Finally, in the first column of page 51, we have the complete normal date 4 Ahau 8 Cumhu (9 Ix). But below this, between red numerals denoting the 1,578,988 mentioned above, there is set down in black the number 1,268,800. This corresponds to the date IV Ahau 3 Zip (2 Cauac). It may have been formed by adding 16,120 = 62 × 260 to 11 Ahau-Katuns = 1,252,680. It is, however, not only equal to 4880 × 260, but also to 158,600 × 8, therefore also divisible by the interval between IV Ahau XII Lamat, as well as by 104 = 8 × 13, while on the contrary it is not as we should expect, divisible by 11,960. I have changed the 11, in the 20 × 11, to 8 by omitting one line and adding two dots, for otherwise the result would not be the one required. The magnitude of the number recalls the one on page 31, which is only 260 less, and that on page 62. Finally it should be noted that the two large numbers on page 51 are separated from one another by 310,188 days = 849 years and 303 days, which corresponds exactly to the dates given for each. One may be situated as far in the future as the other is in the past, but this does not necessarily mean that the present coincides exactly with 1,423,894. Pages 51--58. Thus far we have examined only the upper halves of pages 51 and 52 and have still to consider the lower, but not until we have finished the upper parts of pages 53-58 of which the former are the continuation. We have first to consider the series, then the pictures and lastly the hieroglyphs. As on page 24 we found multiples of the number 2920 (= 8 × 365 = 5 × 584), while on pages 46-50 it was divided into four unequal parts, so on pages 51-52 we find multiples of the number 11,960 (104 × 115 = 46 × 260) while on pages 53-58 it is divided into 69 unequal parts. On pages 51-52 it was the aim to combine only the Mercury course with the Tonalamatl, but here we are confronted with the additional problem of bringing the lunar revolution into accord with these two. The lunar revolution, which we assume to be 29.53 days, of course requires fractional computation, of which the Mayas either were ignorant or which they timorously avoided; like the ancient Egyptians, who were acquainted only with fractions having 1 as numerator, or beyond these at most with 2/3 (see Hultsch, "Die Elemente der ägyptischen Teilungsrechnung," 1895, page 16).