AbstractIn 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. Paper I (de Jong in Arch Hist Exact Sci 73:1–37, 2019) was devoted to a study of system A theory of the outer planets. In this second paper, I will study system A theory of the planet Venus. All presently known ephemerides of Venus appear to have been written after 200 BC so that the development of system A theory of Venus may have been a late development. On the other hand, there are several earlier texts in which the motion of Venus, going from one synodic phenomenon to the next one, is parametrized in quite some detail. At least six computational systems of Venus are known of which only two, systems A0 and A3, are genuine type-A systems. Both are based on the hypothesis that in exactly 1151 years Venus experiences 720 synodic phenomena of the same kind and that after this period of 1151 years Venus returns to exactly the same position in the sky. This period relation was probably derived from the observational fact that after 8 years Venus returns to a position in the zodiac that falls on average 2.5° short of its previous position. The study of the Babylonian planetary theory of Venus presented here will be primarily based on the system A3 theory of Venus because it is the most complete of the two systems. The parameters of the four step functions characterizing the system A3 model of the first and last appearances of Venus are known from previous studies of tablet BM 32599 (ACT 1050). Based on a database of synthetic observations of the first and last appearances (morning first, morning last, evening first and evening last) and the two stationary points (morning station and evening station) of Venus between 315 BC and 50 BC, I first discuss the observational material from the point of view of a Babylonian astronomer. This involves deriving synodic time intervals and synodic arcs from the observed dates and longitudes of Venus. Both vary with the position of Venus in the zodiac. This variation shows up most purely in the synodic arcs and synodic time intervals of Venus at its stations. The variation pattern of the synodic arcs and of the synodic time intervals at the first and last appearances of Venus differ considerably because they are strongly affected by atmospheric extinction and by the ecliptic latitude of Venus. A comparison of predictions of the first and last appearances of Venus computed according to system A3 with observations in the synthetic database shows that the models provide fairly accurate fits to the observations of evening last and morning first with typical standard deviations of about 2° but rather poor fits to the observations of morning last and evening first with typical standard deviations of about 7° and 4°. Quite surprisingly, it turns out that the morning first A3 model fits the observed longitudes of Venus at its stations almost perfectly, with standard deviations of about 1°. This suggests that observations of the stations of Venus must have been available to the Babylonian astronomer(s) who constructed the system A3 model of Venus and who computed the longitudes of Venus at its first and last appearances preserved in tablet BM 32599. This is quite puzzling because observations of the stations of Venus were definitely not part of the standard observing program of the Babylonian astronomers as we know it from the Astronomical Diaries. To resolve this problem, I propose that observations could have been made by some individual astronomer for his own use but that these observations never became part of the tradition of what was regularly observed and recorded in the Astronomical Diaries. It turns out that 25 years of observations of Venus (here illustrated for the years 203 BC to 178 BC) is sufficient for the construction of a system A model that perfectly fits the variation of the synodic arcs of Venus at its stations. By combining observed time intervals and known velocities of Venus, observations over this limited timespan can also be used to construct longitudes of Venus at its first and last appearances from previous known positions of Venus based on Normal Stars passages or from known positions at its stations. This is the only way in which positions of Venus at its first and last appearances can be determined because at its first and last appearances the sky is too bright for Normal Stars to be visible. I propose that the Babylonian system A3 models of Venus were based on longitudes constructed in this way. I finally suggest that the tablet, of which BM 32599 is the remainder, was filled line by line starting with a set of initial values in the bottom left-hand corner of the tablet and that the choice of these initial values was based on an observation of Venus at its evening station when it was exceptionally close to the Normal Star η Piscium in February 179 BC. Based on this choice of initial conditions, the tablet covers the years 403 BC to 170 BC and it may have been composed around 170 BC. It was probably computed with the purpose of creating a raster of longitudes with mazes of about 2.5° that could be used to predict future positions of Venus at its first or last appearance once a position at a previous first or last appearance was available.

中文翻译：

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巴比伦行星理论研究 II. 金星

摘要在这一系列论文中，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（de Jong in Arch Hist Exact Sci 73:1-37, 2019）致力于研究外行星系统 A 理论。在第二篇论文中，我将研究金星的系统 A 理论。所有目前已知的金星星历似乎都是在公元前 200 年之后编写的，因此金星系统 A 理论的发展可能是较晚的发展。另一方面，在一些早期的文本中，金星的​​运动，从一个会合现象到下一个现象，被相当详细地参数化。金星至少有六个计算系统是已知的，其中只有两个系统 A0 和 A3 是真正的 A 型系统。两者都是基于这样的假设，即金星在整整 1151 年内经历了 720 次相同类型的会合现象，并且在这 1151 年之后金星返回到完全相同的天空位置。这种周期关系可能源于观测事实，即金星在 8 年后返回到黄道带中的位置，该位置比之前的位置平均低 2.5°。这里介绍的巴比伦金星行星理论的研究将主要基于金星的系统 A3 理论，因为它是两个系统中最完整的。从先前对平板电脑 BM 32599 (ACT 1050) 的研究可知，表征金星首次和最后一次出现的系统 A3 模型的四个阶跃函数的参数。基于首次和最后一次出现的综合观察数据库（早上第一次，早晚、早晚和晚晚）和公元前315年至公元前50年间金星的两个驻点（早站和晚站），我首先从巴比伦天文学家的角度讨论观测材料。这涉及从观测到的金星日期和经度推导出会合时间间隔和会合弧。两者都随着金星在黄道十二宫的位置而变化。这种变化最纯粹地出现在金星在其站点的会合弧和会合时间间隔中。金星首次和最后一次出现时的会合弧和会合时间间隔的变化模式有很大差异，因为它们受到大气消光和金星黄道纬度的强烈影响。将根据系统 A3 计算的金星第一次和最后一次出现的预测与合成数据库中的观测值进行比较表明，这些模型对最后一个晚上和第一次早上的观测结果提供了相当准确的拟合，典型的标准偏差约为 2°，但与早上最后一次和晚上第一次的观察结果不符，典型的标准偏差约为 7° 和 4°。令人惊讶的是，事实证明，早上第一个 A3 模型几乎完美地拟合了在其站点上观测到的金星经度，标准偏差约为 1°。这表明，巴比伦天文学家肯定已经获得了金星台站的观测结果，他们构建了金星系统 A3 模型，并计算了金星首次和最后一次出现时的经度，保存在 BM 32599 平板电脑中。令人费解，因为对金星站的观测绝对不是我们从《天文日记》中了解到的巴比伦天文学家标准观测计划的一部分。为了解决这个问题，我建议某些天文学家可以进行观测以供他自己使用，但这些观测从未成为天文日记中定期观测和记录的传统的一部分。事实证明，对金星的 25 年观测（此处说明了公元前 203 年至公元前 178 年）足以构建一个系统 A 模型，该模型完全适合金星各站会合弧的变化。通过将观测到的时间间隔和金星的已知速度相结合，在这个有限的时间跨度内的观测也可用于从基于正常恒星通道的金星先前已知位置或从其站点的已知位置构建金星首次和最后一次出现时的经度。这是确定金星第一次和最后一次出现时位置的唯一方法，因为在它第一次和最后一次出现时，天空太亮，普通星星无法看到。我建议金星的巴比伦系统 A3 模型基于以这种方式构建的经度。我最后建议以 BM 32599 为余数的数位板从数位板左下角的一组初始值开始逐行填充，并且这些初始值的选择基于观察公元前 179 年 2 月，当金星离正常星 η Piscium 格外近时，金星在它的晚间站。基于这种初始条件的选择，该片涵盖了公元前 403 年至公元前 170 年，可能是在公元前 170 年左右创作的。计算它的目的可能是创建一个具有大约 2.5° 迷宫的经度栅格，一旦有前一次或最后一次出现的位置可用，可用于预测金星在其首次或最后一次出现时的未来位置。从数位板左下角的一组初始值开始一行一行地填充，并且这些初始值的选择是基于对金星在其傍晚站的观察，当时金星异常靠近正常星公元前 179 年 2 月的 η Piscium。基于这种初始条件的选择，该片涵盖了公元前 403 年至公元前 170 年，可能是在公元前 170 年左右创作的。计算它的目的可能是创建一个具有大约 2.5° 迷宫的经度栅格，一旦有前一次或最后一次出现的位置可用，可用于预测金星在其首次或最后一次出现时的未来位置。从数位板左下角的一组初始值开始一行一行地填充，并且这些初始值的选择是基于对金星在其傍晚站的观察，当时金星异常靠近正常星公元前 179 年 2 月的 η Piscium。基于这种初始条件的选择，该片涵盖了公元前 403 年至公元前 170 年，可能是在公元前 170 年左右创作的。计算它的目的可能是创建一个具有大约 2.5° 迷宫的经度栅格，一旦有前一次或最后一次出现的位置可用，可用于预测金星在其首次或最后一次出现时的未来位置。