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KRONOS Vol. I, Issue 3


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 117.5-7.5 = 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 breath-taking, 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 luni-solar 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.

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