Now the Mayas had determined the lunar revolution so exactly that they perceived the incompatibility of the period of 11,960 days with a multiple of lunar revolutions. They found that 405 lunar revolutions amounted approximately to 11,958 days, which is, in fact, the largest number on the second half of page 58. In order not to drop the significant 11,960 altogether, they made use of a very shrewd artifice. They took as the starting-point the day XII Lamat, corresponding to the number 11,960, and set down XI Manik before it and XIII Muluc after it. Now if the count began with XIII Muluc and ended with XI Manik, it actually resulted in 11,958. Therefore what the Manuscript presents here is, in the first place, the series, which is this time to be read from left to right. Below it are the three days belonging to each member of the series and then a number for each member stating the interval between it and the preceding one. The members, the days and the differences must correspond with one another. It is, therefore, no longer necessary to pay especial attention to the two latter. They will serve merely to control and to correct the manifold errors. The entire period of 11,958 days was doubtless first divided into three equal periods of 3986 days. And in order still further to subdivide these shorter periods, the term of 177 days was employed as far as it would go; 177, however, is the half of a lunar year of 354 days, made up of 6 months of 30 days and 6 of 29 days, thus allowing 29.5 days in round numbers for each month. 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 29-2/3 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.