A Marvellous Connection: Longomontanus’ Battle With the Latitudes of Mars

Introduction

A historian of astronomy who wants to do research on the sources Ptolemy used to compose his Almagest will soon find it a difficult enterprise. For there are not many sources extant in the forms in which their authors left them. A big part of the responsibility for this historical void is due to the Almagest itself. As Toomer puts it, “[. . .] its success contributed to the loss of most of the work of Ptolemy’s scientific predecessors [. . .] by the end of antiquity, because, being obsolete, they ceased to be copied.” ¹ This seems to be the case not only for the Almagest, but for most great works in the history of science: nobody knows for certain how much of Euclid’s Elements is due to its author’s genius, and how much to – again – unknown sources. Similarly, Euclid’s and Ptolemy’s contemporaries – those mathematicians who must have undoubtedly been working on the same subjects as these giants were – are also mostly lost to history, as are their works: we know who won the race, but we have almost no idea against whom the winners were competing.

Kepler’s Astronomia Nova enjoys the same status as the Elements and the Almagest. A founding work in the modern understanding of the solar system, it is itself a sun whose brightness outshines all the surrounding stars. Published in 1622, barely a decade after Astronomia Nova, Longomontanus’ Astronomia Danica is one of those stars that, as dim as it may seem from our historical point of view, it is nevertheless in itself an enormous and impressive body which shines in its own right.

This paper will examine one aspect of Astronomia Danica: the theory of latitudes of Mars. In doing so, I am falling in the same historical trap I mentioned earlier, because the main reason for focusing on this part of the work is the very fact that it was during his fight with Mars’ latitudes that Kepler finally decided to go back over his steps and reject the non-bisected hypothesis – the famous vicaria – he had been working on for some years, and try a Ptolemaic bisected eccentricity which would ultimately suggest the elliptical nature of the orbits. Such a study, though, can serve two purposes: on the one hand, it can help us to more fully appreciate the context in which Kepler made his breakthrough: the way astronomers were tackling the same problems he was, and how they came up with novel solutions to them. On the other, we will be able to see an excellent example of the way in which the pre-Keplerian astronomy made one final attempt to attack a truly difficult problem, that is, how to account for planetary observations with one sixth of Ptolemy’s observational errors, while at the same time maintaining the same basic mathematical and theoretical tools that Ptolemy had used. Regarding Longomontanus’ work on the planets, there is little published work: a brief mention by Delambre ² and a paper by Swerdlow. ³ Recently, Carman ⁴ has explained the geometry behind Longomontanus’ second solar anomaly and the causes of its success in tackling Mars’ longitudes. As it will become apparent, my work draws heavily on this last one, and focuses on some interesting aspects of the consequences that the longitude solution in Astronomia Danica has for the latitude problems which caused Tycho, Longomontanus, and Kepler, so much trouble.

The first section of the article explains the main features of the Martian model that Tycho and Longomontanus were working on when Kepler arrived at Betnaky, focusing particularly on the reasons why that model failed at predicting Mars’ longitudes outside oppositions and its latitudes through its entire synodic period. The second section gives a short account on how Longomontanus solved the longitude problems in the Astronomia Danica. The third part, which constitutes the original section of the paper, explains how Longomontanus’ longitude solution affected the parameters which were at the heart of the latitude problem, and why these changes provide an unexpected solution to the failure of the 1600 model in latitude predictions. The last two sections are the Conclusion, and an Appendix 1 which deals with a technical aspect of Longomontanus’ latitude calculation.

The 1600 model of Tycho and Longomontanus and its problems with longitudes and latitudes

In 1597 Kepler published his Mysterium Cosmographicum. There he proposed some bold theories regarding planetary orbits, theories which, when presented to Tycho Brahe, were found by him as a “[. . .] very clever and polished speculation [. . .],” ⁵ but ones which needed to be checked against precise observations, such as the ones he had been carrying out himself for some 30 years. Spurred by the possibility of finding a more secure observational foundation for his theories, ⁶ and under the pressure of counter-reformers in Graz, Kepler finally met with Tycho on February of 1600. ⁷ When he arrived, he found himself in the midst of a research institution suc as he had not witnessed before. Although it was but a shadow of the facility Tycho had built at Uraniborg, the observatory at Betnaky, near Prague, was a fine place to study the heavens, not only because of the instruments and observational records Tycho had brought there, but also because of the highly trained staff which was employed in the many tasks the Danish astronomer had organized.

Among these co-workers was Longomontanus. As Kepler tells us, when he arrived “[. . .] the work which his [i.e. Tycho’s] aide Christian Severinus [i.e. Longomontanus] had in hand was the theory of Mars [. . .] Had Christian been treating a different planet, I would have started on it as well.” ⁸ As it is well known, Tycho’s decision to put Kepler under Longomontanus’ supervision was a momentous one, which shaped the history of astronomy. Kepler himself attributed this coincidence to Divine Providence, since “[. . .] [Mars’] motions provide the only possible access to the hidden secrets of astronomy [. . .].” ⁹

When Kepler began working on Mars alongside Longomontanus, the Danish astronomers had already developed a model: “A table of mean oppositions was worked out, starting with the year 1580. A hypothesis was invented which, it was proclaimed, represented all these oppositions within a distance of two minutes in longitude. [. . .] It was only in the latitude at achronycal positions and also the parallax of the annual orb that Christian got stuck. There was, actually, a hypothesis and table for the latitudes, but they failed to elicit the observed latitude.” ¹⁰ When Kepler talks about “the parallax of the annual orb,” he is simply refering to all Mars-mean Sun-Earth configurations outside opposition. Refer to Figure 1. Here, we have a simplified heliocentric description of the situation:

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Figure 1.

Simplified representation of the system. The mean Sun S_m is at the center of the system. Earth E lies on the line determined by S_m and Mars M, that is, determines a situation of opposition for Mars. A second position for the Earth at E₁ determines a moment outside of opposition. In those situations the heliocentric and geocentric longitudes do not coincide.

According to Kepler, the model Longomontanus and Tycho were working with in 1600 only predicted good longitudes when Mars was at mean opposition. When the Earth was, in the course of its annual motion, to either side of line S_mM, causing a parallactic effect on Mars’ longitude observations, then the model didn’t work. As the diagram also shows, at opposition the heliocentric and geocentric longitudes of Mars coincide. Thus, a model that correctly predicts longitudes at opposition can be understood as a model that correctly predicts heliocentric longitudes throughout the planet’s period. But while the model had some success regarding longitudes, it was a complete failure when it came to latitudes: not even at “achronycal positions,” that is, at oppositions, did it yield good predictions. The reasons for this lie deep in the geometric structure of the 1600 model, and are connected with the model’s failure to predict correct longitudes outside opposition.

Refer to Figure 2. There we have the 1600 model of Mars, where the mean Sun moves on its orbit centered on the Earth E, and the true Sun moves on its eccentric orbit with center B. Mars’ center of uniform motion, its equant point, lies at Q. We know that the true center of Mars’ orbit lies at D, which bisects the eccentricity SQ. This yields the solid orbit. But, as Kepler tells us, in order to achive good heliocentric predictions – or, as they said, good predictions of mean oppositions – Tycho and Longomontanus had abandoned Ptolemy’s bisectionof the eccentricity, and had instead posed a center of the orbit closer to the equant. This would give us a new center of the orbit at F, and the dashed orbit. Finally, as we mentioned, the 1600 model was still being referenced to the mean Sun. So the model’s center of the deferent would lie at G, with FG and SS _m being parallel, thus making S F F G = S m G G Q . This would give the dotted orbit.

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Figure 2.

Schematic representation of the 1600 model for Mars. The mean Sun Sm moves on an orbit centered on Earth E. The true Sun moves on its orbit, centered on B. The center of uniform motion of Mars lies at Q. The true center of Mars’ orbit lies at D, which bisects SQ. Nevertheless, Tycho and Longomontanus did not bisect the eccentricity but rather decided to put the center of the orbit closer to Q. This would yield a center of Mars’ orbit at F. But since they referenced the apsidal line to the mean Sun, their center of the orbit is actually at G. Points M, N, and O are the three positions of Mars for the three centers, in the same order.

As it can be seen, both deviations from the true position of the center of the orbit imply two different sources of error in the geocentric prediction of Mars’ orbit, given a heliocentric longitude. First, because the eccentricity is not bisected, it will predict a position N instead of the true position M. Second, because the line of apsides is referred to the mean Sun instead of the true Sun, it will add a new error, predicting a position at O. It should be noted that while the fact that they use the mean Sun instead of the true Sun is a mistake which they are still not clearly aware of, the fact that they do not bisect the eccentricity is a voluntary modification to solve the problems which, as Kepler later showed, were caused by the fact that they were using a circle and an equant ¹¹ instead of an ellipse and the area law.

While the 1600 model had some limited success regarding longitudes, it utterly failed at predicting latitudes. The reason for it can also be found in Figure 2. The change in the position of the center of the orbit not only affected the longitude of the planet by changing the line of sight of the observer from EM to EO. It also affected the distance between the mean Sun and the planet. While the real distance is S _m M, the predicted distance is S _m O. This change is essential to explaining the failure of the model to predict accurate latitudes. Refer to Figure 3.

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Figure 3.

Heliocentric representation of the latitude problems in the 1600 model. The mean Sun S_m is the center of the system, which is viewed from the side, at the plane of the ecliptic. The Earth is located at E, and Mars is at mean opposition. Points M and O represent the true and predicted positions of Mars.

In the diagram we are looking at a heliocentric representation of our problem, from the point of view of an observer on the plane of the ecliptic determined by the observer himself, the Earth E and the mean Sun S _m . The plane of Mars’ orbit is determined by the observer, Sm and M – or O. For simplicity, we will assume that Mars is at mean opposition, so Mars, the Earth, and the mean Sun lie on the same plane orthogonal to the plane of the ecliptic. It is easy to see that a variation in the distance from Mars to the mean Sun will result in a variation of the predicted latitude: while it is β if we assume a correct distance MS _m , it will be α if we assume the incorrect 1600 model distance OS _m . Given that the two distances will almost never coincide, then the model will yield very bad latitude predictions.

The introduction of the second solar anomaly to solve the longitude problems

When Longomontanus published his Astronomia Danica in 1622, Kepler’s Astronomia Nova had been in circulation for 13 years. Although prompted by the intention of presenting an alternative, more “orthodox,” solution to the problems which arose from Tycho’s observations than the one his former colleague had proposed in his epoch-making book, Longomontanus incorporates some Keplerian features, the first of which is to reference his system to the true Sun instead of the mean Sun. Now, given that the orbit of Mars was in this way fixed to the true Sun, a proper theory of the Sun was needed to account for the Earth-Sun distance. In order to achieve this, Longomontanus incorporated another Keplerian characteristic, and decided to bisect the solar orbit. ¹²

Refer to Figure 4. Longomontanus’ decision to refer the system to the true Sun meant that he now had located the center of Mars’ orbit on the true apsidal line SQ. But because he still had to somehow account for the problems caused by the use of an equant and a circular orbit, he could not locate it at the bisecting point D, which is where it really is, but at F. ¹³ This non-bisected model still had the capacity to predict fairly good – now true – oppositions, but continued to fail in the predictions of geocentric longitudes for the rest of the synodic period and for latitudes almost throughout the entire period. And these failures where for the exact same reasons as before: the fact that the center of the planet’s orbit is dislocated from D to F means that the final position of Mars will not be M, but N. Thus the line of sight from the observer at E will be EN instead of EM.

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Figure 4.

Representation of Longomontanus’ solution to the geocentric longitudes problem. When the true Sun is at S, Mars’ position is N. When it is at S′, Mars’ position is M′. The opposite happens when the true Sun moves to a position closer to E than S. The period of the Sun’s linear motion is that of Mars’ synodic period. This has the effect of making Mars to be located on line EM, thus yielding correct geocentric longitudes while maintaining the same heliocentric longitudes.

As before, the problem with longitudes is related with the problem regarding latitudes, only that now it is the distance from the true Sun to the planet the one that is being modified for the worse, from SM to SN. The effects of this modification are the same as in the 1600 model.

It is clear that the problems with the longitude prediction of the AD model have their cause in the fact that F does not bisect the eccentricity of the equant. But Longomontanus could not move the center of the orbit to D, because this would bring about other problems in the heliocentric longitudes related to the use of an equant and a circular orbit. His solution to the dilemma was very ingenious. Since his problem was that Mars was seen on line EN, all he had to do was to move the planet a distance equal to NM′ so that it would lie on line EM. To do that, he had to move the entire line of apsides in the same way, ending up with a Martian equant Q′ instead of Q, a center of the orbit F′ instead of F, and a Sun S′ instead of S. This was achieved by introducing a new solar anomaly, such that the true Sun would move farther and closer to E, with a motion coupled to the Martian synodic period, so that the line of sight EM′ would always coincide with EM. In this way, Longomontanus preserved the good heliocentric longitudes his non-bisected model predicted and at the same time solved the problem that this non-bisection produced when it came to geocentric longitudes. The model’s results are not bad at all: for the 1580 to 1600 period, which is the period when the observations the model is based on where made, the maximum error is close to 30′. ¹⁴ This means an improvement not only with respect to the Ptolemaic model with optimized parameters, which had a maximum error of almost 1°, ¹⁵ but also to the Alphonsine and Prutenic tables, which had maximum errors of about 5° and 4;30°, respectively. ¹⁶ It is nevertheless less accurate than the Rudolphine tables Kepler was going to publish some years later, which were much more accurate, and would eventually be adopted by the astronomical community. ¹⁷

Though the introduction of a second solar anomaly was indeed a major improvement in Longomontanus’ theory for the longitudes of Mars, it did nothing to correct the Sun-Mars distance problem which was at the root of the bad latitude predictions of the 1600 model. In fact, the whole idea behind the second solar anomaly was to solve the longitude problem without touching the Sun-Mars distance, because the non-bisected eccentricity, while being the cause of the bad distance, was also the cause of the good heliocentric predictions of the model, and so of its success at opposition. Thus, regarding the geocentric longitudes, the second solar anomaly solves a problem created by a bad Sun-Mars distance not through a correction of this distance, but through the introduction of a bad Earth-Sun distance.

Up to here, we have followed Carman’s study of the geometry of Longomontanus’ second solar anomaly and its implications in longitude predictions. His solution carried an error in planetary distances which helped to solve the problems he had encountered with respect to that aspect of the Martian motion. As we will see, this error will have a major impact in the latitude predictions of the model.

The longitude solution and its consequences for the prediction of latitudes

In chapter 14 of the second book of the pars altera of the Astronomia Danica, Longomontanus presents his final theory for the latitudes of the planets. He first shows how to derive the inclination of the orbit of the three superior planets with respect to the ecliptic. The data for this calculation are the values of the maximum northern and southern latitudes which each planet manifest. He then proceeds to derive the latitude his model predicts for Mars at two different times: August 10 and 24, 1593. ¹⁸ The second date corresponds to a Martian opposition registered at Uraniborg. ¹⁹ His input data for calculating the latitude for the first date are

Additionally, using the maximum latitude observations, he produces a table for the three superior planets where he indicates that

The information from (2) through (4) is given by his tables, while (5) is calculated from observation, using the value at (6).

He then produces the diagrams shown in Figure 5, which represent the situation for those dates in a helicentric framework. ²¹

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Figure 5.

Heliocentric projection of the three-dimensional situation on August 20 1593 (left) and September 3 1593 (right). Point A is the center of the orbit of the Earth, which is located close to opposition at F (left) or at opposition at E (right). Point B is the ascending node, C is Mars, and D is the orthogonal projection of C onto the ecliptic.

The diagrams in Figure 5 are combined in the three-dimensional Figure 6.

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Figure 6.

Representation of Longomontanus’ diagrams in Figure 5. The Sun is at the center A of the system. The Earth is at F for August 10, and at E for August 24. Mars C is on its inclined orbit, and its orthogonal projection on the ecliptic is D. Point B is the ascending node.

Longomontanus’ diagram is clearly a simplification, because it assumes that A is the true Sun, and makes no reference to the two complex anomalies which he had introduced for the Sun – or in this case, for the Earth –, nor to the non-bisected hypothesis he is working with for Mars. Line EA should be interpreted as the distance from the Earth to the true Sun at each date, as should line AC, but for the Mars-true Sun distance.

From (1) we know that

and from (5) we know that

(8)  a r c B C = 75 ; 18 ° .

By construction we know that

So, from these three data we can solve the spherical triangle CBD and obtain

Assuming

we obtain that

yielding a

But from (2) we know that A C = 889 , 000 , so we get

Longomontanus then says that in the plane CDA, which is orthogonal to the ecliptic, we have the right-angled triangle CDA. Given that ∠ C A D = a r c C D , then from (10) we get

And from that, and the value from (2) we can solve the triangle, and get

This value could also be obtained via Pythagoras’ theorem from (2) and (14).

Note that up to (14) Longomontanus has assumed that AC = AD, that is, that both C and D lie on a circle with center A and the radius from (2). Thanks to this assumption he can calculate the length of arcCD. But then he says that triangle CDA is a right-angled triangle, where of course the hypotenuse AC is greater than the cathetus AD. In fact, by construction we know that the latter is the correct assumption, and that the former is just used as an approximation, which becomes better the smaller ∠CAD is. Given that the inclination of Mars’ orbit is very small, the error it produces is negligible.

From (3) we get that

And given that ∠FAD is the supplementary angle to the anomaly of the annual orb ²³ then from (4) we get

So, for triangle FAD we have two sides and one angle at (16) to (18), so we can solve it and get

Lastly, for the right-angled triangle CFD we know sides CD at (14) ²⁴ and FD at (19), so we can solve it and get

which is the angle of geocentric southern latitude. The difference with Tycho’s register is just 8″! ²⁵

Afterwards Longomontanus, using the necessary input data, makes a similar calculation ²⁶ for the opposition on September 3, and obtains a southern latitude of 6;5°, while Tycho had observed 6;5,30°. ²⁷

So, it seems that Longomontanus’ second solar anomaly not only fixed the problems for the geocentric longitudes outside oppositions, but also fixed the completely useless latitude theory he had inherited from his colaboration with Tycho: in fact, for the period 1580 to 1620 the maximum error of the model with both solar anomalies is 12′. It is possible that Longomontanus himself was surprised by this success, since he qualifies the relation between his longitude and latitude theory as a a marvellous connection. ²⁸

This success is truly unexpected, since in his final AD model Longomontanus has not fixed the source of the problem the 1600 model had regarding latitudes: the incorrect Mars-Sun distance. In fact, if anything, he has seemingly introduced a new problem which was not there in the first place, that is, a very bad Earth-Sun distance prediction.

To understand the reason behind the model’s success, we will begin considering the situation for the opposition on the September 3. Refer back to Figure 6. At the moment of opposition, Earth is at E, AE is the Earth-Sun distance, AC the Mars-Sun distance, and ∠CED equals the southern latitude. Since points C, D, E, and A all lie on the same plane, it is easy to see that ∠CED is the supplementary angle to ∠CEA. For the triangle CAE we know the value of two sides AE and AC, which are given, and of ∠CAE, which was given in (1). Thus, we can solve the triangle, obtain ∠CEA and from it ∠CED, the latitude for that opposition.

Now, because what we need in triangle CAE is to get is a correct ∠CEA, it is not relevant if the values of the sides AE and AC are correct, but only the proportion between them. So a successful model for calculating the latitude at opposition doesn’t need to use as input correct values for the Earth-Sun distance and the Mars-Sun distance, but only values which hold the correct proportion between them.

For the situations outside opposition this is also the case. If we look at steps (11) to (20) of Longomontanus’ calculation of the latitude for August 20, we can express them in the following way:

If it is true that only K is important, where

and thus

then if we replace AF with the corresponding expression, we should be able to remove all instances of AC. So we begin with

then we get

and from there we get

We then distribute the square root over the members of the denominator and we get

Finally, we cancel AC, and we get the expression with the proportion K between the two planetary distances to the Sun:

So, as we can see, Longomontanus’ model doesn’t need good values for the Mars-Sun nor for the Earth-Sun distance in order to make good predictions: it only needs a good proportion between them. As we shall see, the introduction of the second solar anomaly indeed provides a correct proportion, while leaving the original – incorrect – Mars-Sun distance, thus retaining the predictive advantages that value yielded regarding Martian longitudes.

Given that the reason behind the good A F A C proportion lies within the modifications which the second solar anomaly brought about, then it is appropriate to look again in more detail the geometric relations behind the new anomaly. Refer to Figure 7.

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Figure 7.

Explanation of the second solar anomaly’s success in producing correct geocentric latitudes. Because the Sun is moved from S to S’, then Mars moves from N to M’, thus lying on line EM. This not only produces a correct geocentric longitude but it also makes that the distances between the Sun, the Earth and Mars have good proportions. The reason is that, thanks to the fact that lines S′M′ and SM are parallel, triangles EMS and EM′S′ are similar.

This is the same as Figure 4, only that here I have determined line S′E. I also removed some elements which are not necessary here. Given that the true center of Mars’ orbit is at D, and thus the true position of Mars is M, then lines SM, SE, and EM represent the correct distances between Mars, the Earth and the Sun. But we know that lines M′N and SS′ are parallel and of the same length. So lines NS and M′S′ are also parallel. This means that the triangles ESM – which is determined by the three relevant distances – and M′S′E – which is determined by the distances after the incorporation of the second anomaly – are similar. Thus, the proportions between their sides are the same. This explains why the second solar anomaly, while seeking to correct an error in the geocentric longitudes of Mars, also corrects the problem regarding the Earth-Sun-Mars distances, and thus its latitude problem.

Conclusion

In Astronomia Danica Longomontanus shows that he was a master of the mathematical astronomy of his times. Equipped with Tycho’s observational records, he probably did the best that could be done at the time without making a shift in the foundations of theoretical astronomy. His work is not only, as we said, a testament to the ingenuity of the author and the community around him, but also a good example of how a theoretical paradigm can be stretched in order to account for the phenomena, and survive in an adverse climate. Published 13 years after Astronomia Nova, Longomontanus’ Astronomia Danica continued an age-old tradition of uniform, circular motions, one that possessed both historical momentum and mathematical advantages. These allowed it to continue being used, taught and improved in astronomical circles several decades after Kepler’s reformation had been published and turned into tabular form. ²⁹ Longomontanus is, without a doubt, one of the most talented astronomers that took part in this effort to carry on with the old theoretical basis of astronomy. His attempt was, nevertheless, ultimately doomed to fail, because in Kepler there was not only another mathematician just as skilful as himself, but – and more importantly – the founder of a new way of doing astronomy, a way which Longomontanus was not ready – nor willing – to pursue.