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Open letter to science editors

KRONOS Vol. I, Issue 3
GREEK ESTIMATES OF THE SYNODIC MONTH
Livio STECCHINI
The problem that Meton intended to solve was – which is the smallest number of solar years
than can be divided exactly into a series of more or less alternating months of 30 and 29 days?
He knew that solar years are about 365.25 days and that a lunar month is about 29.5 days. He
counted that 19 solar years are 19 x 365.25 = 6939.75 days. He assumed that 19 solar years
are 6940 days, either because he did not take 365.25 as an exact figure or because he chose to
disregard a difference of 0.25 days. By dividing he found that in 6940 days there are 235 lunar
months of 29.5 days, with a remainder of 7.5 days. IŁ there had been no remainder he would
have divided the 6940 days into 117 1/2 months of 30 days and 117 1/2 months of 29 days;
but since there was a remainder of 7.5 days, he increased the number of months of 30 days to
117.5 + 7.5= 125. The number of months of 29 days had to be reduced to 117.57.5 = 110.
One hundred years later, Callippus objected to the system of Meton on the ground that the
solar year should be calculated as exactly 365.25 days. Since, according to this reckoning, the
19 years of the Metonic cycle are 6939.75 days, he quadrupled the years of this cycle to 76
years, in order to obtain a round figure of 27,759 days. According to the Metonic cycle this
period would contain:
4 x 125 = 500 months of 30 days
4 x 110 = 440 months of 29 days
Since Callippus had found an excess of one day in 76 years, he changed the pattern to:
499 months of 30 days
441 months of 29 days
Hipparchus (around 150 B.C.), since he knew that a solar year is somewhat shorter than
365.25 days, proposed that the cycle of Callippus be quadrupled to 304 years, but deducting
one day. He assumed that 304 solar years are (304 x 365.25) 1 = 111,035 days, which makes
a solar year equal to 365.24671 days. Calculating correctly, 304 years are 111,033.6 days. As
to the lunar months Hipparchus limited himself to quadrupling the figures of the cycle of
Callippus:
4 x 499 months of 30 days
4 x 441 months of 29 days
If we average the length of the months according to the three cycles, we have:
Meton 29.531915 solar days
Callippus 29.530851
Hipparchus 29.530581
(correct figure – 29.530588)
Meton was Comet to the second decimal figure,Callippusto the third, and Hipparchus to the
fifth. The datum of Hipparchus is breathtaking, since it differs by a second from the correct
one, whereas he was off by about 7 minutes in calculating the length of the solar year.
The precision achieved in calculating the duration of the orrect month is not difficult to
explain. The basic problem was simple: it was a matter of counting how many new moons
occur in a period of solar years. The observations could have been made simply by recording
at each summer solstice how much sooner was the preceding new moon and how much later
was the following new moon. In a few years one could arrive at a good datum for the length
of lunar month. It is true that in marking the date of new moons there was a constant danger
of erring by a day, but in the long run these errors would even out and the very development
of lunisolar calendars would call the errors to attention. Several cultures adopted
independently the Metonic calendar, because calendars were used not only to regulate political
and economic activities, but also to record the occurrence of eclipses. The date of eclipses was
not a matter of mere scientific interest and the ability to predict them had great social value.
With the Metonic calendar, the good recording of eclipses and their prediction became an
elementary operation. Eclipses repeat themselves according to the same pattern after 223 lunar
months, that is, about 18 solar years and 11 days. They occur in the same part of the sky in
three cycles of 223 months. If the lunar month were to be calculated with an accuracy of less
than 29.53 days, in less than 6 years one would notice that eclipses occur not only at the
wrong time, but also on the wrong solar day. Because the Metonic cycle was used to calculate
the date of eclipses, the Greeks were driven to introduce refinements into it. Hipparchus
proposed that for the sake of predicting eclipses there be adopted a cycle of 19 x 223 lunar
months. According to him, this was the shortest period in which a series of lunar months
equals a whole number of solar days. He assumed that 19 x 223 lunar months are exactly
125,121 solar days or 342 solar years and 208 days. Modern figures give 342 years and
208.17 days. Textbooks repeat that Hipparchus reckoned the solar year as 365.24666 days,
but, although he mentioned this figure, he must have known better ones, since in the case
before us he reckoned the solar year as 365.24 days (correct datum 365.242199). All this
proves that the calculation of the ratio between lunar month and solar year did not involve
elaborate observational procedures, so that it could result in the gathering of extremely
accurate data.
