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.     

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.     

The conceptual import of Carnot's theorem to the discovery of the entropy

Summary Among many other things, Carnot stated a principle and proved a theorem. In 1850, Clausius corrected Carnot's theory, modifying it according to Joule's principle. He might have considered a corollary of the theorem as the mathematical formulation of Carnot's principle. We challenge the corollary: it is based on hidden assumptions, nor is it true for all cycles. Clausius realized the corollary's lack of generality, but on different grounds. In 1854, he generalized the theorem, and gave an (other) expression to Carnot's principle. We analyze Clapeyron's account of Carnot's theory, Thomson's account of 1849 and some of Clausius belated comments on his 1850 paper, as well Clausius' paper of 1854. We hope that they shed light on the corollary's tacit hypotheses and on the meaning of Carnot's principle. It is our contention: Clausius took seriously a contemporary meaning of the principle, and looked for a condition of integrability that could express recovery of the initial conditions of the reservoirs. Furthermore, he seems to have had some prior knowledge of the form the expression of the principle should take. Actually, this was the theory's natural candidate.     

Epicycles,eccentrics, and ellipses: The predictive capabilities of Copernican planetary models

A theoretical analysis of the potential accuracy of early modern planetary models employing compound circles suggests that fairly simple extensions of those models can be sufficiently accurate to meet the demands of Tycho Brahe's observations in both ecliptic longitude and latitude. Some of these extensions, such as the substitution of the true sun for the mean sun, had already been taken by Kepler before he abandoned circular models. Other extensions, involving one or two extra epicycles, were well within the mathematical capabilities of sixteenth-century and seventeenth-century astronomers. Hence neither the failure of astronomers before Kepler to correct errors in planetary positions nor Kepler's decision to abandon circular models was a consequence of inherent limitations in those models.     

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).     

Der Eigenartige Einfluß Witelos auf die Entwicklung der Dioptrik

Summary Witelo's Perspectiva, which was printed three times in the sixteenth century, profoundly influenced the science of dioptrics until the Age of Newton. Above all, the optical authors were interested in the so-called Vitellian tables, which Witelo must have copied from the nearly forgotten optical Sermones of Claudius Ptolemy. Research work was often based on these tables. Thus Kepler relied on the Vitellian tables when he invented his law of refraction. Several later authors adopted Kepler's law, not always because they believed it to be true, but because they did not know of any better law. Also Harriot used the Vitellian tables until his own experiments convinced him that Witelo's angles were grossly inaccurate. Unfortunately Harriot kept his results and his sine law for himself and for a few friends. The sine law was not published until 1637, by Descartes, who gave an indirect proof of it. Although this proof consisted in the first correct calculation of both rainbows, accomplished by means of the sine law, the Jesuits Kircher (Ars Magna, 1646) and Schott (Magia Optica, 1656) did not mention the sine law. Marci (Thaumantias, 1648) did not know of it, and Fabri (Synopsis Opticæ, 1667) rejected it. It is true that the sine law was accepted by authors like Maignan (Perspectiva Horaria, 1648) and Grimaldi (Physico-Mathesis, 1665), but since they used the erroneous Vitellian angles for computing the refractive index, they discredited the sine law by inaccurate and even ludicrous results.That even experimental determinations might be unduly biased by the Vitellian angles is evident from the author's graphs of seventeenth century refractive angles. These graphs also show how difficult it was to measure such angles accurately, and how the Jesuit authors of the 1640's adapted their experimental angles to the traditional Vitellian ones. Witelo's famous angles, instead of furthering the progress of dioptrics, delayed it. Their disastrous influence may be traced for nearly thirty years after Descartes had published the correct law of refraction.

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Vorgelegt von C. Truesdell     

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.     

Gauss's geodesy and the axiom of parallels

It is a myth that Gauss measured a certain large triangle specifically to determine its angle sum; he did so in order to link his triangulation of Hanover with contiguous ones. The sum of the angles differed from 180° by less than two thirds of a second; he is known to have mentioned in conversation that this constituted an approximate verification of the axiom of parallels (which he regarded as an empirical matter because his studies of hyperbolic trigonometry had led him to recognize the possibility of logical alternatives to Kant and Euclid). However, he never doubted Euclidean geometry in his geodetic work. On the contrary, he continually used 180° angle sums as a powerful check for observational errors, which helped him to achieve standards of precision equivalent to today's. Nor did he ever plan an empirical investigation of the geometrical structure of space.     

Geminus and the concept of mean motion in Greco-Latin astronomy

Conclusion Geminus account of lunar motion in chapter 18 of hisIntroductio astronomiae is, in our view, an important contribution to Greco-Latin astronomy because, in attempting to reconstruct arithmetically (the parameters of) the Moon's motion in longitude, he undermines the task astronomers had hitherto set for themselves. This undermining of a commonly acknowledged view of the purpose of astronomy is articulated in a whole new set of questions concerning the nature and place of both observation and mathematical reasoning in the science of the heavens. Yet, one must not overlook the fact thatGeminus reconstruction also indicates resources for addressing these questions. Of these resources, the most powerful proved to be the idea that irregular motion could be quantified as a systematic departure from a mean motion, and the idea that observational data could be organized and structured by means of genetic arithmetical reconstructions.But, since we limit our attention to extant treatises and decline to speculate about works or parts of works that have not survived, we must say that it would takePtolemy to discern the new direction for astronomy thatGeminus opened up and to pursue it. In part, this involved straightening out the conflated conception of mean motion in chapter 18 — the qua arithmetic mean daily displacement can only be anapparent lunar motion in longitude and not one the Moonreally makes, but the same need not be true of the qua periodic mean daily displacement — and determining its proper relation to real and apparent planetary motion. Indeed,Ptolemy's genius lay, we think, in seeing that even though, in assimilating Babylonian astronomy, earlier and contemporary Greco-Latin writers betrayed a confused, inconsistent, and unsophisticated grasp of the proper role of arithmetic, geometry, and observation in astronomical argument [seeBowen 1994], the solution lay in a mathematical reconstruction of the observed celestial motions, in which mean motion played an essential role.     

Effects of glycolytic metabolites on preservation of high energy phosphate level and synaptic transmission in the granule cells of guinea pig hippocampal slices

The present study was undertaken to investigate whether neural activity of hippocampal slices can be preserved after replacingd-glucose with glycolytic intermediate metabolites such as lactate, pyruvate and citrate or with other sugars such as fructose, mannose, maltose, glucosamine, sucrose and galactose. As an index of neural activity, population spikes (PS) were recorded in the granule cell layers after electrical stimulation to the perforant path of guinea pig hippocampal slices. In addition, we determined the levels of ATP and creatine phosphate (CrP) in each slice after the replacement ofd-glucose with these substrates, and correlated it with the neural activity. Substrates other thand-glucose could not maintain the PS for even 20 min although the slices perfused with medium containing lactate, pyruvate, galactose, fructose and maltose maintained similar levels of ATP and CrP as in slices incubated in thed-glucose-containing medium. These results indicate thatd-glucose is essential for the preservation of synaptic activity in addition to its main role as the substrate for energy production to maintain the levels of high energy phosphates.     

On de Movire's solutions of the problem of duration of play 1708–1718

Summary I attempt to solve the enigma of how de Moivre derived his recursion formulae and the formulae for the probability of a duration of exactly n games. I believe that he obtained the latter probability by differencing Nicholas Bernoulli's formula for the probability of ruin and that he obtained the recursion formulae by a simple algebraic method and induction.     

Rapporto fra “metodo” e “contenuti matematici” in P. Gesualdo Melacrinò da Reggio Calabria

Summary Father Gesualdo Melacrinò (1725–1803), from Reggio Calabria (Italy), is an unknown Capuchin philosopher and theologian, who produced several works at the time he was teaching (only five years, from 1748–53); these works contained an original approach to the foundations and philosophy of mathematics. His main purpose was to reconciliate the classical traditions with the reality of his time. For him, this included a critical examination of the scholastic curriculum and a new orientation towards the methodological relevance of mathematics for all other sciences, especially for philosophy. Concerning mathematics, he emphasized the necessity of a basic revision and logical reconstruction of its foundations. This paper provides a comparative examination of Melacrinò's work with reference to its cultural and historical environment.     

Greek angles from Babylonian numbers

Models of planetary motion as observed from Earth must account for two principal anomalies: the nonuniform speed of the planet as it circles the zodiac, and the correlation of the planet’s position with the position of the Sun. In the context of the geometrical models used by the Greeks, the practical difficulty is to somehow isolate the motion of the epicycle center on the deferent from the motion of the planet on its epicycle. One way to isolate the motion of the epicycle center is to determine the longitude and time of oppositions of the planet with the mean Sun. A Greek astronomer might have realized that the predictions of mean oppositions by Babylonian models could serve as useful proxies for real empirical observations. It is shown that a Greek astronomer with a reasonable understanding of Babylonian System A models for the outer planets and the Sun–Moon could have used those models to estimate approximate values for the eccentricity e and longitude of apogee A required for geometrical models. The same method would work for the inner planets if conjunctions were observable, but they are not, and the variation of the observable synodic events—first and last morning and evening visibilities—is dominated more by the motion of the planet in latitude than the nonuniform motion of the epicycle center.     

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.     

Paroxysmal discharges in the electroencephalogram of the El mouse

Summary The spike discharges in the EEG of the El mouse, a seizure-susceptible strain, were recorded during convulsive seizures. This fact provides evidence that those seizures are really epileptic convulsions.I thank Dr.K. Imaizumi and Dr.K. Nakano of National Institute of Health for their sharing the strain of El mouse to us and MissY. Nakamoto and Mr.N. Ozawa for their technical assistance and Mr.K. Moriyama for his care for the mice, and also thank Dr. H.Narabayashi for his criticism on this paper.     

Newton's early computational method for dynamics

On December 13, 1679Newton sent a letter toHooke on orbital motion for central forces, which contains a drawing showing an orbit for a constant value of the force. This letter is of great importance, because it reveals the state ofNewton's development of dynamics at that time. Since the first publication of this letter in 1929,Newton's method of constructing this orbit has remained a puzzle particularly because he apparently made a considerable error in the angle between successive apogees of this orbit. In fact, it is shown here thatNewton's implicitcomputation of this orbit is quite good, and that the error in the angle is due mainly toan error of drawing in joining two segments of the oribit, whichNewton related by areflection symmetry. In addition, in the letterNewton describes quite correctly the geometrical nature of orbits under the action of central forces (accelerations) which increase with decreasing distance from the center. An iterative computational method to evaluate orbits for central forces is described, which is based onNewton's mathematical development of the concept of curvature started in 1664. This method accounts very well for the orbit obtained byNewton for a constant central force, and it gives convergent results even for forces which diverge at the center, which are discussed correctly inNewton's letterwithout usingKepler's law of areas.Newton found the relation of this law to general central forces only after his correspondence withHooke. The curvature method leads to an equation of motion whichNewton could have solvedanalytically to find that motion on a conic section with a radial force directed towards a focus implies an inverse square force, and that motion on a logarithmic spiral implies an inverse cube force.     

Boltzmann's ergodic hypothesis

Summary Boltzmann's ergodic hypothesis is usually understood as the assumption that the trajectory of an isolated mechanical system runs through all states compatible with the total energy of the system. This understanding of Boltzmann stems from the Ehrenfests' review of the foundations of statistical mechanics in 1911. If Boltzmann's work is read with any attention, it becomes impossible to ascribe to him the claim that one single trajectory would fill the whole of state space. He admitted a continuous number of different possible mechanical trajectories. Ergodicity was formulated as the condition that only one integral of motion, the total energy, is preserved in time. The two reasons for this are external disturbing forces and collisions within the system. Boltzmann found it difficult to ascribe ergodic behavior to a single system where the theoretical dependence on initial conditions, though never observed, has to be admitted as possible. To circumvent the dependence, he invented the concept of a microcanonical ensemble.     

A proto-Normal Star Almanac dating to the reign of Artaxerxes III: BM 65156

Babylonian methods for predicting planetary phenomena using the so-called goal-year periods are well known. Texts known as Goal-Year Texts contain collections of the observational data needed to make predictions for a given year. The predictions were then recorded in Normal Star Almanacs and Almanacs. Large numbers of Goal-Year Texts, Normal Star Almanacs and Almanacs are attested from the early third century BC onward. A small number of texts dating from before the third century present procedures for using the goal-year periods to predict planetary phenomena. In addition, two texts, one dating to the late sixth century BC and the other to the late fifth century BC, contain planetary data which was probably predicted using these methods. In this article, I discuss a further example of a tablet dating from before the third century BC which contains planetary data predicted using the goal-year periods. I show that the planetary phenomena contained in this tablet can be dated to the twelfth year of the reign of Artaxerxes III (347/6 BC) and that they were predicted using goal-year periods without the application of the kind of corrections which were used in the third century BC texts in order to produce more accurate predictions.

     

Essais de solubilisation et de détoxification de la Tyrothricine

Summary As Tyrothricin —Dubos's discovery of the antibiotic agent extracted from cultures ofB. brevis — does not give true aqueous solutions and retains, in the colloïdal state of the usually employed solutions, toxic properties for polymorphonuclear blood cells, we have attempted to effect the solubilization and detoxification of this compound. Treatment with formalin in given proportions results in making it water-soluble. The action of the formalin-treated compound on blood cells suspended in Ringer's solution seems to show a loss of toxicity. Subcutaneous and intramuscular injections in mice do not give visible general toxic effects, and the experiments incite to further investigations in this direction.