Abstract
When at the beginning of 1600, Kepler arrived to work with Tycho Brahe, Longomontanus, Tycho’s principal assistant, who was working with a model for Mars that predicted with remarkable accuracy its longitudes at oppositions. According to Kepler, this hypothesis “represented all these oppositions within a distance of two minutes in longitude.” The model, however, was unsuccessful in predicting longitudes at other elongations from the Sun, and latitudes even at opposition. Much has been said on how Kepler developed his model after this meeting, arriving finally at the so-called first two laws published in his Astronomia Nova in 1609. By contrast, Longomontanus’ attempt, published as a final model in his Astronomia Danica, has received little scholarly attention. In this paper, I will systematically analyse and explain this model. Even if Longomontanus’ solution is not as elegant as Kepler’s, it deserves scholarly attention, both because it solves the problems posed by the model that he and Tycho were working on when Kepler arrived and because it offers an interesting though heterodox solution that, by contrast, helps to highlight the elegance and simplicity of Kepler’s own solution.
Keywords
Introduction
When at the beginning of 1600, Kepler arrived to work with Tycho Brahe, he was hoping to make use of Tycho’s observations for checking the planetary distances he proposed in his Mysterium Cosmographicum. 1 Instead, Tycho put Kepler to work on Mars, under the supervision of his principal assistant, Christen Sørensen from Lomborg, better known as Longomontanus. Tycho was very jealous of his observations, and Kepler suspected that this was a way of preventing his access to the whole corpus of Tycho’s observations. Kepler was certainly disappointed at the beginning of his time with Tycho. 2 Years later, however, when he wrote his Astronomia Nova, he seemed to have changed his mind, thanking Divine Providence for the fact that Longomontanus was working on Mars, because the motion of Mars “provides the only possible access to the hidden secrets of astronomy, without which we would remain forever ignorant of those secrets.” 3 Longomontanus was actually working with a model for Mars that predicted with remarkable accuracy its longitudes at oppositions. According to Kepler, this hypothesis “represented all these oppositions within a distance of two minutes in longitude.” 4
This model, however, was only successful in predicting longitudes at oppositions. Kepler tells us that Longomontanus got stuck in his work because the model did not correctly predict the longitudes at other elongations from the Sun besides opposition and, even in opposition it only correctly predicted longitudes, but not latitudes. Kepler says that “the parallax of the annual orb” was a problem for this model, that is, while it accurately predicted the heliocentric longitude of Mars, it seemed that the parallax due to the fact that the observer is not at the Sun but at the Earth was a problem. 5
Scholars have abundantly explored how Kepler developed his model after this meeting and arrived finally at the so-called first two laws, 6 published in his Astronomia Nova in 1609. Little has been said, however, about Longomontanus’ decision to fight his own war against Mars, armed with a different model. Even if his battle was not as successful as Kepler’s, it deserves scholarly attention, mainly because it solves the problem by offering an interesting though heterodox solution that, by contrast, helps to highlight the elegance and simplicity of Kepler’s own solution. In this paper, I will therefore describe how Longomontanus solved the problem “of the parallax of the annual orb,” analysing the model that he offered in his Astronomia Danica (AD for now on). 7 As far as I know, this is the first public attempt at a systematic description of Longomontanus’ model for Mars. The only previous references I was able to find are one paragraph in Delambre’s Histoire de l’astronomie modern that does not describe the model, and a paper by Swerdlow that describes in detail only the first anomaly of model for Mars in AD. 8
Longomontanus worked in a Tychonic framework in which the planets revolve around the Sun, which in turn revolves around a fixed Earth. Therefore, I will first briefly describe Longomontanus’ model for the Sun and then Longomontanus’ model for Mars in detail. When describing the model for Mars, I will first analyse how Longomontanus handled the first anomaly of Mars, and then how he dealt with the second anomaly. This last section will show the most original part of Longomontanus’ model. I will highlight the elements of the model that Longomontanus borrowed from Kepler, and those Keplerian contributions that Longomontanus refused to incorporate. Finally, I will analyse the accuracy of the predictions of the model. The paper will end with two technical appendixes.
Theory of the Sun
Longomontanus develops his model for the Sun in AD, Book I, Chapter 5, p. 225-235. I will only describe this model briefly because Moesgaard analysed it in detail. 9 The model is not particularly original. Longomontanus assumes an epicycle system (AD, 231), with the centre of the epicycle revolving with the mean longitude period and the Sun revolving on the epicycle with the anomalistic period. As in Copernicus’ model, the value for the annual anomalistic motion is not exactly the same as the tropical year. The difference represents the motion of the solar apsidal line. The eccentricity of the solar orbit is constant over time. The value for the eccentricity is 1/28, that is, 0.0357. The correct value for 1600 is (2e =) 0.0337, implying that the maximum value of the solar anomaly predicted by the model will be a bit greater. This difference produces a maximum error in the solar longitude of around ±8.’ It is worth noting that, as Moesgaard shows, 10 the value for the solar eccentricity features one of the perfect numbers that are part of Longomontanus’ philosophical foundation.
Model for Mars
Longomontanus develops his model for Mars in AD, Bk. 2, Ch. 8-9, p. 341-363. It is a Tychonic model, like the model of the other planets. Mars revolves around the Sun, which in turns revolves around the fixed Earth. As I will show shortly, when Kepler arrived to work with Tycho, Longomontanus’ model for Mars assumed a planet revolving about the mean Sun, following Copernicus. But in the model of AD, the fundamental reference point of the theory is not the mean but the true Sun. This change evidences a clear Keplerian influence, explicitly recognized by Longomontanus (AD, 342-343). 11
The planetary models have two anomalies. In a Ptolemaic model, for example, the first anomaly affects the motion of the centre of the epicycle and is explained by introducing an eccentric deferent and an equant point, while the second anomaly is manifested in the retrogradations and is explained by the motion of the planet around the epicycle. In a Copernican model, the first anomaly affects the motion of the planet around the mean Sun, while the second is explained with the motion of the Earth also around the mean Sun. In the case of Longomontanus’ model, the first anomaly affects the motion of Mars around the true Sun, while the second anomaly is explained by the motion of the true Sun around the Earth. I will analyse in detail below how Longomontanus’ model deals with each anomaly.
First anomaly: the anomaly in the heliocentric motion of Mars
Ptolemy accounted for the first anomaly by introducing an eccentric and an equant point. That is, the centre of the circle on which the epicycle’s centre moves is not the Earth (eccentric), and the centre of uniform motion of the epicycle’s centre is neither the Earth nor the eccentric point (equant point). For all the planets except Mercury, Ptolemy bisects the eccentricity: the centre of the deferent is located midway between the equant point and the Earth. Copernicus rejects the equant point and replaces it either by two epicycles with a concentric deferent in the Commentariolus, 12 or by one epicycle revolving on an eccentric deferent in De Revolutionibus, with both models being geometrically equivalent and approximating very well the equant model. 13 Longomontanus will use his own version of the two-epicycle model. 14
Longomontanus’ model for the first anomaly is represented in Figure 1. S is the true Sun and M is Mars. The Earth is not represented in the diagram. SA is the apsidal line. A is the aphelion of Mars. C moves uniformly around S with the mean motion of Mars. C is the centre of an epicycle with radius CE = r1. E revolves around C so that CE is always parallel to the apsidal line. SD is equal to CE. Consequently, E revolves uniformly around D. The epicycle with centre in C simply replaces the eccentricity. M revolves around E in a small epicycle of radius EM = r2 in the same direction that C around S but with double speed, so that CEM = 2ASC. The initial configuration is such that when E is at the apsidal line, both in aphelion and perihelion, M will be pointing “down” in the figure, and so the two positions of Mars on the apsidal line will be equidistant not from D, but from F, which is below D by a distance equal to r2. The eccentric dotted line represents the trajectory of Mars. F is the effective centre of the trajectory. The path that results from the combination of the deferent and two epicycles is no longer circular, but slightly oblong, with the long axis at right angles to the apsidal line. However, F is the point exactly midway between the aphelion and perihelion along the apsidal line, so I will refer to it as the effective centre. Distance SF = e1 = r1 – r2. DQ = DF = r2, so that D bisects QF. It can be demonstrated that the motion of M is uniform measured from Q (Evans, 1988), which, thus, works like a hidden equant point. Because CE is always equal in magnitude and parallel to SD, also SC and ED are equal and parallel. Therefore, because C moves uniformly as seen from S, E also moves uniformly as seen from D. Now, angles QDE and DEM are equal 15 and QD and EM are equal. Consequently, QM is always parallel to DE and, so, M moves uniformly from Q. Distance FQ = e2 = 2r2. Finally, e1 + e2 = r1 + r2.16
The two-epicycle model for the first anomaly in AD. S is the Sun; M, Mars; and A is the aphelion. C revolves around S uniformly at the rate of the mean longitude of Mars. E revolves around C with the same speed but opposite direction that C around S, and M revolves around E with a speed that doubles the speed of C around S. The model is such that D is the centre of uniform motion of E; Q, the hidden equant, is the centre of uniform motion of M; and F is midway between aphelion and perihelion on the apsidal line so that SF is the real eccentricity.
As I mentioned, this model is very similar to that of the Commentariolus, but in the case of Copernicus’ model, the radii of the epicycles are selected so that e1 is equal to e2, fulfilling the Ptolemaic bisection. If e1 = e2, then r1 = 3r2. In the case of the other two outer planets, Longomontanus also assumes the proportion 3 to 1 for the radii of the epicycles. Nevertheless, in the case of Mars, Longomontanus says that the proportion is 4 to 1. He puts r1 + r2 = 18,550 parts, where the radius of the orbit of Mars, R, is 100,000 parts. (From now on, every time I introduce parts (p) I assume that True = 100,000 p). Because the proportion is 4 to 1, r1 will be 14,840 p and r2, 3710 p. The fact that the ratio r1/r2 is not 3 to 1 requires that F does not bisect SQ, that is, that e1 ≠ e2. In the AD model, therefore, the distance between the Sun and the hidden equant point (SQ) is not bisected by the effective centre of Mars trajectory (F).
This not-bisected model has a precursor in the hypothesis that Kepler says that Tycho and Longomontanus were working on when he arrived to work with Tycho at Prague. 17 I will call this model the 1600 model, for it was the model on which Longomontanus was working in 1600.
As I mentioned above, Kepler enumerates the parameters of the 1600 model. He says that Longomontanus assumed (1) r1 = 16,380 p, (2) r1 + r2 = 20,160 p, (3) the longitude of the apogee of Mars in 1585 = 143° 45,’ and (4) the mean longitude is obtained adding 1.75’ to the value of the Prutenic Tables. 18 The first two data confirm that Longomontanus had already abandoned the bisection when Kepler arrived because if r1 + r2 = 20,160 p and r1 = 16,380 p, then r2 = 3780 p and r1/r2 = 4.5, not 3.
The value for the longitude of the apogee that Kepler gives shows that Longomontanus was still working with the mean and not the true Sun, because the longitude of the apogee when the apsidal line goes through the mean Sun for 1585 was 143,38°, while the longitude of the apogee when the apsidal line goes through the true Sun was 148,42°.
In AD, Longomontanus still keeps the non-bisected eccentricity, but changes the radii of the epicycles and the proportion between them. He finds that r1 = 14,840 p and r2 = 3710 p, so r1 + r2 = 18,550 p and r1/r2 = 4 19 (see Table 1). The main reason for the reduction of the total eccentricity must be looked for in the change of the apsidal line, from the mean Sun in the 1600 model to the true Sun in AD model.
Values of the radii of the two epicycles and their corresponding relations and values of the eccentricities. All the values are given in parts (except the ratios).
There is a mistake in J. Kepler, W.H. Donahue, op. cit. (Note 3), p. 251. The value is 13,680 p instead of 16,380 p. The 3780 p is correct, but then he obtains 9900 p for e1 instead of 12,600 p. The correct values are given in p. 185.
These are the values of J. Kepler, W.H. Donahue, op. cit. (Note 3), p. 269. In the following page Kepler introduced slightly modified values using the tangent function instead of the sine: 14,988 p and 3,628 p. He explicitly says that the values are close to those of the Tychonic model.
See Figure 2: T is the Earth; MS, the mean Sun revolving around the Earth; S is the true Sun revolving in an epicycle with centre at MS; M is Mars, and O is the centre of the orbit of Mars. Therefore, OMS is the apsidal line referred to the mean Sun and OS the apsidal line related to the true Sun. Finding the longitude of OS is a simple exercise in trigonometry. Actually, it seems that this very exercise was the first that Kepler undertook, not without troubles, under Longomontanus’ supervision early in 1600. 20
The two possible apsidal lines depending on taking the true or the mean Sun as the centre of the motion of the planet. T is the Earth; M, Mars that revolves around O. MS is the mean Sun revolving around the Earth, and S is the true Sun revolving in an epicycle with centre in MS. OS is the apsidal line that goes through the true Sun and OMS the apsidal line that goes through the mean Sun.
Introducing the values that Kepler used in this calculation: (1) distance S-MS (solar eccentricity with respect to the mean Sun = 2350 p), (2) distance O-MS (Mars eccentricity with the mean Sun = 20,160 p), and (3) the angle at MS between the two apsidal lines (around 48°), one obtains that OS is 18,669 p, the same value that Kepler found in his first exercise, and close to the 18,550 that Longomontanus proposed in AD. But the shift of the apsidal line cannot explain the change in the ratio. There must be another unknown reason, either based on new observations, or in some aesthetic relation between the numbers. In any case, it is a shame that Longomontanus changed the ratio, because the ratio of the 1600 model was better than that of AD. Actually, the 1600 model ratio was almost optimal. 21 According to Maeyama, 22 the optimal ratio e1/e2 for matching the longitudes of oppositions all around the orbit is 4.99/3 while that of the 1600 model ratio was 5/3. According to Kepler’s own testimony, it took him several years and around 70 iterations to obtain the values for his vicarious hypothesis. 23 The ratio of Kepler’s values (e1/e2 = 4.7/3) is a bit better than the value of AD. But it is interesting to note that it is worse than the value of the 1600 model, a value that he knew as soon as he arrived to work with Tycho.
Analysis of the accuracy of the first anomaly
See the graph in Figure 3. The maximum error produced in the heliocentric longitude of Mars, assuming the two-epicycle model and the values proposed in AD, is around 6.65’. This error is similar to the one produced by the values of the Keplerian vicarious hypothesis (assuming, again, a two-epicycle model, even if Kepler always worked with an equant model), which is 5.72’. The maximum error with optimal values (taken from Maeyama 24 ) is around 2’, very similar to the error produced by the 1600 model. Now, 1’ of this error is due to the fact that the two-epicycle model and the equant model are not perfectly equivalent. 25 Therefore, of the 6’ of maximum error that the values of AD model produce, 1’ is due to the fact that Longomontanus uses a two-epicycle model instead of an equant model, making the 1’ error unavoidable. The remaining 4’ error is caused by the not-optimal values used, an error that, if he had retained the proportion of the 1600 model, would have disappeared.
Error in heliocentric longitude of Mars expressed in minutes of arc as a function of the mean anomaly of Mars assuming the values of Table 1 and a two-epicycle model.
Second anomaly: the anomaly due to the annual motion
As I have mentioned previously, the retrograde motion of Mars and the other outer planets is explained in a Tychonic model by the annual motion of the Sun around the Earth. Therefore, in principle, if one added the annual motion of the Sun to the two-epicycle model, one should obtain the geocentric longitude of Mars. Figure 4 shows the error in the geocentric longitudes of Mars from 1580 to 1600 calculated using the Alfonsine tables, the Prutenic tables and the model of Longomontanus just described. Longomontanus’ values are certainly not much better than the values calculated using the Alfonsine or Prutenic tables. Nevertheless, the error in opposition is almost 0, when the error of both the Prutenic and Alfonsine tables is the biggest. This is something expected, because the model predicts the heliocentric longitudes very well.
Error in the longitude of Mars from 1580 to 1600 calculated using the Alfonsine tables, the Prutenic tables, and the model of Longomontanus without introducing corrections to the second anomaly.
Figure 5 plots the same error but as a function of the mean anomaly of Mars. The graph shows that the maximum error is sometimes as big as 3°, which could certainly be improved upon. For example, even a purely Ptolemaic model for Mars with optimally determined parameters has a maximum error of around 1° if applied in Ptolemy’s time. And even Ptolemy’s own Mars theory, if corrected for the approximately 1o-error in his fundamental equinox, does nearly as well. 26 According to Longomontanus, however, two corrections must be introduced to the second anomaly in the case of Mars. Both corrections introduce a change in the Earth-Sun distance (TS); the first correction depends on the solar anomaly, while the second, on the anomaly of Mars.
The same error of Longomontanus’ model of Figure 4, but now plotted as a function of the mean anomaly of Mars. For calculating the error, I compare the longitude of the model with modern values at 994 instances, 8 days apart, starting at 2 January 1580, and finishing at 12 October 1601 (old calendar).
On the first correction: the correction due to the solar anomaly
As I have already mentioned, the model refers the mean motion of Mars to the true Sun. This implies two things when one wants to calculate the geocentric longitude of Mars. On the one hand, one must consider the true longitude of the Sun. On the other, one must take into account the true Earth-Sun distance, because, while the distance to the mean Sun is always the same, this is not the case for the distance to the true Sun. As I have said, Longomontanus assumed that the radius of the solar epicycle is 1/28 of the solar orbit. Because the radius of the deferent of the Sun is 65,495 p (AD, 347), the radius of the solar epicycle is 2339 p. Therefore, the distance of the true Sun could vary ± 2339 p from the mean distance. Still, Longomontanus finds that the variation is just ±1175 p. The variation in distance is around half of the eccentricity needed for producing the correct solar longitude. In some sense, Longomontanus is bisecting the eccentricity of the solar orbit. He says that Kepler has demonstrated that the eccentricity of the solar orbit is bisected using observations of Mars and he, again, borrows this idea from Kepler (AD, 342-343). This is the second element he borrowed from Kepler. For Longomontanus this bisection of the solar eccentricity would affect all the planets, but is only detectable in the case of Mars.
On the second correction: the correction due to the anomaly of Mars
As I have said, in the AD model, the effective centre of Mars’ trajectory (F) not bisect the distance between the Sun and the hidden equant (SQ). See Figure 6: Q is the hidden equant point and F, the effective centre of Mars trajectory. Q is approximately at the right distance (compare in Table 1 the 18,550 p vs the optimal value according to Mayeama: 18,648 p). F, however, is closer to Q than to S. This means that at the aphelion (A) Mars will be farther from the true Sun than what should be, and the opposite will happen at perihelion. The distance from the Sun to the real eccentricity in AD model is 11,130 p (see Table 1). Around 1600, Mars eccentricity (e) was 9302 p, therefore, in aphelion Mars will be around 1828 p farther than it should be and in perihelion, 1828 closer.
The two-epicycle model for the first anomaly in AD. The references are the same as those in Figure 1.
Consequently, the non-bisected-eccentricity model produces better heliocentric longitudes for Mars, but at the cost of predicting worse values for heliocentric distance of Mars. These wrong distances do not affect the heliocentric longitudes, but do affect the geocentric longitudes (except when Mars is in opposition), because they change the apparent size of the orbit of Mars. This is the cause of the problem with the “annual parallax,” which, as Kepler mentioned, got Longomontanus stuck. In AD Longomontanus suggested a solution compensating this effect by the introduction of a new variation in the Earth-Sun distance. At first sight, this is a very odd move because, in order to solve a problem in the size of the orbit of Mars related to the position of Mars in its own orbit, he introduces a variation not in the orbit of Mars but in the solar orbit.
Longomontanus explains how he obtained the parameters of the variation of the Earth-Sun distance in AD, 345-347. Kremer 27 offers a detailed explanation of Longomontanus’ calculation. But, following Longmontanus’ rationale, it is hard to see why a change in the Earth-Solar distance would solve a problem produced by a change in heliocentric distance of Mars. So, let me offer my own explanation. See Figure 7: T is the Earth and S the true Sun; A is the real centre of the orbit of Mars (in dashed lines), so that the aphelion is in J and the perihelion in K. The eccentricity of AD model, however, is greater, say at B (intentionally exaggerated) and its orbit is the dotted circle. Suppose that Mars is seen at M. Therefore, the model will predict it being at L, which is clearly wrong. Angle LTM measures the error in the geocentric longitude. A possible solution is to move the whole solar model closer to Earth, so that the Sun is now at point D, the new real eccentricity of Mars will be C, and the planet will now be at E. DE is parallel to SL, therefore, the heliocentric longitude is the same and DC is equal to SB, so that the eccentricity be the same. Therefore, at perihelion the Earth-Sun distance must contract by the segment SD. In the same way, according to the model, if the planet is at F, close to aphelion, it should be seen at N if the Sun is at S. One has to move the Sun to H and the real eccentricity to G for the planet to be at I, keeping the correct heliocentric longitude and obtaining the correct geocentric longitude. At aphelion, therefore, the Earth-Sun distance must increase by the segment HS. This explains of why Longomontanus introduces a variation in TS – increasing TS at the aphelion of Mars and diminishing it at its perihelion – in order to compensate an incorrect value in the Mars-Sun distance.
A change in the Earth-Sun distance can compensate the error in the Mars-Sun distance. S is the Sun, T is the Earth. A is the correct eccentricity of the orbit of Mars and B is the eccentricity according to the AD model (intentionally exaggerated). J is the aphelion and K the perihelion. When Mars is at M, the model predicts that it is at L, and one would have to move the Sun to D and the eccentricity to C, diminishing the Earth-Sun distance, in order to see Mars at the correct geocentric longitude, E. When Mars is at F, the model predicts that it is at N, and one would have to move the Sun to H and the eccentricity to G, increasing the Earth-Sun distance, in order to see Mars at the correct geocentric longitude, F.
Therefore, the AD model for Mars introduces two modifications in TS, depending on two variables: the solar anomaly and the anomaly of Mars. It is easy to show that the influence of each anomaly is approximately the same: we know that the maximum correction due to the solar eccentricity must be around 1175 p. We also found that the maximum error in Sun-Mars distance is approximately 1828 p in the apsidal line.
See Figure 8: S is the Sun and T is the Earth. M-M1 is the difference between the correct position of Mars (M) and the predicted position (M1) at the apsidal line, 1828 p. In order to see Mars in line TM, we must move the predicted Mars from M1 to M2 and, consequently, the Sun from S to S1. We must then calculate line SS1. We know that M1M2 = S1 S. and that MM1/ M1M2 is equal to SM/ST, that is, the ratio between the mean distance of Mars and the mean distance of the Earth: 100,000 p/65,495 p. Thus, S1 S is 1197 p. Consequently, the influence of Mars eccentricity is slightly greater than the influence of the solar anomaly. The value that Longomontanus found is also slightly greater, 1292,5 p. The proportion between the maximum variation of TS due to the anomaly of Mars and the maximum variation due to the solar anomaly in AD model is exactly 11/10.
How to calculate the influence in the Earth-Sun distance of the error in Mars-Sun distance. S is the Sun, T is the Earth, and M is Mars. If the model predicts that Mars is at M1, the Sun must be at S1 in order to predict its geocentric longitude correctly.
Curiously, Longomontanus does not offer a geometrical model producing the change of the radius of the annual orb; instead, he only offers a table for calculating the radius, depending on both anomalies. 28
The two-purpose table and the model behind it
Taking advantage to the fact that the proportion between the maximum variation of TS due to the anomaly of Mars and that due to the solar anomaly is exactly 11/10, Longomontanus elaborates one single table for obtaining both corrections to be applied to the solar distance.
The mean value of TS is 65,495 p. So, the minimum size that TS can reach is 65,495 p – 1175 p – 1292,5 p = 63,027.5 p. Longomontanus elaborates a table with just one output column but that allows you to obtain the two values to add to this minimum value for TS. If you want to know the value that you must add to introduce the correction for the solar distance depending on the solar anomaly, you use the solar anomaly as input and take the output value of the table. But if you want to know the value that you must add to introduce the correction of the solar distance depending on the anomaly of Mars, you must use this anomaly as input and multiply the output value by 11/10. In this way, with just one table, you can obtain both corrections.
The implicit formula in the table is thus
where av is the true anomaly. 29 Even if Longomontanus does not offer a geometrical device for explaining the change in the size of TS, the values of the table are consistent not with the anomaly produced by an eccentric or epicycle model, but with a Tusi couple (see Appendix 1).
Therefore, TS is affected by two corrections. See the graph in Figure 9 representing both corrections over time. The correction due to the solar anomaly has an annual period and a maximum value of 2350 p, 30 while the correction due to the anomaly of Mars has a period of around 2 years and a maximum value of 2585 p. The combination of the two sinusoidal curves produces an irregular shape in the solar distance. The graph in Figure 10 compares the real variation of TS to the variation deduced by the AD model.
The two corrections to the Earth-Sun distance proposed by AD model as a function of time. The amplitude is similar, being a bit bigger the correction depending on the anomaly of Mars and the period of the last one almost duplicates that of the correction depending on the solar anomaly.
Comparison of the correct Earth-Sun distance and the Earth-Sun distance predicted by the AD model.
Once one has the corrected TS, it is matter of pure trigonometry to calculate the geocentric longitude. The procedure is explained in Appendix 2. I have already showed that before the corrections are applied to TS, the geocentric longitudes of Mars produce errors that can reach up to 3°. I will now analyse in detail the accuracy of the final model (with the two corrections to TS incorporated).
Analysis of the accuracy of the second anomaly
The modification of the solar distance due to the solar anomaly
I compared (A) TS calculated using the AD model but only introducing the correction depending on the solar anomaly that intends to reflect the changes in the solar distance due to the solar eccentricity to (B) the solar distance according to modern values. Their maximum difference is less than 75 p. Most of the error is due to the wrong solar eccentricity that the AD model assumes (a slightly greater value, as I have said above). If we introduce the correct eccentricity (1105 p instead of 1175 p), the error has a peak only of 30 p. See Figure 11: The remaining error with peaks at quadratures can be explained remembering that the implicit model used for calculating the correction in the table is a Tusi couple, which produces a trajectory slightly wider at quadratures, while the ellipse is narrower at them.
Error in parts of the Earth-Sun distance predicted by AD model (introducing only the correction related with the solar eccentricity) and the same error assuming the correct value for e, the solar eccentricity, as a function of the mean solar anomaly. For calculating the error, I compare the longitude of the model with modern values at 994 instances, 8 days apart, starting at 2 January 1580, and finishing at 12 October 1601 (old calendar).
The introduction of the correction of TS due to the solar eccentricity improves the accuracy of the predictions to around 1.5°. See the graph in Figure 12.
Error in the longitude of Mars of the model of Longomontanus without introducing corrections to the second anomaly and introducing the correction depending on the solar anomaly as a function of the mean anomaly of Mars. For calculating the error, I compare the longitude of the model with modern values at 994 instances, 8 days apart, starting at 2 January 1580, and finishing at 12 October 1601 (old calendar).
The modification of the solar distance due to the anomaly of Mars
It makes little sense to compare the value of TS that includes the correction due to the anomaly of Mars to the real solar distance because, as we know, there is no variation in the Earth-Solar distance due to the anomaly of Mars. See again Figure 10. Still, some comparison can be made. To analyse the term that should be added to TS for compensating the error in the Mars-Sun distance see again Figure 7. TS is the correct TS and SM is the correct Mars-Sun distance when Mars is at M. SL is the Mars-Sun distance predicted by the model. SL is equal to DE. We want to obtain the value of TD that makes E to be in the line of sight of TM and compare it to the value for TD predicted by the AD model. Now, triangles TSM and TDE are similar, therefore
When we compare TD obtained in this way with the TD obtained by the AD model, we find an error between 300 and -200 p. The graph in Figure 13 shows that the AD model produces too big a TS distance at the apsidal line and a too small one at quadratures, being approximately correct at the octants. The fact that the mean error is approximately zero shows that there is not a significant problem with the parameters. The curve shows again that the problem is the Tusi-couple implied in the calculation of the correction that does not reflect the real trajectory that TS should follow in order to compensate the error in Mars-Sun distance.
Difference between the optimal correction to the Earth-Sun distance in order to compensate the error in the Mars-Sun distance and the correction predicted by the AD model. For calculating the difference I compare the values calculated at 994 instances, 8 days apart, starting at 2 January 1580, and finishing at 12 October 1601 (old calendar).
The introduction of this second correction to TS significantly improves the model by around 1° with respect to the model without this correction. See Figure 14: The final AD model for Mars, therefore, has a maximum error of only 36.22’ and a standard deviation of 10’. The main error is around oppositions. Of the maximum error, around one third is due to the fact that the values for r2 and r1 of the orbit of Mars are not optimal, another third to the second correction of TS, and the remaining third to the solar theory (mainly the big eccentricity).
Error in the longitude of Mars of the model of Longomontanus without introducing corrections to the second anomaly and introducing the two corrections as a function of Mars mean anomaly. For calculating the error I compare the longitude of the model with modern values at 994 instances, 8 days apart, starting at 2 January 1580, and finishing at 12 October 1601 (old calendar).
The accuracy in the geocentric longitudes of Mars of the final AD model is better than that of the Alfonsine tables and the Prutenic tables as Figure 15 shows. The main improvement is due to the fact that Longomontanus incorporates a correct longitude of the apogee. Moreover, the AD model is also an improvement in accuracy with respect to the optimal Ptolemaic model (which has a maximum error of 1°). See Figure 16: This is mainly due to the incorporation of the true Sun and the bisection of the solar eccentricity, the two elements that Longomontanus borrowed from Kepler. Nevertheless, it is certainly poorer than the Rudolphine Tables, which Kepler published 7 years after the first edition of AD.
Error in the longitude of Mars from 1580 to 1600 calculated using the Alfonsine tables, the Prutenic tables, and the final AD model.
Error in the longitude of Mars from 1580 to 1600 calculated using a Ptolemaic model with optimal parameters and the final AD model.
Conclusion
The model for the longitude of Mars described in AD represents Longomontanus’ final solution to the problem with which he was stuck when Kepler arrived early in 1600. In some sense, it also represents Tycho’s final model. Even if Tycho did not live to see it, and therefore, we cannot be sure what Tycho would have come up with the solution that his disciple offered, Longomontanus’ model is clearly a development of Tycho’s research project. After all, as Moesgaard
31
reminds us, Astronomia Danica was labelled as a Tychonian Almagest. Longomontanus is probably one of the most ardent defenders of circular and uniform motion: the circularity of the orbs and uniformity of the motion were, for him, simply nonnegotiable in astronomy. In AD, after mentioning the Keplerian solution to the anomalies of Mars that introduces oval or elliptical figures and implies that equal arcs of revolution will not correspond to equal times, he emphatically describes his own position: But we, moved by very serious causes, in astronomy give maximum value to and almost only contemplate Copernicus’s dictum, that the movement of the celestial bodies is perpetually equal and circular, or composed of circular movements. And we certainly cannot think otherwise because of those anomalies of Mars or some other bodies. Nor should we do it, before necessity has imposed this; since the circular figures – either simple things or the most, complicated and compound – can describe all the curved figures of this type, and even straight lines; we should not think that – I said – they have abandoned their function with respect to the all-embracing representation of celestial phenomena. Moreover, although we do not make them material, we do not hesitate to affirm that such things are real, which, armed with the strength and virtue of the centers, fulfill their innumerable conversions in equal times. (AD, 343)
His faithfulness to circular and uniform motion does not prevent him to accepting all Keplerian innovations that do not contradict the Copernicus’s dictum, as he called it. He, therefore, borrows from Kepler the idea of using the true Sun as the centre of the orbits and the bisection of the solar eccentricity. This strategy improves the accuracy of his model, as I have analysed. It could be argued that Longomontanus was as Keplerian as the Copernicus’s dictum allowed him to be.
The most original part of his model, however, is also the weakest. The non-bisected-eccentricity model introduces an error in the heliocentric distance of Mars. This error depends on the anomaly of Mars. To solve this problem, Longomontanus introduces a variation not in the Mars-Sun distance, but in the Earth-Sun distance. This modification of the solar distance is exclusive for the model of Mars. If it were applied to the models of the other planets, it would produce undesirable errors. Therefore, the true Sun will be at a certain distance from the Earth when working with the models of all planets except Mars, and, at the same, at a different distance when working with the model of Mars. This constitutes a hard to accept inconsistency in a realistic approach to the Tychonic model in which all the planets revolve around the Sun. But we have to remember that, as Swerdlow and Kremer 32 assert, Longomontanus is more concerned with offering algorithms that fit Tycho’s observations properly than with building consistent geometric models.
Anyway, even if Longomontanus’ sole goal was to find a working algorithm, it is hard to explain why, if the error depends on the anomaly of Mars, he did not introduce a modification in the orbit of Mars, and decided, instead, to modify the solar orbit. This latter would be the natural strategy because Longomontanus knows that the anomaly found depends on the position of Mars around the Sun. He could have suggested some algorithm changing the Mars-Sun distance (based on, for example, a Tusi-couple), while keeping the correct heliocentric longitude, in the very same way that he actually did for the solar distance. Other astronomers before and after AD offered similar strategies. Giovanni Magini, for example, developed a method in 1614 in which he calculates the heliocentric longitude of Mars using the vicarious hypothesis and the Mars-Sun distance from a pseudo-ellipse. 33 And Nicolaus Mercator, in his Hypothesis Nova of 1664, proposes using a vicarious hypothesis for calculating the heliocentric longitudes and an ellipse for the orbit of Mars. 34 The reason for deciding to modify the solar model instead of the Martian model might have been his conviction that the accuracy that his non-bisected two-epicycle model achieved in the prediction of longitudes at oppositions proved that he already had the correct heliocentric model for Mars and, consequently, the error could only be in the solar theory. Still, this is a mistake, for the success in the prediction of longitudes at oppositions only proves that you have a heliocentric theory for Mars that adequately predicts the heliocentric longitudes. Nothing is said, however, about the distance of Mars from the Sun, because at oppositions the Mars-Sun distance is irrelevant, for they are aligned with the Earth. Kepler was clever in realizing that the problem in the vicarious hypothesis was not in the solar distance of the Earth, but in that of Mars, restoring, in the first place, the bisection of the eccentricity, and then, changing the shape of the orbit from circular to elliptical. If this mistake is not the reason for Longomontanus’ second correction to the Earth-Sun distance, the explanation might be looked for in his desire to depart from Kepler’s solution.
In sum, Longomontanus’ model for Mars can be considered a step forward with respect to a Ptolemaic or even a Copernican model, when analysing the accuracy of the predictions only. Part of the improvement of the accuracy is certainly due to the introduction of some Keplerian innovations, while another important part depends on the adoption of the non-bisected-eccentricity model that produces accurate heliocentric longitudes for Mars. The particular way in which Longomontanus solved the problems that a non-bisected model poses in the Sun-Mars distance is probably the most deluding part of his proposal. It serves, however, by contrast to underscore the ingenuity, elegance, and simplicity of the solution proposed by the other astronomer that tried to solve the same problem after the meeting at Prague of 1600, Johannes Kepler.