This work is a reasearch article that highlights the importance of planet Jupiter in ancient Mesopotamian astronomy and astrology and provides an in-depth interpretation of omens related to this planet. Studies have generally approached the omen series from the point of view of astronomical science and have not focused much on what the omens actually meant. The omens sampled in this study are analyzed in detail and different information about ancient Mesopotamia is obtained. Jupiter is regarded as the largest planet by the ancient Mesopotamians and referred to as King Star, associated with the god Marduk, a figure of central importance in Mesopotamian belief systems. The movements of Jupiter were carefully observed by diviners, and omens were formulated accordingly. While the steady position of Jupiter was interpreted as favorable, its appearance in the same omen as a malefic planet was generally considered unfavorable. In addition to these interpretations, Jupiter’s movements were also used in relation to issues such as participation in military campaigns, regulation of market commodities, weather forecasting, and omens based on the planet’s color. These interpretations were recorded in a comprehensive series of celestial omens known as Enūma Anu Enlil. Comprising more than seven thousand omens, this series made a substantial contribution to the development of astronomical knowledge and has survived to the present day. Although the fragmented condition of the cuneiform tablets and the complex nature of the omen texts posed certain challenges, familiarity with ancient Mesopotamian history, culture, and religion, alongside basic astronomical knowledge, facilitated a clearer understanding of the material.

Babylonian astral science of the 1st millennium BC was a multi-faceted practice including detailed and systematic astronomical observation, the development and use of various computational methods for predicting the positions and phenomena of the sun, moon, and planets, descriptions using simple numerical schemes of regular changes in the celestial realm, and various forms of astrology. Despite the detailed study of these practices over the past 150 years or so, historians of science remain divided as to whether Babylonian astronomy should be classified as a ‘science’. Specialists of Babylonian astronomy and astrology uniformly agree that it is science, whereas some non-specialists argue that science is a uniquely Greek invention. Much of this disagreement comes down to the question of whether Babylonian astral science is ‘theoretical’ or (merely) the fitting of numbers to empirical data. In this paper, I explore whether we can identify ‘theories’ within Babylonian astral science and if so where these are to be found.

This paper presents initial findings on the interaction between astronomical procedures on Babylonian tablets. Using algorithms as a lens, this research investigates the relationship between procedures, data provided on the tablet, and the representations of methods within the tablets. This paper first provides a critical analysis of the term "algorithm" in a historical, Mesopotamian context and how algorithms may be used as an analytical tool for thinking about the relationship between astronomical procedures. Following this description, the paper moves on to discuss a case study from the planetary procedure texts that showcases how an algorithm-based analysis can inform reinterpretations of how the individual procedures interact on a given tablet. The ultimate goal of this work is to shed new light on how the authors and users of these astronomical tablets may have interacted with them.

Abstract Observations of the first visibility of the new moon crescent by ancient Babylonian astronomers often refer to the moon “wearing a crown”. In this article, we undertake a detailed analysis of all Babylonian reports of the observation of the new moon crescent in order to test the common assumption that this phrase refers to the phenomenon of Earthshine. We conclude that the assumption is correct and further explore the observational circumstances that made Earthshine visible to the Babylonian astronomers—and by analogy to any experienced naked‐eye observer. We demonstrate that the most significant factor in whether or not Earthshine is seen is not, as might be expected, the age of the moon but instead the prevailing sky darkness. We also show that the Babylonian observations display a seasonal variation that is consistent with modern observations of and models for the Earth's albedo.

Modern scholarship asserts that the Babylonians were able to determine the synodic periods (discussed below) of the outer planets with exceptional accuracy. We experimented with a simple Maple mathematical software model to see if we could mathematically re-create the synodic cycles of Mars determined by Babylonian astronomers starting from around 400 BC. We sought first to understand whether unique planetary cycles known as Goal-Year periods which the Babylonians gleaned from centuries of observation were mathematically inevitable. We wanted to determine their accuracy quantitatively and find out if other plausible Goal-Year choices were available to them. The Babylonians also invented a method of using Goal-Years in error-canceling combinations to create long-term ‘exact’ periods that markedly improved the accuracy of their planetary predictions for centuries ahead. We tested whether the mathematical method surmised by scholars made sense in terms of our own program and whether we could replicate it. This paper is a report on the success of those efforts.

A simple mathematical scheme to represent the variation in the length of the shadow cast by a vertical gnomon at different times of day and in different months of the year is presented in the early astronomical compendium MUL.APIN. A small number of texts composed in the Late Babylonian period investigate and expand this scheme. These texts have previously been studied and understood as part of Babylonian astronomy. In this article, I suggest that two of these later texts can be better understood as mathematical texts. As such they provide evidence for the influence of astronomy on Late Babylonian mathematics, either or both as the context for simple mathematical problems and/or as a topic of mathematical investigation.

Late Babylonian astronomy and astrology

Time series, i.e. data collected at various times. The data collection segments may differ de-pending on the task. Time series are used for decision making. Time series analysis allows you to get some result that will determine the format of the decision. Time series analysis was carried out in very ancient times, for example, various calendars became a consequence of the analysis. Later, time series analysis was applied to study and forecast economic, social and other systems. Time se-ries appeared a long time ago. Once upon a time, ancient Babylonian astronomers, studying the po-sition of the stars, discovered the frequency of eclipses, which allowed them to predict their appearance in the future. Later, the analysis of time series, in a similar way, led to the creation of various calen-dars, for example, harvest calendars. In the future, in addition to natural areas, social and economic ones were added. Aim. Search for classification patterns of time series, allowing to understand whether it is possible to apply the ARIMA model for their short-term (3 counts) forecast. Materials and methods. Special software with ARIMA implementation and all need services is made. We examined 59 data sets with a short length and step equal a year, less than 20 values in the paper. The data was processed using Python libraries: Statsmodels and Pandas. The Dickey – Fuller test was used to de-termine the stationarity of the series. The stationarity of the time series allows for better forecasting. The Akaike information criterion was used to select the best model. Recommendations for a rea-sonable selection of parameters for adjusting ARIMA models are obtained. The dependence of the settings on the category of annual data set is shown. Conclusion. After processing the data, four categories (patterns) of year data sets were identified. Depending on the category ranges of parame-ters were selected for tuning ARIMA models. The suggested ranges will allow to determine the starting parameters for exploring similar datasets. Recommendations for improving the quality of post-forecast and forecast using the ARIMA model by adjusting the settings are given.

The Antikythera Mechanism, an ancient Greek astronomical calculator, has challenged researchers since its discovery in 1901. Now split into 82 fragments, only a third of the original survives, including 30 corroded bronze gearwheels. Microfocus X-ray Computed Tomography (X-ray CT) in 2005 decoded the structure of the rear of the machine but the front remained largely unresolved. X-ray CT also revealed inscriptions describing the motions of the Sun, Moon and all five planets known in antiquity and how they were displayed at the front as an ancient Greek Cosmos. Inscriptions specifying complex planetary periods forced new thinking on the mechanization of this Cosmos, but no previous reconstruction has come close to matching the data. Our discoveries lead to a new model, satisfying and explaining the evidence. Solving this complex 3D puzzle reveals a creation of genius—combining cycles from Babylonian astronomy, mathematics from Plato’s Academy and ancient Greek astronomical theories.

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. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve Ephemerides and seven Procedure Texts. Three computational systems of Mercury are known, all of system A. System A1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A2 by two Ephemerides covering the years 310 to 290 BC. System A3 is known from a Procedure Text and from Text M, an Ephemeris of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the Astronomical Diaries and Planetary Texts we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase.

Education

Of the whole volume on Babylonian Astronomical Diaries, that has recently appeared on the shelves by Brill, Tuplin’s contribution is perhaps the most useful among the several papers to help the not experts in orientating though the topic. Tuplin presents an exhaustive and detailed analysis of the historical entries connected to the diaries from the Achaemenid one to the Parthian period. This is the first step in order to be able to appreciate the complexity of the diaries and their value as a...

BM 76829, a fragment from the mid-section of a small tablet from Sippar in Late Babylonian script, preserves what remains of two new unparalleled pieces from the cuneiform astronomical repertoire relating to the zodiac. The text on the obverse assigns numerical values to sectors assigned to zodiacal signs, while the text on the reverse seems to relate zodiacal signs with specific days or intervals of days. The system used on the obverse also presents a new way of representing the concept of numerical ‘zero’ in cuneiform, and for the first time in cuneiform, a system for dividing the horizon into six arcs in the east and six arcs in the west akin to that used in the Astronomical Book of Enoch. Both the obverse and the reverse may describe the periodical courses of the sun and moon, in a similar way to what is found in astronomical texts from Qumran, thus adding to our knowledge of the scientific relationship between the two cultures.

Abstract The astronomical compendium MUL.APIN was composed sometime before the seventh century BC and is known from many copies dating between the Neo-Assyrian and the Seleucid period. In this paper I argue that MUL.APIN continued to be read and understood throughout this period, and that there was still an active tradition of astronomy founded upon MUL.APIN until the end of cuneiform astronomy.

The most discussed and mysterious column within the Babylonian astronomy is column Φ. It is closely connected to the lunar velocity and to the duration of the Saros. This paper presents new ideas for the development and interpretation of column Φ. It combines the excellent Goal-Year method (for the prediction of Lunar Six time intervals) with old ideas and practices from the “schematic astronomy”. Inspired by the old “TU11” rule for prediction of times of lunar eclipses, it proposes that column Φ, in a similar way, used the sum of the Lunar Four to predict times of lunar eclipses as well as the duration of one, 6, and 12 months by means of what usually is called “R–S” schemes. It also explains fully the structure and development of such schemes, a fact that strongly supports the new interpretation of column Φ.

Significant progress has been made in understanding Mesopotamian astronomy and astrology since the decipherment of cuneiform tablets containing astronomical and astrological texts in the late nineteenth century. However, until now few attempts have been made to write adetailed history of the Mesopotamian astral sciences as opposed to detailed studies of particular texts and types of astronomy or astrology. My aim in this paper is to present some ideas of how such ahistory should be written and in particular to consider if and how visual evidence can be brought alongside textual evidence in the writing of this history.

O texto propõe estudo da concepção de Terra esférica a partir de alguns modelos da filosofia grega, com destaque para Anaximandro e Platão. São estabelecidos apontamentos sobre a filosofia pré-socrática, cujo estudo da natureza é marcado por mutações em relação às mitologias, bem como por abordagem inovadora da astronomia babilônica. Ressalta-se que a atividade intelectual dos filósofos da natureza fez parte de contexto histórico específico, com a inserção de questões políticas, como o advento da pólis e a organização espacial decorrente. São apresentadas concepções sobre o formato da terra nas mitologias de Homero e Hesíodo, prosseguindo com a contextualização das cosmologias pré-socrática e platônica. Finalizamos apontando o caráter transcendental das imaginações sobre a Terra esférica no contexto da filosofia e sua influência no autor romano Cícero.

Current understanding of astronomy attributes the earliest geometric models to the Greeks. Yet there remains substantial uncertainty about the Mesopotamian origins of the classical Greek constellations. It is here shown how clues famously given by Plato in his Timaeus provide the key to understanding the original geometric design framework. Having allocated the four elements (Water, Earth, Fire and Air) to regular polyhedra, Plato assigned a fifth figure to the cosmos, traditionally identified as the dodecahedron. Based on geometrical and philosophical arguments, it is here proposed that Plato also had in mind the orbicular elevated dodecahedron, consisting of 360 fundamental Platonic scalene triangles. In mapping it out as a convenient approximation to the celestial sphere, we discover that it offers a geometric framework for the Paths of Anu, Enlil and Ea of Mesopotamian astronomy, while explaining the enigma of why the constellations of the zodiac are not equally distributed along the ecliptic. Three rings with partial ten-way rotational symmetry are also identified that appear to have been used in the design framework. The conclusions emphasize Plato’s debt to earlier astronomers, while transforming our understanding of the constellations so familiar today.

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. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve Ephemerides and seven Procedure Texts. Three computational systems of Mercury are known, all of system A. System A1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A2 by two Ephemerides covering the years 310 to 290 BC. System A3 is known from a Procedure Text and from Text M, an Ephemeris of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the Astronomical Diaries and Planetary Texts we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase.

The Sovereign Pole System ( Huang ji li 皇極曆, a.d. 600) is the first Chinese astronomical system to include a solar equation table for calculating the motion of the Sun. From then on, each Chinese system took the solar equation of centre into consideration. Some scholars have argued that Chinese solar theory in the Sui and Tang dynasties was developed independently by Chinese astronomers. However, other scholars have speculated that this theory was ultimately influenced by Babylonian astronomy, through the medium of Indian astronomy. In this paper, we compare Chinese solar theory in the Sui and early Tang periods with Indian solar theory in the sixth century. First, we discuss the content, meaning, and accuracy of the solar equation tables in the Sovereign Pole System, the Great Patrimony System ( Da ye li 大業曆, a.d. 607), and the Great Expansion System ( Dan yan li 大衍曆, a.d. 727), three representative Chinese systems of the Sui and early Tang periods. Then, we discuss the speed of the Sun, the accuracy of the solar equation of centre, and the difference between the motions of the true Sun and mean Sun in the Vāsiṣṭha Siddhānta, Pauliśa Siddhānta, and Romaka Siddhānta, three Indian astronomical works collected in the Pañcasiddhāntikā in the sixth century. Finally, we discuss the similarities and differences between Indian and Chinese solar theories in the Sui and early Tang periods. The results show that while Chinese solar theory in the Sui and early Tang periods was very similar to Indian solar theory in the sixth century, Indian astronomers had a more sophisticated view of solar theory than their Chinese contemporaries, both in relation to the difference between the mean Sun and true Sun and in the selection of the longitude of perigee of the Sun. At the same time, we must note that the influence of Indian solar theory on Chinese solar theory was very limited, even if Chinese solar theory in the Sui and Tang dynasties was possibly influenced by Indian solar theory.