have the same length of shadow at the same apparent solar time.

The ancient Mesopotamians produced ephemerides or tables of the position and motion of the Sun for each day. There were essentially two methods of computing the ephemerides of the Sun: Systems A and B. System A used a step function for the solar speed (i.e., its change in position per day): one value for the first half of the year, and another value for the second half. System B used different values for each month in first increasing and then decreasing series. The effect is what Neugebauer (1983, p. 28) has referred to as a linear zig-zag function. See Figure 7.14 for an illustration. A table of the solar longitude is, in effect, a solar calendar, and although Mesopotamians used a lunar calendar, tabulation of the progress of the seasons with their changing temperatures and rainfall is an important concern of any civilization. The development of a mechanism by which the lunar calendar could be regularly coordinated with the solar calendar was an important result. Systems A and B were used also to obtain the length of daylight throughout the year. Their main utility seems to have been to regulate the beginning of the month and to predict eclipses. Van der Waerden (1974) thinks that these clever systems were most likely created between ~540 and 440 b.c. They are discussed further in §7.1.

Even more ancient are the megalithic solar observatories, the alignments of which indicate solar calendar activity. The mechanism here is the variation of azimuth of rise or set of the Sun as the solar declination changes over the year. A similar kind of calendrics is seen in at least some of the Medicine Wheels of North America (§6.3), some of which may have been used for a much more extensive interval of time (that at Majorville for more than 4000 years). In addition to Medicine Wheels, spirals carved or painted on rock in the southwestern United States, and elsewhere, are seen to be so placed that a dagger of sunlight created by the passage of sunlight through crevices in intervening rocks, marked critical solsticial and/or equinoctial times of the year. The oldest known astronomically aligned sites are passage graves, such as those at Newgrange, County Meath, Ireland. Some of the monuments at this site and the site of Gav'rinis have spiral engravings, and sunlight at midwinter sunrise illuminated them (see §6 for an extensive discussion of Megalithic sites). Again in the New World, marked sunwatchers' stations are places from which observers studied the December solsticial Sun for indications that it was returning northward again to renew and warm the earth.

Related to the determination of solar dates is the determination of the length of the tropical year: the time for the Sun to reappear at the same celestial longitude (e.g., to return again to the vernal equinox). In the Almagest,

Ptolemy gives the value obtained by Hipparchos11: 365 + 1/4 - 1/300 = 365.24667 mean solar days.12 Compared with the correct value for his time, 365.2422 days, his result was too long by a little under 6.5 minutes. Ptolemy repeated the calculation using the same data and added his own recent observations with the same result. Pedersen (1974, p. 131) attributes the difference from the modern value (365d24219) to instrumental error and refraction.

There is more than one way to talk about the length of the year. Table 4.2 lists four lengths of the year according to different criteria. The value is for the epoch 1900.0, and the variation term is the change in days/century, with T given in Julian centuries from 1900. Note that T is negative for dates prior to 1900. The anomalistic year is the time for the Sun to return to perigee (i.e., for the Earth to return to perihelion), the sidereal year is the time for the Sun to return to a line to a particular distant star, and the eclipse year is the period for the Sun to return to the same node of the Moon's orbit. Note that the sidereal period is the true period of revolution of the Earth around the Sun, with respect to a line to a distant star. The tropical year, measured from successive passages of the Sun through the vernal equinox, is shorter than the sidereal year because of the westward precession of the equinoxes, whereas the anomalistic year is longer than the sidereal year because of the advance (eastward motion) of the major axis of the orbital ellipse. The eclipse year is very much shorter than is the sidereal year because of the rapid regression (westward) of the lunar nodes (see §1.3.4 and §5.2).

Many cultures have used a 365d civil year, and some, like our own, have modified it by strategic, well-planned intercalation, to keep the civil calendar in step with the tropical year. Intercalation has not been a universal concern, however, and for particular purposes, different units were adopted. Egypt and Mesoamerica both had a 365d year that cycled through the seasons. The simplicity of such a scheme was, and still is, important in the calculation of the number of days between, say, two New Year's days N years apart. In this calendar, it is merely N X 365d, with no worries about which intervening years had intercalations. For this reason, the Egyptian 365d year was called the "astronomers' year," and Ptolemy among others used it. Neugebauer (1957/1969,

11 Whose own discussions of the length of the year were contained in two books, now both lost: On the Length of the Year, and On Intercalary Months and Days.

12 When we write a "d" following a number or in superscript above a decimal point, it can be understood to indicate units of the mean solar day, the time interval between two successive tranits by the mean sun.

p. 140) states that in Babylonian texts, the term "year" always refers to a sidereal year. He also notes that Ptolemy is the first to define the "year" as the tropical year. The term "year" has had other interpretations also: The Book of Enoch, known from Ethiopic sources and the Dead Sea Scrolls, cites a 364d year, called the "year of Enoch." The number 7 divides evenly into such a year; so it is useful in finding the day of the week corresponding to a particular date. Modern scientists use Julian day numbers to solve both types of problems for which the Egyptian and Ethiopic years provided solutions. See §8.1 and §8.3 for further discussion of time measurement and the astronomy of these cultures.

A tie-in between the civil date and the equinoxes is provided by the stars. The heliacal rising of a particular star, e.g., Sirius was sufficient to tell ancient Egyptians what season it was, regardless of the date of the local calendar, and thus when to expect such seasonally linked occurrences as the flooding of the Nile. Ultimately, there are very nearly 366 sidereal days to each 365 solar-day year.13

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