Abstract
Tycho Brahe was not just an observer; he was a skilled theoretical astronomer, as his lunar and solar models show. Still, even if he is recognized for proposing the Geoheliocentric system, little do we know of the technical details of his planetary models, probably because he died before publishing the last two volumes of his Astronomiae Instaurandae Progymnasmata, which he planned to devote to the planets. As it happens, however, there are some extant drafts of his calculations in Dreyer’s edition of Tycho’s Opera Omnia under the name Calculi ad Corrigenda Elementa orbitae Saturni, which, to the best of my knowledge, have not yet been analyzed before. In these manuscripts, Tycho starts with calculations based on the Prutenic Tables and makes a series of adjustments to the mean longitude, the longitude of the apogee, and the eccentricity to fit a series of observations of oppositions. In doing that, Tycho (1) describes and applies a new method for obtaining accurate values for the parameters of the superior planets, he (2) develops a divided eccentricity (not bisected) model of Saturn, similar to the one we know Longomontanus and Kepler applied to Mars, and finally (3) he realizes that the true position of the Sun somehow affects the motion of Saturn around the zodiac and develops a method to correct the position of Saturn as a function of solar equation of anomaly. So, a close analysis of the calculations reveals details of the Tychonic planetary models unknown until now. The present study analyzes these drafts.
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Notes
From now on, a Roman number followed by Arabic numbers refers to Dreyer, Tychonis Brahe Dani opera omnia. The Roman number indicates the volume; page numbers are given after the comma, and line numbers after the colon (both in Arabic numbers).
I shall call a simple epicycle model an epicyclic model, as for the Earth or the Sun, equivalent to an eccentric model in which the center of the deferent and the center of uniform motion coincide, which I call an undivided eccentricity model. If the center of the deferent and the center of uniform motion do not coincide, i.e., if there is an equant point, I shall name it a divided eccentricity model. If the center of the deferent bisects the distance between the observer and the equant point, I shall call it a bisected eccentricity model, or simply, a bisected model. Consequently, from now on, I will reserve the use of the term divided eccentricity model for those divided eccentricity models that are not bisected. Kepler’s Vicarious Hypothesis has a divided eccentricity that is not bisected, but I will restrict the term “vicarious hypothesis” to Kepler’s model in the Astronomia Nova.
For example, at V, 248:14, Tycho adds “in locis mediis 4 Minuta desideratum”; at V, 249:5 Tycho adds: “non longe a maxima eccentricitate et maxima latitudine” (and another longer note from lines 22–25). To see exactly which texts are written by Tycho’s own hand and which not, see V, 333. In several places, appears the terms limitata and limitatio, for referring to some kind of procedure to get a corrected or more accurate quantity. See, for example, V, 238:23 and V, 254:14. These terms are used by Longomontanus (Swerdlow 2010, 181–83) so it could be a clue to attribute at least some of these texts to Longomontanus. But a closer examination of the manuscripts is mandatory.
It is hard to know whether the divided eccentricity model was Tycho’s or Longomontanus’s idea. In ch. 7 of his Astronomia Nova, Kepler says that when he arrived to work with Tycho, it was Longomontanus who was working with the theory of Mars (Donahue 2015, 134), but in chapter 16 he attributes to Tycho the divided eccentricity model (Donahue 2015, 186). Additionally, in two letters from 1598, one to Kepler (VIII, 44–45) and the other to Magini (VIII, 121), Tycho explicitly says that the eccentricity is divided but not bisected. It is possible, however, that Tycho was attributing to himself what was actually his assistants' idea. After all, according to (Swerdlow 2009) something similar happened with the Lunar theory. As Swerdlow says, “the master receives the benefit from the work of his servant” (p. 10).
This constraint is not arbitrary because Ptolemy found the bisection of the eccentricity by the method described in Almagest 10.6.
I do not know when Tycho started to suspect that the iterative method was not trustworthy. I only found a reference, as earlier as 1585, when he proposes to use the iterative method for obtaining accurate values of the longitude of the apogee and the size of the eccentricity of Mars (see V, 284). The first calculations of Calculi ad Corrigenda that already used another method probably date from 1593, because he used all the oppositions until 1593 and not oppositions after that date. If the calculation had been made years later, there would be no reason not to have incorporated the oppositions of the following years. So, the date should be sometime between 1585 and 1593. If the reason for Tycho’s distrust of the iterative method is that it wrongly assumes the bisection, the use of the new method in 1593 would imply that Tycho was aware of the divided eccentricity hypothesis as early as 1593. He could have empirically discovered, however, the instability of the method before finding a reason for that. So, it would be too risky to say that he invented the divided eccentricity hypothesis around 1593. As I will show, in Calculi ad Corrigenda, the first calculations including the divided eccentricity are around 1599, but he mentions the hypothesis to Kepler on April 1598.
My analysis was carried out by programming an excel spreadsheet with Duke’s equations. I checked the equations making a step-by-step derivation following Ptolemy’s procedure and I obtained the same result. In the Table published in Astronomia Nova, Tycho offers the longitudes with respect to Mars’s Circle and the longitude with respect to the ecliptic. When I use the second one, the values obtained are closer to those using modern data. Modern data are taken from Carman and Duke (2019).
This means that you must subtract from the Prutenic eccentricity 1 of the 250 parts in which the eccentricity can be divided, i.e., 11,390 p/250 = 45;34 p. His corrected value would be 11344;26 p. But what Tycho usually does in Calculi ad Corrigenda is to directly subtract 1/250 part from the value of the equation of center for each opposition obtained from the Prutenic table.
At the head of the calculations, Tycho adds the modifications of the parameters used. Next to the value of 1/39 p that must be subtracted from the eccentricity, he adds: “vel potius 36a” (i.e., or rather 1/36 parts), but the calculations were made using − 1/39. In the table, I omitted opposition of 1570, because usually the difference between the observed and calculated longitude at the opposition of 1570 is enormous when compared with the others. The value of the observations is not particularly wrong. The main reason for the difference is probably the gravitational perturbation that Jupiter produces on the orbit of Saturn. Saturn is so far away from the Sun and Jupiter so large and so close to Saturn that the gravitational influence of Jupiter on Saturn is not negligible. The perturbation is maximum when Saturn and Jupiter are in heliocentric conjunctions (Wilson 1985; Carman 2020b). In July 1583, a Saturn–Jupiter heliocentric conjunction took place that considerably altered the parameters of Saturn. This explains why Tycho mentioned on several occasions that the opposition of 1570 does not fit with his models. See, for example, V, 254: 8–11.
Besides Kepler’s own words in Astronomia Nova (Donahue 2015, 52–54), I only found evidence in Tycho’s work about the non-bisection hypothesis applied to Mars in V, 269–275. A set of diagrams related to the latitude of Mars is accompanied by the enumeration of the parameters. It says that while the whole eccentricity (Eccentrictitas tota) is 12;6 p if the radius of the orbit of Mars is 60 p, the radius of the small epicycle is 2;15 p. The distance between the eccentricity of the center and the equant point is twice the radius of the small epicycle, i.e., 4;30 p. If the total eccentricity is 12;6 p, then the distance between the mean Sun and center of the orbit is 7;36 p. The proportion between both eccentricities is 121/76, close to 5/3. In the cases of Saturn (V, 257, 260) and Jupiter (V, 263, 266), there is a mistake and in place of the radius of the small epicycle, the value reported is twice the radius, i.e., e2. Taking into account these corrections, the eccentricities in these models are still bisected.
It is difficult not to perceive the similarity with the steps that Ptolemy describes in the Almagest dealing with the lunar model, in which he first elaborates a model that works at syzygies, then proposes a modification to correct the error at quadratures, and, finally, a new modification to solve the error at octants (Pedersen and Jones 2011, 167–195).
In a brief note in his notebook of observations, referring to the calculation of the opposition of 1600 (XIII, 212), he says. “You remember, certainly, that the alteration of the eccentricity [produced by the modification of] the small circle has not yet been added; this [alteration] can introduce one minute or at most a minute and a half. We will see about that later.” The opposition of 1601 is also registered in his observational notebook (XIII, 262), in which he did not, however, calculate the longitude; he only registered the observation.
On Longomontanus’s solution to this problem in the case of Mars, see Carman (2020a).
There is another method developed by Jābir ibn Aflah, but the conditions imposed on the input observations are so demanding that, in practice, cannot be applied. See Swerdlow (1987).
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Acknowledgements
I would like to thank Noel Swerdlow, Dennis Duke, Anibal Szapiro, Diego Pelegrin, and Gonzalo Recio for discussing previous versions of this paper.
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Carman, C.C. Tycho Brahe’s Calculi ad Corrigenda Elementa Orbitae Saturni and the technical aspects of his planetary model of Saturn. Arch. Hist. Exact Sci. 74, 565–586 (2020). https://doi.org/10.1007/s00407-020-00253-0
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DOI: https://doi.org/10.1007/s00407-020-00253-0