CHAPTER ONE CHRONOLOGY The astronomical interest in chronology is, in the first instance, to enable one to establish the number of days between two observations so that they may be used to determine the parameters of a model. Reports of observations are given with a date in some calendar, and the study of chronology is intended to allow one to find the corre- sponding date in the calendar preferred by the astronomer who wishes to use them. In the Almagest Ptolemy includes observational reports in various calendars used by observers and then converts those dates to the dates in his calendar: Egyptian years each of which contains 365 days and an epoch, “Era Nabonassar,” that corresponds to Feb. 26, 747 BC. For example, in Almagest X.9 Ptolemy cites an observation of Mars “in the 13th year of the calendar of Dionysius, Aigon 25, at dawn . i.e., in the 476th year from Nabonassar, Athyr [month III] 20/21 in the Egyptian calendar,” corresponding to Jan. 17/18, 272 BC (Toomer 1984, p. 502). However, Ptolemy does not describe any calen- dar and does not explain how he made these conversions. Moreover, Ptolemy decided to begin the day (daytime and nighttime together, i.e., our 24-hour day) at noon, in contrast to the Egyptians who began the day at sunrise and the Babylonians who began the day at sunset. This convention of starting the astronomical day at noon continued in the Middle Ages despite the usages in the civil calendars on which they were based. To explain the dating of observations and the con- version of dates from one calendar to another, astronomers from the 9th century on included a section on chronology in sets of astronomi- cal tables. Each calendar was defined, giving the names of the months, their lengths, the length of the year, cycles for leap years, and the epoch (the initial day of year 1 of the calendar). In this context a calendar was intended to serve as a device for counting days so that one could easily determine the number of days between two dates. This meant that observation did not play a role in defining the calendar and that a civil calendar based on phases of the Moon (e.g., New Moon) had to be recast as a “mean” calendar which allowed for a simple counting of days from the epoch. This applies specifically to the Hijra calendar 14 chapter one used by Muslims. In the civil Hijra calendar the day began at sunset and some months were determined by observation of the New Moon. But in the astronomical Hijra calendar the day began at noon, and the pattern of month lengths and leap years was fixed. The most exten- sive study of calendars in the Middle Ages is al-Bīrūnī’s Chronology of Ancient Nations (tr. Sachau 1879). The next issue to be considered is the procedure for converting dates from one calendar to another. The modern solution is to convert a date in one calendar into the corresponding Julian Day Number (where the day begins at noon), a simple day count, with no division into months or years, from an arbitrary epoch, and then to convert the Julian Day Number into the date in some other calendar. The medieval solution, however, was to display different tables for converting the date in one calendar to the corresponding date in another calendar, i.e., each pair of calendars was treated separately. In the course of the Middle Ages the original astronomical rationale for including a section on chronol- ogy in sets of astronomical tables was largely forgotten, but tables for chronology persisted (in some cases for astrological purposes or for establishing a fixed chronology for Biblical events). There are, however, instances where a medieval astronomer sought to revise a parameter by comparing his own observation with one taken from the Almagest (e.g., to determine the length of the mean synodic month, Levi ben Gerson compared a lunar eclipse that he observed in 1335 with one observed by Ptolemy in 134: see Goldstein 2003, pp. 70–71). 1. Epochs and Intervals The compilers of medieval sets of tables referred to many calendars which use different epochs and specified the intervals in days between the various epochs. These differences in days, sometimes expressed in sexagesimal form, were presented in the form of a list, of which there are a great many variations (see, e.g., Suter 1914, p. 109; Ratdolt 1483, f. c8r). In Table 1.1A below, we display the names of the most fre- quently used epochs, together with their corresponding dates in the Julian calendar and weekdays. We indicate their Julian Day Number (JDN), based on a modern counting of days, beginning arbitrarily in year –4712; note that the Julian Day begins at noon and it has noth- ing to do with the Julian calendar. We also accept the convention of negative years in the Julian calendar: for instance, –4712 corresponds to 4713 BC..