## The Perception and Measurement of Time

Time carries us along, willing or not, into, through, and out of the world relentlessly, without "time-out" to recover our breaths, wits, or fortunes. The perceived arrow of time points always in one direction, from the past to the possible, from the known to the unknown. This implicit nature of time is characterized in many different ways in different languages: Some recognize the past, present, and futures with conditional and subjunctive, preterite, and pluperfect nuances. On the other hand, among the Hopi, for example, the important distinction is between "near" and "far" time, whether past, present, or future. The perception of the meaning of time has changed much through history and across cultures, but the experience of time as an enslaving tyranny is common to many. Whatever its ultimate meaning or importance, the measurement of time has practical importance for many areas of human endeavor, enabling individuals and groups to coordinate their activities and thereby keep their societies functioning. This was fully recognized among ancient cultures too.

Our 24-hour day, sexagesimal system of measuring seconds and minutes, and our use of decimal fractions of seconds, record a curious blend of heritage from ancient civilizations, specifically, Egypt and Mesopotamia. The 24-hour day, we now believe, originated in Egypt, with the use of the decans (Neugebauer 1955,1969 p. 81ff): During the shorter summer nights, a total of only about 12 decans (each roughly

1 St. Anne's, 5th verse;Isaac Watts [1674-1748].

10° apart on the ecliptic; see §3.3) could be seen. As each decan appeared at the southeastern horizon, it could serve as a marker of time. The scheme may have been extended to the winter nights and then to the days as well, resulting in the two 12-hour periods that characterize day and night, a derivative form of which is our ante- and post-meridian time-keeping.

Time units shorter than a day were mentioned in the Babylonian diaries (Sachs and Hunger 1988): the US (pronounced 'oosh' and meaning 'length') or the "time degree," which measured the time for the sky to turn through 1° in right ascension; and the NINDA, 1/60 of an US. The US therefore marked four minutes of our time, and the NINDA, corresponding to a minute of arc, marked four seconds of time. According to Neugebauer (1941, 1983, p. 16), the US had its origin as 1/30 of a unit of length, the danna, equivalent to about seven miles, and which was in use in Mesopotamia as early as 2400 b.c. The usage of a time-equivalent distance—a distance equivalent to the time it takes to travel this distance—is not unfamiliar today.2 The danna was equivalent to a time interval of two hours, sometimes called a "double hour." The equivalence is, then, 12 double hours = 360 US = 1 day. These units from ancient Babylon, then, are the origin of our "degrees" of angle, and of the astronomical usage of time units to measure distance on the celestial sphere. The base 60, and the use of sexagesimal fractions, used both for our minutes and seconds of time as well as for minutes and seconds of arc, has also a Babylonian origin. The 24h/d, 60m/h, and 60s/m combination was in use by Hellenistic times.

Time could be measured by the length of equatorial arc (i.e., the length of arc along the equator as delineated by two

2 For example, we say that such a place is 10 minutes away by car. Time is usually a more important quantity than is distance for many travelers, so that the time to travel may be more meaningful for a given set of traveling conditions (traffic, road/track quality, weather, etc.) of a region.

or more stars) that had traversed a particular spot in the observer's horizon system. The spot could be on the meridian or at the east or west points of the horizon. Such measurements yielded what were called equatorial times (%povot iohimeptvoi), and the unit was the "time-degree," 1/15 of an hour or four minutes of time. This measure of time is directly analogous to our sidereal time, which marks the passage of stars across the celestial meridian and is defined formally as the hour angle of the vernal equinox:

where LST is the local sidereal time because the meridian is different for each observer. This is equivalent to the sum of the hour angle and the right ascension of any object in the sky; i.e.,

When time was to be expressed explicitly in hours in the ancient world, solar times were used. As we discuss below, the Sun's motion is not constant during the year, but a uniform measure was sometimes used: Astronomers of antiquity used equinoctial hours as the Greek term (copat tohieptvai) is translated (Toomer 1984, p. 23). The equinoctial hour was 1/24 of the length of a solar day. It was roughly the equivalent of one of our hours of mean solar time, but not exactly the same, because the modern definition involves the mean rate of the Sun averaged over the year. The measure of time that we use is traditionally defined as the hour angle of a fictitious "mean sun" traveling on the celestial equator at average rate plus 12 hours:

where LMSolT is the local mean solar time and HAMS is the hour angle of the mean sun. The 12h in (4.3) is arbitrary. It is a convenience for the modern world to have the day start at midnight rather than at noon. In ancient Mesopotamia, the day started at sundown.

In more common use in the ancient world were seasonal hours (in Greek, copat KatptKai) or civil hours (Toomer 1984, p. 23). Each such hour was 1/12 of the actual length of daylight at a given place and, therefore, varied in length with the season. They were obtained from the motion of the actual Sun, usually by means of a sundial. Ptolemy's Almagest (cf. Toomer 1984, p. 104ff and Appendix A) contains instructions for the conversion of equinoctial hours into seasonal hours and vice versa. Because the number of hours of daylight varies with latitude every day of the year except at the equinoxes, so did the length of a seasonal hour.

In the ancient world, time was local and immediate, and reckoned exclusively either by hour angle (of the Sun) or by reference to horizon phenomena or celestial meridian passage (of the Sun or stars). Ptolemy (Almagest, Book II, §9; Toomer 1984, p. 104) gives instructions for converting from one type to another. To convert from seasonal to equinoctial hours, multiply the time interval in seasonal hours by the length in time-degrees and divide by 15. For example (taken from the Almagest, Appendix A; Toomer 1984, p. 650), if the length of a seasonal hour at a given date and site is, say, 18;7 time-degrees, 5V2 seasonal hours is

5V2 x 18;7/15 = 6;38 equinoctial hours.3 To convert from equinoctial hours to seasonal hours, multiply by 15 and divide by the length of the hour of the relevant time interval in time-degrees. For example (p. 649), given the longitude of the Sun as _28;18° (that is, 28°18' in the sign of Sagittarius) and a site with the terrestrial latitude of Rhodes (14;30), find the length of a seasonal hour in the night. To do this, Ptolemy uses a look-up table of rising times (Book II, §8) that gives the time-degrees corresponding to 10° intervals on the ecliptic and totals accumulated since Y0°.

For Rhodes, the accumulated rise time for _20 is 277;29° and that for its opposite (in the night sky), £20°, is 94;18°. The arc for a 10° stretch on the ecliptic is 11;16° between _20° and _30°, and 10;34° between £20° and £30°. The interval 8;18° is 0.833 of the 10° of zodiacal sign or 0.833 of these intervals, or 9;21,5 and 8;46,13 respectively. Adding these to the accumulated rise times at _20° and £20°, we get 286;50,5 and 69;27,13, respectively. The difference, A, between these values, corresponding to exactly one half-day, is 217;22,52. Dividing A by 12, we get the length of a seasonal hour, 217.3811/12 = 18.115 = 18;7. Dividing A by 15, we get the length of the night in equinoctial hours: 217.3811/15 = 14.492 = 14;29. Further division of the latter by 12 gives the length of one seasonal hour in equinoctial hours at Rhodes when the Sun is near the end of the sign of Sagittarius: 1.2077 = 1;12,28.

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