XIX
GREEK MONTHS, YEARS, AND CYCLES1
ALTHOUGH there is controversy as to whether in the earliest times (e. g. with Homer and Hesiod) the day was supposed to begin with the morning or evening, it may be taken as established that in historic times the day, for the purpose of the calendar, began with the evening, both at Athens and in Greece generally. As regards Athens the fact is stated by Gellius on the authority of Varro, who, in describing the usage of different nations in this respect, said that the Athenians reckoned as one day the whole period from one sunset to the next sunset;2 the testimony of Pliny3 and Censorinus4 is to the same effect. The practice of regarding the day as beginning with the evening is natural with a system of reckoning time by the moon’s appearances; for a day would naturally be supposed to begin with the time at which the light of the new moon first became visible, i. e. at evening.
There is no doubt that, from the earliest times, the Greek month 
was lunar, that is, a month based on the moon’s apparent motion. But from the first there began to be felt, among the Greeks as among most civilized peoples, a desire to bring the reckoning of time by the moon into correspondence with the seasons of the year, for the sake of regulating the times of sacrifices to the gods which had to be offered at certain periods in the year; hence there was from the beginning a motive for striving after the settlement of a luni-solar year. The luni-solar year thus had a religious origin. This is attested by Geminus, who says:5
‘The ancients had before them the problem of reckoning the months by the moon, but the years by the sun. For the legal and oracular prescription that sacrifices should be offered after the manner of their forefathers was interpreted by all Greeks as meaning that they should keep the years in agreement with the sun and the days and months with the moon. Now reckoning the years according to the sun means performing the same sacrifices to the gods at the same seasons in the year, that is to say, performing the spring sacrifice always in the spring, the summer sacrifice in the summer, and similarly offering the same sacrifices from year to year at the other definite periods of the year when they fell due. For they apprehended that this was welcome and pleasing to the gods. The object in view, then, could not be secured in any other way than by contriving that the solstices and the equinoxes should occur in the same months from year to year. Reckoning the days according to the moon means contriving that the names of the days of the month shall follow the phases of the moon.’
At first the month would be simply regarded as lasting from the first appearance of the thin crescent at any new moon till the next similar first appearance. From this would gradually be evolved a notion of the length of a moon-year. A rough moon-year would be 12 moon-months averaging 
days; but it was necessary that a month should contain an exact number of days, and it was therefore natural to take the months as having alternately 29 and 30 days. These ‘hollow’ and ‘full’ months are commonly supposed to have been introduced at Athens by Solon (who was archon in 594/3 B.C.), since he is said to have ‘taught the Athenians to reckon days by the moon’.6 But it can hardly be doubted that ‘full’ and ‘hollow’ months were in use before Solon’s time; Ginzel therefore thinks that Solon’s reform was something different. We shall revert to this point later.
At the same time, alongside the ‘full’ and ‘hollow’ months of the calendar, popular parlance invented a month of 30 days, as being convenient to reckon with. Hippocrates makes 280 days = 9 months 10 days;7 Aristotle speaks of 72 days as 1/5th of a year;8 the riddle of Cleobulus implies 12 months of 30 days each;9 the original division of the Athenian people into 4 
, 12 
, and 360 
is explained by Philochorus as corresponding to the seasons, months, and days of the year.10 In the Courts a month was reckoned at 30 days, and wages were reckoned on this basis, e. g. daily pay of 2 drachmae makes for 13 months 780 drachmae (2 × 30 × 13).11 From such indications as these it has been inferred that the Greeks had at one time years of 360 days and 390 days respectively. Indeed, Geminus says that ‘the ancients made the months 30 days each, and added the intercalary months in alternate years 
’.12 Censorinus has a similar remark; when, he says, the ancient city-states in Greece noticed that, while the sun in its annual course is describing its circle, the new moon sometimes rises thirteen times, and that this often happens in alternate years, they inferred that 
months corresponded to the natural year, and they therefore fixed their civil years in such a way that they made years of 12 months and years of 13 months alternate, calling each of such years ‘annus vertens’ and both years together a great year.13 Again, Herodotus14 represents Solon as saying that the 70 years of a man’s life mean 25,200 days, without reckoning intercalary months, but, if alternate years are lengthened by a month, there are 35 of these extra months in 70 years, making 1,050 days more and increasing the total number of days to 26,250. But under this system the two-years period (called nevertheless, according to Censorinus, trieieris because the intercalation took place ‘every third year’) would be more than 7 days too long in comparison with the sun, and in 20 years the calendar would be about 
months wrong in relation to the seasons. This divergence is so glaring that Ginzel concludes that the system cannot have existed in practice. He suggests, in explanation of Geminus’s remark, that Geminus is not to be taken literally, but is in this case merely using popular language (cf. his remark that 9c days = 3 months15); he regards Censorinus’s story as suspicious because in the following sentence Censorinus says that the next change was to a pentaëteris of four years each, which involves the supposition that the Greeks of, say, the eighth or ninth century B.C., had already anticipated the Julian system; moreover, Geminus says nothing of a four-years period at all (whether called tetraëteris or pentaëteris) but passes directly to the octaëteris which, according to him, was the first period that the ancients constructed.
On the alternation of ‘full’ and ‘hollow’ months an apparently interpolated passage in Geminus says:16
‘The moon-year has 354 days. Consequently they took the lunar month to be 
days and the double month to be 59 days. Hence it is that they have hollow and full months alternately, namely because the two-months period according to the moon is 59 days. Therefore there are in the year six full and six hollow months. This then is the reason why they make the months full and hollow alternately.’
The octaëteris.
Geminus’s account of the eight-years cycle follows directly on what he says of the supposed ancient system of alternating years of 12 and 13 months of 30 days each.
‘Observation having speedily proved this procedure to be inconsistent with the true facts, inasmuch as the days and the months did not agree with the moon nor the years keep pace with the sun, they sought for a period which should, as regards the years, agree with the sun, and, as regards the months and the days, with the moon, and should contain a whole number of months, a whole number of days, and a whole number of years. The first period they constructed was the period of the octaëteris (or eight years) which contains 99 months, of which three are intercalary, 2922 days, and 8 years. And they constructed it in this way. Since the year according to the sun has 
days, and the year according to the moon 354 days, they took the excess by which the year according to the sun exceeds the year according to the moon. This is 
days. If then we reckon the months in the year according to the moon, we shall fall behind by days in comparison with the solar year. They inquired therefore how many times this number of days must be multiplied in order to complete a whole number of days and a whole number of months. Now the number [] multiplied by 8 makes 90 days, that is, three months. Since then we fall behind by days in the year in comparison with the sun, it is manifest that in 8 years we shall fall behind by 90 days, that is, by 3 months, in comparison with the sun. Accordingly, in each period of 8 years, three intercalary 
months are reckoned, in order that the deficiency which arises in each year in comparison with the sun may be made good, and so, when 8 years have passed from the beginning of the period, the festivals are again brought into accord with the seasons in the year. When this system is followed, the sacrifices will always be offered to the gods at the same seasons of the year.
‘They now disposed the intercalary months in such a way as to spread them as nearly as possible evenly. For we must not wait until the divergence from the observed phenomena amounts to a whole month, nor yet must we get a whole month ahead of the sun’s course. Accordingly they decided to introduce the intercalary months in the third, fifth, and eighth years, so that two of the said months were in years following two ordinary years, and only one followed after an interval of one year.17 But it is a matter of indifference if, while preserving the same disposition of (i.e. intervals between) the intercalary months, you put them in other years.’18
Here then we have an account which purports to show how the octaëteris was first arrived at, the supposition being that it was based on a solar year of 
days. Ginzel, however, thinks it impossible that this can have been the real method, because the evaluation of the solar year at days could hardly have been known to the Greeks of, say, the 9th and 8th centuries B.C.; this, he thinks, is proved by the erroneous estimates of the length of the solar year which continued to be put forward much later.
Ginzel considers that the octaëteris was first evolved as the result of observation of the moon’s motion, which was of course easier to approximate to within a reasonable time. The alternation of 6 full with 6 hollow months gives a moon-year of 354 days; but the true moon-year exceeds this by 0·36707 day, and hence, after about 
moon-years, a day would have to be added in order to keep the months in harmony with the phases; that is to say, at such intervals, there would have to be a year of 355 days. Now this rate of intercalation corresponds nearly to the addition of 3 days in a period of 8 moon-years, i. e. to a cycle of 8 moon-years in which 5 have 354, and 3 have 355 days, each. (And, as a matter of fact, the same proportion of 5 : 3 serves very roughly to bring the moon-year into agreement with the solar year, for we have only to reckon, in a cycle of 8 solar years, 5 moon-years of 354 days and 3 of 384 days.)19 Ginzel cites evidence showing that particular years actually had 355 days and 384 days, e.g. 01. 88, 3 = 355 days, 01. 88, 4 = 354 days, 01. 89, 1 = 384 days, and 01. 89, 2 = 355 days.20 The method by which the octaëteris was evolved is, he thinks, something of this sort. Having from observation of the moon constructed an 8-years period containing 5 moon-years of 354 days and 3 intercalated years of 355 days each, making a total of 2,835 days, the Greeks, by further continual observation directed to fixing the duration of the phases exactly, would at last come to notice that, after 8 returns of the sun to the same azimuth-point on the horizon, the phases fell nearly on the same days once more, and also that the sun returned to the same azimuth-point for the eighth time after about 99 lunar months. Now, if the ancients had divided the 2,835 days of 8 moon-years by 96, they would have found the average lunar month to contain 
days; and again, if they had multiplied this by 99, they would have obtained 
or nearly 
days. But the first inventors of the octaëteris certainly did not make the 8 solar years contain 
days; this, we are told, was a later improvement on the 2,922 days which, according to Geminus, the first octaëteris contained. No doubt the first discoverers of it would notice that 99 times 
days is 
days, that is to say, approximately 8 years of 365 days (= 2,920 days). This may have been what led them to construct a luni-solar octaëeteris. But why did they give it 2,922 days? Ginzel suggests that, as the octaëeteris was thus shown to be very useful for the purpose of bringing into harmony the motions of the sun and moon, the Greeks would be encouraged to try to obtain a more accurate estimate of the average length of the lunar month. If then, for example, they had assumed 
days as the average length, they would have found, at the end of an octaëteris, that they were only wrong by 0–3 of a day relatively to the moon, but were nearly two days ahead in relation to the sun.21 This might perhaps lead them to conjecture that the solar year was a little longer than 365 days; and they may have hit upon 
days by a sort of guess. This would give 
days as the length of the lunar month. Ginzel thinks that the gradual process by which the Greeks arrived at the 2,922 days may have lasted from the 9th or 8th century into the 7th.22 This, he suggests, may explain the fact that we find mentions or indications of eight-years periods going back as far as the mythical age. Thus Cadmus passed an ‘eternal’ 
year (i.e. says Ginzel, an 8-year year) in servitude for having slain the dragon of Ares; similarly Apollo served 8 years with Admetus after he killed the dragon Python. The Daphne-phoria were celebrated every 8 years; in the procession connected with the celebration an olive staff was carried with a sphere above (the sun), a smaller one below (the moon), and still smaller spheres representing other stars, while 365 purple bands or ribbons were also attached, representing the days of the solar year. The Pythian games were also, at the beginning, eight-yearly. Kingships were offices held for eight years (thus Minos spoke with Zeus, the great God, ‘nine-yearly’).23 According to Plutarch the heaven was observed at Sparta by the Ephors on a clear night once in eight years.24 These cases, however, though showing that 8-years periods were recognized and used in various connexions, scarcely suffice, I think, to prove the existence in such very early times of an accurately measured period of 2,922 days. Ginzel, in arguing for so early a discovery of the octaëteris of 2,922 days, departs considerably from the views of earlier authorities on chronology. Boeckh thought that the octaëteris was introduced by Solon, and that the first such period actually began with the beginning of the year at the first new moon after the summer solstice in Ol. 46, 3, i. e. 7th July, 594 B.C.25 As regards the period before Solon, Boeckh went, it is true, so far as to suggest that, as early as 642 B.C., there may have been a rough octaëteris in vogue which was not actually fixed or exactly observed; this, however, was only a conjecture. Ideler26 argued that the octaëteris could not be as old as Solon’s time (594/3 B.C.) or even as old as Ol. 59 (544–540), because so accurate a conception is in too strong a contrast to what we know of the state of astronomical knowledge in Greece at that time. As regards Solon’s reforms, we are told27 that he prescribed that the day in the course of which the actual conjunction at the new moon took place should be called 
, the ‘last and new’ or ‘old-new’, and that he called the following day 
(new moon), which therefore was the first day belonging wholly to the new month. Diogenes Laertius says that Solon taught the Athenians ‘to reckon the days according to the moon’;28 and Theodorus Gaza, a late writer, it is true, says that Solon ‘ordered everything in connexion with the year generally better’.29 Boeckh, as already stated, thought that Solon’s reform consisted in the introduction of the octaéteris. Ginzel, however, holding as he does that the octaeteris of 2,922 days was discovered much earlier, considers that Solon’s reform had to do with the improvement on this figure by which 99 lunations were found to amount to 
days, a discovery which led to the formulation of the 16-years and 160-years periods presently to be mentioned; this may be inferred, according to Ginzel, from the fact that the accounts show Solon’s object to have been the bringing of the calendar specially into accordance with the moon. But it is difficult to accept Ginzel’s view of the nature of Solon’s reform in the face of another statement as to the authors of the octaéteris. Censorinus says:
‘This octaeteris is commonly attributed to Eudoxus, but others say that Cleostratus of Tenedos first framed it, and that it was modified afterwards by others who put forward their octaeterides with variations in the intercalations of the months, as did Harpalus, Nauteles, Menestratus, and others also, among whom is Dositheus, who is most generally identified with the octaeteris of Eudoxus.’30
Now we know nothing of the date of Cleostratus, except that he came after Anaximander; for Pliny says that Anaximander is credited by tradition with having discovered the obliquity of the zodiac in Ol. 58 (548–544 B.C.), after which (deinde) Cleostratus distinguished the signs in it.31 Thus Cleostratus may have lived soon after 544 B.C. Ginzel seems to admit that Cleostratus was the actual founder (‘eigentliche Begründer’) of the octaëteris.32 Of Harpalus, who was later than Cleostratus but before Meton (432 B.C.), we only know that he formed a period which brought the moon into agreement with the sun after the latter had revolved ‘through nine winters’,33 which statement must, as Ideler says, be due to a misapprehension of the meaning of the words ‘nono quoque anno’. According to Censorinus, Harpalus made the solar year consist of 365 days and 13 equinoctial hours.34 Eudoxus’s variation will be mentioned later.
The 16-years and 160-years cycles.
After describing the octaëteris of 2,922 days, Geminus proceeds thus:
‘If now it had only been necessary for us to keep in agreement with the solar years, it would have sufficed to use the aforesaid period in order to be in agreement with the phenomena. But as we must not only reckon the years according to the sun, but also the days and months according to the moon, they considered how this also could be achieved. Thus the lunar month, accurately measured, having 
days, while the octaeteris contains, with the intercalary months, 99 months in all, they multiplied the 
days of the month by the 99 months; the result is 
days. Therefore in eight solar years there should be reckoned 
days according to the moon. But the solar year has 
days, and eight solar years contain 2922 days, this being the number of days obtained by multiplying by 8 the number of days in the year. Inasmuch then as we found the number of days according to the moon which are contained in the 8 years to be 
we shall, in each octaeteris, fall behind by 
days in comparison with the moon. Therefore in 16 years we shall be behind by 3 days in comparison with the moon. It follows that in each period of 16 years three days have to be added, having regard to the moon’s motion, in order that we may reckon the years according to the sun, and the months and days according to the moon. But, when this correction is made, another error supervenes. For the three days according to the moon which are added in the 16 years give, in ten periods of 16 years, an excess (with reference to the sun) of 30 days, that is to say, a month. Consequently, at intervals of 160 years, one of the intercalary months is omitted from 
one of 
the octaeterides; that is, instead of the three (intercalary) months which fall to be reckoned in the eight years, only two are actually introduced. Hence, when the month is thus eliminated, we start again in agreement with the moon as regards the months and days, and with the sun as regards the years.’35
This passage explains itself; it is only necessary to add that there is no proof that the 16-years period was actually used. The 160-years period was, however, presupposed in Eudoxus’s octaeteris, the first of which, according to Boeckh, may have begun in 381 or 373 B.C. (01. 99, 4 or Ol. 101, 4) on 22/23 July, the ‘first day of Leo’. i. e. the day on which the sun entered the sign of Leo; the effect was that, after 20 octaeterides and the dropping of 30 days, the beginning of the solar year was again on ‘the first of Leo’. In Eudoxus’s system, then, the luni-solar reckoning was independent of the solstices.36 According to the Eudoxus-Papyrus (Ars Eudoxi) the intercalary months came in the 3rd, 6th, and 8th years ot Eudoxus’s octaëteris,
Meton’s cycle.
Curiously enough, Meton is not mentioned by Geminus as the author of the 19-years cycle; his connexion with it is, however, clearly established by other evidence. Diodorus has the following remark with regard to the year of the archonship of Apseudes (01. 86, 4 = 433/2 B.C.).
‘In Athens Meton, the son of Pausanias, and famous in astronomy, put forward the so-called 19-years period 
; he started 
from the 13th of the Athenian month Skirophorion.’37
Aelian says that Meton discovered the Great Year, and ‘reckoned it at 19 years’,38 and also that ‘the astronomer Meton erected pillars and noted on them the solstices’. Censorinus, too, says that Meton constructed a Great Year of 19 years, which was accordingly called enneadecaeteris.39 Euctemon, whom Geminus does mention, assisted Meton in the matter of this cycle.
Geminus’s account of the cycle shall be quoted in full:
‘Accordingly, as the octaeteris was found to be in all respects incorrect, the astronomers Euctemon, Philippus, and Callippus [the phrase is 
., as usual] constructed another period, that of 19 years. For they found by observation that in 19 years there were contained 6940 days and 235 months, including the intercalary months, of which, in the 19 years, there are 7. [According to this reckoning the year comes to have 
days.] And of the 235 months they made 110 hollow and 125 full, so that hollow and full months did not always follow one another alternately, but sometimes there would be two full months in succession. For the natural course of the phenomena in regard to the moon admits of this, whereas there was no such thing in the octaeteris. And they included 110 hollow months in the 235 months for the following reason. As there are 235 months in the 19 years, they began by assuming each of the months to have 30 days; this gives 7,050 days; Thus, when all the months are taken at 30 days, the 7,050 days are in excess of the 6,940 days; the difference is 110 days, and accordingly they make 110 months hollow in order to complete, in the 235 months, the 6,940 days of the 19-years period. But, in order that the days to be eliminated might be distributed as evenly as possible, they divided the 6,940 days by 110; this gives 63 days.40 It is necessary therefore to eliminate the [one] day after intervals of 63 days in this cycle. Thus it is not always the 30th day of the month which is eliminated, but it is the day falling after each interval of 63 days which is called 
.’41
The figure of 
days = 365 days 6h 18m 56·9s and is still 30 minutes 11 seconds too long in comparison with the mean tropic year; but the mean lunar month of Meton is 29 days 
, which differs from the true mean lunar month by not quite 1 minute 54 seconds. When Diodorus says that, in putting forward his 19-years cycle, Meton started from the 13th of Skirophorion (which was the 13th of the last month of Apseudes’ year= 27th June, 432), he does not mean that the first year of the period began on that date; this would have been contrary to the established practice. The beginning of the first year (the 1st Hekatombaion of that year) would be the day of the first visibility of the new moon next after the summer solstice, i.e. in this case 16th July, 432. The 13th Skirophorion was the day of the solstice, and we have several allusions to Meton’s observation of this;42 presumably, therefore, Diodorus meant, not that the first year of Meton’s cycle began on that day, but that it was on that day that Meton began his parapegma (or calendar).43 Ginzel44 gives full details of the many divergent views as to the date from which Meton’s cycle was actually introduced at Athens. Boeckh put it in Ol. 112, 3 = 330/29 B.C, Unger between Ol. 109, 3 (342/1 B.C.) and Ol. 111, 1(336/5 B.C). Schmidt holds that Meton’s cycle was introduced in 342 B.C., but in a modified form. The 235 months of the 19-years cycle contained, according to the true mean motion of the moon, 235 × 29·53059 days, or 6,939 days and about 
hours. Consequently after 4 cycles there was an excess of four times the difference between 6,940 days and 6,939 days hours, or an excess of 1 day 6 hours; after 10 cycles an excess of 3 days 3 hours, and so on. The Athenians, therefore, according to Schmidt, struck out one day in the 4th, 7th, 10th, 13th, 16th, 20th, and 23rd cycles, making these cycles 6,939 days each. But, as Ginzel points out, the confusions in the calendar which occurred subsequently tell against the supposition that such a principle as that assumed by Schmidt was steadily followed in Athens from 342 B.C.
Callippus’s cycle of 76 years.
Geminus follows up his explanation with regard to the Metonic cycle thus:
‘In this cycle [the Metonic] the months appear to be correctly taken, and the intercalary months to be distributed so as to secure agreement with the phenomena; but the length of the year as taken is not in agreement with the phenomena. For the length of the year is admitted, on the basis of observations extending over many years, to contain 
days, whereas the year which is obtained from the 19-year period has 
days, which number of days exceeds 
by 
th of a day. On this ground Callippus and the astronomers of his school corrected this excess of a fraction of a day and constructed the 76-years period 
out of four periods of 19 years, which contain in all 940 months, including 28 intercalary, and 27,759 days. They adopted the same arrangement of the intercalary months. And this period appears to agree the best of all with the observed phenomena.’45
With Meton’s year of 
days (6,940 divided by 19), four periods of 19 years amount of course to 27,760 days, and the effect of Callippus’s change was to reduce this number of days by one. 27,759 days divided by 940 gives, for the mean lunar month, 29 days 
, only 22 seconds in excess of the true mean length.
Callippus was probably born about 370 B.C.; he came to Athens about 334 B.C.; the first year of the first of his cycles of 76 years was Ol. 112, 3 = 330/29 B.C, and probably began on the 29th or 28th of June. His cycles never apparently came into practical use, but they were employed by individual astronomers or chronologists for fixing dates; Ptolemy, for example, gives various dates both according to Egyptian reckoning and in terms of Callippic cycles.46
Hipparchus’s cycle.
It is only necessary to add that yet another improvement was made by Hipparchus about 125 B.C. Ptolemy says of him:
‘Again, in his work on intercalary months and days, after premising that the length of the year is, according to Meton and Euctemon, 
days, and according to Callippus 
days only, he continues in these words: “We find that the number of whole months contained in the 19 years is the same as they make it, but that the year in actual fact contains less by 
th of a day than the odd 
of a day which they give it, so that in 300 years there is a deficiency, in comparison with Meton’s figure, of 5 days, and in comparison with Callippus’s figure, of one day.” Then, summing up his own views in the course of the enumeration of his own works, he says: “I have also discussed the length of the year in one book, in which I prove that the solar year—that is, the length of time in which the sun passes from a solstice to the same solstice again, or from an equinox to the same equinox—contains 365 days and 
, less very nearly 
th of a day and night, and not the exact which the mathematicians suppose it to have in addition to the said whole number of days.”’47
Censorinus gives Hipparchus’s period as 304 years, in which there are 112 intercalary months.48 Presumably, therefore, Hipparchus took four times Callippus’s cycle (76 × 4 = 304) and gave the period 111,035 days instead of 111,036 (= 27,759 × 4). This gives, as the length of the year, 365 days 5h 55m 15·8s while 
days = 365 days 5h 55m 12s, the excess over the true mean tropic year being about 
minutes. The number of months in the 304 years is 304 × 12 + 28 × 4 = 3,760, whence the mean lunar month is, according to Hipparchus, 29 days 
, which is very nearly correct, being less than a second out in comparison with the present accepted figure of 29·53059 days !
.
.
days (against 2923.528, the correct figure); 8 solar years have 8 × 365.2422 = 2921.938 days.
the solstice fell on 14th Skirophorion or the Egyptian 11th Payni. Diels showed from another fragment that the archon must have been Polykleitos (110/109 B.C.), so that the second observation of the solstice mentioned in the fragment must have been on 27th June, 109, i.e. in the last (19th) year of the 17th Metonic cycle (Ginzel, ii, pp. 423, 424).
was a posted record 
a sort of almanac giving, for a series of years, the movements of the sun, the dates of the phases of the moon, the risings and settings of certain stars, besides 
or weather indications.