Modern astronomy shows that the synodical revolution of the moon consumes 29.53059 days, about .03 days more than 29½ days. Since a calendar must be based on whole days the natural method of combining the months would be to alternate one of 29 days with one of 30 days. At the end of two months or 59 days the true synodical month would be in advance of the calendrical month by .06118 days. Every two months this error is doubled so that at the end of 34 months the calendar would have completed 1003 days and the synodical month 1004.04 days. (See Table VIII, p. 19.) One method of correcting this would be to make the last month a 30-day month instead of one of 29 days as it would be by simple alternation. This 34-month period could then be repeated as a cycle with an accumulating error of .04 days at every repetition.
Such a series utterly disregards, however, all other phenomena such as eclipses, seasons, etc. As soon as eclipses are considered the arrangement of the months must be altered in order to use the periodicity of eclipses in the calendar. Eclipses occur at regular seasons, approximately six months apart. The average interval between eclipse seasons is 173.310 days, 3.874 days less than six synodical lunar months. In Table IX (p. 20) the eclipse season is compared with the nearest synodical lunar month. It will be noticed that the difference increases between the two series until it is necessary to use five synodical months for one interval instead of six to keep the difference less than half a month. It is necessary to do this three times in 135 synodical months, or 3986.630 days, which exceed 23 eclipse seasons, or 3986.131 days, by practically one half-day. It would be most logical to drop these extra months out of the set of six, during that group in which the difference tends to become most nearly half a month. That would be just before the 23d, 70th, and 117th month, that is, 47 months apart, requiring 41 months to complete the 135-month period.
This series of 135 lunar months, or 23 eclipse seasons, can be repeated almost indefinitely, alternating 3986 and 3987 days to the series and still keep the synodical month in accord with the eclipse season. But another factor must also be considered. Months of 29 and 30 days cannot be simply alternated and either conform with the true synodical month or complete the ecliptic series mentioned, for 3986 contains three more days than sixty-eight 30-day months, and sixty-seven 29-day months. Therefore in the 3986 series three of the 29-day months must be changed to 30-day months, and in the 3987 series four must be changed. The position of these changes is arbitrary. They can, for example, be the 34th, 68th, and 102d months, and when necessary, the 134th.
| Number of month | Number of days in month | Elapsed days calendar month | Elapsed days synodical month | Error |
|---|---|---|---|---|
| 1 | 30 | 30 | 29.53 | -0.47 |
| 2 | 29 | 59 | 59.06 | 0.06 |
| 3 | 30 | 89 | 88.59 | -0.41 |
| 4 | 29 | 118 | 118.12 | 0.12 |
| 5 | 30 | 148 | 147.65 | -0.35 |
| 6 | 29 | 177 | 177.18 | 0.18 |
| 7 | 30 | 207 | 206.71 | -0.29 |
| 8 | 29 | 236 | 236.24 | 0.24 |
| 9 | 30 | 266 | 265.78 | -0.22 |
| 10 | 29 | 295 | 295.31 | 0.31 |
| 11 | 30 | 325 | 324.84 | -0.16 |
| 12 | 29 | 354 | 354.37 | 0.37 |
| 13 | 30 | 384 | 383.90 | -0.10 |
| 14 | 29 | 413 | 413.43 | 0.43 |
| 15 | 30 | 443 | 442.96 | -0.04 |
| 16 | 29 | 472 | 472.49 | 0.49 |
| 17 | 30 | 502 | 502.02 | 0.02 |
| 18 | 29 | 531 | 531.55 | 0.55 |
| 19 | 30 | 561 | 561.08 | 0.08 |
| 20 | 29 | 590 | 590.61 | 0.61 |
| 21 | 30 | 620 | 620.14 | 0.14 |
| 22 | 29 | 649 | 649.67 | 0.67 |
| 23 | 30 | 679 | 679.20 | 0.20 |
| 24 | 29 | 708 | 708.73 | 0.73 |
| 25 | 30 | 738 | 738.26 | 0.26 |
| 26 | 29 | 767 | 767.80 | 0.8 |
| 27 | 30 | 797 | 797.33 | 0.33 |
| 28 | 29 | 826 | 826.86 | 0.86 |
| 29 | 30 | 856 | 856.39 | 0.39 |
| 30 | 29 | 885 | 885.92 | 0.92 |
| 31 | 30 | 915 | 915.45 | 0.45 |
| 32 | 29 | 944 | 944.98 | 0.98 |
| 33 | 30 | 974 | 974.51 | 0.51 |
| 34 | 29 | 1003 | 1004.04 | 1.04 |
| Eclipse season | Synodic month | |||
|---|---|---|---|---|
| Number | Days | Number | Days | Difference |
| 1 | 173.310 | 6 | 177.184 | 3.874 |
| 2 | 346.620 | 12 | 354.367 | 7.747 |
| 3 | 519.930 | 18 | 531.551 | 11.621 |
| 4 | 693.240 | 23 | 679.204 | -14.036 |
| 5 | 866.550 | 29 | 856.387 | -10.163 |
| 6 | 1039.860 | 35 | 1033.571 | -6.289 |
| 7 | 1213.170 | 41 | 1210.754 | -2.416 |
| 8 | 1386.480 | 47 | 1387.938 | 1.458 |
| 9 | 1559.790 | 53 | 1565.121 | 5.331 |
| 10 | 1733.100 | 59 | 1742.305 | 9.205 |
| 11 | 1906.411 | 65 | 1919.489 | 13.078 |
| 12 | 2079.721 | 70 | 2067.141 | -12.580 |
| 13 | 2253.031 | 76 | 2244.325 | -8.706 |
| 14 | 2426.341 | 82 | 2421.508 | -4.833 |
| 15 | 2599.651 | 88 | 2598.692 | -0.959 |
| 16 | 2772.961 | 94 | 2775.875 | 2.914 |
| 17 | 2946.271 | 100 | 2953.059 | 6.788 |
| 18 | 3119.581 | 106 | 3130.243 | 10.662 |
| 19 | 3292.891 | 112 | 3307.426 | 14.535 |
| 20 | 3466.201 | 117 | 3455.079 | -11.122 |
| 21 | 3639.511 | 123 | 3632.263 | -7.248 |
| 22 | 3812.821 | 129 | 3809.446 | -3.375 |
| 23 | 3986.131 | 135 | 3986.630 | 0.499 |
| Upper number | Month number | Interval | Groups | |
|---|---|---|---|---|
| 502 | 17 | |||
| 2244 | 76 | 59 |  | 94 (47 + 47) |
| 3278 | 111 | 35 | ||
| 4488 | 152 | 41 | 41 | |
| 6230 | 211 | 59 | | 94 (47 + 47) |
| 7264 | 246 | 35 | ||
| 8474 | 287 | 41 | 41 | |
| 10039 | 340 | 53 | | 94 (47 + 47) |
| 11250 | 381 | 41 | ||
The next logical step is a comparison between the theoretical calendars just described and the manuscript. A study of the manuscript reveals that: (1) the series recorded represents 405 lunar months or three times 135 months, and that the series naturally falls into three great subdivisions of 3986 days each; (2) each third consists of 23 columns or unequal subdivisions; (3) the intervals between the 178-day groups are 47 and 88 months; (4) the 148-day groups fall approximately at 47 and 41 month intervals (see Table X); (5) the first 178-day group in each third occurs between the 30th and 35th month inclusive, and the other 178-day group of the third comes 47 months later. Since the number 178 is composed of four 30-and two 29-day months, an extra day must have been added, that is, a 30-day month was substituted for one of the 29-day months, if the manuscript represents a regular alternating series.
The obvious conclusions to be drawn from these facts are: (1) that the series was divided into three groups of 3986 days each in order to associate the lunar calendar closely with the ecliptic cycle of the same length; (2) that the 23 columns in each third may represent the twenty-three eclipse seasons in each eclipse period of 3986 days; (3) that groups of 47 and 41 months were used in some way in the series, for the 178-day groups are separated by 47 and 88 months and 88 is composed of 47 and 41, the two periods so closely associated with the recurrence of eclipses; (4) that the six months period was changed to one of five months of 148 days approximately every 47 and 41 months, which is the method already advanced in the theoretical ecliptic lunar series for keeping the synodical months and ecliptic seasons together; (5) that one extra day was added to the alternating 29-and 30-day months, between the 30th and 35th month inclusive of each third, in accordance with the theoretical necessity for so doing already brought out, and that another of the three extra days was added 47 months later.