The planetary increase of brightness during retrograde motion: An explanandum constructed ad explanantem

In Ancient Greek two models were proposed for explaining the planetary motion: the homocentric spheres of Eudoxus and the Epicycle and Deferent System. At least in a qualitative way, both models could explain the retrograde motion, the most challenging phenomenon to be explained using circular motions. Nevertheless, there is another explanandum: during retrograde motion the planets increase their brightness. It is natural to interpret a change of brightness, i.e., of apparent size, as a change in distance. Now, while according to the Eudoxian model the planet is always equidistant from the earth, according to the epicycle and deferent system, the planet changes its distance from the earth, approaching to it during retrograde motion, just as observed. So, it is usually affirmed that the main reason for the rejection of Eudoxus' homocentric spheres in favor of the epicycle and deferent system was that the first cannot explain the manifest planetary increase of brightness during retrograde motion, while the second can. In this paper I will show that this historical hypothesis is not as firmly founded as it is usually believed to be.     

Celestial Measurement in Babylonian Astronomy

Late Babylonian astronomical texts contain frequent measurements of the positions of the Moon and planets. These measurements include distances of the Moon or a planet from a reference star and measurements of the position of celestial bodies within a sign of the zodiac. In this paper, I investigate the relationship between these two measurement systems and propose a new understanding of the concepts of celestial longitude and latitude in Babylonian astronomy. I argue that the Babylonians did not define latitude using the ecliptic but instead considered the Moon and each planet to move up or down within its own band as it travelled around the zodiac.     

A Study of Babylonian Observations of Planets Near Normal Stars

The present paper is an attempt to describe the observational practices behind a large and homogeneous body of Babylonian observation reports involving planets and certain bright stars near the ecliptic (Normal Stars). The reports in question are the only precise positional observations of planets in the Babylonian texts, and while we do not know their original purpose, they may have had a part in the development of predictive models for planetary phenomena in the second half of the first millennium B.C. The paper is organized according to the following topics: (I) Sections 1–3 review the format of the observations and the texts in which they are found; (II) Sections 4–6 discuss the composition of the Normal Star list; (III) Sections 7–8 concern the orientation of the reported celestial directions from star to planet; (IV) Sect. 9 concerns the relationship between the reported distances and the actual angular distances between planet and star; and (V) Sect. 10 discusses the reports of planetary stations, which are the most common reports giving precise locations of planets when they are not near their closest approach to stars, and draws some brief general conclusions about the utility of the Babylonian observations for estimating planetary longitudes and calibrating models in antiquity.I wish to thank Lis Brack-Bernsen, John Britton, Peter Huber, Hermann Hunger, Teije de Jong, Norbert Roughton, John Steele, and Noel Swerdlow for comments on drafts of the paper, for access to work before publication, and for help in various forms.     

The Mathematics of the Area Law: Kepler's Successful Proof in Epitome Astronomiae Copernicanae (1621)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (Received April 19, 2002) Published online April 2, 2003 Communicated by N. M. Swerdlow     

Mean Motions in Ptolemy’s <Emphasis Type="Italic">Planetary Hypotheses</Emphasis>

In the Planetary Hypotheses, Ptolemy summarizes the planetary models that he discusses in great detail in the Almagest, but he changes the mean motions to account for more prolonged comparison of observations. He gives the mean motions in two different forms: first, in terms of ‘simple, unmixed’ periods and next, in terms of ‘particular, complex’ periods, which are approximations to linear combinations of the simple periods. As a consequence, all of the epoch values for the Moon and the planets are different at era Philip. This is in part a consequence of the changes in the mean motions and in part due to changes in Ptolemy’s time in the anomaly, but not the longitude or latitude, of the Moon, the mean longitude of Saturn and Jupiter, but not Mars, and the anomaly of Venus and Mercury, the former a large change, the latter a small one. The pattern of parameter changes we see suggests that the analyses that yielded the Planetary Hypotheses parameters were not the elegant trio analyses of the Almagest but some sort of serial determinations of the parameters based on sequences of independent observations.     

Writing tea’s empire

The Almagest of Ptolemy (mid-second century ad) contains eleven dated reports of observations of the positions of planets made during the third century bc in Babylon and Hellenistic Egypt. The present paper investigates the character, purpose, and conventions of the observational programmes from which these reports derive, the channels of their transmission to Ptolemy's time, and the fidelity of Ptolemy's presentation of them. Like the Babylonian observational programme, about which we have considerable knowledge through cuneiform documents, the Greco-Egyptian ones were not directed towards the deduction of mathematical models of celestial motion but appear to have investigated patterns, correlations, and periodicities of phenomena. Ptolemy's immediate sources most likely were not the original series of observational records, but treatises by various astronomers of the intervening four centuries, including Hipparchus. While Ptolemy does not appear to have tampered with the wording of the reports, he faced difficulties and uncertainties in interpreting them; critically, he lacked sufficiently detailed information about the ancient calendars to be able to convert the reported dates accurately into his own chronological framework based on the Egyptian calendar.     

Aristotle, Copernicus, Bruno: centrality, the principle of movement and the extension of the Universe

This paper¹ studies the different conceptions of both centrality and the principle or starting point of motion in the Universe held by Aristotle and later on by Copernicanism until Kepler and Bruno. According to Aristotle, the true centre of the Universe is the sphere of the fixed stars. This is also the starting point of motion. From this point of view, the diurnal motion is the fundamental one. Our analysis gives pride of place to De caelo II, 10, a chapter of Aristotle’s text which curiously allows an ‘Alpetragian’ reading of the transmission of motion.In Copernicus and the Copernicans, natural centrality is identified with the geometrical centre and, therefore, the Sun is acknowledged as the body through which the Deity acts on the world and it also plays the role of the principle and starting point of cosmic motion. This motion, however, is no longer diurnal motion, but the annual periodical motion of the planets. Within this context, we pose the question of to what extent it is possible to think that, before Kepler, there is a tacit attribution of a dynamic or motive role to the Sun by Copernicus, Rheticus, and Digges.For Bruno, since the Universe is infinite and homogeneous and the relationship of the Deity with it is one of indifferent presence everywhere, the Universe has no absolute centre, for any point is a centre. By the same token, there is no place that enjoys the prerogative of being—as being the seat of God—the motionless principle and starting point of motion.     

Babylonian solar theory on the Antikythera mechanism

This article analyzes the angular spacing of the degree marks on the zodiac scale of the Antikythera mechanism and demonstrates that over the entire preserved 88° of the zodiac, the marks are systematically placed too close together to be consistent with a uniform distribution over 360°. Thus, in some other part of the zodiac scale (not preserved), the degree marks have been spaced farther apart. By contrast, the day marks on the Egyptian calendar scale are spaced uniformly, apart from minor errors. A solar equation of center is apparent which rises by nearly 2.7° over the preserved portion of the zodiac. The placement of the degree marks indicates that, in the preserved portion of the zodiac, the Sun was considered to run at a uniform pace of about 30° per synodic month, which is consistent with the Sun’s speed in the fast zone of the Babylonian solar theory of System A.

     

Graphical Choices and Geometrical Thought in the Transmission of Theodosius’ Spherics from Antiquity to the Renaissance

Spherical geometry studies the sphere not simply as a solid object in itself, but chiefly as the spatial context of the elements which interact on it in a complex three-dimensional arrangement. This compels to establish graphical conventions appropriate for rendering on the same plane—the plane of the diagram itself—the spatial arrangement of the objects under consideration. We will investigate such “graphical choices” made in the Theodosius’ Spherics from antiquity to the Renaissance. Rather than undertaking a minute analysis of every particular element or single variant, we will try to uncover the more general message each author attempted to convey through his particular graphical choices. From this analysis, it emerges that the different kinds of representation are not the result of merely formal requirements but mirror substantial geometrical requirements expressing different ways of interpreting the sphere and testify to different ways of reasoning about the elements that interact on it.     

The Concept of Existence and the Role of Constructions in Euclid's Elements

 This paper examines the widely accepted contention that geometrical constructions serve in Greek mathematics as proofs of the existence of the constructed figures. In particular, I consider the following two questions: first, whether the evidence taken from Aristotle's philosophy does support the modern existential interpretation of geometrical constructions; and second, whether Euclid's Elements presupposes Aristotle's concept of being. With regard to the first question, I argue that Aristotle's ontology cannot serve as evidence to support the existential interpretation, since Aristotle's ontological discussions address the question of the relation between the whole and its parts, while the modern discussions of mathematical existence consider the question of the validity of a concept. In considering the second question, I analyze two syllogistic reformulations of Euclidean proofs. This analysis leads to two conclusions: first, it discloses the discrepancy between Aristotle's view of mathematical objects and Euclid's practice, whereby it will cast doubt on the historical and theoretical adequacy of the existential interpretation. Second, it sets the conceptual background for an alternative interpretation of geometrical constructions. I argue, on the basis of this analysis that geometrical constructions do not serve in the Elements as a means of ascertaining the existence of geometrical objects, but rather as a means of exhibiting spatial relations between geometrical figures. (Received January 1, 2002) Communicated by A. JONES Dedicated to the memory of Yonathan Begin     

Studies on Babylonian goal-year astronomy II: the Babylonian calendar and goal-year methods of prediction

This paper is the second part of an investigation into Babylonian non-mathematical astronomical texts and the relationships between Babylonian observational and predicted astronomical data. Part I (Gray and Steele 2008) showed that the predictions found in the Almanacs and Normal Star Almanacs were almost certainly made by applying Goal-Year periods to observations recorded in the Goal-Year Texts. The paper showed that the differences in dates of records between the Goal-Year Texts and the Almanacs or Normal Star Almanacs were consistent with the date corrections of a few days which, according to theoretical calculations, should be added to allow for the inexactness of Goal-Year periods. The current paper follows on from our earlier study to consider the effect of the Babylonian calendar on Goal-Year methods of prediction. Due to the fact that the Babylonian calendar year can contain either 12 or 13 months, a Goal-Year period can occasionally be month longer or shorter than usual. This suggests that there should in theory be certain points in the Metonic intercalation cycle where a predicted event occurs one Babylonian month earlier or later than the corresponding event a Goal-Year period later. By comparing dates of lunar and planetary records in the Astronomical Diaries, Goal-Year Texts, Almanacs and Normal Star Almanacs, we show that these month differences between the observational records and the predictions occur in the expected years. This lends further support to the theory that the Almanacs’ and Normal Star Almanacs’ predictions originated from records in the Goal-Year Texts, and clarifies how the Goal-Year periods were used in practice.     

An evolutionarily conserved mechanism for presynaptic trapping

Presynaptic differentiation takes place over three interrelated acts involving the biogenesis and trafficking of molecular complexes of active zone material, the “trapping” or stabilization of active zone sites, and the subsequent development of mature synapses. Although the identities of proteins involved with establishing presynaptic specializations have been increasingly delineated, the exact functional mechanisms by which the active zone is assembled remain poorly understood. Here, we discuss a theoretical model for how the trapping stage of presynaptic differentiation might occur in developing neurons. We suggest that subsets of active zone proteins containing polyglutamine domains undergo concentration-dependent prion-like conversions as they accumulate at the plasma membrane. This conversion might serve to aggregate the proteins into a singular structure, which is then able to recruit scaffolding agents necessary for regulated synaptic transmission. A brief informatics analysis in support of this ‘Q’ assembly hypothesis—across commonly used models of synaptogenesis—is presented.     

Geometric division problems, quadratic equations, and recursive geometric algorithms in Mesopotamian mathematics

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1).     

The Keplerian and mean motions. A geometrical study

Summary The point of reference to which the mean motion of the planets against Keplerian motion can optimally be applied and the behavior of its astronomical functions are analysed mathematically. To the results of our problem, which was solved by the first and second laws of Kepler, virtually all models of planetary motions seem to be related.     

Pluto and the platypus: An odd ball and an odd duck - On classificatory norms

Many astronomers seem to believe that we have discovered that Pluto is not a planet. I contest this assessment. Recent discoveries of trans-Neptunian Pluto-sized objects do not militate for Pluto's expulsion from the planets unless we have prior reason for not simply counting these newly-discovered objects among the planets. I argue that this classificatory controversy — which I compare to the controversy about the classification of the platypus — illustrates how our classificatory practices are laden with normative commitments of a distinctive kind. I conclude with a discussion of the relevance of such “norm-ladenness” to other controversies in the metaphysics of classification, such as the monism/pluralism debate.     

Newton's Propositions on Comets: Steps in Transition, 1681–84

Isaac Newton's closest approach to a system of the world in the critical period 1681–84 is provided in a set of untitled propositions concerning comets. They drastically revise his position maintained against Flamsteed in 1681 and may signal his adoption of a single comet solution for the appearances of 1680/1. Points of agreement and difference with the key pre-Principia texts of 1684–85 are analysed. He shows substantial control of the phenomena of tails which change very little in mechanical detail throughout his subsequent work. An emerging theory of gravitation brings planets, their satellites, and comets under the same laws of motion, yet retains a celestial vortex and includes a singular proposition in lieu of the usual formulation of Keplers area law. The analysis raises questions on a number of issues of recent Newtonian scholarship ranging from his achievement following his correspondence with Robert Hooke in 1679 to his veneration of the wisdom of the ancients. (Received September 7, 1999)     

The meaning of πλασμαтιкό<Emphasis Type="Italic">ν</Emphasis> in Diophantus’ <Emphasis Type="Italic">Arithmetica</Emphasis>

Three problems in book I of Diophantus’ Arithmetica contain the adjective plasmatikon, that appears to qualify an implicit reference to some theorems in Elements, book II. The translation and meaning of the adjective sparked a long-lasting controversy that has become a nonnegligible aspect of the debate about the possibility of interpreting Diophantus’ approach and, more generally, Greek mathematics in algebraic terms. The correct interpretation of the word, a technical term in the Greek rhetorical tradition that perfectly fits the context in which it is inserted in the Arithmetica, entails that Diophantus’ text contained no (implicit) reference to Euclid’s Elements. The clause containing the adjective turns out to be a later interpolation, that cannot be used to support any algebraic interpretation of the Arithmetica.     

Proposition 10, Book 2, in the <Emphasis Type="Italic">Principia</Emphasis>, revisited

In Proposition 10, Book 2 of the Principia, Newton applied his geometrical calculus and power series expansion to calculate motion in a resistive medium under the action of gravity. In the first edition of the Principia, however, he made an error in his treatment which lead to a faulty solution that was noticed by Johann Bernoulli and communicated to him while the second edition was already at the printer. This episode has been discussed in the past, and the source of Newton’s initial error, which Bernoulli was unable to find, has been clarified by Lagrange and is reviewed here. But there are also problems in Newton’s corrected version in the second edition of the Principia that have been ignored in the past, which are discussed in detail here.     

Rounding numbers: Ptolemy’s calculation of the Earth–Sun distance

In this article, I analyze the coincidence of the prediction of the Earth–Sun distance carried out by Ptolemy in his Almagest and the one he carried out, with another method, in the Planetary Hypotheses. In both cases, the values obtained for the Earth–Sun distance are very similar, so that the great majority of historians have suspected that Ptolemy altered or at least selected the data in order to obtain this agreement. In this article, I will provide a reconstruction of some way in which Ptolemy could have altered or selected the data and subsequently will try to argue in favor of its historical plausibility.