In this article, we aim at presenting a thorough and comprehensive explanation of the mathematical and theoretical relation between all the aspects of Ptolemaic planetary models and their counterparts which are built according to Kepler’s first two laws. Our article also analyzes the predictive differences which arise from comparing Ptolemaic and these ideal Keplerian models, making clear distinctions between those differences which must be attributed to the structural variations between the models, and those which are due to the specific parameters (...) Ptolemy determined in the Almagest. We expect that our work will be a contribution for a better understanding not only of the Ptolemaic theories for planetary longitudes through a clearer perception of the way in which Keplerian features are present—or absent—in Ptolemy’s models, but also for a more balanced judgement of different aspects of the contribution of the first two laws of Kepler to the modern astronomical revolution. (shrink)

Resumen: Usualmente se ha asumido que una única distinción puede dar cuenta del rol que cumplen los conceptos en una teoría respecto de la contrastación y respecto de la explicación. Intentaremos mostrar que esta asunción es incorrecta. Por una parte, no hay razones para considerar que esta coincidencia deba darse, y por otra, como se intentará mostrar a partir de varios ejemplos, de hecho, no se da. La base de contrastación de una teoría no tiene por qué coincidir con el (...) explanandum de la teoría. Para defender este punto se asumirá el estructuralismo metateórico. Se extraerán consecuencias para la concepción metateórica presupuesta. Palabras claves: Explicación; contrastación; T-teoricidad; Estructuralismo metateórico; Distinción teórico observacional -/- Title: Explain and test Abstract: It is usually held that one distinction can account for the role that concepts play in a theory regarding both test and explanation. We will demonstrate that this assumption is incorrect. On the one hand, there is no reason to think that this coincidence should exist. On the other, this is not the case, as we will show analysing several examples. The testing basis of a theory does not have to coincide with the explanandum of the theory. To defend this point we will endorse de metatheoretical structuralism. In addition, we will consider some repercussions that this discussion has for the assumed metatheoretical framework. Key-words: Explanation; Test; T-theoreticity; Metatheoretical Structuralism; Theoretical observational distinction . (shrink)

The gravitational influence of Jupiter on Saturn produces, among other things, non-negligible changes in the eccentricity of Saturn that affect the magnitude of error of Ptolemaic astronomy. The value that Ptolemy obtained for the eccentricity of Saturn is a good approximation of the real eccentricity—including the perturbation of Jupiter—that Saturn had during the time of Ptolemy's planetary observations or a bit earlier. Therefore, it seems more probable that the observations used for obtaining the eccentricity of Saturn were done near Ptolemy’s (...) time, and rather unlikely earlier than the first century AD. Even if this is not quite a demonstration that Ptolemy used observations of his own, my argument increases its probability and practically discards the idea that Ptolemy borrowed values or observations from astronomers further back than the first century AD, such as Hipparchus or the Babylonians. (shrink)

In the third century b.c.e., Eratosthenes of Cyrene made a famous measurement of the circumference of the Earth. This was not the first such measurement, but it is the earliest for which significant details are preserved. Cleomedes gives a short account of Eratosthenes’ method, his numerical assumptions, and the final result of 250,000 stades. However, many ancient sources attribute to Eratosthenes a result of 252,000 stades. Historians have attempted to explain the second result by supposing that Eratosthenes later made better (...) measurements and revised his estimate or that the original result was simply rounded to 252,000 to have a number conveniently divisible by 60 or by 360. These explanations are speculative and untestable. However, Eratosthenes’ estimates of the distances of the Sun and Moon from the Earth are preserved in the doxographical literature. This essay shows that Eratosthenes’ result of 252,000 stades for the Earth’s circumference follows from a solar distance that is attributed to him. Thus it appears that Eratosthenes computed not only a lower limit for the size of the Earth, based on the assumption that the Sun is at infinity, but also an upper limit, based on the assumption that the Sun is at a finite distance. The essay discusses the consequences for our understanding of his program. (shrink)

Diagrams in medieval manuscripts of Greek mathematical and astronomical works can seem peculiar for a modern reader, given their persistent and widespread tendency to represent more geometric regularity than the argument requires and their usual visual inaccuracy in depicting the mathematical objects discussed in the text. Although most scholars believe that these tendencies go back to the original Greek authors, in a recent paper I argued that these odd features should not be attributed to Greek authors, but to transmission. My (...) main argument there is based on experiments made with university students who played the role of copyists. In this paper, I show that the vestiges of the changes caused by transmission are present in the very diagrams of the extant manuscripts. I analyze four diagrams taken from Aristarchus's On Sizes. (shrink)

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. (shrink)

The book of Aristarchus of Samos, On the distances and sizes of the sun and moon, is one of the few pre-Ptolemaic astronomical works that have come down to us in complete or nearly complete form. The simplicity and cleverness of the basic ideas behind the calculations are often obscured in the reading of the treatise by the complexity of the calculations and reasoning. Part of the complexity could be explained by the lack of trigonometry and part by the fact (...) that Aristarchus appears unwilling to make some simplifications that could be simply taken for granted. But an important part of the complexities is due to some unnecessary inconsistencies, as recently discovered by Berggren and Sidoli (Arch Hist Exact Sci 61:213–254, 2007). In the first part of this paper, I will try to show that some of these inconsistencies are just apparent. But the complexity of the calculations and reasoning is not the only reason that could disturb a reader of the treatise. The great inaccuracy—even for the measurement methods and instruments available at those times—of one of the three input values of the treatise is really astonishing. In the sixth and last hypothesis, Aristarchus states that the moon’s apparent size is equal to 2∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}, while the correct value is one-fourth of that. Some attempts have been made in order to explain such a big value, but all of them have problems. In the second part of this paper, I will propose a new speculative but plausible explanation of the origin of this value. (shrink)

The eclipse predictor (or Saros dial) of the Antikythera mechanism provides a wealth of astronomical information and offers practically the only possibility for a close astronomical dating of the mechanism. We apply a series of constraints, in a sort of sieve of Eratosthenes, to sequentially eliminate possibilities for the epoch date. We find that the solar eclipse of month 13 of the Saros dial almost certainly belongs to solar Saros series 44. And the eclipse predictor would work best if the (...) full Moon of month 1 of the Saros dial corresponds to May 12, 205 BCE, with the exeligmos dial set at 0. We also examine some possibilities for the theory that underlies the eclipse times on the Saros dial and find that a Babylonian-style arithmetical scheme employing an equation of center and daily velocities would match the inscribed times of day quite well. Indeed, an arithmetic scheme for the eclipse times matches the evidence somewhat better than does a trigonometric model. (shrink)

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. (shrink)

David Fabricius, a Reformed pastor in Ostfriesland, was highly regarded by Kepler as an exceptional observer, second only to Tycho Brahe. From 1596 to 1609, Fabricius engaged in extensive correspondence, exchanging numerous letters with Brahe and subsequently with Kepler. These communications also provided values for direct observations on meridian altitudes of planets and stars, as well as elongations between a planet and a star or between two stars. We provide a detailed summary of Fabricius’s observations and compare them with the (...) prediction of twenty-first-century models. The analysis indicates that under specific conditions, his observations exhibit sub-arcminute deviations in relation to those calculated from modern theories. Our findings preliminarily indicate that Fabricius’ astronomical observational abilities were comparable to, an occasionally superior to, those of Brahe himself. We provide machine-readable tables of his observations. (shrink)

Intelligent Design (ID) argues for the existence of a designer, postulating it as a theoretical entity of a scientific theory aiming to explain specific characteristics in nature that seems to show design. It is commonly accepted within the Scientific Realism debate, however, that asserting that a scientific theory is successful is not enough for accepting the extramental existence of the entities it postulates. Instead, scientific theories must fulfill additional epistemic requirements, one of which is that they must show successful novel (...) predictions. Evolutionists typically attack ID by offering cases of bad design, such as the inverted retina of vertebrates. ID defenders defend their position affirming that the inversion of the retina must be a detail of design for an as of yet unknown function. The recent discovery of such a function is celebrated by ID defenders as a triumph over evolutionists. The inverted retina case is a good candidate for a novel prediction in favor of ID. In this paper, I analyze whether this is the case. (shrink)

According to selective, retentive, scientific realism, past empirical success may be explained only by the parts of past theories that are responsible of their successful predictions being approximately true, and thus theoretically retained, or approximated, by the parts of posterior theories responsible of the same successful predictions. In this article, we present as case study the transit from Ptolemy’s to Kepler’s astronomy, and their successful predictions for Mars’ orbit. We present an account of Ptolemy’s successful prediction of Mars’ orbit from (...) Kepler’s perspective, and scrutinize whether the theoretical elements responsible for Ptolemy’s empirical success are approximately retained in Kepler. In order to give to the realist the best chances, we try different strategies. We conclude that all fail and thereby this case constitutes a prima facie strong anomaly for selective retentive realism. Structural realists may call preservation of structure to the rescue, but the existing notions of structure do not work. In absence of a new notion that works, the burden of the proof lies on the realist side. (shrink)

To assert that the ancient planetary theory proposed by Ptolemy was irrefutable – at least until the telescope discovery – is a bit of a cliché. The aim of this paper is to analyze in what sense it could be said that the epicycle and deferent model proposed by Ptolemy to explain the planetary movement is irrefutable and in what sense it is not. To do this, we will use the conceptual framework developed by the Structuralist Conception, and in particular, (...) the Moulines’ analysis of the “guiding principles”. • DOI:10.5007/1808-1711.2010v14n2p211. (shrink)

The eighth book of Martianus Capella’s famous De Nuptiis Philologiae et Mercurii deserves a prominent place in the history of astronomy because it is the oldest source that came down to us unambiguously postulating the heliocentrism of the inner planets. Just after the paragraph in which Capella asserts that Mercury and Venus revolve around the Sun, he describes a method for calculating the size of the Moon, as well as the proportion between the size of its orbit and the size (...) of the Earth. It is possible to find some descriptions of the argument in general histories of astronomy or in books dedicated to Capella’s work, but usually they do not try to make sense of the argument. Rather, they limit themselves to describe or paraphrase what Capella says. As far as I know, there is no single study of the argument itself. The explanation for this absence is simple: the calculation offers many difficulties in its interpretation, for it shows obvious inconsistencies in the steps of the argument and apparent arbitrariness in the selection of the data used. In this article, I offer an interpretation that tries to discover, behind Capella’s confusing presentation, a well-sound argument for calculating the Moon’s absolute size. Interestingly, we have no records of this argument in other sources, at least in the form described by Capella. (shrink)

Tycho's star catalog enjoyed enormous prestige for centuries due to its accuracy. The entire catalog depends on the coordinates of one single star, Hamal (α Arietis), which explains why Tycho was so scrupulous in determining its coordinates using two different methods applied to more than 50 observations, as he described in his Progymnasmata. One of them proposed an ingenious way of dealing with refraction and parallax, two factors that he knew he could not control. Selecting particular observations, he was able (...) to cancel out the effects of both refraction and parallax. Still, the entire calculation starts from the coordinates of the Sun calculated from his solar model. But Tycho's solar model assumes too large of an eccentricity, producing errors in the predictions of the solar longitude that can reach up to 8'. In this paper, I analyze Tycho's method for calculating the coordinates of α Arietis and explain how the method he proposed unintentionally avoided transferring the error of his solar model to his catalog. (shrink)

In 1975, Imre Lakatos and Elie Zahar claimed that the determination of planetary distances represents excess empirical content of Copernicus's theory over that of Ptolemy. This claim provoked an interesting discussion during the first half of the 1980s. The discussion started when Alan Chalmers affirmed that it is not correct to attribute this advantage to the Copernican system over the Ptolemaic. Other scholars criticized Chalmers's assertion, reaffirming the position of Lakatos and Zahar: one went even further, asserting that Copernicus has (...) not one but two methods for calculating distances, even though this claim was subsequently also criticized. But all participants assumed that Ptolemy has no method for calculating planetary distances. In this article, I argue that this is not correct. I argue, in fact, that Ptolemy has two independent methods for calculating the distances of some of the planets and, therefore, as far as the calculation of planetary distances is concerned, Ptolemy's system surpasses that of Copernicus. (shrink)

Tycho Brahe was not just an observer; he was a skilled theoretical astronomer, as his lunar and solar models show. Still, even if he is recognized for proposing the Geoheliocentric system, little do we know of the technical details of his planetary models, probably because he died before publishing the last two volumes of his Astronomiae Instaurandae Progymnasmata, which he planned to devote to the planets. As it happens, however, there are some extant drafts of his calculations in Dreyer’s edition (...) of Tycho’s Opera Omnia under the name Calculi ad Corrigenda Elementa orbitae Saturni, which, to the best of my knowledge, have not yet been analyzed before. In these manuscripts, Tycho starts with calculations based on the Prutenic Tables and makes a series of adjustments to the mean longitude, the longitude of the apogee, and the eccentricity to fit a series of observations of oppositions. In doing that, Tycho describes and applies a new method for obtaining accurate values for the parameters of the superior planets, he develops a divided eccentricity model of Saturn, similar to the one we know Longomontanus and Kepler applied to Mars, and finally he realizes that the true position of the Sun somehow affects the motion of Saturn around the zodiac and develops a method to correct the position of Saturn as a function of solar equation of anomaly. So, a close analysis of the calculations reveals details of the Tychonic planetary models unknown until now. The present study analyzes these drafts. (shrink)

It is well known that heliocentrism was proposed in ancient times, at least by Aristarchus of Samos. Given that ancient astronomers were perfectly capable of understanding the great advantages of heliocentrism over geocentrism—i.e., to offer a non-ad hoc explanation of the retrograde motion of the planets and to order unequivocally all the planets while even allowing one to know their relative distances—it seems difficult to explain why heliocentrism did not triumph over geocentrism or even compete significantly with it before Copernicus. (...) Usually, scholars refer to explanations of sociological character. In this paper, I offer a different explanation: that the pre-Copernican heliocentrism was essentially different from the Copernican heliocentrism, in such a way that the adduced advantages of heliocentrism can only be attributed to Copernican heliocentrism, but not to pre-Copernican heliocentrism proposals. (shrink)

Turner [The past vs. the tiny: Historical science and the abductive arguments for realism. Studies in History and Philosophy of Science 35A 1] claims that the arguments in favor of realism do not support with the same force both classes of realism, since they supply stronger reasons for experimental realism than for historical realism. I would like to make two comments, which should be seen as amplifications inspired by his proposal, rather than as a criticism. First, it is important to (...) highlight that Turner’s distinction between ‘tiny’ and ‘past unobservables’ is neither excluding nor exhaustive. Second, even if we agreed with everything that Turner says regarding the arguments for realism and their relative weight in order to justify the experimental or historical version, there is an aspect that Turner does not consider and that renders historical realism less problematic than experimental realism.Keywords: Scientific realism; Derek Turner; Rom Harré; Historical science; Ian Hacking. (shrink)