determination.

The values of the solar eccentricities which would result from accurate observations, following their procedures described above (p 0 , r 0 = 0), are equal to the values given by the two astronomers to the third place after the comma: Table 5.3. 1(t3 − t1 ) 1l1,3 Copernicus al-Battānı̄ ◦ ±0.051 ±0.010 d h −0.104 = −2.49 −0.020 = −0.49 1k k0 −0.00089 −0.00017 (3.1), (4.6) 0.0329 0.0323 0.0342 0.03465 Tab. 3.1., 3−, 4−o, −p records In short, Copernicus’ ignorance of refraction caused too great a colatitude [ϕ̄0 > ϕ̄, (5.10)], forcing him to measure too low declinations of the celestial bodies [d 0 < d (e.g., = 0 for the Sun at the equinoxes)], thus further to determine, e.g. too short a time interval between two equinoxes (Tab. 4.1., 4-q). This gave him too small an eccentricity of the solar orbit (Tab. 3.1., 4-n ).5.7 Hence Copernicus’ enigmatically small eccentricity of the Sun was simply caused by the effect of refraction. This actually occurred, 1) comparing to al-Battānı̄, both of them ignorant of refraction, by the high geographical latitude of Copernicus’ observing place, and 2) comparing to Tycho, both having equally high latitudes, by Copernicus’ determination of his colatitude not as an independent but as a dependent parameter by means of the solar altitudes, as shown above [cf. (5.10)]. All this can be clearly explained by means of few equations, (4.6) –(4.8), (4.12) (Figs. 4.2–4.5). It is now clear why numerous Muslim astronomers succeeded in determining the solar orbit accurately:5.8 1. They generally had high observational accuracy, 2. They neglected the (traditional) solar parallax, 3. They observed high solar altitudes at the meridian due to low geographical latitudes thus their observations were scarcely affected by refractions. Al-Battānı̄’s case is simply an example of our argument above. Two fundamental parameters of the Sun, the obliquity of the ecliptic and the eccentricity, which had tacitly been considered as astronomical constants, decreased from the time of Hipparchus and Ptolemy over al-Battānı̄’s down to Copernicus’ epoch much in the same manner. Copernicus’ diligent and accurate determination of these parameters [cf. (5.10), Tab. 5.3] must have been for him clear evidence for the necessity of a new theory asserting that both parameters changed as a function of time. Together with those different values for the precessional constant of the equinoxes as historically recorded, he was thus led, as if by an invitatio divina, to his theory of the 1717 year-cycle.5.9 From our above statements it follows that all this can be reduced to the problem of his determination of the colatitude by means of the solar altitudes. The colatitude was too

high by 2 minutes of arc, caused by the effect of refraction, of which Copernicus, like all other astronomers of that time, had no quantitative knowledge. And this is eventually attributable to the high geographical latitude where he had to observe the minimum solar altitude (ca. 12◦ ). My early conjecture that Copernicus had favoured his own theory of the 1717 yearcycle by deliberately choosing a very low solar altitude, 19◦ , at l2 = 225◦ is in reality ungrounded. This choice gave no numerical contribution to that theory, since the 1δ-term for the determination of the eccentricity in (3.1) is negligibly small. [cf. (4.2)]. Thus, the neglect of the traditional solar parallax of 30 , which favoured the accuracy of the solar determination by al-Battānı̄ and many other Muslim astronomers, was spoiled in the case of Copernicus by his use of the Sun’s low position hmin ≈ 12◦ due to the effect of refraction. 5.4. Tycho Brahe Tycho’s astronomical procedures, unlike those of his predecessors, can precisely be followed with no ambiguities (a high degree of testability), a fact which is certainly more significant than the question of observational accuracy. We can in fact for the first time give our deep insight into an astronomer’s daily activity. By simply applying his data to his procedures as he describes them we could reproduce his underlying time determinations at t1 , t2 and t3 , and his two time intervals (t2 − t1 ) and (t3 − t1 ), to fractions of an hour between 0.13h = 8m and 0.55h = 33m (Tab. 4.1., 5-q, -r). The neglect of solar parallax and refraction (p0 , r 0 = 0) changes the time interval between the two equinoxes approximately by 1(t3 − t1 )p0 ,r0 ≈ 2 × 2.5 × (−p 0 cos h1,3 ; +r 0 ) ≈ (−0.21d )p0 , (+0.06d )r0 respect. [(4.8)], hence his error of +0.27d would improve to +0.12d : 1(t3 − t1 ) = +0.27 (Tab. 4.1., 5−q, or + 0.25, Tab. 3.1., 5−h) − 0.21 + 0.06 ≈ +0.12d ≈ +2.9h (Tab. 4.1., Remarks). (5.14) Figs. 5.4–5.5 show Tycho’s solar observations again in relation to the two fundamental, independent parameters, solar parallax and refraction, and Fig. 5.6 illustrates the origin of his error, +6 hours, in the length of the half-year in contrast to the error in the tropical year, −3 seconds.

Fig. 5.4. Deviations of Tycho’s solar altitudes over the horizon, immediately observed (hob ) and corrected for solar parallax and refraction (hT ), from the theoretical values (h): cf. Figs. 4.1, 5.5, 5.6. Tycho’s immediate observations: 1hob = hob − h = −p cos h + r cot h, Tycho’s errors after correction for p0 , r 0 : 1hT = hT − h = 1p cos h + r cot h − r 0 . Note: In Tycho’s observational range for his solar determination his errors are primarily covered by his correction for the solar parallax, p0 cos h. The irregular curve of Tycho’s errors after correction originates from his erroneous values for refraction in which in turn the erroneous value for the solar parallax 30 is fully involved. From our statements above it follows that, unlike the case of his predecessors, there are no ambiguities in Tycho’s last stages. In particular he corrected all his solar observations for solar parallax and refraction, and criticized his predecessors for their neglect of these two problems: With all these facts he himself did not determine the solar parallax but always accepted and employed the traditional value 30 for the mean horizontal solar parallax.5.10 As for the refraction he determined it not as an independent but as a dependent parameter,

Fig. 5.5. Deviations of Tycho’s time intervals from the modern values as a function of the solar longitude: 1(t − t1 )(h ); t1 at l1 = 0, t at l = 0 − 360◦ (cf. Figs. 4.2, 4.5, 5.6) Tycho’s errors: 1. the total error 1(t − t1 ) = 24(1l − 1l1 ) ω = 241l ω + 3.21h 1l1 = −0.132◦ (Tab. 4.1., 5−o; cf. also Tab. 5.1.), 1l according to (4.6). 2. the error caused by p0 = 30 1(t − t1 )p , computed from (4.6), (4.8) only with 1p. ◦ : Tycho’s records (Op. om. II, 15, 19–24) - modern values [Maeyama (1988)]: cf. curves 1. Note: Tycho’s errors are accurately reproduced by the curves 1. In the observational range for his solar determination Tycho’s errors of the time intervals are primarily caused by his erroneous solar parallax, p 0 = 30 (curve 2). Irregular parts of the curves are directly connected, as indicated, with Tycho’s erroneous values for refraction caused again by those for the solar parallax. that is, he deduced the deviations of the observed solar altitudes from those given by his solar theory without the term for refraction, and then identified the deviations as the solar refractions. Like other parameters, such as the obliquity of the ecliptic [(4.10)], his solar refractions are therefore distorted by the erroneous value of the traditional solar parallax.5.11

Fig. 5.6. Tycho’s determination of the equinoxes (Tab. 5.1., Figs. 4.1, 4.2, 4.5, 5.5) Tycho determined the equinoxes by observing the meridional altitude of the apparent Sun (hob ), corrected for the solar parallax and refraction (hT ) such that this altitude is equal to his colatitude: hT = hob + p 0 cos h − r 0 = ϕ̄0 . Tycho’s resulting time interval between the two equinoxes relating to the theoretical altitude of the true Sun h0 in the ecliptic is therefore too long by 1t = 1t3 − 1t1 ≈ +6h , corresponding to 1l = 1l3 − 1l1 ≈ +0.26◦ (Tab. 3.1., 5−h; 4.1, 5−o). His tropical year, measured at the same position in the ecliptic, is accurate to 1t = −2.6s = −0.0007h , an accuracy which he obtained by comparing his observations with the old ones made 100 years earlier by B. Walther (Op. om. II, 42–44). In order to minimize the effects of the parallax and of refraction Tycho determined the geographical latitude by the fixed stars (Op. om. X, 297). Using the thus independently determined (and therefore accurate) colatitude and the maximum observable solar altitude (hob )max corrected for the solar parallax (rh0 = 0), he then determined his final obliquity of the ecliptic: ε0 = (hob )max + p0 cos hmax − ϕ̄0 = (hT )max − ϕ̄0 ; 1ε ≈ p0 cos hmax = 1.610 [≈ 1ε2 = 2.160 (4.10); Tycho = 23;31,30; X, 293]. This result was therefore too large, the difference being nearly equal to the deviation caused by his erroneous solar parallax [Maeyama (1974), 40–42]. Note: Tycho’s determinational error of the length of the half-year, 6h , and that of a year, 0.0007h , caused by his solar parallax, 30 , are typical examples of his rigor; his observational accuracy was constant, but he strictly kept to the – erroneous, traditional – solar parallax, the only astronomical quantity which he accepted without his own determination. In his later stage all solar observations were made under the same conditions (cf. Fig. 4.1). Here is where Kepler begins to struggle for his new astronomy, using Tycho’s most accurate immediate observations of the Sun but corrected for the traditional solar parallax and the refraction according to Tycho.5.12

Summary From Hipparchus down to the time of Kepler’s Astronomia nova (1609) the Sun’s orbit was determined by means of the solar hypothesis according to Hipparchus and the length of two seasons measured by three observations of the Sun. In the so determined solar elements we therefore have two error sources: 1. the error of the solar hypothesis, 2. the error of the observations. We have first deduced some simple formulae showing the theoretical deviations of the solar elements determined by Hipparchus’ hypothesis from the modern ones and shown that these deviations are negligibly small compared with our very varying historical records of the solar elements (ch. 2). We have then deduced other formulae capable of reproducing the historical records of the solar elements as a function of the underlying observational errors (ch. 3). Since all this functioned precisely enough, we then proceeded to clarify the basic conditions under which the recorded observations were actually made. This could be done sufficiently well by means of our analysis of Tycho’s observations into which we were able, thanks to his consistency of description and conduct, to give our deep insight (chs. 4, 5). An outstanding example thereof is the marvelous agreement between his independently determined geographical latitude and his instrumentally fixed colatitude as deduced by us from his various lengths of the seasons to ca. ±10 seconds of arc, which means an agreement of his three independent operations - determination of the latitude, installation of the instrument and observation of the Sun. Being convinced that, apart from the problem of accuracy, Tycho’s observational procedures were in principle the same as those of earlier astronomers, we then applied our discovery about Tycho’s observations to our other astronomers. In the case of Ptolemy and al-Battānı̄ we thus came to find a systematic error, each in different magnitudes, which seems to clear up all their apparent inconsistencies and uncertainties, such as the obliquity of the ecliptic (Ptolemy) and the geographical latitude (Ptolemy, al-Battānı̄), to a permissible minimum. Thus we have eventually arrived at our aim of connecting the historically determined Sun’s orbits with their immediate observations. The results obtained strongly vary from case to case but are associated with individual circumstances and characteristics of their times in history. Those who neglected the erroneous traditional solar parallax were capable of determining the Sun’s orbit – depending on the order of accuracy of the immediate observations and on their frequency – accurately within the effect of refraction. A contrast is therefore clearly visible between al-Battānı̄ and Copernicus due to the greatly different latitudes of their observing places (refraction). Contrary to all his predecessors Tycho consistently took the solar parallax into account and therefore shows us ubiquitously his errors arising from the traditional quantity of 30 . Tycho’s length of the half-year between two equinoxes was erroneous by 6 hours, but that of the tropical year was accurate to 3 seconds of time. These facts can now

be simply explained. In each determination the same rigorous observation of the Sun occurred twice, equally displaced by the solar parallax; twice inversely in the former and twice the same in the latter. As a result the effect of Tycho’s erroneous value for the solar parallax upon the measured intervals in time and position turned out to be double in the former, whereas it disappeared without trace in the latter. This most remarkable phenomenon was later easily recognized by those astronomers, who took the solar parallax as a fundamental independent parameter, such as Horrocks (ca. 1638), Flamsteed (ca. 1675) and Cassini (ca. 1684) [Maeyama (1974), 51, n.6]. However dramatic our conclusion may sound, much corroborating evidence can be found. One example may suffice here. Tycho’s solar tables, computed with his solar eccentricity 0.03584 (too large by the error of the half-year, i.e. the double of the solar parallax) were the corner stone for Kepler’s Astronomia nova (1609). Moreover, Kepler introduced the corresponding rounded bisected eccentricity 0.018 (too large by the erroneous solar parallax of 30 ) into his final Tabulae Rudolphinae (1627). Exactly on this basis – the most extensive solar observations ever made in number and accuracy but corrected for the solar parallax and refraction all according to Tycho – Kepler had to begin with his struggle for a new astronomy of the solar system. As we have frequently shown, many questions of astronomy at that time can therefore ultimately be reduced to the problem of the solar parallax and refraction. Notes 1. Cf. Tab. 3.1., cols. a, p and ch. 5.2 below. 2. E.g. Delambre, Hist. Astr. ancienne, vol. 2, 117ff.; Hist. Astr. moyen âge, 35f.; Hist. Astr. mod., 113f, 152 ff.; Neugebauer 57f.; Pedersen 144ff.; Petersen-Schmidt; Rome (1943); Swerdlow-Neugebauer 150ff.; Thoren 220ff. Those who analysed the observational errors will be dealt with below. 3. Although Ptolemy does not say that Hipparchus invented the solar model with the anomaly (Alm. III, 4), we call it below for simplification „Hipparchus’ solar hypothesis“. 1.1. E.g. Kennedy (1963); Hartner-Schramm, 208f. 2.1. Cf. e.g. Brouwer-Clemence, 65; Maeyama (1974), 45. 3.1. Some statements of Petersen and Schmidt, e.g. 1/4d versus 14–15◦ error-interval, are therefore not incorrect (pp. 74, 80, 83, cf. also Toomer, 154) but the problem is more complex (cf. ch. 5.1 below). 4.1. For eqs. (4.1), (4.2) cf. eq. (5.1) below. 4.2. Cf. Maeyama (1974), 39f. 4.3. Thoren, e.g. 234f.; Maeyama (1974), e.g. 36f. 4.4. Tycho Op. om. II, 64, 136, 287; Cleomedes, Czwalina 78; Plinius, Nat. hist. II, 57. For Alhazen, Walther etc. cf. Thoren 227ff. 4.5. For eqs. (4.12), (4.13) cf. eq. (5.1) below.

4.6. Tab. 3.1., 5-p; Tycho Op. om. II 21–24; Kepler, Astr. nova, cap. 23–28, Tab. Rudol. 44ff; Maeyama (1974), 50–59. 4.7. Maeyama (1974), 58, Tab.8; (1990.1), 88, Tab. 5.1.; (1990.2), 136f., Figs. 1, 3. 4.8. For (4.18) cf. (5.1) below. 5.1. Ptolemy’s allegedly more accurate values are in reality less accurate, cf. Fig. 5.2 below (cf. also e.g. Pedersen 148; Toomer 154, n. 47). 5.2. E.g. Britton (1969) has shown that the error of Ptolemy’s obliquity of the ecliptic can be explained by the deviation of his observation from the meridian [also (1992), 1ff.]; cf. also Newton (1977), 96ff.; (1982), 31ff., supported by v. d. Waerden, 258. If our statement above is probable, Newtons probability such as „1 out of 1092 “ [(1977), 97] will be totally rejected. 5.3. That Hipparchus may have taken his 94 21 days for the spring from the Babylonian sources (Bowen-Goldstein, Jones 118) – and that, instead of 92 43 days for the summer or 187 41 days for the total from the same sources (cf. Kugler 84ff.), he took 92 21 and 187 days – and further that three centuries later Ptolemy reconfirmed, knowing all this or not, the correctness of Hipparchus’ whole solar theory with those fundamental underlying lengths of the seasons – though he was in possession of his own, alledgedly more accurate lengths (Alm III, 4; H 234; Toomer 154, n. 47) -, all this is beyond my understanding. In view of his observation of the stars and determination of the length of the year, etc. Hipparchus’ determination of the Sun’s orbit does not belong to his early stage soon after his Commentary (around −140 ∼ −150) but undoubtedly to his later epoch (around −130) [Maeyama (1984), Figs. 7, 8, 301–305]. Britton (1992, 12ff.) tends to accept −134 for Hipparchus’ determination of the length of the year. There is much evidence of Hipparchus’ (early) dependence on the Babylonian sources (esp. Jones) and many questions can be raised about accuracy, precedents, and similarities with other sources. Yet his numerous observations of the Sun indicate that he tried to obtain all the solar data he needed and was confident about what he had. We accept Hipparchus’ lengths of the seasons as his own; similarly we accept Ptolemy’s claim to have measured the Sun’s two extreme positions at the meridian (cf. ch. 5.1). 5.4. Cf. the ambigous claim of Petersen and Schmidt about Hipparchus’ accurate apogee-determination as a result of „coincidence“ (p. 74, 83). A good example may be given: If at t ≈ −1400 (v1 = 135◦ ) we take the observational errors of ±1d [1γ, 1δ in (3.7)], 1A will be ±40◦ (Tab. 3.2, 3-d) and negligibly small (≈ 0) respectively. Precisely based on their premise, 1t1,2,3 = 1t (p. 82), but not |1t1,2,3 | = |1t|, so their claim only relates to the maximum possible errors (cf. also Pedersen 148f; Toomer 154, n. 47). 5.5. Tab. 3.1., l. 6. 5.6. Cf. e.g. Swerdlow-Neugebauer, 131.

5.7. The Commentariolus gives the eccentricity 1/25 = 0.04, a conveniently rounded figure of Ptolemy’s value 1/24 = 0.0416 . . . , certainly for facilitating his numerical transformation of the Ptolemaic world-system. 5.8. Cf. e.g. Ibn al- Ac lam (d. 985), max. equat. of center= 2;0,10◦ , Kennedy (1977), 21f.; al-Maghribi (d. 1283), k = 2;5,57–2;5,59p , and Ibn al-Shāt.ir (d. 1375), max. eq. = 2;2,6◦ , Saliba (1985), 117–120, and (1987), 39ff.. Al-Bı̄rūnı̄ mentions in his „Qānūn“ (Treatise 6, ch. 7) several similar solar elements, (I owe this information to Dr. Benno van Dalen). Cf. also van Dalen’s contribution to Encycl. of Islam, al-Shams, the Sun. 5.9. Cf. Rheticus’ Narratio prima; De rev. III, 6; also Moesgaard. 5.10. Cf. e.g. Thoren, 227. 5.11. Maeyama (1974), 46ff. 5.12. Esp. Astronomia nova, chs. 15–19; Wilson; Maeyama (1990.1). References Al-Battānı̄: Op. astr. Al-Battānı̄ sive Albatenii Opus astronomicum. Ed. C. A. Nallino. 3 pts. Milano 1899–1907. Al-Bı̄rūnı̄: Al-Qānūn al-Mas’ūdı̄, Mas’ūdic Canon. Bowen, A. C. - Goldstein, B. R.: Meton of Athens and astronomy in the late fifth century B.C. A Scientific Humanist: Studies in memory of Abraham Sachs. E. Leichty, M. de J. Ellis, P. Gerardi (ed.), Philadelphia 1988, 39–81. Britton, J. P. (1969): Ptolemy’s determination of the obliquity of the ecliptic. Centaurus 14, 29–41. – (1992): Models and Precision: The quality of Ptolemy’s observations and parameters. New York-London. Brouwer, D. - Clemence, G. M.: Methods of celestial mechanics. New York-London 1961. Caspar, M.: cf. Kepler. Clemence, G. M.: cf. Brouwer-Clemence, Woolard-Clemence. Cleomedes: De motu circulari corporum caelestium. A. Czwalina: Kleomedes, Die Kreisbewegung der Gestirne. Leipzig 1927. Connaissance des temps, ou des mouvements célestes pour l’an ... Bureau des Longitudes, Paris 1953-. Copernicus, N.: Commentariolus. – (1543): De revolutionibus orbium coelestium, libri VI. Czwalina, A.: cf. Cleomedes. Dalen, B. v.: Al-Shams, the Sun. To appear in Encycl. of Islam. Delambre, J. B. J. (1817): Histoire de l’astronomie ancienne. 2 vols. Paris. – (1819): Histoire de l’astronomie du moyen âge. Paris. – (1821): Histoire de l’astronomie moderne. 2 vols. Paris. Explanatory supplement to the astronomical ephemeris and the American ephemeris and nautical almanac. London 1961. Fotheringham, J. K.: The secular acceleration of the Sun as determined from Hipparchus’ equinox observations; with a note on Ptolemy’s false equinox. Monthly Notices of the Royal Astr. Soc. 78 (1918), 406–423. Geminus: Isagoge. Gemini Elementa Astronomiae. Ed. and German transl. C. Manitius, Leipzig 1898. Goldstein, B. R.: cf. Bowen-Goldstein.

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Schramm, M.: cf. Hartner-Schramm. Swerdlow, N. M.: Al-Battānı̄’s determination of the solar distance. Centaurus 17 (1973), 97–105. Swerdlow, N. M.- Neugebauer, O.: Mathematical Astronomy in Copernicus’s De Revolutionibus. 2pts. New York-Berlin-Heidelberg-Tokyo 1984. Thoren, V. E.: The Lord of Uraniborg. A biography of Tycho Brahe. Cambridge 1990. Toomer: cf. Ptolemy. Tuckerman, B.: Planetary, lunar, and solar positions. 2 vols. Philadelphia 1962, 1964. Van der Waerden, B. L.: Die Astronomie der Griechen. Darmstadt 1988. Vieta, F.: Francisci Vietae Apollonius Gallus. Appendicula II. Paris 1600. Repr. François Viète, Opera mathematica. Hildesheim-New York 1970, 343–346. Wilson, C.: The error in Kepler’s acronychal data for Mars. Centaurus 13 (1969), 263–268. Woolard, E. W. - Clemence, G. M.: Spherical astronomy. New York-London 1966. Institut für Geschichte der Naturwissenschaften Goethe Universität Postfach 111932 60054 Frankfurt am Main Germany (Received April 2, 1996)