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 A 1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A 2 by two Ephemerides covering the years 310 to 290 BC. System A 3 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 A 3 . 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.

中文翻译：

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巴比伦行星理论研究 III．水星行星

在这一系列论文中，我试图回答巴比伦学者如何得出行星运动数学理论的问题。论文 I 和 II 专门讨论外行星和金星的 A 系统理论。在这第三篇也是最后一篇论文中，我将研究系统 A 水星的理论。我们对巴比伦水星理论的了解目前基于十二个星历表和七个程序文本。已知水星的三个计算系统，所有系统 A。系统 A 1 由涵盖公元前 190 年至公元前 100 年的九个星历表表示，系统 A 2 由涵盖公元前 310 年至 290 年的两个星历表表示。系统 A 3 是从程序文本和文本 M 中得知的，文本 M 是公元前 424 年至 403 年水星最后一晚能见度的星历。为了能够研究巴比伦学者构建他们系统的方式 A 水星模型从可用的观测材料中创建了一个合成观测数据库，通过计算所有首次和最后一次出现以及所有静止点的日期和黄道经度公元前 450 至 50 年间巴比伦的水星。在构建星历会合时间间隔 Δt 所需的数据中，可以直接从观测日期导出，但黄道经度和会合弧 Δλ 必须以其他方式确定。因为对于水星相对于正常恒星的位置很少能在它第一次或最后一次出现时确定，我建议巴比伦学者使用关系 Δλ = Δt -3; 39,40，它是从周期关系得出的，从观测到的会合时间间隔计算水星的会合弧。构建系统 A 阶跃函数的另一个困难是大多数振幅大于相关的区域长度，因此在计算水星会合相位的经度时，两个区域边界经常交叉。这种复杂性使人们难以理解巴比伦学者如何设法为水星构建系统 A 模型，该模型非常适合观测结果，因为它需要大量的计算工作才能在复杂的试错拟合过程中找到最佳可能的阶跃函数四个或五个自由参数。为了规避这个困难，我建议巴比伦学者使用另一种更直接的方法将系统 A 型模型拟合到水星的观测数据中。这种替代方法基于这样一个事实，即在三个会合间隔后，水星返回到天空中经度平均仅小 17.4° 的位置。使用大约 14°-25° 的减小幅度但保持相同的区域边界，我将称之为 3-synarc 系统 A 水星模型的计算得到显着简化。然后可以通过组合使用 3-synarc 阶跃函数计算的三列经度来组成水星会合相位的完整星历，每列以水星的经度开始，相隔一个会合事件。确认巴比伦天文学家确实使用了这种方法来自文本 M (BM 36551+)，公元前 424 年至 403 年水星傍晚最后一次出现的非常早期星历，根据系统 A 3 以三列计算。根据对文本 MI 的分析表明，大约在公元前 400 年，水星系统 A 建模中的初始方法可能是针对为系统 A 阶跃函数的振幅选择“好的”六十进制数，而在后来的最终模型中，可追溯到大约公元前 300 年以后，更多的重点放在选择振幅的数值上，以便它们通过简单的比率相关联。在后来的模型中，水星的四个会合相位中的每一个都使用了不同的星历周期，这一事实可能与为每个会合相位选择一组最合适的系统 A 阶跃函数幅度有关。