Now 177 is = 3 × 29 + 3 × 30. The average, 29.5, however, is too short for the length of the lunar revolution. In order to raise it as nearly as possible to the exact time, two other numbers were introduced at certain points of the series, viz:—148 = 2 × 29 + 3 × 30, 178 = 2 × 29 + 4 × 30. 148 = 5 months of 29.6 days, while 178 = 6 months of 29⅔ days. Now let us see in what proportion these 148 and 178 days were distributed among the periods of 177.
First we see that the period of 3986 days (i.e., a third of the whole) was divided into 3 sections of 1742, 1034 and 1210 days, as follows:—
| 1742 = 8 × 177 + 148 + 178 |
| 1034 = 4 × 177 + 148 + 178 |
| 1210 = 6 × 177 + 148 |
| ———————————— |
| 3986 = 18 × 177 + 3 × 148 + 2 × 178. |
This is equal to 135 months of 29.526 days each. Now the question arises how did the Mayas express this fraction?
Perhaps some time in the future it will be found, that following their vigesimal system, they designated it approximately thus:—
29 + ½ + 1⁄40 + 1⁄800.
The whole period of 11,958 days was divided as follows:—
| 3 × 1742 = 24 × 177 + 3 × 148 + 3 × 178 |
| 3 × 1034 = 12 × 177 + 3 × 148 + 3 × 178 |
| 3 × 1210 = 18 × 177 + 3 × 148 |
| ————————————————- |
| 3 × 3986 = 54 × 177 + 9 × 148 + 6 × 178. |
Thus for every 6 parts of 177 days there was consequently 1 of 148 and to every 9 parts of 177, 1 of 178.