Search

In Chapter 16 of Astronomia nova, Kepler describes and applies a method for finding the parameters of what he will call the vicarious hypothesis: a model that still assumes circular orbits and an equant point, but does not assume the bisection of the eccentricity, that is, that the center of the orbit is halfway between the equant point and the Sun. The method allows Kepler to find independently both centers in a very elegant way, but its application is tedious. He confesses that he had to apply it seventy times over a period of 5 years to obtain trustable results. Years earlier, when Kepler arrived to work with Tycho, he found that Tycho and Longomontanus had rejected bisection and somehow had obtained a ratio between eccentricities that, as Kepler himself highlights, happened to be very close to the one Kepler would later find after so much effort. Kepler does not say how Tycho and Longomontanus obtained their parameters and, to the best of my knowledge, there is no single published work that attempts to answer this question. Still, it is a very interesting question to ask how they arrived at values so close to those that took so much pain for Kepler to obtain. Recently, I published a paper describing a method Tycho used for finding Saturn’s parameters. In this paper, I show that by applying this method to the data of Tychonic observations of oppositions, it is possible to arrive at parameters very close to those that we know Tycho found. In this way, I argue that this is the method Tycho applied for obtaining Mars’s parameters. The simplicity of the Tychonic method contrasts with the complexity of Kepler’s.

This paper discusses measurements of the apparent diameter and parallax of the star Sirius, made in the early 18th century by Jacques Cassini, and how those measurements were discussed by other writers. Of particular interest is how other writers accepted Cassini’s measurements, but then discussed Sirius and other stars as though they were all the same size as the sun. Cassini’s measurements, by contrast, required Sirius and other stars to dwarf the sun—something Cassini explicitly noted, and something that echoed the ideas of Johannes Kepler more than a century earlier.

The article explores Johannes Kepler’s abortive attempts to produce an opulent, decorative art object to accompany the publication of his first treatise, Mysterium Cosmographicum (1596). It was Kepler’s hope that this Credentzbecher, so-called because it was designed to resemble a large, ceremonial chalice, would valorize the significance of what he believed to be an epoch-defining discovery concerning the proportional nature of the planetary intervals and serve as a personal introduction to his local sovereign, Duke Friedrich I of Württemberg (1557–1608). The correspondences of Kepler and his circle, some of which have been reproduced and translated here for the first time, reveal in excruciating detail the struggles to negotiate the demands, and exacting standards, of the Stuttgart court and Kepler’s difficulty working with the local goldsmiths employed by the court to enact his vision. Though met with skepticism and destined for failure, the model, its design, and the misunderstandings its failure revealed, poignantly display the sometimes-insurmountable gap between artisanal knowledge and scientific ambition.

The State Archives of Belgium, in particular, the archives of Archduke Albert VII of Austria, hold a letter which possibly is a copy of the letter, or an excerpt of the letter, of Kepler to Emperor Rudolf II of October 1604 on SN1604, that is, the first letter of Kepler on the subject. Together with this letter, there are other letters on SN1604, written by Johannes Brengger and Michiel Coignet. In one of these letters, the very observation by Brengger which Kepler cites can be found. The letters are in Albert’s archive because he asked his Court Mathematician Coignet about the phenomenon. It is less clear why Albert was interested in the phenomenon, given the lack of interest in science at his court.

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.

This article discusses the reception of Kepler’s work in the earliest interpretations of the history of astronomy, which appeared in the seventeenth and eighteenth centuries. The focus is not on the reception of Kepler’s work among astronomers themselves but instead on its significance for the history of science as seen by early historians of mathematics and astronomy. The first section discusses the evaluation of Kepler in the so-called “Prefatory Histories” of astronomy that appeared in various astronomical works during the seventeenth century. In these, Kepler was considered mainly to be the person who brought the work of Tycho Brahe to completion, rather than an original astronomer. The second section is devoted to the evaluation of Kepler in interpretations of the history of astronomy that appeared in the eighteenth century (often as part of the history of mathematics). In these works, Kepler is regarded as a genius who deserves tremendous credit for the advancement of the human spirit. Both sections also devote attention to Copernicus and Tycho Brahe because this facilitates the explanation of how Kepler’s contribution was judged. By studying the reception of Johannes Kepler’s work, we may gain greater insight into the transition from a cyclical perception of the history of science to the progressive model.

The article addresses the genesis and visualization of the capstone image to Kepler’s polyhedral hypothesis of the planetary intervals from his first major work, Mysterium Cosmographicum (1596). The contention is that the famous Tabula III was directed less by Kepler than it was an initiative spearheaded by Georg Gruppenbach, the printer of Mysterium, and Kepler’s mentor Michael Mäistlin, who sought to produce a marketable broadsheet that would appeal to the contemporary German fashion for illustrations of polyhedral geometry. More generally, the article seeks to redefine the key role played by the printing workshop and the decorative arts in the theory’s inception and ultimate graphic manifestation.

This paper is devoted to Laurentius Eichstadt, a Baltic astronomer of the generation between Tycho and Hevelius. As a calendar-maker, Eichstadt used and tested the astronomical tables and the planetary theories of his elder contemporaries, Longomontanus and Kepler; as a town physician and gymnasium professor, he taught mathematics and astronomy alongside medicine and natural philosophy in Stettin and Gdańsk. Eichstadt’s indefatigable engagement with theory, practice, and teaching is marked by his continuous reassessment, adjustment, and revision of views in astronomy, physics, and metaphysics, aimed at bringing these fields in better agreement with each other and with empirical observation. Eichstadt’s critical attitude did not prevent him from remaining committed to his scholastic legacy. As a matter of fact, his creative reworking and teaching of astronomy and philosophy bear witness to the long vitality of the northern European scientific tradition rooted in Melanchthonian literacy and Aristotelian philosophy. The work and conceptions of this participant in the astronomical debates of the early seventeenth century offers us an insight into the complex interplay of technical astronomy and metaphysical discourse in a time of transition from a geometrical approach to planetary theory resting on Aristotelian metaphysics to a post-Keplerian physical–mathematical science unifying heavens and earth.

Time and celestial motion: the source of Kepler's beliefsKepler (1571-1630) was rigidly educated in the Greek classical tradition at the University of Tubingen, in preparation for his intended career as a Lutheran minister. From this tradition, he recognized and adopted the fundamental dichotomy between celestial and terrestrial matters that was instigated by Plato and promulgated with the authority of Aristotle: in this paper, we shall be concerned with the celestial sphere alone.The influence of Aristotle extended well into the Renaissance; in outline, Aristotle's world-system, as set out mainly in De Caelo Books I and II, consisted of a spherical universe in rotation (about the Earth at its centre), having the sphere of the stars as its outermost circumference. It is essential for our present purpose to emphasize that Kepler regarded the uniform rotation of the superlunary spheres as a primary feature of the heavens, envisaged as an attribute of nature (to Kepler, it would have seemed a provision of the Deity). Indeed, circular motion was closely identified with celestial perfection, and it must have come as something of a shock to ancient astronomers when they noticed, quite early on, that there were a few heavenly bodies (notably the planets, or wanderers) that did not appear to rotate uniformly in circles, but instead moved in a less ordered way against the background of the stars. This later led to increasingly sophisticated representations of planetary motion (not necessarily closer to reality) by combinations of circles in order to satisfy observations ("save the phenomena"), until Kepler's conversion to Copernicanism, when his adoption of a rigorously heliocentric approach enabled him to tackle the problems with a framework of maximum simplicity derived from the work of Ptolemy, on the basis laid out in [Figure 2].Kepler was a devout (though somewhat unorthodox) Lutheran. He was, moreover, motivated by the conviction that God's creation could be understood only through the geometry of Euclid. Like his contemporaries, he believed that the power of God was infinite, but with the additional Copernican slant that he expected the divine power to be exerted through the Sun fixed at the centre of the universe. However, since God's universe had to be comprehensible to humankind, it was simply common sense - or alternatively, in accordance with physical reality - to limit the function of the Sun in generating planetary motion to the only two powers that a single fixed body could effectively exert. These were the powers which produced motions in the two perpendicular directions regarded by Aristotle as natural: a pair of motions taking place in a plane, around a circle about its centre, and along a straight line towards or away from that centre (the Sun in this case). Moreover, these two directions happened to be precisely those that could be drawn by the only tools (compasses and straightedge) that were permitted by Euclidean convention to be used in geometrical constructions, so that we are able to employ a two-dimensional structure and restrict consideration to an exclusively kinematical account. Indeed, the treatment of motion by resolution into components was something that ensured Kepler's eventual success, because it enabled him, informally, to apply the Principle of Orthogonal Independence to deal separately with the two laws for an individual planet: Law II stating the time-measure and Law I stating the distance, which specified the path. Though both constituents were vital to the eventual synthesis - set out in the final section - it is the measurement of time (the subject of Law II), associated with circular motion, which is the topic of our interest - up to the end of the penultimate section - in this paper.We take the opportunity to explain that, nowadays, circular motion can be interpreted mathematically with equal validity, either as angular motion measured about a centre or as linear motion round the circumference of a circle. …

This essay looks at the roles in the history of the sciences of empathetic approaches to past authors. It focuses on the ways in which Arthur Koestler and other historians have engaged with Kepler and his works, and in particular with his Astronomia nova. I argue that here, as with many other historical sources, close attention to the process of composition and conventions of genre is crucial for adequate interpretation.

1. PreambleThere is an extensive secondary literature on Kepler's Rudolphine tables, including a substantial body of work on its frontispiece depiction of a temple of Urania, Muse of Astronomy (Figure 1).' By contrast, Hebenstreit's poetic description of the temple has received relatively little scholarly attention.2 In his classic history of astronomy Jean-Baptiste Delambre described it as being "in mediocre verses";3 and in his edition of the Rudolphine tables Franz Hammer remarked that if Delambre is correct in his judgement, it should be bom in mind that Hebenstreit was given little time to complete the poem.4 Why has this portion of so famous a work been so little studied? The answer, we strongly suspect, has to do not with its quality but with its difficulty. As with much poetry of the period his hexameters are replete with learned classical allusions and echoes of ancient Roman poets. Its sentence structure is sometimes convoluted. As for the interpretation of the image that it offers, it is not only complex and often ambiguous, but full of oblique references to technical aspects of Kepler's astronomy.A full study of the significances of this demanding poem would be a major undertaking. Our primary purpose here is more modest: to provide a literal translation with sufficient contextualisation and annotation to render the piece accessible to modern readers conversant with the history of astronomy. Accordingly, we first look at the circumstances of production of the poem. We next run briefly through it, pausing over some passages that cast light on its aims and strategies. Then, after a discussion of the genres of the image and poem, we offer some tentative remarks on the messages conveyed, both openly and cryptically, in this remarkable composition.2. The Context of the PoemThe Rudolphine tables were initiated by Tycho Brahe, shortly before his death in October 1601.5 The task of completing the Tables then passed to Kepler, Tycho's successor as Imperial Mathematician to Rudolph n. A host of factors conspired to delay their completion. To start with one of Tycho's heirs, his son-in-law Tengnagel, contested Kepler's free access to Tycho's observations; this was settled only in 1604. But the heirs continued to make difficulties for Kepler over the issue of authorship and profits, and in 1612 he wrote to them committing himself to acknowledgement of Tycho as first author and to submission of the manuscript to them for approval prior to printing.6 Other sources of delay included Kepler's projects for the establishment of his own Copemican world system. Then there were arrears in his salary and difficulties in obtaining funds for production of the Tables from successive emperors, that is, Rudolph until 1611, then Matthias up to 1619, then Ferdinand. From 1617 to 1621 Kepler was distracted by the protracted trial of his mother for witchcraft. Moreover, in 1617 there occurred what he described as "that happy calamity", his encounter with Napier's logarithms, which prompted him to develop his own form of logarithms for use in the Tables.1 From 1618 there was the chaos of the Thirty Years War. But for all that, by 1624 the work was effectively complete, and for the next three years Kepler battled on, first in Linz and then in Ulm, struggling to obtain imperial finance, a sufficient stock of good paper, and a competent imperially approved printer. Kepler's relations with the printer finally chosen, Jonas Saur, were stormy: in mid-production of the work Kepler accused him of dishonest and threatening attempts to extort money from him; and he seriously considered taking legal action against Saur and transferring the printing to Tubingen.8 Finally, in mid-September 1627 a few copies appeared, rushed out for the Frankfurt Book Fair.In the final phases of production Kepler ran into further difficulties with Tycho's heirs, difficulties that are directly relevant to the frontispiece image and the poem. …

In the three last centuries, especially in the nineteenth century, Johannes Kepler has played an eminent part in the German history of ideas. Largely ignored for more than 50 years after his death, due in part to the aftermath of the Thirty Years' War and in part to increasing interest in new astronomical discoveries in the solar system, Kepler would not be comprehensively considered before G. W. Leibniz reviewed his work. A century later, G. W. F. Hegel appreciated Kepler's idea of planetary harmony as an expression of belief that the universe is reigned by reason but not by chance. At that time, around 1800 idealistic philosophers, especially F. W. Schelling (1775-1854), assigned an extraordinary position to Kepler in German history.This book edited by Paul Ziche (University of Utrecht) and Petr Rezvykh (Research Higher School of Economics, Moscow) presents two main traditions of the Kepler reception of the nineteenth century: the veneration of Kepler in Wurttemberg in connection with the awaking German national identity, and the high regard of Kepler as a representative of speculative natural philosophy of German Idealism. These two moments are recognized by the authors as essential motives for the first edition of Kepler's collected works by Christian Frisch (in eight volumes, 1858-1871). Schelling and other philosophers praised Kepler's holistic view of nature in his search for the structure of the Mysterium cosmographicum , in contrast to Newton who was viewed as a representative of mechanical science with analysing and splitting procedures. This distinction between speculative and inductive science in Germany would not fade before the mid-nineteenth century, when the inductive science of William Whewell was absorbed especially by the philologist Joseph Kopp (1788-1842) in Erlangen, in context of the planned Kepler edition (pp. 115-54).Frisch's edition, Joannis Kepleri Astronomi opera omnia (KOO), is the central point of interest in the book. This first essential Kepler edition was prepared by an invisible circle of congenial contemporaries described by Ziche as a "virtual Kepler commission," referring evidently to the subsequent Kepler-Kommission (1934-2003) of the Bavarian Academy of Sciences. Key source material is the presentation of 30 documents, particularly taken from members of the "virtual Kepler commission" in the years 1839-1855 (pp. 177-270). The planning and realization of the Frisch edition is well illustrated in both scientific and political aspects. The "virtual Kepler commission" collaborated informally, a modus operandi nicely expressed by the term Sygkepleriazein in the title of the book. This expression comes from a letter of Joseph Kopp in the year 1839, inviting his brother-in-law Christian Frisch to the Kepler project: "Wir wollen so recht gemeinschaftlich keplerianisieren" ("We really want to keplerianize mutually") (p. …

Several comets and new stars drew European eyes between 1572 and 1618, a period of new worlds, warring Empires, and Christian schism. Historians of astronomy have, understandably, centred on luminaries like Kepler and Galileo to see how objects like the supernova of 1604 cracked the crystalline purity of the Aristotelian worldview. More recently, following Sarah Schechner's study of popular interest in comets (1997), we have begun to learn how such celestial novelties mattered throughout European culture. Tessicini and Boner's collection of essays spans these two approaches, connecting the history of celestial novelties with broader intellectual, political, and religious histories of early modern Europe.One theme is the ways early moderns turned to history to make sense of comets. Adam Mosley opens with a study of how the historical record was used to debate the worth of astrology. Antoine Mizauld and those inspired by Melanchthon found that history supports the inference that comets do, in fact, portend human affairs. Others, from Thomas Erastus to Thaddaeus Hegacius, disagreed, based on the same evidence as extracted from the new genre of comet catalogues. Not that astrologers grew scarce during the period; Tayra M.C. Lanuza Navarro and Victor Navarro Brotons show that commentaries on comets written in Spain were driven by astrological assumptions; history showed the validity of astrological prediction and heavenly conjunctions in turn explained the great turning points in history. As Nick Jardine argues, Christoph Rothman's Dialexis on the comet of 1585 exemplifies historia as collections of observationes .History also mattered in political and confessional battles. Boner and Francesco Barreca together unfold the rhetorical layers of Kepler's dedicatory letters to Rudolph II, showing how Kepler coupled the comet of 1604 with the rise and fall of war with the Ottomans and the Hungarian Rebellion, with the comet and conflicts all fading by 1606. Celestial novelties and their meaning for empire gave Kepler a chance to defend, at one stroke, his salary and the new Copernican theory. The science of the stars divided along confessional lines, as Isabelle Pantin shows for the Carmelite Francesco Giuntini, suspected of Lutheran sympathies for his astrology. To answer suspicions, he shored up positions associated with Gemma Frisius and Melanchthon using a longer tradition of ancient and medieval authorities, thus blurring confessional lines. Elide Casali similarly finds Christian motivations for astronomy across the period, despite great changes in the nature of astronomy. In response to the comet of 1577, Italian comet literature presented the interpreter of the stars as the Christian sapiens . But after 1618--after the study of the stars had been mathematized and mechanized, Casali contends--Italian comet literature evinced the same 'Christian astrology', so that 'religious and devotional sentiments' drove accounts of comets no less than before (131).Several contributions describe what celestial novelties meant for medicine, natural philosophy, and planetary theory. John Henry describes the French physician Jean Fernel's De abditis rerum causis , which underscored the ongoing significance of celestial influences for reformist approaches to medicine. …

Consider for a moment the problem that the inferior planets posed to early cos- mologists such as Ptolemy or Copernicus. In principle, observations of Mercury or Venus at elongation could establish with fair accuracy the size and eccentricity of their orbits, although in reality the difficulty of observing Mercury at certain critical elongations made this determination somewhat problematic, and the fairly close commensurability of the period of Venus to that of the Earth (5/8) made it almost impossible over extended periods to observe satisfactory configurations. Finding the heliocentric longitude of Mercury or Venus (equivalent to the actual position in the epicycle) was even more difficult, however, because as the planet rounded the curve near elongation it was moving almost directly toward or away from the observer. The angle of elongation gave an excellent fix on the size of the orbit (or epicycle), but the distance from the observer that could have determined the longitude of the planet in its orbit was practically unobtainable. In contrast, an observation of a transit could provide strong leverage for establishing the heliocentric longitude and could furnish a check on the other parameters as well.Transits in the seventeenth century could therefore provide the astronomer with a significant test of planetary theories, and in particular could have revealed that Kepler's astronomy, as expressed in his Tabulae Rudolphinae, was substantially superior to the alternatives.' In principle the Keplerian physics, melded with the large base of Tychonic observations, provide a sophisticated interpolating scheme for the previously unobserved parts of the orbits. One historical fact especially sup- ports the notion that transits played a critical role here. Noel Duret (Natalis Durret), a little-known French astronomer who published a set of ephemerides for 1637 to 1651, started out using the 1632 tables of the Dutch astronomer Philips Lansbergen.2 He was not quite halfway through computing his ephemerides when he noticed that the Lansbergenian-based retro-calculations for the transit of Mercury observed by Gassendi in 1631 were not nearly as satisfactory as those reckoned from Kepler's tables. So, in midstream, Duret switched the basis of his calculations, and when his ephemerides were published in 1641, he prominently explained why Kepler's methods were to be preferred.3Two more publications, from the end of the century, seem to confirm the role of the transits by using and comparing multiple tables for specific transits. Ulrich Junius in Ulm (Figure 1) calculated the circumstances of the Mercury transit of 1697 from eight different tables (including Kepler and Lansbergen), and he also computed the circumstances for earlier transits including the one in 1631 predicted by Kepler and observed by Gassendi.4 Matthew Honold followed suit for the transit of 1721, calculating in advance from six tables, but conspicuously omitting Lansbergen and the Prutenics.5 He dedicated his thesis to Junius.The archives of the Paris Observatory contain the greatest depository of seventeenth-century transit material. These records were systematically collected and arranged by Joseph-Nicolas Delisle in the 1740s during his two decades in St Petersburg. Unlike the majority of his French colleagues, Delisle was a convinced Newtonian, and a fan of Edmond Halley, who had proposed that planetary transits could help determine the solar parallax.6 Delisle hoped that by gathering the data from the Mercury transits of the previous century, he might single-handedly be able to improve the numerical value of the solar parallax. Not only that, but as a Halley disciple he had the advance printed pages of Halley's tables, which would not be finally published until 1752, well after Halley's death. Thus Delisle was not particularly interested in validating Kepler's century-old tables, but in checking out the Halley tables. Nevertheless, his well-organized files showed who had observed the various transits, even if they rarely demonstrated that those observers had any interest in checking up on Kepler. …

1. INTRODUCTIONThe late sixteenth century is one of the most interesting and exciting periods in the history of astronomy. It is a period of both consolidation and transition, as Richard Jarrell has appropriately described it: "consolidation of the mathematical techniques of Copernicus and transition from the purely mathematical account of planetary motions to a wider discussion of the actual nature of the universe."1 During the years following the publication of Copernicus's De revolutionibus in 1543, the most important developments in astronomy and cosmology took place in Northern Europe, particularly in Protestant Germany and Bohemia. Jarrell further classifies the serious practitioners of astronomy at the end of the century into three categories: "those loyal to the Ptolemaic order of the universe but willing to introduce Copernican mathematical techniques; those who had adopted some variation of the geo-heliocentric framework; and those who were confirmed Copernicans."2The humanistic educational reform initiated by Philip Melanchthon encouraged the study of mathematics and astronomy, and the University of Wittenberg became a sort of Mecca for the mathematical astronomers of the time. It was the place where Georg Rheticus, Copernicus's only direct disciple, had taught, and it became the place where Copernican astronomy was most seriously and thoroughly studied. Among those young astronomers who undertook the pilgrimage to Wittenberg was Tycho Brahe (1546-1601). But it was in Bohemia where the latter's encounter and subsequent brief period of collaboration with Johannes Kepler (1571-1630) took place. This was certainly one of the most significant events in the history of astronomy, for it involved an unprecedented fusion of theory and observation: Kepler was one of the ablest mathematicians in Europe and Tycho certainly the best observational astronomer.For our present purposes, what interests us is that both Tycho and Kepler came to work together under the patronage of Emperor Rudolph ? in Prague and, at the same time, both men worked under the shadow of a third character: Nicolaus Reimers Baer (1551-1600), also known by his Latin nickname of Ursus ('The Bear"), who had been the former Imperial Mathematician, a post in which first Tycho and then Kepler were to succeed him. During the last decade of the sixteenth century, Tycho and Ursus became engaged in a fierce and bitter priority dispute over the invention of the geo-heliocentric system of the world, a compromise between Ptolemaic geocentrism and Copernican heliocentrism, and Kepler, who worked for Tycho, was also reluctantly drawn into it.Shortly after Tycho's first diagram of the geo-heliocentric system appeared in print, Ursus published a book of his own, containing a very similar cosmological arrangement. When Tycho learned about it, he accused Ursus of having stolen the system from him in a visit the latter made to Uraniborg, his observatory on the island of Hven. Tycho initially aired his accusation only in private correspondence, but when he decided to publish a selection of his letters, the matter went public. When Ursus in turn found himself labelled a plagiarist in print, he decided to respond, and he did so by publishing a second book attacking Tycho's claims to priority and originality in the discovery of the geo-hehocentric system, and also attacking him personally. As a result, Tycho decided to take legal action against Ursus, on the grounds of plagiarism and defamation. When arrangements were being made for the upcoming trial, and relevant circumstantial evidence was being gathered by Tycho, Ursus died, so the trial never took place. Nonetheless, Tycho persevered in his pursuit of Ursus beyond the grave. Now that the author was gone, the enemy was his book, so Tycho launched a campaign against Ursus's book and also planned to publish a volume of his own containing all the relevant documents, with the aim of settling matters forever: establishing his priority, restoring his honour and reputation, and refuting Ursus's claims. …

(ProQuest: ... denotes formulae omitted.)In the work1 of Kepler (1 57 1-1630), an archetype meant some pattern or structure that was ordained by God, or, that was in accordance with the divine plan. Kepler's belief that astronomy depended above all on the primacy of observations is well-known and has been frequently applauded - notably when he abandoned an especially promising solution on his way to finding the path of Mars just because it did not match the level of accuracy of the best available observations.2 By contrast, Kepler's cosmologica! principles often depended on archetypes, which were proposed in circumstances when observations were not feasible or measurements were impractical.The purpose of this Note is to draw attention to an investigation in Kepler's Epitome Book IV (1620) in which the reasoning was entirely founded on archetypes, in a situation in which it has always been natural to suppose that Kepler had relied on observation - natural, because all his predecessors had attempted to derive similar results using observational methods (however inadequate). These attempts had arisen in connection with the sizing of the Sun-Earth-Moon system. It was only after Kepler's era, simply as a result of the technical improvement of instruments, that the problem indeed became amenable to observation, and later astronomers were able to determine the correct value for the solar parallax.3 This value was the single measurement that provided the foundation for the accurate determination of the Earth-Sun distance (and eventually for the dimensions of the universe itself).The reasoning that we present here was set out in a self-contained sub-section in Epitome IV, I, 4 (KGW, vii, p. 276 line 40 - p. 281 line 26). Kepler's own diagram of KGW, vii, p. 278 (repeated p. 280), is reproduced in Figure 1, using solid lines. We can appropriately describe it as the 'Eclipse Diagram', since it represents the situation in which the Moon exactly eclipses the Sun, viewed from the Earth - so that the radii of the Sun and the Moon appear precisely equal.It may not previously have been appreciated that the Eclipse Diagram had two separate functions, as an archetype, as well as being an observational tool. For Kepler, though the diagram was indeed drawn in accordance with observations, its archetypal significance was the over-riding consideration: he believed that the universe was the embodiment of divine principles, incorporating observations that agreed with the archetype ordained by God. Kepler set out many arguments to explain the underlying plan of the universe (KGW, vii, p. 276 line 41 - p. 277 line 28). Compellingly, it seemed to him self-evident that the positions of the Sun and Moon relative to the Earth had been arranged by the Creator for the benefit of humankind, so that "the contemplative creature" (or alternatively, "the measuring creature") would be able to calculate the dimensions of the universe (KGW, vii, p. 280 lines 1-2). Kepler set out to do just this - and the results, with citations of his own words, are collected for ease of reference in Table 1.During this limited investigation Kepler avoided the discussion of detailed measurements and concerned himself with considerations of relative size alone (KGW, vii, p. 277 lines 13-17). For instance, the question of whether the distances of the Sun or the Moon from the Earth were apogeal, or perigeal, or intermediate, was irrelevant; instead, Kepler stated what they ought to be, implying a (subsidiary) archetypal reason (KGW, vii, p. 280 lines 2-4).Throughout, we refer to "the distance between bodies" rather than "the semidiameter of the sphere" and to "the radius" (a term that was seldom, if ever, used by Kepler) instead of "the semidiameter [of the body]". This is simply to avoid the inaccuracies that have been found in translations and commentaries when 'semidiameter' is mistyped in haste as 'diameter' .4For the convenience of modern readers we shall use symbolic notation (as Kepler did not) to write the proportions as fractions, and to express Kepler's intentions more concisely by using indices to replace his extended verbal descriptions involving powers and roots. …

We construct in this paper the following argument:(1) Johannes Kepler (1571-1630) was critical of the traditional method for calculating the solar eccentricity; it was not sufficiently accurate for his needs.(2) He therefore developed his own method which was based on the variation of the apparent diameter of the Sun.(3) The results of the application of his novel method gave him confidence that the magnitude of the solar eccentricity, which Tycho Brahe (1546-1601) assumed, had to be halved.(4) This was the first substantial change in the calculated magnitude of the solar eccentricity since the time of Ptolemy.Kepler's new value of the eccentricity of the Sun is an essential piece in the revolutionary scheme of his New astronomy (1609).1In July 1600, in a letter to Herwart von Hohenburg, Kepler argued that in order to confirm his polyhedral hypothesis he needed accurate values of eccentricities and positional data.2 Kepler's insistence on accurate observational records was one of the reasons for his meeting with Tycho at Benatky Castle in April 1600. However, Tycho' s calculations of the planetary eccentricities and especially that of the Sun's circle were not sufficiently accurate for Kepler's purpose. Moreover, he realized that Tycho, like Ptolemy and Copernicus, had considered the mean motion of the Sun around the centre of its eccentric circle and not the true motion.3 Already in his Mysterium cosmographicum (1596) Kepler expressed his concerns about the disagreements between astronomical predictions resulted from the equations of the solid-sphere theories and astronomical observations. He remarked,For though the approach to the whole history of the heavenly motions is slippery, and requires lengthy and difficult observations, yet it is particularly apparent that in establishing the eccentricities and the positions of the apogee, the solar (or terrestrial) eccentricity should be the most accurately known of all.4Kepler believed that the main obstacle, which had prevented him from attaining knowledge of the exact planetary motions [exactam motuum cognitionem], was due to errors in the calculations of the eccentricities.5 He therefore decided to reconsider the value of the solar eccentricity.Kepler published his novel calculations of the eccentricity of the Sun in chapters 10 and 1 1 of his Astronomiae pars optica (1604). He made the following revealing statement which we quote at length:Ptolemy was deceived when he raised the planetary epicycles too high on the one side and pushed them down on the other, because the slowness of the one place, and the speed of the other, seemed to require as much. But the error was immediately obvious from the apparent magnitude, for the epicycles increased less at perigee than accords with such a close approach, for which reason another cause for slowing down was seized upon, which, as I have just said, Ptolemy ascribed to the circle of the equant. In the Sun, no epicycle was needed, and as a consequence this error has remained to this day. It was, however, first discovered by me, through an exact observation of the visible diameter [of the Sun], as I shall say below [in the Astronomiae pars optica], and then by Tycho's most precise observations taken of the star Mars, as I shall make plain at the proper time and place [appeared in 1609 as New astronomy ... treated by means of commentaries on the motions of the star Mars]. By both arguments it is established that the Sun recedes from us by only half of the eccentricity that was attributed to it by Albategnius and Tycho, and thus that an equant circle governs the motion in the Sun too.6Three issues are worthy of emphasis. In the first place, Kepler noted that in the Ptolemaic and, following it, in the Copernican, system the Sun's circle does not require an epicycle. Kepler explained the reason for this claim: he discovered that the apparent solar diameter required by the standard solar theory was greater than the apparent diameter of the Sun, measured at perigee, which he obtained in his observations of the Sun's disc through the year (to be discussed further by Kepler in the optical treatise of 1604, chaps. …

BEGINNINGS, QUESTIONS AND APPENDICESIn his 1973 article, Anthony Grafton described the intensive collaborative work undertaken between Johannes Kepler and his former university professor, Michael Maestlin, during the writing of Kepler's Mysterium cosmographicum. Kepler had left the University of Tubingen, Germany, to work as a mathematics teacher in Graz, Austria, where he had immediately begun to conduct his first independent research in astronomy. The book took the form of a collection of works by contributors: besides Kepler's main contribution, it contains the Narrano prima of Rheticus, which consisted of a comprehensive representation of the doctrine of Nicolaus Copernicus, with corrections and additions, followed by an appendix by Maestlin in which the latter presented an advanced version of Copernican planetary theory, the improved parameters of which he had reconstructed from Erasmus Reinhold's Prutenic tables. Grafton writes:For in the Mysterium Kepler began the work which led to his discovery of the true nature of planetary motion; it was there that he first posed the questions to which his "three laws" were the answers. Yet no one seems to have discussed the treatise which Kepler's teacher, Michael Maestlin, added to the Mysterium when he edited his student's work for publication - namely, Maestlin's appendix "On the Dimensions of the Heavenly Circles and Spheres, according to the Prutenic Tables, after the theory of Nicolaus Copernicus".1Clearly, praise is due to Kepler as the author of the Mysterium, but were Maestlin's contributions confined to the appendix, leaving it ambiguous to what extent Maestlin 'edited' Kepler's text? Kepler was able to answer his own questions only in his later works, Astronomia nova and Harmonics. And were the research questions, whose pursuit led later to the so-called 'Kepler's laws', perhaps answered differently in the Mysterium!The intense exchange of letters between Kepler and Maestlin in fact tells us more about Kepler's ignorance of astronomy than of his knowledge, and highlights the indispensable role that Maestlin played in Kepler's research. Maestlin is more than just the author of an appendix: he was responsible for working out the underlying astronomical theory of Kepler's first major work. Maestlin can, therefore, be rightly regarded as the co-author of the Mysterium, a fact that Kepler himself acknowledged in one of his letters to his former teacher.DIMENSIONS OF THE COSMOSThe fact that the Copernican Revolution changed the astronomical and physical conception of the world in many different ways is often overlooked. Displacing the Earth from the centre of the universe was only a minor change; it involved no more than a small geometrical reconfiguration, the consequences of which had been discussed in Antiquity. The consequences, of course, were crucial. Of dynamical importance is the fact that, from a geometrical viewpoint, the Earth rotates around its own axis every 24 hours. Kepler, as well as Maestlin, indicated that this alternative theory of a daily rotating celestial sphere implied that the rotating bodies would have far greater velocities. Grave difficulties then arose when it came to laying the foundations of classical mechanics, which had to describe and explain the movement of heavy bodies on the Earth's rotating surface under the influence of the Earth's heaviness. Copernicus recognized these consequences in De revolutionibus orbium coelestium (1543); however, he stressed that it was up to future researchers to deal with them. Neither Kepler nor Maestlin addressed these issues as such in the Mysterium; they examined the consequences, which were astronomically far more radical.The idea - that one can obtain much more empirical data on the positions of the planets on their way around the Sun from the movement of the Earth and from observers on the Earth - was clear to all mathematical astronomers of the time. It was impossible to deduce the sizes of the planetary orbits on the basis of empirical astronomical observations using the Ptolemaic view of the world, as the projected positions of the planets moving around the Earth (with an observer at the centre of the universe) were independent of the presumed distances of the planets from the Earth. …

David Tost, better known as Origanus (Glatz, 1558 - Frankfurt an der Oder, 1628), Professor of Greek and mathematics at the Lutheran University of Frankfurt an der Oder from 1586, was renowned in his time for his ephemerides, Ephemerides novae (Frankfurt an der Oder, 1599), for the years up to 1630, and Ephemerides Brandeburgicae (Frankfurt an der Oder, 1609), for the years up to 1655. His contribution to astronomical theory and cosmology was less important than his calculations, and it was overshadowed by the achievements of the major Copernicans of his day, Johannes Kepler and Galileo Galilei. For this reason, his name appears only rarely in the history of astronomy. Nonetheless, his views are worthy of closer consideration, in particular those concerning the planetary system and the motion of the Earth. It should be remarked that the time when he investigated these issues was an extraordinary period in the history of astronomy: the peak of his scientific activity lies between the publication of Tycho Brahe's De mundi aetherei recentioribus phaenomenis (Uraniborg, 1588) and Nicolaus Raimarus Ursus's Fundamentum astronomicum (Strasbourg, 1588), on the one hand, and Kepler's Astronomia nova (Prague, 1609) and Galileo's Sidereus nuncius (Venice, 1610), on the other.Origanus received his education in Wroclaw (Breslau) and, beginning in 1578, at the University of Frankfurt.1 Hence, it is very likely that he was well acquainted with the Wroclaw humanist milieu attached to the Italo-Hungarian man of letters Andreas Dudith-Sbardellati, the physician Crato von Krafftheim and the mathematician Paul Wittich, all of whom occupy an acknowledged place in the history of early modern scientific culture. Several British mathematicians belonged to this sodalitas as well, notably the Englishmen Henry Savile and Robert Sidney, and the Scotsmen Duncan liddel and John Craig. The last two were both professors at Frankfurt for a period and discussed astronomy not only with Dudith and Wittich but also with Tycho.2 Hence, Origanus should be seen as a participant in the intensive German debate on post-Copernican astronomy. At this stage Frankfurt was closely tied with other Lutheran universities (Wittenberg, Leipzig, Jena, Rostock, Helmstedt) that played an important role in the first dissemination of Copernicus 's astronomy3 In addition, Origanus maintained a correspondence with the imperial mathematician Kepler. For all these reasons, his cosmological considerations are to be seen in the light of the contemporary collective work on planetary theory, cosmology and celestial physics.Origanus 's Change of Mind on Astronomical HypothesesOriganus began his computations in 1592 and had completed them by 1599 when he published his Ephemerides novae.4 He relied on Reinhold's Copernican tables, Prutenicae tabulae (Tubingen, 1551), and presented his ephemerides as a useful implementation of the works of two renowned Copernicans, Johannes Stadius and Michael Maestlin. In contrast to his forerunners, Origanus was not inclined to accept the heliocentric hypotheses as true and was merely interested in the precision of heavenly predictions. He declared it openly:The only thing that helps astronomers to deal with [celestial] motions is the exact derivation of ephemerides from the tables that the most prominent scholars in this discipline recommend. In actual fact, none are presently more correct than the Prussian.5This interpretation of the Copernican hypotheses as merely mathematical tools for astronomical prediction was endorsed by scholars belonging to the so-called 'Wittenberg school', among them Oslander, Melanchthon, Reinhold and Peucer, to mention only a few.6 Origanus thought he could employ Copernican tables for the sake of exact computations and embrace the Aristotelian-Ptolemaic cosmology at the same time. In the dedicatory epistle he referred explicitly to the motion of the Sun:Who does not see that the twofold motion of the Sun (annual and daily) produces seasons (winter, spring, summer and fall), days and nights (from which nature greatly benefits) according to its varying relation and height toward the Earth? …

A CONTEMPORARY OF GALILEO Ilario Altobelli: Scienziato, Teologo, Corrispondente di Galileo Galilei. Alessandro Giostra, Francesco Merletti and William Shea (empatiabooks, Camerano, Italy, 2011). Pp. 143. euro15. ISBN 978-88-906165-0-1.This is an interesting little book making a modest contribution to our knowledge of early modern science. It contains three chapters, each more or less concerned with Ilario Altobelli, a Franciscan scholar with astronomical and astrological interests who was also, as the title states, an occasional writer of letters to major figures of his day, including Galileo, Clavius, and Magini. It is unclear how many of those letters were ever answered, making it difficult to argue for Altobelli's significance for the history of his time, but exploring his thoughts on the subjects of, first, Kepler's supernova of 1604 and, second, the publication of Galileo's Sidereus nuncius (1610), offers insights into how a minor, but well-informed and opinionated, character sought to contribute to the major issues current in his world.The first chapter, by Merletti, surveys Altobelli's life, publications, and other writings. A considerable expansion on the biographical material available in the Altobelli entry of the Dizionario biografico degli Italiani, this short chapter is a useful contribution to the literature. Its oddly discursive nature (it devotes a short section to a biography of Mauro Saraceni, who might or might not have been a teacher of Altobelli) sets the tone for the other contributions, which occasionally end up wandering far from their titles.The second chapter, by Shea, is a short essay on the early history of the telescope. Although a competent account for the most part, it is almost entirely superseded by Van Helden et al. (eds), The origins of the telescope (Amsterdam, 2010). It also has effectively no connection to Altobelli aside from the fact that after the publication of Sidereus nuncius, Altobelli asked Galileo to send him a set of lenses so that he could make his own instrument. However, Galileo never sent him lenses and there is no evidence that Altobelli ever made or used a telescope. So Shea's essay lays a background certainly relevant to Galileo's reception of Altobelli's request, but not for a scene in which Altobelli himself ever makes an entrance, despite his desire to do so. The odd sense of Altobelli's conspicuous absence, almost a negative presence, so to speak, builds. Giostra's chapter constitutes the bulk of the volume. He examines Altobelli's brief reports to Galileo on the nova of 1604 in which the former details his observations and defends the priority of his very early sighting in the evening sky of 9 October. Altobelli expresses sympathy with Galileo's public lectures in Padua on the nova and his energetic demolition of Aristotelian explanations thereof. Giostra also draws on Altobelli's much later (1610) congratulatory letter to Galileo about the discoveries detailed in the Sidereus nuncius. …

FICTIONAL TOOLS IN SEVENTEENTH-CENTURY SCIENCE Fictions of the Cosmos: Science and Literature in the Seventeenth Century. Frederique Ait-Touati, translated by Susan Emanuel (University of Chicago Press, Chicago, 201 1). Pp. ? + 261. $45. ISBN 978-0-226-01 122-6.It is a truism that nothing can exasperate us like the things or people we love. And there is much indeed to love in a book by an observant scholar immersed in some of the most bracing seventeenth-century writing about space travel, lunar and stellar prospects, and a universe in which a planetary Earth is truly conceived to move. Thus I shall begin here by admiring Frederique Ait-Touati's contributions to our understanding of literature and the history of astronomy, only afterwards glancing briefly at her book's exasperating faults.Ait-Touati surveys how the fictional tools employed by Johannes Kepler, John Wilkins, William Godwin, Cyrano de Bergerac, Bernard le Bouvier de Fontenelle, Christiaan Huygens, Robert Hooke, and Margaret Cavendish enriched and to a degree enabled science in the seventeenth century. Recognizing "the incertitude inherent in cosmological discourse", Ait-Touati sketches how fictional "poetic responses . . . were adopted to accredit a discourse with a fragile epistemological and ontological status" (p. 191). The "poetic responses" to which she refers are closely akin to thought experiments such as Einstein's famous imaginative account a century ago of "the man in the accelerated chest", which illustrates, as well as lending credibility to, his theory of relativity (p. 191). For her part, Ait-Touati aims to describe the role played in the seventeenth century by what she denominates as imaginary, theoretical, and optical voyages, principal among them the lunar tale recounted by Kepler, story-teller par excellence.Kepler's posthumous Somnium (or Dream), according to Ait-Touati, functions as a "hypotyposis" - a "vivid description of a scene, event, or situation, bringing it, as it were, before the eyes of the hearer or reader" (OED). That imagined voyage to the Moon delivers a vision of the moving Earth, an "autopsy" in that word's etymological sense of "firsthand observation". Thus the fiction of the Dream servesto dramatize, to visualize, an experiment that cannot be performed on Earth, by imagining the physical consequences of the displacement of a body in space. In this sense, the Dream's fable is a thought-experiment. The role of lunar fiction is then a textual test . . .: What would be the consequences of sending a human body into space? What would the traveler see from the Moon? . . . While it can be realized only in the form of a game of fiction, this is a game played according to the laws of physics (p. …

L'A. compare les tables astronomiques hebraiques a celles de Kepler concernant l'elongation de la lune a partir de la position du soleil

L'erreur de calcul relative a la distance de la planete Mars au soleil etablie par Kepler est en fait due a des methodes de calculs approximatives dans lesquelles Kepler n'a pas tenu compte de la forme elliptique de la planete