The year 2009 marks the 400th anniversary of the publication of one of the most revolutionary scientific texts ever written. In this book, appropriately entitled, Astronomia nova, Johannes Kepler developed an astronomical theory which departs fundamentally from the systems of Ptolemy and Copernicus. One of the great innovations of this theory is its dependence on the science of optics. The declared goal of Kepler in his earlier publication, Paralipomena to Witelo whereby The Optical Part of Astronomy is Treated, was to (...) solve difficulties and expose illusions astronomers face when conducting astronomical observations with optical instruments. To avoid observational errors that had plagued the antiquated measuring techniques for calculating the apparent diameter and angular position of the luminaries, Kepler designed a novel device: the ecliptic instrument. In this paper we seek to shed light on the role optical instruments play in Kepler's scheme: they impose constraints on theory, but at the same time render astronomical knowledge secure. To get a comprehensive grasp of Kepler's astonishing achievements it is required to widen the approach to his writings and study Kepler not only as a mathematico-physical astronomer, but also as a designer of instruments and a practicing observer. (shrink)

The year 2009 marks the 400th anniversary of the publication of one of the most revolutionary scientific texts ever written. In this book, appropriately entitled, Astronomia nova, Johannes Kepler developed an astronomical theory which departs fundamentally from the systems of Ptolemy and Copernicus. One of the great innovations of this theory is its dependence on the science of optics. The declared goal of Kepler in his earlier publication, Paralipomena to Witelo whereby The Optical Part of Astronomy is Treated , was (...) to solve difficulties and expose illusions astronomers face when conducting astronomical observations with optical instruments. To avoid observational errors that had plagued the antiquated measuring techniques for calculating the apparent diameter and angular position of the luminaries, Kepler designed a novel device: the ecliptic instrument. In this paper we seek to shed light on the role optical instruments play in Kepler's scheme: they impose constraints on theory, but at the same time render astronomical knowledge secure. To get a comprehensive grasp of Kepler's astonishing achievements it is required to widen the approach to his writings and study Kepler not only as a mathematico-physical astronomer, but also as a designer of instruments and a practicing observer. (shrink)

In this series of papers, I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Paper I (de Jong in Arch Hist Exact Sci 73:1–37, 2019) was devoted to a study of system A theory of the outer planets. In this second paper, I will study system A theory of the planet Venus. All presently known ephemerides of Venus appear to have been written after 200 BC so that the (...) development of system A theory of Venus may have been a late development. On the other hand, there are several earlier texts in which the motion of Venus, going from one synodic phenomenon to the next one, is parametrized in quite some detail. At least six computational systems of Venus are known of which only two, systems A0 and A3, are genuine type-A systems. Both are based on the hypothesis that in exactly 1151 years Venus experiences 720 synodic phenomena of the same kind and that after this period of 1151 years Venus returns to exactly the same position in the sky. This period relation was probably derived from the observational fact that after 8 years Venus returns to a position in the zodiac that falls on average 2.5° short of its previous position. The study of the Babylonian planetary theory of Venus presented here will be primarily based on the system A3 theory of Venus because it is the most complete of the two systems. The parameters of the four step functions characterizing the system A3 model of the first and last appearances of Venus are known from previous studies of tablet BM 32599 (ACT 1050). Based on a database of synthetic observations of the first and last appearances (morning first, morning last, evening first and evening last) and the two stationary points (morning station and evening station) of Venus between 315 BC and 50 BC, I first discuss the observational material from the point of view of a Babylonian astronomer. This involves deriving synodic time intervals and synodic arcs from the observed dates and longitudes of Venus. Both vary with the position of Venus in the zodiac. This variation shows up most purely in the synodic arcs and synodic time intervals of Venus at its stations. The variation pattern of the synodic arcs and of the synodic time intervals at the first and last appearances of Venus differ considerably because they are strongly affected by atmospheric extinction and by the ecliptic latitude of Venus. A comparison of predictions of the first and last appearances of Venus computed according to system A3 with observations in the synthetic database shows that the models provide fairly accurate fits to the observations of evening last and morning first with typical standard deviations of about 2° but rather poor fits to the observations of morning last and evening first with typical standard deviations of about 7° and 4°. Quite surprisingly, it turns out that the morning first A3 model fits the observed longitudes of Venus at its stations almost perfectly, with standard deviations of about 1°. This suggests that observations of the stations of Venus must have been available to the Babylonian astronomer(s) who constructed the system A3 model of Venus and who computed the longitudes of Venus at its first and last appearances preserved in tablet BM 32599. This is quite puzzling because observations of the stations of Venus were definitely not part of the standard observing program of the Babylonian astronomers as we know it from the Astronomical Diaries. To resolve this problem, I propose that observations could have been made by some individual astronomer for his own use but that these observations never became part of the tradition of what was regularly observed and recorded in the Astronomical Diaries. It turns out that 25 years of observations of Venus (here illustrated for the years 203 BC to 178 BC) is sufficient for the construction of a system A model that perfectly fits the variation of the synodic arcs of Venus at its stations. By combining observed time intervals and known velocities of Venus, observations over this limited timespan can also be used to construct longitudes of Venus at its first and last appearances from previous known positions of Venus based on Normal Stars passages or from known positions at its stations. This is the only way in which positions of Venus at its first and last appearances can be determined because at its first and last appearances the sky is too bright for Normal Stars to be visible. I propose that the Babylonian system A3 models of Venus were based on longitudes constructed in this way. I finally suggest that the tablet, of which BM 32599 is the remainder, was filled line by line starting with a set of initial values in the bottom left-hand corner of the tablet and that the choice of these initial values was based on an observation of Venus at its evening station when it was exceptionally close to the Normal Star η Piscium in February 179 BC. Based on this choice of initial conditions, the tablet covers the years 403 BC to 170 BC and it may have been composed around 170 BC. It was probably computed with the purpose of creating a raster of longitudes with mazes of about 2.5° that could be used to predict future positions of Venus at its first or last appearance once a position at a previous first or last appearance was available. (shrink)

Giovanni Bianchini’s fifteenth-century Tabulae primi mobilis is a collection of 50 pages of canons and 100 pages of tables of spherical astronomy and mathematical astrology, beginning with a treatment of the conversion of stellar coordinates from ecliptic to equatorial. His new method corrects a long-standing error made by a number of his antecedents, and with his tables the computations are much more efficient than in Ptolemy’s Almagest. The completely novel structure of Bianchini’s tables, here and in his Tabulae magistrales, (...) was taken over by Regiomontanus in the latter’s Tabulae directionum. One of the tables Regiomontanus imported from Bianchini contains the first appearance of the tangent function in Latin Europe, which both used as an auxiliary quantity for the calculation of stellar coordinates. (shrink)

An Andalusian tradition of zījes seems to have been predominant in the Maghrib due to the popularity of the zīj of Ibn Is[hdotu]āq al-Tūnisī and derived texts compiled in the fourteenth century. This tradition computed sidereal planetary longitudes and allowed the calculation of tropical longitudes by using trepidation tables based on models designed in al-Andalus by Abū Is[hdotu]āq ibn al-Zarqālluh. This tradition also used Ibn al-Zarqālluh's model to calculate the obliquity of the ecliptic, which implied that this angle had (...) a cyclical period of oscillation between a maximum of 23;53° and a minimum of 23;33°: after reaching this minimum value the obliquity of the ecliptic was bound to increase. This paper argues that some new Maghribi sources give information on observations made in the Maghrib in the fourteenth and beginning of the fifteenth centuries that imply that precession had increased beyond the limits allowed by the Zarqāllian trepidation theory, while the obliquity of the ecliptic had diminished below the level accepted by astronomers who followed Ibn al-Zarqālluh. This explains the introduction in the Maghrib of Eastern zījes, which computed directly tropical longitudes and did not accept the cyclical variation of the obliquity. Information about observations of dawn and twilight made both in Egypt and in the Maghrib in the fourteenth and fifteenth centuries is also presented. (shrink)

Abstract.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. (shrink)

In this series of papers, I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Paper I (de Jong in Arch Hist Exact Sci 73:1–37, 2019) was devoted to a study of system A theory of the outer planets. In this second paper, I will study system A theory of the planet Venus. All presently known ephemerides of Venus appear to have been written after 200 BC so that the (...) development of system A theory of Venus may have been a late development. On the other hand, there are several earlier texts in which the motion of Venus, going from one synodic phenomenon to the next one, is parametrized in quite some detail. At least six computational systems of Venus are known of which only two, systems A0 and A3, are genuine type-A systems. Both are based on the hypothesis that in exactly 1151 years Venus experiences 720 synodic phenomena of the same kind and that after this period of 1151 years Venus returns to exactly the same position in the sky. This period relation was probably derived from the observational fact that after 8 years Venus returns to a position in the zodiac that falls on average 2.5° short of its previous position. The study of the Babylonian planetary theory of Venus presented here will be primarily based on the system A3 theory of Venus because it is the most complete of the two systems. The parameters of the four step functions characterizing the system A3 model of the first and last appearances of Venus are known from previous studies of tablet BM 32599 (ACT 1050). Based on a database of synthetic observations of the first and last appearances (morning first, morning last, evening first and evening last) and the two stationary points (morning station and evening station) of Venus between 315 BC and 50 BC, I first discuss the observational material from the point of view of a Babylonian astronomer. This involves deriving synodic time intervals and synodic arcs from the observed dates and longitudes of Venus. Both vary with the position of Venus in the zodiac. This variation shows up most purely in the synodic arcs and synodic time intervals of Venus at its stations. The variation pattern of the synodic arcs and of the synodic time intervals at the first and last appearances of Venus differ considerably because they are strongly affected by atmospheric extinction and by the ecliptic latitude of Venus. A comparison of predictions of the first and last appearances of Venus computed according to system A3 with observations in the synthetic database shows that the models provide fairly accurate fits to the observations of evening last and morning first with typical standard deviations of about 2° but rather poor fits to the observations of morning last and evening first with typical standard deviations of about 7° and 4°. Quite surprisingly, it turns out that the morning first A3 model fits the observed longitudes of Venus at its stations almost perfectly, with standard deviations of about 1°. This suggests that observations of the stations of Venus must have been available to the Babylonian astronomer(s) who constructed the system A3 model of Venus and who computed the longitudes of Venus at its first and last appearances preserved in tablet BM 32599. This is quite puzzling because observations of the stations of Venus were definitely not part of the standard observing program of the Babylonian astronomers as we know it from the Astronomical Diaries. To resolve this problem, I propose that observations could have been made by some individual astronomer for his own use but that these observations never became part of the tradition of what was regularly observed and recorded in the Astronomical Diaries. It turns out that 25 years of observations of Venus (here illustrated for the years 203 BC to 178 BC) is sufficient for the construction of a system A model that perfectly fits the variation of the synodic arcs of Venus at its stations. By combining observed time intervals and known velocities of Venus, observations over this limited timespan can also be used to construct longitudes of Venus at its first and last appearances from previous known positions of Venus based on Normal Stars passages or from known positions at its stations. This is the only way in which positions of Venus at its first and last appearances can be determined because at its first and last appearances the sky is too bright for Normal Stars to be visible. I propose that the Babylonian system A3 models of Venus were based on longitudes constructed in this way. I finally suggest that the tablet, of which BM 32599 is the remainder, was filled line by line starting with a set of initial values in the bottom left-hand corner of the tablet and that the choice of these initial values was based on an observation of Venus at its evening station when it was exceptionally close to the Normal Star η Piscium in February 179 BC. Based on this choice of initial conditions, the tablet covers the years 403 BC to 170 BC and it may have been composed around 170 BC. It was probably computed with the purpose of creating a raster of longitudes with mazes of about 2.5° that could be used to predict future positions of Venus at its first or last appearance once a position at a previous first or last appearance was available. (shrink)

Summary The conical sundial from the museum Thyrrheion is found to be designed with cardinal parametersgeographical latitude ϕ = arc tan(3/5) = 30°57′50″half cone angle α = arc tan(4/9) = 23°57′45″radius at equinox r0 = 4 unciae = 98.7mm (pes monetalis)position of the cone tip h = 18 unciae = 444.3 mmThe half cone angle is equal to the angle of the ecliptic which leads to the special case of a conical sundial with the associated sphere being tangent at (...) the day line of the winter solstice. The ratio of sides in the generating triangle 4:9 may thus be interpreted as an approximation for the angle of ecliptic.The place of finding (operation) is 10° North of the intended latitude (design). The error of the sundial's indications due to the displacement are analyzed. (shrink)

Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure representing (...) the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics. Some implications of the use of geometric methods by Babylonian astronomers are also explored. (shrink)

Summary 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. (shrink)

This paper analyses a kinematic model for the solar motion by Quṭb al-Dīn al-Shīrāzī, a thirteenth-century Iranian astronomer at the Marāgha observatory in northwestern Iran. The purpose of this model is to account for the continuous decrease of the obliquity of the ecliptic and the solar eccentricity since the time of Ptolemy. Shīrāzī puts forward different versions of the model in his three major cosmographical works. In the final version, in his Tuḥfa, the mean ecliptic is defined by (...) an eccentric of fixed mean eccentricity and a mean obliquity fixed with respect to the celestial equator, and the center of the epicycle, which is inclined to the eccentric, moves on the eccentric with an annual period. By an additional slow motion of the sun on the epicycle, the true eccentricity of the solar deferent, defined by the annual motion of the sun, and the sun’s extreme declination from the equator change, accounting for the reduction of the eccentricity and the obliquity of the ecliptic since the time of Ptolemy. (shrink)

Giovanni Bianchini’s fifteenth-century Tabulae primi mobilis is a collection of 50 pages of canons and 100 pages of tables of spherical astronomy and mathematical astrology, beginning with a treatment of the conversion of stellar coordinates from ecliptic to equatorial. His new method corrects a long-standing error made by a number of his antecedents, and with his tables the computations are much more efficient than in Ptolemy’s Almagest. The completely novel structure of Bianchini’s tables, here and in his Tabulae magistrales, (...) was taken over by Regiomontanus in the latter’s Tabulae directionum. One of the tables Regiomontanus imported from Bianchini contains the first appearance of the tangent function in Latin Europe, which both used as an auxiliary quantity for the calculation of stellar coordinates. (shrink)

ArgumentIn theAlmagest, Ptolemy finds that the apogee of Mercury moves progressively at a speed equal to his value for the rate of precession, namely one degree per century, in the tropical reference system of the ecliptic coordinates. He generalizes this to the other planets, so that the motions of the apogees of all five planets are assumed to be equal, while the solar apsidal line is taken to be fixed. In medieval Islamic astronomy, one change in this general proposition (...) took place because of the discovery of the motion of the solar apogee in the ninth century, which gave rise to lengthy discussions on the speed of its motion. Initially Bīrūnī and later Ibn al-Zarqālluh assigned a proper motion to it, although at different rates. Nevertheless, appealing to the Ptolemaic generalization and interpreting it as a methodological axiom, the dominant idea became to extend it in order to include the motion of the solar apogee as well. Another change occurred after correctly making a distinction between the motion of the apogees and the rate of precession. Some Western Islamic astronomers generalized Ibn al-Zarqālluh's proper motion of the solar apogee to the apogees of the planets. Analogously, Ibn al-Shāṭir maintained that the motion of the apogees is faster than precession. Nevertheless, the Ptolemaic generalization in the case of the equality of the motions of the apogees remained untouchable, despite the notable development of planetary astronomy, in both theoretical and observational aspects, in the late Islamic period. (shrink)

: Scientific observation is determined by the human sensory system, which generally relies on instruments that serve as mediators between the world and the senses. Instruments came in the shape of Heron's Dioptra, Levi Ben Gerson's Cross-staff, Egnatio Danti's Torqvetto Astronomico, Tycho's Quadrant, Galileo's Geometric Military Compass, or Kepler's Ecliptic Instrument. At the beginning of the seventeenth century, however, it was unclear how an instrument such as the telescope could be employed to acquire new information and expand knowledge about (...) the world. To exploit the telescope as a device for astronomical observations Galileo had to: 1. establish that telescopic images are not optical defects, imperfections in the eye of the observer, or illusions caused by lenses; 2. develop procedures for systematically handling errors that may occur during observation and measurement and methods of processing data. Galileo made it clear that in order to measure and interpret natural phenomena accurately, a suitable method and instrument would need to be developed. It is intriguing, therefore, to regard the Galilean telescope in this light and to discover the linkage established by Galileo among theory, method, and instrument—the telescope. Although the telescope was not invented through science, it is instructive to see how Galileo used optics to employ a theory-laden instrument for bridging the gulf between picture and scientific language, between drawing and reporting physical facts, and between merely sketching the world and actually describing it. (shrink)

With the introduction of Greco-Islamic science and natural philosophy, medieval natural philosophers were confronted with three distinct astronomical systems: Aristotelian, Ptolemaic, and the system of al-Bitruji. A fundamental problem that each had to confront was how to explain simultaneous contrary motions in the heavens -for example, the sun's motion, which moves east to west with a daily motion while simultaneously moving west to east along the ecliptic- within an Aristotelian physical system that assumed that a simple body could have (...) only one proper motion. How medieval natural philosophers resolved this problem is the focus of the article. (shrink)

With the introduction of Greco-Islamic science and natural philosophy, medieval natural philosophers were confronted with three distinct astronomical systems: Aristotelian, Ptolemaic, and the system of al-Bitruji. A fundamental problem that each had to confront was how to explain simultaneous contrary motions in the heavens -for example, the sun's motion, which moves east to west with a daily motion while simultaneously moving west to east along the ecliptic- within an Aristotelian physical system that assumed that a simple body could have (...) only one proper motion. How medieval natural philosophers resolved this problem is the focus of the article. (shrink)

In Book 8 of his Geographike Hyphegesis Ptolemy gives coordinates for ca. 360 so-called noteworthy cities. These coordinates are the time difference to Alexandria, the length of the longest day, and partly the ecliptic distance from the summer solstice. The supposable original conversions between the coordinates in Book 8 and the geographical coordinates in the location catalogue of Books 2–7 including the underlying parameters and tabulations are here reconstructed. The results document the differences between the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega}$$\end{document} - and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Xi}$$\end{document} -recension. The known difference in the longitude of Alexandria underlying the conversion of the longitudes is examined more closely. For the ecliptic distances from the summer solstice of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega}$$\end{document} -recension, it is revealed that they were originally computed by means of a so far undiscovered approximate, linear conversion. Further it is shown that the lengths of the longest day could be based on a linear interpolation of the data in the Mathematike Syntaxis 2.6. (shrink)

The assumption that Ptolemy adopted star coordinates from a star catalog by Hipparchus is investigated based on Hipparchus’ equatorial star coordinates in his Commentary on the phenomena of Aratus and Eudoxus. Since Hipparchus’ catalog was presumably based on an equatorial coordinate system, his star positions must have been converted into the ecliptical system of Ptolemy’s catalog in his Almagest. By means of a statistical analysis method, data groups consistent with this conversion of coordinates are identified. The found groups show a (...) high degree of agreement between Hipparchus’ and Ptolemy’s data. The value of the obliquity of the ecliptic underlying the conversion is estimated by adjustment and statistically agrees with Ptolemy’s value of this parameter. The results allow the assumption that Ptolemy’s coordinates were determined from Hipparchus’ coordinates by an accurate star globe or even by calculation. For a calculative derivation of ecliptical coordinates from equatorial ones, possible calculation methods are discussed considering the mathematics of the Almagest. (shrink)

ABSTRACT This article is devoted to a thirteenth-century Latin text on how to construct, set up, and use a version of the so-called armillary instrument (instrumentum armillarum), which was first described in Ptolemy’s Almagest as a tool for measuring ecliptic coordinates. Written in 1264 by Guillaume des Moustiers, bishop of Laon, this hitherto unstudied Tractatus super armillas survives in a single manuscript, where it is accompanied by a copious set of glosses. The text and its glosses jointly offer an (...) unusually detailed account of the instrument’s material aspects and methods of assembly. In addition, they reflect a keen awareness of the potential sources of error that may arise in the context of astronomical observation, while making suggestions on how these errors may be minimized or avoided. The Tractatus super armillas accordingly is a valuable source on the observational side of medieval European astronomy, which has often been minimized in modern historical accounts. (shrink)

Recent investigations have thrown new light on such topics as the early Greek belief in heliocentricity, the relation between Greek and Babylonian astronomy, the reliability of Ptolemy's Syntaxis, Hipparchus's theory of motion for the sun, Hipparchus's value for the obliquity of the ecliptic, and Eratosthenes' estimate of the size of the earth. Some claims resulting from these investigations are controversial, especially the reevaluation of Ptolemy (though it is notable that no one any longer uses data from the Syntaxis for (...) investigating such things as the spin of the earth). This essay presents the evidence for these claims; it makes no pretense of presenting the evidence against them. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:242 HISTORY OF PHILOSOPHY his political and religious predispositions prodded him to demonstrate that the roots of modern science were in the Christian Middle Ages. Sarton's particular foibles are best understood by referring them to his pacifist commitments and the moralistic assumption that the values of science are transferable to other human endeavors. Categories such as inductivism, conventionalism and Popperianism are of little help in gaining historical understanding. For (...) this, they are in fact Platonic ideals which have little heuristic value. I would venture that a truly historical historiography of science would aid us in spotting errors in the past and help us in the future. My guess is that a good analysis would reveal a major source of trouble to have stemmed from preconceived notions which the history of science was meant to illustrate. Surely some of them have been philosophical in character, and even Baconian or conventionalist. But until recent times, most of these preconceptions have stemmed from religious, nationalistic, or personal motivations, or from scientists who wanted support for a particular concept. I agree with Agassi only in the suggestion that a parti pris was at the root of our troubles. Why, then, substitute a Popperian preconception for a conventionalist one? Why not, on the contrary, combat their distorting influence and reduce their impact? This is in fact what has happened in recent times. As the history of science has fallen into the hands of a gradually organized profession, the blatant motivations --of any type--which colored past historical writings have slowly receded. By a constant and critical confrontation of opposing views among specialists, the purpose for examining the history of science has slowly been transformed. The older external motivations have given way to an interest in the subject for its own sake and a gradual transformation has been in the making for the last few decades. As happens in other human endeavors, professionalization has already raised the standards in the discipline and there is every reason to think it will continue to do so in the future. We need not really be as pessimistic as Agassi, nor embrace his particular salvation. ROGEa HAm~ University o] Cali]ornia, Berkeley A MEDIEVAL ANALYSIS OF INFINITY IN THE COURSEof arguing that time must have had a beginning, St. Bonaventure (1221-74) put forward a demonstration which in some ways seems to anticipate Cantor's definition of infinity.1 Bonaventure pointed out that, if the world were infinitely old, then there would have been an infinite number of annual revolutions of the sun around the ecliptic. But during each such period there occur (roughly) twelve revolutions of the moon, i.e. lunar months or lunations. Thus there would be one infinity which was twelve times another--which, says Bonaventure, is impossible. This attempted proof might be formulated as the following For other adumbrationsof this definition,see I. Thomas,"A TwelfthCentury Paradoxof the Infinite,"Journal o] Symbolic Logic, XXIII (1958), 133-134, and W. & M. Kneale, The Development of Logic (Oxford,1962),p. 440. NOTES AND DISCUSSIONS 243 reductio ad absurdum, in which a revolution of the sun is specified as the period between successive vernal equinoxes. If there could be an infinite set of past lunations, then clearly it could be put in a one-to-one correspondence with a proper sub-set of itself--viz., with the set of past lunar months during which vernal equinoxes occurred. But this consequence is preposterous; no set could be so correlated with its own sub-set. Therefore the set of past lunar months cannot be infinite, and the world must have had a beginning. The passage, which is found in Bonaventure's commentary on the Sentences of Peter Lombard, reads as follows :2 Si mundus est aeternus, revolutiones solis in orbe suo sunt infmitae ; rursus, pro una revolutions solis necesse est fuisse duodecim ipsius lunae: ergo plus revoluta est luna quam sol; et sol infinities: ergo infinitorum ex ea parle, qua infinita sunt, est reperire excessum. Hoe autem est impossibile : ergo etc. PA~-r~asoN BROWN Harpur College, State University o] New York Sententiarum L/b. H, Dist. I, P.I, Art. I, Quaest. II, as found in S. Bonaventurme, Opera Omnla... (shrink)

Zodiacal and Planetary ‘decani’. The 36 ecliptical ‘decani’ (sectors of 10°) were distributed either to the twelve zodiacal signs or to the seven planets. The first system has been transmitted only by the Roman didactic poet Manilius, who commits an error at the end of his catalogue that can be explained by comparing it with the more frequent planetary one. Both systems follow the Roman calendar beginning with the Ram respectively Mars. Although the zodiacal system (36 : 12) runs without (...) remainder whereas the planetary one (36 : 7) repeats Mars once more (six times), so that it both begins and ends with this planet, this planetary system prevails. Four possible astrological reasons of this contradiction are discussed including other texts, mostly taken from the ‘Babylonian’ Teucrus (at the latest first century BC), who represents the first witness, if not the inventor of this particular lore. (shrink)

The paper brings into light and discusses a concentric solar model briefly described in Chapter 5 of Section III of ‘Abd al-Raḥmān al-Khāzinī’s On experimental astronomy, a treatise embedded in the prolegomenon of his comprehensive Mu‘tabar zīj, completed about 1121 c.e. In it, the Sun is assumed to rotate on the circumference of a circle concentric with the Earth and coplanar with the ecliptic, but the motion of the vector joining the Earth and Sun is monitored by a small (...) eccentric hypocycle. The ratio between the distance of the hypocycle’s center from the Earth and the hypocycle’s radius is equal to the solar eccentricity in the eccentric model. The model is to account for the constancy of the apparent diameter of the solar disk as held by Ptolemy. The source of the model is unknown. Since the mechanism employed in it clearly resembles the pin-and-slot device, whose use in mechanical astronomical instruments has a long history from the Antikythera Mechanism to the medieval solar, lunar, and planetary equatoria and dials, we argue that the solar model can be positioned within this long-standing tradition and considered the result of the correct understanding of some Byzantine prototype and thus a typical example of the transmission of astronomical ideas via media of the material culture. (shrink)

An outdated geography supplies the bond among the thirty‐one articles in Sur les traces des Cassini. In the seventeenth century, when the Italians Gian Domenico Cassini and his nephew Giacomo Filippo Maraldi were born in Perinaldo, north of Genoa, their birthplace belonged to the County of Nice. Hence the rationale of building a set of papers on astronomy in the south of France around Cassini I and his family, which for four generations ran the Royal Observatory in Paris.Over half the (...) articles concern the Cassinis, mostly Cassini I and his great‐great‐grandson and namesake Cassini IV. There was also a Cassini V, Henri de Cassini, who countered the family genius and stamina by preferring botany and dying early, of the same outbreak of cholera that took the life of Sadi Carnot, without having created Cassini VI. He had already entered the Academy of Sciences with a push from his father. “I dare to beg of you [Cassini IV wrote to his fellow academician A. M. Ampère] to consider whether this unique situation in the history of letters, [a family's] devotion to the sciences for five successive generations and 170 years, ought not add some weight to the scientific credentials of my son.” It is hard to refuse the children of important alumni.The portion of Traces dealing more directly with astronomy in the south of France gets off to a distant start. Pytheas of Marseilles, who lived about 350 b.c., sailed to the Orkneys and the Baltic and earned himself the reputation of a liar back home for his stories of midnight suns and frozen lakes. He measured the latitude of Marseilles, the obliquity of the ecliptic, and the size of the earth. Ptolemy praised him. Strabo did not: “Pytheas lied about everything and covered it up with his knowledge of astronomy and numbers.”No traces worth following up were laid down for just under two thousand years. Then, in 1580, Nicolas Fabri de Peiresc first saw the light of day. He lived in Italy for four years as a very young man, deepening his knowledge of astronomy and human nature and meeting the main future actors in the Galileo affair: Galileo himself, Bellarmine, and Matteo Barberini . At his center in Aix‐la‐Provence, Peiresc made many useful astronomical observations, some in collaboration with Pierre Gassendi. He died in harness, worrying about the change of the obliquity since the days of Pytheas.There follow articles on Provençal astronomers who determined longitude and latitude at sea, on neglected observers in Languedoc who assisted the cause of the Enlightenment, and on modern observatories in the south of France. The political circumstances after the defeat of 1870–1871 favored decentralization of astronomy away from the Paris Observatory. In Italy, too, recent political events—the unification of 1870—made a restructuring of astronomical institutions desirable and possible. But whereas France had too few observatories, Italy had too many. Georges Rayet and Pietro Tacchini, both astrophysicists, compared the circumstances in their countries and made mutually reinforcing proposals to their colleagues and governments. Their respective proposals, most of which were enacted, called for reassigning some Italian observatories to meteorological work, building new observatories in Besançon, Bordeaux, and Lyon, and refurbishing older ones at Marseilles and Toulouse. Once again, as in the days of Cassini I, Italy made a decisive contribution to the practice of astronomy in France.Sur les traces des Cassini mixes slight and weighty work, admits antiquarian and broader approaches, offers new documentation, displays pertinent illustrations, and does it all at a high level of scholarship. Since, because of its title, the book's primary audience probably will be people interested in the Cassinis, its fullest articles about them may usefully be mentioned here: Anna Cassini on Cassini's brief return to Italy, 1694–1696; Claude Teillet on the provincial life and poetry of Cassini IV; Christiane Demeulenaere‐Douyère on the Cassinis and the Académie des Sciences; Fabrizio Bonoli and Alessandro Braccesi on Cassini I's astronomical work in Bologna, with full bibliography; and Monique Pelletier on the Cassini map of France, on which she has written a book . Pytheas and Peiresc are the subjects of collaborative articles by Simone Arzano and Yvon Georgelin. (shrink)