Summary Tycho Brahe's lunar theory, mostly the work of his assistant Christian Longomontanus, published in the Progymnasmata (1602), was the most advanced and accurate lunar theory yet developed. Its principal innovations are: the introduction of equant motion for the first inequality in order to separate the determination of direction and distance; a more accurate limit for the second inequality although requiring a more complex calculation; additional inequalities of the variation and, in place of the annual inequality in Tycho's (...) earlier theory, a reduction in the equation of time; in the latitude theory a variation of the inclination of the orbital plane and an inequality of the motion of the nodes; a reduction in the range of variation of distance, parallax, and apparent diameter. Some of these were already present in Tycho's earlier lunar theory (1599), but all were changed in notable ways. Twenty years later Longomontanus published a modified version of the lunar theory in Astronomia Danica (1622), for the purpose of facilitating the calculation through new correction tables, and also explained his reasons for parts of the theory in the Progymnasmata. This paper is a technical study of both lunar theories. (shrink)

David Farrell Krell reflects on nine writers and philosophers, including Heidegger, Derrida, Blanchot, and Holderlin, in a personal exploration of the meaning of sensual love, language, tragedy, and death. The moon provides a unifying image that guides Krell's development of a new poetics in which literature and philosophy become one. Krell pursues important philosophical motifs such as time, rhythm, and desire, through texts by Nietzsche, Trakl, Empedocles, Kafka, and Garcia Marquez. He surveys instances in which poets or novelists explicitly address (...) philosophical questions, and philosophers confront literary texts--Heidegger's and Derrida's appropriations of Georg Trakl's poetry, Blanchot's obsession with Kafka's tortuous love affairs, and Garcia Marquez's use of Nietzsche's idea of the Eternal Return--all linked by the tragic hero Empedocles. In his search to understand the insatiable desire for completeness that patterns so much art and philosophy, Krell investigates the identification of the lunar voice with woman in various roles--lover, friend, sister, shadow, and narrative voice. (shrink)

Summary of conclusionsThe seven MS lunar and Paschal tables of De ratione paschali fall into two distinct groups which we have classified as Sirmond-type and Padua-type respectively, and from these we have restored the tables of their archetype. The Sirmond-type tables preserve a unique lunar year, which we term the Anatolian lunar year, and they first emerge in the context of a larger computus which was assembled in southern Ireland c. 658, a copy of which Wilfrid had (...) evidently obtained by the time of the Synod of Whitby. The weight of circumstantial evidence supports the hypothesis that it was he who then instigated the corruption of both the tables and the patristic authorities of De ratione paschali, a copy of which subsequently passed to Bede and thence to the Sirmond group of MSS. The Padua-type tables on the other hand are represented only by the Padua MS, and they appear to have originated from within Insular circles on the Continent. These too were crudely corrupted, first by changing their lunar year from Anatolian to Roman and moving their ferial data from January to December and changing some Paschal data, and then secondly by collation with Sirmond-type tables. In the case of both types the objective was clearly to weaken the relationship between the lunar and Paschal tables and to try and obscure the Paschal principles that they preserve and thereby undermine the position of those followers of the Insular latercus who relied on the authority of De ratione paschali for their Pasch. These conclusions naturally give a rather different colour to the events of Whitby from that given by Bede, who places them completely in a theological context. However in material terms what was really at stake at Whitby was the transfer of the patronage of the entire kingdom of Northumbria from the Celtic to the Roman church. Here then was sufficient motivation to justify the most ruthless of expedients. (shrink)

Scientific discovery requires not only that one correctly explain phenomena, but that one persuade one’s peers. Anaxagoras’ theory was accepted by Empedocles, Diogenes of Apollonia, and even Philolaus (who had a radically different model of the heavens), and apparently Democritus. The doxographic tradition shows a mixture of different theories competing with heliophotism and antiphraxis. But when errors are corrected and the theories are put in chronological order (as they are not in doxographies), we see that there were virtually no new (...) theories of lunar light and eclipses proposed after the time of Anaxagoras. Indeed, Aristotle himself uses antiphraxis as a paradigm of successful scientific explanation–even though it fits poorly with his own model of the heavens. Mathematical astronomers from Aristarchus to Ptolemy adopted heliophotism as an axiom and antiphraxis as a theorem. No competing hypothesis was even considered from the time of Ptolemy to the present day. (shrink)

This paper is the second of a multi-part examination of the creation of the Babylonian mathematical lunar theories known as Systems A and B. Part I (Britton 2007) addressed the development of the empirical elements needed to separate the effects of lunar and solar anomaly on the intervals between syzygies. This was accomplished in the construction of the System A lunar theory by an unknown author, almost certainly in the city of Babylon and probably early in the (...) 4th century B.C. The present paper focuses mainly on System A and the likely process of its construction. The first three sections are largely descriptive – first of the basic concepts which underlie the theory; then of the component schemes comprising the theory; and finally of two distinctive texts which suggest how the theory was constructed. The crux of the paper is Sect. 4, which describes how the theory seems likely to have been constructed. Here the crucial insight in separating the effects of lunar and solar anomaly appears to have been recognizing that – of all the measurable intervals bounded by eclipses – only 235 months exhibits a variation due solely to lunar anomaly, and that by means of an elegant mathematical model the amplitudes of 223 and 12 months could be deduced from its amplitude. The rest of the section describes the likely details of the derivation of the Φ ~ Λ scheme, and the extension of the methodology to the other components of the theory. It concludes with a demonstration that Φ and its dependent schemes were anchored through the Φ ~ W scheme to the syzygy on –403 Aug 18 (GN 7391) which concluded the shortest 6-month interval in the first 24 saros cycles since –746 (and in fact in the 900 years separating Nabonassar and Ptolemy). The next three sections address a number of largely technical details and amplifications of the theory, beginning with the schemes describing the variation of lunar velocity (column F) in Sect. 5. Section 6 addresses issues concerning the interpretation of Φ and the so-call Saros Text (BM 36705), while Sect. 7 discusses System B’s corresponding treatment of the effects of lunar anomaly, illustrating both its derivative nature and mathematically less rigorous structure. Section 8 examines the accuracy of the two theories, showing that the System A theory was both remarkably accurate and superior to System B. The final section offers some brief remarks on the power and elegance of the mathematical treatment of the problem by the author of System A. (shrink)

A brief scholion allusion to a “Selenite” community in Arcadia raises a question concerning this epithet and its meaning on the background of similar expressions denoting extreme antiquity. The better known term associated with the Arcadians is Proselēnoi, namely, pre-lunar, people who preceded the moon. This term is examined through several options of understanding. At the core of this analysis stands the Classical tendency to highly appreciate early periods of time and early peoples. This opens up a discussion of (...) autochthony and the concept of extreme antiquity, particularly associated with Arcadia. The result is an etymologically based mythographic study centred on the Arcadians’ existence in relation to the first appearance of the moon. The conclusion offers a new interpretation of a neglected term. (shrink)

This paper is the third of a multi-part examination of the Babylonian mathematical lunar theories known as Systems A and B. Part I (Britton, AHES 61:83–145, 2007) addressed the development of the empirical elements needed to separate the effects of lunar and solar anomaly on the intervals between syzygies, accomplished in the construction of the System A lunar theory early in the fourth century B.C. Part II (Britton, AHES 63:357–431, 2009) examines the accomplishment of this separation by (...) the construction of a successful theory depicting the variations due to lunar anomaly in System A and its subsequent adaptation in System B. The present paper examines the introduction of the uniform zodiac, necessary for any theory depicting variations depending on the position of syzygy. It addresses three questions: (1) In light of all available evidence, what is the magnitude of the constant term in the expression Δλ* = C − 1.3828°Y, describing the difference between the Babylonian sidereal longitudes and modern tropical longitudes? (2) What considerations governed the placement of the Babylonian sidereal zodiac relative to the fixed stars? (3) When was the uniform zodiac introduced? To the first question it finds C = 3.20° ± 0.1°, scarcely different from Huber’s (Centaurus 5:192–208, 1958) estimate of 3.08°, essentially confirming Huber’s result obtained from much less data. For the second it shows that accommodating the three asterisms comprising Taurus limited the placement of the zodiac to within 3°, while the prominence of half sign multiples among the measured intervals between prominent Normal Stars led irresistibly to the choice adopted. Finally, it finds that the zodiac was introduced between −408 and −397 and probably within a very few years of −400. (shrink)

O texto visa traçar um paralelo, uma espécie de irmandade, entre dois filósofos bastante afastados no tempo e que se movimentaram em contextos sociais muito distintos: Blaise Pascal e Miguel de Unamuno. Defende-se que o que aproxima ambos é sua filiação a certa filosofia lunar. Tal conceito é elucidado em contraposição ao que seria a filosofia solar. Enquanto esta última seria racionalista, de jaez iluminista, aquela seria um modo de pensamento que, ainda que não necessariamente irracionalista, identifica nos elementos (...) irracionais ou, em todo caso, inescrutáveis, instâncias determinantes da realidade e da condição humana. (shrink)

Leonhard Euler (1707–1783) published two lunar theories in 1753 and 1772. He also published lunar tables in 1745, 1746, and—anonymously—in 1750. There are notebook records, unpublished manuscripts, and manuscript fragments by Euler reflecting the development of his lunar theories between about 1725 until about 1752. These documents might be used to reconstruct Euler’s theory on which he based his calculations of those lunar tables and to analyze the development of his lunar theories within this time (...) span. The results of this analysis will be published here in three parts representing three stages of Euler’s research on this topic: First approaches (about 1725–1730), developing the methods (about 1730–1744), and the breakthrough (about 1744–1752). In this part, I analyze Euler’s manuscripts and, predominantly, Euler’s records of his first two notebooks written between 1725 and 1730. I found that his early theoretical approach is coined by his development of analytical (rational) mechanics of punctiform bodies moved by central forces. He tried to describe the Moon’s motion in terms of two simultaneously acting centripetal forces, Huygens’ centrifugal theorem, and associated osculating radii. (shrink)

In this paper, three problems that have hardly been noticed or even gone unnoticed in the available literature in the cosmology of Philolaus are addressed. They have to do with the interrelationships of the orbits of the Earth, the Sun, and the Moon around the Central Fire and all three of them constitute potentially insurmountable obstacles within the context of the Philolaic system. The first difficulty is Werner Ekschmitt’s claim that the Philolaic system cannot account for the length of the (...) day (νυχϑήμερον). It is shown that this problem can be solved with the help of the distinction between the synodic day and the sidereal day. The other two problems discussed in this paper are concerned with two hitherto unnoticed deficiencies in the explanation of lunar eclipses in the Philolaic system. The Philolaic system cannot account for long-lasting lunar eclipses and according to the internal logic of the system, during lunar eclipses the Moon enters the shadow of the Earth from the wrong side. It is almost unbelievable that nobody, from the Pythagoreans themselves up to recent authors, has noticed these two serious deficiencies, and especially the latter, in the cosmology of Philolaus the Pythagorean. (shrink)

The analysis of unpublished manuscripts and of the published textbook on mechanics written between about 1730 and 1744 by Euler reveals the invention, application, and establishment of important physical and mathematical principles and procedures. They became central ingredients of an “embryonic” lunar theory that he developed in 1744/1745. The increasing use of equations of motion, although still parametrized by length, became a standard procedure. The principle of the transference of forces was established to set up such equations. Trigonometric series (...) expansions together with the method of undetermined coefficients were introduced to solve these equations approximatively. These insights constitute the milestones achieved in this phase of research, which thus may be characterized as “developing the methods”. The documents reveal the problems Euler was confronted with when setting up the equations of motion. They show why and where he was forced to introduce trigonometric functions and their series expansions into lunar theory. Furthermore, they prove Euler’s early recognition and formulation of the variability of the orbital elements by differential equations, which he previously anticipated with the concept of the osculating ellipse. One may conclude that by 1744 almost all components needed for a technically mature and successful lunar theory were available to Euler. (shrink)

From the early ninth century until about eight centuries later, the Middle East witnessed a series of both simple and systematic astronomical observations for the purpose of testing contemporary astronomical tables and deriving the fundamental solar, lunar, and planetary parameters. Of them, the extensive observations of lunar eclipses available before 1000 AD for testing the ephemeredes computed from the astronomical tables are in a relatively sharp contrast to the twelve lunar observations that are pertained to the four (...) extant accounts of the measurements of the basic parameters of Ptolemaic lunar model. The last of them are Taqī al-Dīn Muḥammad b. Ma‘rūf’s trio of lunar eclipses observed from Istanbul, Cairo, and Thessalonica in 1576–1577 and documented in chapter 2 of book 5 of his famous work, Sidrat muntaha al-afkar fī malakūt al-falak al-dawwār. In this article, we provide a detailed analysis of the accuracy of his solar and lunar observations. (shrink)

SummaryThe celebrated Swedish natural philosopher and visionary theologian Emanuel Swedenborg (1688–1772) devoted major efforts to the establishment of a reliable method for the determination of longitude at sea. He first formulated a method, based on the astronomical observation of lunar position, while in London in 1710–12. He issued various versions of the method, both in Latin and in Swedish, throughout his career. In 1766, at the age of 78, he presented his scheme for judgment by the Board of Longitude (...) in London. The rich archive of Swedenborg's career allows an unusually detailed historical analysis of his longitude project, an analysis rather better documented than that available for the host of contemporary projectors who launched longitude schemes, submitted their proposals to the Board of Longitude, and have too often been ignored or dismissed by historians. This analysis uses the longitude work to illuminate key aspects of Swedenborg's wider enterprises, including his scheme to set up an astronomical observatory in southern Sweden to be devoted to lunar and stellar observation, his complex attitude to astronomical and magnetic cosmology, and his attempt to fit the notion of longitude into his visionary world-view. Swedenborg's programme also helps make better sense of the metropolitan and international networks of diplomatic and natural philosophical communication in which the longitude schemes were developed and judged. It emerges that his longitude method owed much to the established principles of earlier Baroque and Jesuit natural philosophy while his mature cosmology sought a rational and enlightened model of the universe. (shrink)

Name der Zeitschrift: Apeiron Jahrgang: 50 Heft: 4 Seiten: 483-510.

Nevil Maskelyne, the Cambridge-trained mathematician and later Astronomer Royal, was appointed by the Royal Society to observe the 1761 transit of Venus from the Atlantic island of St Helena, assisted by the mathematical practitioner Robert Waddington. Both had experience of measurement and computation within astronomy and they decided to put their outward and return voyages to a further use by trying out the method of finding longitude at sea by lunar distances. The manuscript and printed records they generated in (...) this activity are complemented by the traditional logs and journals kept by the ships’ officers. Together these records show how the mathematicians came to engage with the navigational practices that were already part of shipboard routine and how their experience affected the development of the methods that Maskelyne and Waddington would separately promote on their return. The expedition to St Helena, in particular the part played by Maskelyne, has long been regarded as pivotal to the introduction of the lunar method to British seamen and to the establishment of theNautical Almanac. This study enriches our understanding of the episode by pointing to the significant role played by the established navigational competence among officers of the East India Company. (shrink)

In Asian countries, there is a tradition that a rabbit, known as the Moon rabbit, lives on the Moon. Typically, two reasons are mentioned for the origin of this tradition. The first reason is that the color pattern of the lunar surface resembles the shape of a rabbit. The second reason is that both the Moon and rabbits are symbols of fertility, as the Moon appears and disappears (i.e., waxing and waning) cyclically and rabbits are known for their high (...) fertility. Considering the latter reason, is the color pattern of the lunar surface not similar to a rabbit? Here, the similarity between rabbit and the lunar surface pattern was evaluated using seven AI architectures. In the test conducted with Contrastive Language-Image Pre-Training (CLIP), which can classify images based on given words, it was assumed that people frequently observe the Moon in the early evening. Under this condition, the lunar surface pattern was found to be more similar to a rabbit than a face in low-latitude regions, while it could also be classified as a face as the latitude increases. This result is consistent with that the oldest literatures about the Moon rabbit were written in India and that a tradition of seeing a human face in the Moon exists in Europe. In a 1000-class test using seven AI architectures, ConvNeXt and CLIP sometimes classified the lunar surface pattern as a rabbit with relatively high probabilities. Cultures are generated by our attitude to the environment. Both dynamic and static similarities may be essential to induce our imagination. (shrink)

This article deals with an unstudied criterion for determining lunar crescent visibility, which appears in the Mufrad Z $$\bar{\iota }$$ ι¯ j,. Al-Ṭabar$$\bar{\upiota }$$ ι¯ attributes this circular criterion to Al-B$$\bar{\upiota }$$ ι¯ rūn$$\bar{\upiota }$$ ι¯. Initially, Prof. David King shed light on this criterion in 1987 and explained it briefly. We will examine this criterion by re-computing the underlying numerical values to reconstruct it, in order to demonstrate that it originates from Ḥabash’s simple criterion.

The following article has two parts. The first part recounts the history of a series of discoveries by Otto Neugebauer, Bartel van der Waerden, and Asger Aaboe which step by step uncovered the meaning of Column $$\varPhi $$ Φ, the mysterious leading column in Babylonian System A lunar tables. Their research revealed that Column $$\varPhi $$ Φ gives the length in days of the 223-month Saros eclipse cycle and explained the remarkable algebraic relations connecting Column $$\varPhi $$ Φ to (...) other columns of the lunar tables describing the duration of 1, 6, or 12 synodic months. Part two presents John Britton’s theory of the genesis of Column $$\varPhi $$ Φ and the System A lunar theory starting from a fundamental equation relating the columns discovered by Asger Aaboe. This article is intended to explain and, hopefully, to clarify Britton’s original articles which many readers found difficult to follow. (shrink)

Recent longitudinal observations show that human menstrual cycles, sleep‐wake cycles and manic‐depressive cycles can become synchronized with lunar cycles, but do so in uniquely complex and heterogeneous ways that are unlikely to have been detected by past studies. Past studies’ negative results have given rise to a scientific consensus that human biology and behavior are unaffected by lunar cycles. The recent observations show that synchrony can be temporary, and can occur with more than one type of lunar (...) cycle, more than one phase of a lunar cycle and more than one resonant frequency of a lunar cycle. Given the variability of human responses to lunar cycles, aggregate analyses used in almost all previous studies would likely have cancelled out individuals’ responses and led to false negative results. In light of these observations, the question of lunar influence should be investigated further, with longitudinal observations and case‐by‐case analyses of individuals’ data. (shrink)

In the 1760s, the international debate on the solution to determining longitude at sea is at its acme. Two solutions emerge, the mechanical and the astronomical ones. The Portuguese mathematician and astronomer José Monteiro da Rocha (1734–1819) is well aware of that debate. For him, Harrison’s No. 4 marine timekeeper cannot be seen as a solution. The desirable solution could only be astronomical. In a manuscript from c. 1765, which unfortunately he fails to publish, Monteiro da Rocha is very critical (...) of Lacaille's lunar-distance method (1759) and proposes another one. In this paper, we intend to analyse Monteiro da Rocha’s criticisms and proposals, trying to understand how this manuscript fits into the international longitude debate and the Portuguese scientific scenario at the time. Concurrently, we will re-examine the classical historiography around the English vs. French priority proposal of the lunar-distance method, purging it from its mythologies to shift it towards a more open, less linear history. (shrink)

This paper investigates the methods of eclipse prediction in China before the fourth century AD, with a detailed example of the eclipse theory in the Jing chu li (Luminous Inception System ). As the official calendar of the Jin dynasty and the Kingdom Wei during the three kingdoms period, the Jing chu li was used for more than 200 years after it was adopted in 237 AD. From the San tong li (Triple Concordance System ) of the Western Han to (...) the Jing chu li, methods for predicting eclipses developed in three important ways: (i) from predicting only lunar eclipses to the prediction of both solar and lunar eclipses; (ii) from relying only on the mean periods of the sun and the moon to taking into consideration the variation in lunar velocity; and (iii) from estimating only a rough date to predicting the exact time of eclipses. This paper addresses two questions: first, how did ancient Chinese astronomers use cycles to predict eclipses in the Han dynasty? Second, how did astronomers such as Liu Hong and Yang Wei revise early eclipse prediction methods? The original text of the Jing chu li is analyzed to show how Yang Wei combined lunar velocity theory with the traditional method of predicting eclipses using cycles. (shrink)

The lunar Three-Body problem is a famously intractable problem of Newtonian mechanics. The demand for accurate predictions of lunar motion led to practical approximate solutions of great complexity, constituted by trigonometric series with hundreds of terms. Such considerations meant there was demand for high speed machine computation from astronomers during the earliest stages of computer development. One early innovator in this regard was Wallace J. Eckert, a Columbia University professor of astronomer and IBM researcher. His work illustrates some (...) interesting features of the interaction between computers and astronomy. (shrink)

This paper is a technical study of the systematic observations and computations made by Muḥyī al-Dīn al-Maghribī (d. 1283) at the Maragha observatory (north-western Iran, c. 1259–1320) in order to newly determine the parameters of the Ptolemaic lunar model, as explained in his Talkhīṣ al-majisṭī, “Compendium of the Almagest.” He used three lunar eclipses on March 7, 1262, April 7, 1270, and January 24, 1274, in order to measure the lunar epicycle radius and mean motions; an observation (...) on April 20, 1264, to determine the lunar eccentricity; an observation on August 29, 1264, to test the model; and another on March 15, 1262, for measuring the lunar parallax. In the second period of activity at the Maragha observatory, Shams al-Dīn Muḥammad al-Wābkanawī (c. 1254–1320) adopted all of al-Maghribī’s parameter values in his Zīj, but decreased his value for the mean longitude of the moon at epoch by 0;13,11∘\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}. By comparing the times of the new moons and lunar eclipses in the period of 1270–1320 as computed from the astronomical tables of the Maragha tradition with the true modern ones, it is argued that this correction was very probably the result of actual observations. (shrink)