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)

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)

Some variants in the materials related to the planetary latitudes, including computational procedures, underlying parameters, numerical tables, and so on, may be addressed in the corpus of the astronomical tables preserved from the medieval Islamic period, which have already been classified comprehensively by Van Dalen. Of these, the new values obtained for the planetary inclinations and the longitude of their ascending nodes might have something to do with actual observations in the period in question, which are the main concern of (...) this paper. The paper is in the following sections. In the first section, Ptolemy’s latitude models and their reception in Islamic astronomy are briefly reviewed. In the next section, the medieval non-Ptolemaic values for the inclinations and the longitudes of the nodal lines are introduced. The paper ends with the discussion and some concluding remarks. The derivation of the underlying inclination values from the medieval planetary latitude tables and determining the accuracy of the tables are postponed to “Appendix” in the end of the paper. (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)

The classical theory of errors can be divided into stochastic and determinate parts, or branches. The birth of the first of therse became inevitable after Bradley's idea of cultivating astronomy and natural science in general by “regular series of observations and experiments” became universally accepted. Such scholars as Lambert, Simpson, Lagrange, Daniel Bernoulli and Euler were responsible for the development of the stochastic theory of errors while Laplace and Gauss completed its construction. About fifty or sixty years ago it was (...) included into mathematical statistics. (shrink)

ArgumentThe present paper is an attempt to understand how medieval astronomers working within the Ptolemaic astronomical context in which the annular eclipse is an unjustified and impossible phenomenon, could know, define, justify, and later make attempts that led to success in predicting annular solar eclipses. As a context-based study, it reviews the situation of annular eclipses with regard to the medieval hypotheses applied to the calculation of the angular diameters of the sun and the moon, which was basic for contemplating (...) the possibility of annular eclipses. This gives the premises and the preliminary insights that were necessary to clarify the complex situation of the annular eclipse in the late medieval Islamic period and to explain the historical mechanisms leading to justifying the phenomenon during that period. This was, first due to a convincing and efficient observational evidence which, of course, was available only to a number of medieval astronomers and significantly for only a limited time period, and, second, the result of an amazing interaction amongst various astronomical traditions available to them. At a more general level, the research aims to inspect or, at least, to give some impressions of the essential conditions, i.e., identification of phenomenon, empirical evidence, and the justifying underlying tradition, under which it became possible in the tradition-based science of the `medieval period to permit a not-already-defined and tradition-opposed phenomenon to be posed and justified. (shrink)

Ibn al-Bann1321) is the author of one of the four extant of the unfinished zq (fl. Tunis and Marrakesh ca. 1193j accessible for the computation of planetary longitudes. The present paper studies some modifications of the structure of the tables the purpose of which is to make calculations easier. The tables of the planetary and lunar equations of the centre are ' appears as a clever adapter, who displays a clear ingenuity allowing him to introduce formal modifications which give his (...) work an appearance of novelty which does not correspond to reality. (shrink)