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The ancient civilised peoples appear in history with a fully-developed system of time-reckoning—the Egyptians with the shifting year of 365 days, which comes as nearly as possible to the actual length of the year, counting only whole days and neglecting the additional fraction; the Babylonians and the Greeks with the lunisolar, varying between twelve and thirteen months and arranged by the Greeks from the earliest known period of history in the cycle of the Oktaeteris. It has always been clear that these systems of time-reckoning represent the final stage of a lengthy previous development, but as to the nature of this development the most daring hypotheses have been advanced. Thus, for example, eminent philologists and chronologists have believed the assertion of Censorinus, Ch. 18, and have supposed that the Oktaeteris was preceded by a Tetraeteris, even by a Dieteris. It may indeed at once be asserted that such a hypothesis lacks intrinsic probability. To account for the early development hard facts are needed, and unfortunately these, especially in the case of the Greeks, are extremely few. Where they are required they must be sought elsewhere.

Setting aside all ingenious but uncertain speculations, our only practicable way of proceeding is by means of a comparison with other peoples among whom methods of time-reckoning are still in the primitive stage. This is the ethnological method which is so well-known from the science of comparative religion, but the claims of which have been so vigorously contested upon grounds of no small plausibility. Fortunately this dispute need not be settled in order to prove the validity of the comparative method for an investigation into the origin and development of methods of reckoning time. The gist of the dispute may be expressed as follows:—The ethnological school of students of comparative religion assumes that the intellect of the natural man can only master a certain quite limited number of universal conceptions; from these spring more and more abundantly differentiated and complicated ideas, but the foundation is everywhere the same. Hence our authority for comparing the conceptions of the various peoples of the globe with one another in order to lay bare this foundation. The opponents of the school deny the existence of these fundamental conceptions, and maintain that the points of departure, the primitive ideas of the various peoples, may be as different as the peoples themselves, and that therefore we are not authorised in drawing general conclusions from the comparison or from the fundamental conceptions themselves.

In the matter of the indication and reckoning of time, however, we have not to do with a number of conceptions which may be supposed to be as numerous and as various as we please. At the basis lies an accurately determined and limited and indeed small number of phenomena, which are the same for all peoples all over the globe, and can be combined only in a certain quite small number of ways. These phenomena may be divided into two main groups: (1) the phenomena of the heavens—sun, moon, and stars—and (2) the phases of Nature—the variations of the climate and of plant and animal life, which on their side determine the affairs of men; these, however, depend finally upon one of the heavenly bodies, viz. the sun. The claim that the comparative ethnological method can be justified only when we are dealing with a narrowly circumscribed number of factors is therefore here complied with, owing to the very nature of the subjects treated. The comparative method does not shew how things have happened in a special case in regard to one particular people: it only indicates what may have happened. But much is already gained if we can eliminate the impossibilities, since from the complete result of the development, no less than in other ways, we may obtain a certain basis for our deductions.

For the investigation of primitive methods of time-reckoning no special astronomical or other technical knowledge is needed: in fact, such knowledge has rather played a fatal part by causing attention to be paid exclusively to the system of time-reckoning and leading to constant attempts to discover older and more primitive systems. A priori, indeed, we might venture to state that a system is always based upon previous data: unsystematic indications of time precede the system of time-reckoning. These modest beginnings have been obscured from view by the prejudice in favour of the systematic technical and astronomical chronology. The only absolutely necessary thing is a clear idea of the apparent motions of the heavenly bodies, i. e. the sun, the moon, and the most important of the fixed stars, and of the phases of the climate and the life of animals and plants, which give the units of the time-reckoning.

For a statement of the course and phases of the heavenly bodies and the units of the time-reckoning given by these I refer to the article mentioned in the preface, the pertinent sections of which are here quoted:—

“The units of the time-reckoning are given by the motions of the heavenly bodies (expressed according to the Ptolemaic system), and the more intimately these enter into the life of man, the more important do they become. For this reason only those units which depend upon the sun have asserted themselves in our calendar, those depending upon the moon having been dropped, except for the movable paschal term, which has been kept on religious grounds. The units are the year, the month, and the day. Other units more convenient for time-reckoning play no part in the arrangement of the calendar since they are without importance for practical life. The day (= 24 hours, νυχθήμερον) is determined from the apparent motion of the heavenly bodies about the earth, which is caused by the rotation of the earth on its axis; but since the sun also, on account of the annual revolution of the earth about it, runs through the zodiac in an opposite direction to its daily movement and completes the circle of the ecliptic in a year, a day will be a little longer than a complete rotation of the earth. Or to put it otherwise:—The time between two successive upper culminations of a star, i. e. between the moments at which the star passes through the meridian-line of one and the same place (= attains the zenith), represents an axial rotation: that is a stellar day. The time between two successive culminations of the sun is, on account of the annual motion of the sun (really that of the earth), 3 min. 56.₅ secs. longer than a stellar day: that is a solar day. The number of stellar days in a year is greater by one day than the number of solar days. The stellar day does not follow the variations of light and darkness and therefore does not enter into the calendar. The difference between the actual solar day, which is of slightly varying length, and the mean solar day abstracted from it for the purposes of our clock-regulated time-reckoning has no significance for antiquity. The second unit determined by the sun is the year, the period of a revolution of the earth about the sun. In relation to the apparent motion of the sun it may be defined as the time which the sun takes to come back again to the same fixed star. This is a stellar or sidereal year, the length of which amounts to 365 days 6 hrs. 9 min. 9.₃₄ secs. The tropic year is the time which the sun takes to come back to the crossing point of the equator, viz. the vernal equinox. This is the natural year. Its length varies a little; it is about 20 minutes shorter than the stellar year. The lunar or moon-month is determined from the visible phases of the moon. This term will be used only when it is necessary to make an express distinction between the lunar and our Roman month; the latter is a conventional subdivision of the year which has nothing to do with the moon, and has the name ‘month’ only because it historically arose from the lunar month and in its duration comes fairly near the latter. But when in relation to antiquity—apart from Rome and Egypt—we speak of months, lunar months are as a rule to be understood. The moon revolves around the earth twelve times a year and a little more: consequently it moves backwards in the zodiac much more rapidly than the sun. The interval between two successive moments at which the moon culminates at the same spot at the same time as one and the same star is a sidereal month (cp. the sidereal year); its length is 27 days 7 hrs. 43 min. 11.₄₂ secs., but it does not follow the phases of the moon and is therefore of no consequence for the calendar. The phases of the moon are dependent upon the position of the moon in relation to the sun and the earth. When the three bodies are in a straight line (or rather in a plane perpendicular to the plane of the ecliptic) in such a way that the earth is in the middle, the side of the moon turned towards the earth is completely illuminated and we have full moon: when the moon is in the middle, the side turned towards the earth is completely overshadowed, and that is new moon. In between lie the separate phases of the waxing and waning moon. The synodic month is the interval between two new moons and comprises on an average 29 days 12 hrs. 44 min. 2.₉₈ secs. This is the true lunar month: other varieties of month are of no importance for us.

"The risings and settings of the stars. It has already been remarked that the sun in the course of a year runs through the zodiac backwards, so that one particular star culminates 3 min. 56 secs. earlier every day. Hence it is evident that if we indicate the exact interval of time between the culmination of the sun and that of one particular star, or name the star with which the sun precisely culminates, we can determine the day of the solar year. This is the principle of one method of computing time which was very common among ancient and primitive peoples, but has entirely dropped out of use in modern times owing to our paper calendar. The stars are so to speak the stationary ciphers on the clock-face and the sun is the hand. In practice we naturally have to do not with the invisible culmination of the stars but with the position of the sun and certain neighbouring stars on the edge of the horizon, whereby the matter becomes more complicated on the astronomical side. For this observation the so-called circumpolar stars are singled out, that is to say the stars situated so near the pole that they do not set (e. g. the Great Bear). If the star rises or sets simultaneously with the rising of the sun, this is called the true cosmic rising or setting. If the star rises or sets simultaneously with the setting of the sun, this is termed the true acronychal rising or setting. These risings and settings of the star are not visible, since the sun hides them by its light: the rising and setting are perceptible only when the star stands at some distance from the sun, i. e. only the so-called apparent rising and setting are practically observable. We have already seen that the sun every day drops nearly 4 minutes behind a certain star. Assuming that sun and star rise simultaneously on one day (true cosmic rising), then after a few days have passed—the period varying somewhat according to the latitude of the place of observation, the time of the year, the size and place of the star—there will come a day on which the star rises so early that it is visible in the morning twilight, immediately before the sun appears. This is the heliacal or morning rising. From this day the star will rise earlier and earlier, and will therefore remain visible for a longer and longer period. In the course of half a year, commonly a little sooner or later, the time of rising will have been pushed so far back that it will take place in the evening twilight; when it is pushed still farther back the rays of the setting sun eclipse the star and its rising is no longer visible. The last visible rising of the star in the evening twilight is the apparent acronychal or evening rising. After a few more days the star goes so far back that it rises at the very moment in which the sun sets—the true acronychal rising. The rising, which is advanced constantly further into the light of day, is no longer visible, but on the other hand we now see the setting of the star. If it is assumed that the star is situated on the western horizon, i. e. sets, when the sun is on the eastern horizon, i. e. rises—and incidentally it is to be noted that this position, when the star is not situated in the ecliptic, may be divided by an interval of a larger or smaller number of days from the opposite position, viz. star on the eastern, sun on the western horizon—this is the true cosmic setting. The star moves forward, i. e. its setting takes place earlier in the morning, and after a few days it will be noticed in the morning twilight immediately before it sets, and this is the first visible setting in the morning twilight, the apparent cosmic or morning setting. From this day the setting moves further and further forward into the night and approaches the evening twilight. At length it will be so near sunset that the star no longer sets in the night but in the evening twilight. The last visible setting of the star in the evening twilight is the heliacal or evening setting. After a few days the star has approached still nearer to the sun: both set at the same moment, the true cosmic setting. If the star stands in the ecliptic, the true cosmic setting coincides in date with the true cosmic rising, otherwise these are divided by a greater or smaller number of days (see above). As the star moves on, a heliacal rising follows again, and so on. Between the day of the heliacal setting and that of the heliacal rising the star is invisible, since it stands so near the sun that it is eclipsed by the sun’s rays. It has already been remarked that we can determine the day of the year by indicating the true rising and setting of a star at a certain spot. As far as the apparent rising and setting are concerned this indication can only be approximate, since the visibility of a star depends on several variable factors—the size of the star (because a smaller star, in order to be visible, must move farther from the sun than a brighter one), the transparency of the atmosphere, the keenness of vision of the observer, the geographical latitude of the place of observation (since the farther north or south the sun is, the more slowly, because more obliquely, will it sink below the horizon). In this latter respect, for instance, there is a perceptible difference between Rome and Egypt. Only an approximate indication of time, therefore, can be derived from the rising and setting of the stars”.

The phases of the climate and of plant and animal life cannot be particularly described, since they naturally vary so much in different countries. It can only be remarked that though they depend upon the course of the sun, yet in certain cases, owing to the special climatic conditions of the individual years, they may be to some extent advanced or retarded, and further that the climatic phenomena of many parts of the earth, especially in the Tropics but also in the Mediterranean countries, recur with a far greater regularity than in our northern climes, which are subject to such uncertain weather. Instances are the trade-winds and monsoons, the dry and the rainy seasons.

Upon the above-mentioned units the system of time-reckoning will be based. The days are joined into months and the months into years; only more rarely are the seasons interposed as regular units of time. The system is like a chain the links of which run into one another without gaps: each link is equivalent, or as nearly as possible equivalent, to every other link of the same class, and therefore need only be given a name and counted, not necessarily conceived in the concrete, although this is not excluded. This is the only genuine system, a system of continuous time-reckoning, which excludes all gaps in the chain and all links of indeterminate length. The relation between the larger and the smaller units may be treated in various ways, chiefly on account of the fact that the smaller units do not divide exactly into the larger. Sometimes the smaller units may be fitted into the larger as subdivisions of the latter, so that they constitute the links of the chain formed by the larger unit. The inequality referred to shews then that the units vary to some extent in number or size (year of 365 or 366 days, of 12 or 13 lunar months, lunar month of 29 or 30 days). In that case the beginnings of the larger unit and of the first of the smaller units coincide. Thus in our year New Year’s Day and the first day of the first month coincide, but the length of the months varies somewhat. This is an inheritance from the lunisolar year, in which also New Year’s Day and the first day of the first month coincided and the length of the month varied between 29 and 30 days, but in addition the year varied between 12 and 13 months. This mode of reckoning, in which the smaller units are contained in the larger as subdivisions of them, will be termed the fixed method.

But where the smaller units do not exactly divide into the larger, both may also be counted independently of one another without being equalised. A case in point is our week, which is reckoned without reference to the year, so that every year begins with a different day of the week. This method of reckoning we shall term the shifting method. It is less systematic than the fixed method, and we shall therefore expect to find it play a greater part in earlier times than at the present day.

The system of time-reckoning, the continuous counting of the time-units, represents the final point of the development. It is our object to investigate the preceding stages, both systematic and unsystematic. Certain important ideas which frequently recur must however first be clearly set down. The time-reckoning in the proper sense of the term is preceded by time-indications which are related to concrete phenomena of the heavens and of Nature. Since these indications depend upon the concrete phenomenon, their duration fluctuates with the latter, or rather the duration does not stand out by itself but the phenomenon as such is exclusively regarded: the time-indication is not durative, like the link in any system of time-reckoning, but indefinite, or, to borrow a grammatical term, aoristic. And setting aside these finer distinctions we also find that the phenomena to which the time-indications are related are of fluctuating and very unequal duration. Since the duration is indeterminate and fluctuating, and the time-indications are not limited one by the other but overlap and leave gaps, they cannot be numerically grouped together. Here we ought really to speak not of a time-reckoning in the proper sense, but only of time-indications. But since the word ‘time-reckoning’ has become naturalised, this method may be described as the discontinuous system of time-reckoning, because the time-indications do not stand in direct relation to other time-indications but are related only to a concrete phenomenon, and through that to other time-indications, so that they are of indeterminate length and cannot be numerically grouped together.

If the number of dawns, suns, autumns, or snows that has passed since a certain event took place, or will elapse before a certain event is to take place, be indicated, the time that has passed or is to pass will be defined, because the dawn or the sun recurs once in the day, and an autumn or a snow, i. e. winter, once in the year. This is the oldest mode of counting time. It is not the units as a whole that are counted, since the unit as such had not yet been conceived, but a concrete phenomenon recurring only once within this unit. It is the pars pro toto method so extensively used in chronology, and by this name we shall call it[1].

Since it must now be regarded as the natural course of development that the systematic has gradually arisen out of the unsystematic, and that the indication of concrete phenomena following one another in the regular succession of Nature has preceded the abstract numerical indication of time offered by our calendars, the origin of the time-reckoning must be sought not in any one system, however simple, but in the discontinuous or pars pro toto time-indications which are related to concrete phenomena.

Our task is now to make clear the nature of these discontinuous and pars pro toto time-indications, since from them proceeds, as order is ever evolved out of chaos, the continuous time-reckoning, the calendar.