The Lunar Theories of Tycho Brahe and Christian Longomontanus in the Progymnasmata and Astronomia Danica

Summary

Tycho Brahe's lunar theory, mostly the work of his assistant Christian Longomontanus, published in the Progymnasmata (1602), was the most advanced and accurate lunar theory yet developed. Its principal innovations are: the introduction of equant motion for the first inequality in order to separate the determination of direction and distance; a more accurate limit for the second inequality although requiring a more complex calculation; additional inequalities of the variation and, in place of the annual inequality in Tycho's earlier theory, a reduction in the equation of time; in the latitude theory a variation of the inclination of the orbital plane and an inequality of the motion of the nodes; a reduction in the range of variation of distance, parallax, and apparent diameter. Some of these were already present in Tycho's earlier lunar theory (1599), but all were changed in notable ways. Twenty years later Longomontanus published a modified version of the lunar theory in Astronomia Danica (1622), for the purpose of facilitating the calculation through new correction tables, and also explained his reasons for parts of the theory in the Progymnasmata. This paper is a technical study of both lunar theories.

Notes

¹KGW 15.343.37–44. (Aluit, ‘employed’, could mean ‘supported’ or ‘encouraged’.) The comment is in a letter of 30 September 1606 (15.342–47) to Caspar Odontius, who had written to Kepler on 8 August (15.333–38) with a computational problem of his own, due to misunderstanding the epoch of Tycho's lunar tables, and objections to Tycho's lunar theory by Johannes Praetorius, who he quotes literally. Kepler's remark concerns Praetorius's reasonable if exasperated objection to the reduced equation of time (n. 10 below), which he answers at length. Both letters are of interest, and raise some of the points considered in this paper, as the violation of uniform circular motion and the presence of equant motion in the lunar model. Kepler returns to Tycho's lunar theory in detail in a letter to Johann Georg Herwart von Hohenburg of January 1607, answering ‘Some doubts concerning the lunar hypothesis of Tycho Brahe’, a slightly different statement of Praetorius's objections, although Praetorius is not mentioned (15.397–404), with further comments in a letter to Herwart of April 1607 (15.459–62). I had once intended to consider these letters here, but they require separate, detailed treatment, and this paper is long enough already.

²The importance of Jöstellius's publication, Lunae deliquium ad Hunc Christi annum 1599 . . ., that it contains a lunar theory differing from the theory in the Progymnasmata, was first recognized by Thoren, who published a facsimile of it with an introduction and notes (1972). There is biography of Jöstellius (1559–1611) by Christianson (2000), 297–98, and an analysis of the lunar theory and the computation of the eclipse in Swerdlow (2004). Two corrections to that paper: p. 19 l. 5, 21;10,42? should be 1;10,42?; p. 36 l. 2 from bottom, ‘hypothesis’ should probably be ‘drawing’ or ‘plan’ (hypotyposis) although ‘hypothesis’, a model, is possible.

³ Opera 8.150.14–31; the observational record of the eclipse of 31 January 1599 is in 13.162–64 and of 19 January 1581 in 10.94–99 with the use of the eclipse to find the coordinates of ? and ? Leonis. There is a biography of Longomontanus (1562–1647) by Christianson (2000), 313–19, with references to earlier literature. He is mentioned frequently in biographies of Tycho: Dreyer (1890), Thoren (1990), Christianson (2000), upon which I have relied for biographical information on Tycho and Longomontanus. The most extensive treatment of Longomontanus's work is, not surprisingly, Delambre (1821), 1.262–87, and there are two interesting articles by Moesgaard (1972, 1975).

⁴The Restoration of the fixed stars is published in Opera 3.335–77, and was the basis of the catalogue of stars in the Rudolphine Tables (KGW 10. Tables 105–29). There is a list of eighteen known manuscripts of the star catalogue and an account of its first publication by Francesco Pifferi in his Italian translation of the De sphaera of Sacrobosco with commentary in 1604 by G. Truffa (2002). The Prolegomena is printed from the single known copy in Opera 5.165–89, which contains corrections and could not have been a presentation copy; whether the work was ever received by the Emperor is not certain. A specimen of the diary for March 1599 (5.193–95) contains: (1) the mean longitude of the Sun and the Moon's mean elongation, anomaly, and argument of latitude; (2) the total lunar equation, true longitude, day to day difference, and latitude; (3) longitude of the ascending node, which moves nonuniformly, time in hours and minutes of conjunction, opposition, trine, quartile, sextile, and octant. There are also columns giving the differences of longitude, latitude, longitude of node, and aspects from the Alfonsine Tables and Copernicus, meaning the Prutenic Tables. The lunar theory is presumably the same as that used for the computation of the eclipse; I have not tested it, but the nonuniform motion of the node can only come from Tycho's theory.

⁵Kepler makes use of Longomontanus's theory of Mars, which he usually calls Tycho's theory, in various places in the Astronomia nova; the principal references are chapters 4, 7, and 8.

⁶The rule for converting from the divided epicycle to the divided eccentricity is e ₁=r ₁−r ₂ and e ₂=2r ₂. To convert from a divided eccentricity to a divided epicycle, r ₁=e ₁+ e ₂ and r ₂=% e ₂. In Astronomia Danica, for Saturn, Jupiter, and Venus Longomontanus uses the standard division r ₁/r ₂=3/1, so e ₁=e ₂ and (e ₁+e ₂)/e ₁=2/1, a bisected eccentricity. For Mars r ₁=14840 and r ₂=3710, that is, r ₁/r ₂=4/1, so e ₁/e ₂=3/2 and (e ₁+e ₂)/e ₁=5/3. For Mercury, which uses an eccentricity and epicycle, if we nevertheless call the eccentricity r ₁ since its function is the same as the larger epicycle, r ₁=5685 and r ₂=1137, so that r ₁/r ₂=5/1, and thus, e ₁/e ₂=2/1, (e ₁+e ₂)/e ₁=3/2. The ratios of small number suggest that the divisions were guesses confirmed by trial using Tycho's observations, as was probably true of the earlier division of 8/5 for Mars.

⁷After the descriptions in the Progymnasmata and Astronomia Danica, the principal literature on Tycho's lunar theory is: Riccioli (1651), 1.262–64, 282–83; Delambre (1821), 1.162–74; Anschütz (1886–87); Dreyer (1890), 337–45; Herz (1894), 107–16; Herz (1897), 68–71; Thoren (1990), 312–33, 486–96, and Thoren's articles listed in our references. I have not seen Thoren's dissertation, Tycho Brahe on the Lunar Theory (Indiana University, 1965). There are valuable notes by Dreyer in his edition of Tycho's works.

⁸It is true that in Ptolemy's complete lunar model the uniform motion of the centre of the epicycle takes place around the centre of the Earth rather than the centre of the eccentric, but this separation is only for the second inequality and is not equant motion in the sense of distinguishing the eccentricities that determine direction and distance. Ptolemy's model for the first inequality alone uses only an epicycle or eccentricity.

⁹Delambre (1821), 1.159–60, recomputed Tycho's Equation of Natural Days from its components at intervals of 2°, finding the values in the table consistently less by 0;7^h or 0;8^h plus seconds although the table is only to 0;0^h. This systematic reduction of about 0;7^h=^h—Delambre computes a mean of 0;7,36^h, which seems false precision—not from the components separately, but from the total ΔE=c _α +c _s was intentional; for some reason, Tycho believed that it was required. Since 0;7,30^h=1;52,30?, the reduction is substantial, nearly as great as c _sm=2;3,15°=0;8,13^h. The table precedes Jöstellius's computation of the eclipse, as he uses it and refers to ‘a special table of the equation of the Moon for the correction of time, in which the equations of natural days with respect to the Sun are also included’. This appears to describe a table containing the reduced ΔE, which he uses, and possibly also an unreduced ΔE=c _α +c _s, although it may describe only the reduced ΔE used for both the Moon and the Sun.

¹⁰KGW 15.343.57–63. (Adhuc abit, ‘still deviates’, might mean ‘deviates still more’.) Praetorius's objection quoted in Odontius's letter is also worth quoting (15.336.133–41): ‘He (Tycho) thinks that apparent time should be reduced to uniform (mean) time in one way for calculating the motion of the Sun, in another way for the motion of the Moon; as in the example presented by him (Opera 2.118), apparent time for the Sun is equated (reduced to mean time) by the subtraction of 0;7^h, for the Moon by the addition of 0;7,36^h, so the difference of both is 0;14,36^h. What such a thing may be, I cannot divine in whichever way I turn. By many observations (he says), I have learned etc. Why does he not apply astronomical remedies to reconciling anomalies? Why does he try to cover over either his or heaven's errors by such absurd plastering? No sensible astronomer will approve this, unless perhaps I do not know what the equation of days is and from whence it arises’.

¹¹Since the residual 0;11°–0;4;30°=0;6,30° is close to the error in the solar inequality of 0;7,10°, it does appear that it is in fact that error, showing why 0;4,30° is sufficient to compensate the entire effect of the annual equation. This may be so in computing the mean time of opposition in a lunar eclipse. However, in computing an eclipse, the equation of time is applied at the end to convert mean to apparent time, which reverses the sign of the omission of c _s so that the effect of the omission becomes opposite to the sign of c _a; consequently, the difference between c _a and c _s can then reach 0;11°–(−0;4,30?)=0;15,30°, very far from 0;7,10°. There may be a way of reconciling these relations, but I have been unable to find it. Hence, if Longomontanus found somehow that c _a should be reduced to about 0;4°, and decided to omit c _s from the equation of time in order to do so, he was creating, not solving, a problem. Another curious coincidence of numbers: the difference, or error, in measuring c _a from Leo 0° in the earlier theory instead of the solar apogee also reaches about 0;4,30°.

¹²Kepler proposed this explanation in a letter of 29 January 1599 to Herwart von Hohenburg (KGW 13.282–85) after eliminating possible causes in the motion of the Sun and Moon. He then believed it an effect on the speed of the diurnal rotation of the Earth of the solar ‘virtue’, which causes the Earth to move about the Sun. He later considered it an effect of the solar ‘illumination’, which also accounts for other inequalities in his lunar theory. However, the errors in the times of the eclipses, over an hour, were much larger than the later physical equation of time, and were due to errors in the Prutenic Tables and perhaps in Kepler's computations.

¹³What if one begins with mean time , as kept by an accurate clock (which no astronomer had)? Then, one should subtract the component due to the solar inequality, , which Tycho does not tabulate, although Kepler does in the Rudolphine Tables (KGW 10.Tables 32), where it is called the ‘astronomical part’ of the equation of time. Or one could subtract the equation of time to find the apparent time and then find the reduced mean time as before from . Note that since Tycho uses the entire ΔE for the Sun, he does consider it correct; it is only for the Moon that it should be reduced to c _?. In Astronomia Danica none of this would apply since c _s does not exist, so always

¹⁴There is in manuscript ‘Instructions for the computation of the true longitude of the Moon from modern tables’ with an example for (Julian) 9 January 1587 (Opera 9.245–47; 2.446–47 n. 117,1) based upon a form of the table in which and are tabulated, c _v called the ‘equation of the center of the epicycle’ and the ‘small equation’ (aequatiuncula), and R′ and e′ sexagesimal to three places. Allowing for differences of a few seconds in the computation, the parameters appear to be the same as in the Progymnasmata.

¹⁵Since c′+c ₂=γ or 360°– γ, the procedure described in the instructions is equivalent to the common application of the law of tangents in the formThe more complicated form in the instructions, including its specific terms, in which is called ‘inventum primum’, ‘inventum secundum’ and or ‘inventum tertium’, is that of the solution to a triangle given two sides and the included angle in Dogma IV of Tycho's De triangulis planis compendium dogmata septem (Opera 1.285–88), prepared, along with a compendium for spherical trigonometry, in 1591 for the use of his assistants at Uraniborg.

¹⁶Tycho would solve the first part, the law of cosines, using the prosthaphaeretic formula, This appears as Dogma VI, given two sides and the included angle, to find the remaining side, of his De triangulis sphaericis compendium continens dogmata novem (Opera 1.288–93).

¹⁷There are some errors of 1” in the parallaxes and 1;2,39° is a correction of 1;1,39° in the text. The distances in Tycho's earlier theory are based upon a mean distance of 59;30r _e and include the small circle of the variation. The distances from De revolutionibus have been recomputed in the Progymnasmata, but the differences are very small; the mean distance 60;29 in the text should be 60;19. The distances from the Almagest are not really comparable as all distances at quadrature are far less than at syzygy. But the point of the comparison is that the range at syzygy is 10;20r _e, about the same as De revolutionibus, and between greatest distance at syzygy 64;10r _e and least distance at quadrature 33;33r _e, the difference is no less than 30;37r _e, a variation of nearly two to one.

¹⁸ Opera 2.136; the tables for the Sun and stars are at 2.64 and 2.287. The same tables are in Astronomia Danica, 159, and the Rudolphine Tables, KGW 10.Tables 142. The modern refractions are from Newcomb (1906), 433. There is a comparison for the Sun by Delambre (1821), 1.156, which is much the same. Tycho's treatment of refraction is discussed by Thoren (1990), 226–35, who dates the investigations between 1584 for the Sun to after 1595 for the Moon. There appears to be no reason to attribute the table of lunar refraction to Longomontanus.

²⁰ Opera 2.147.4–10. Thoren (1990) 495–96, notes that the apparent reduction in solar eclipses is an effect of the solar corona, extending beyond the disc of the Sun, so that the Moon does not appear to cover the entire Sun and the eclipse appears annular. With the reduced semidiameter of the Moon, there would be no total solar eclipses, all central eclipses would be annular. It should be noted that in the period 1570–1600, no central total or annular eclipse was visible at Tycho's locations. At Wandesburg he observed an eclipse—total to the north-west, in Scotland—the morning of (Julian) 25 February 1598 until the Sun was eclipsed about 7 digits (12 is total), after which clouds covered the Sun, and he received reports of observations of from 9 to 11 digits (Opera 13.117–21). This is one of the eclipses that led Kepler to suspect an annual inequality that eventually became his physical equation of time.

¹⁹He was in Prague briefly in February 1601 (Christianson (2000), 316). The examples of eclipses using the table are a lunar eclipse of (Julian) 29 November and a solar eclipse of 14 December 1601(Opera 2.141–49), after Tycho's death on 24 October, although they were probably computed prior to the eclipses. Kepler says in the Appendix to the Progymnasmata that the apparent diameters of the Sun and Moon used in computing the solar eclipse are defective (Opera 3.321.42–322.4).

²¹ Astronomia Danica, 172–76. As part of his treatment of eclipses, Longomontanus investigates the distance, parallax, and apparent diameters of the Sun, Moon, and shadow; his changes in the table in the Progymnasmata are the reduction in the range of the Moon at opposition, the elimination of the column for the Moon at conjunction, and the addition of the small table just described. As noted above (n. 19), Kepler also considers the reduction in the apparent diameter at conjunction to be erroneous. In the Rudolphine Tables (KGW 10. Tables 92–93, 98), the range of the semidiameter of the Sun is 0;15° to 0;15,32° and of the Moon 0;15° to 0;16,22°.

²²The methods of computing and tabulating the apparent semidiameter of the shadow and its variation are explained in Swerdlow and Neugebauer (1984), 251–56, which also shows the standard methods of computing the apparent semidiameters of the Sun and Moon.

²³I have been unable to recompute the second column, ‘hourly (motion) beyond (conjunction and opposition)’, which appears to include in some way the second inequality and variation; using different methods, I have found exact agreement for some entries but differences as large as 1′ for others, and nothing is consistent. The instructions specify the first column for finding the interval between mean and true syzygy, and the second column is never mentioned. But in the specimen eclipses in 1601, the second column is used even though the interval is but 0;11,30^h in the lunar eclipse and 1;44,22^h in the solar eclipse with an iteration. Kepler corrects these to the first column in a letter to Herwart von Hohenburg of 12 November 1602 (KGW 14.305), and also corrects the seconds of the last five entries in the second column, which may show that he knows how the column is computed. Dreyer places Kepler's corrections in the table and reports the original numbers in his notes (Opera 2.448). The second column is used in the examples for finding the time of immersion and, in the solar eclipse, the interval between true and apparent conjunction, as Kepler explicitly mentions. Kepler also notes that in the instructions for confirming the interval between mean and true syzygy the phrase ob implicationem annuam (2.139.13), which he places in parentheses, is a vestige of the earlier annual equation, although its context refers only to the right ascension component of the equation of time.

²⁴The text is curiously flawed by miscopying the total equations from the primary model and then using them for the comparisons. In the first case the total equation is given as 4;40,36° and the difference as +0;0,44°; in the second as 7;25,19° and the difference as +0;0,47°. In fact, there are many errors in numbers in Astronomia Danica, some computational, some typographical, so just about everything has to be checked.