The number 236 denotes the time of the western elongation, when Venus is the morning star, 90 the time of the invisibility of the planet, during its superior conjunction, 250 that of its eastern elongation, when Venus is the evening star, and 8 the time of its invisibility during inferior conjunction. The disproportion between 236 and 250 is somewhat striking. These periods which need not of course be exactly equal are usually computed at 243 days. The short period of eight days is only calculated for very sharp eyes; we actually find in the Anales del Museo Nacional de Mexico II, 341 (Mex. 1882), that the Aztecs calculated only eight days for the invisibility of Venus, and this period is also mentioned in the Anales de Quauhtitlan. The repetition of the cardinal points in the 15th and 20th lines of the extract given above refer to these periods; in the upper line to their beginning and in the lower to their close. Hence in the lower line the cardinal points must advance one place and the gods belonging to them in the 16th and 19th lines must follow the same course. The numbers in the 17th line indicate to which day of the period of 2920 days the position has advanced. But now we see that the indication of days in the lines 1-13, the indication of months in lines 14, 18 and 21, and the numbers in line 17 are separated from those directly to the right of them by a number of days equal to the numbers given in the lowest line. From this it follows that each day of the thirteen top lines is joined to each of the month dates placed just below them, forming a complete calendar date. Therefore from the III Cib on the left upper corner of page 46 a III Cib 4 Yaxkin, a III Cib 8 Zac, a III Cib 19 Kayab must be formed. All the 4 × 13 × 5 = 260 day indications combined with three month indications each, show therefore that this whole passage is a huge abbreviation for 780 calendar dates and that the whole refers to 3 × 37,960 days = 113,880 days. But 37,960 which we already found on page 24, is equal to 146 × 260, 104 × 365, 65 × 584, 13 × 2920. I am inclined to think that I also found 113,880 on page 24. But the 3 × 37,960 = 113,880 days do not form the entire period treated of here. For the three periods begin and end with the days:-- I Ahau 13 Mac (10 Muluc), I Ahau 18 Kayab (3 Kan), I Ahau 3 Xul (4 Cauac). Hence these three dates, the second of which was found on page 24, prove that the three periods of 37,960 are not consecutive, but that there is an interval between them. Now between the first and second of the three dates the interval is 19 years + 85 days = 7020 days, and between the second and third, the interval is 26 years + 130 days = 9620 days. If these two periods be added to the 113,880 days, the sum is the whole period treated of here, viz:-- 130,520 = 502 × 260 days. But a truly surprising result is obtained, if, as must often be the case with series, we begin not with the upper of the three dates, but with the lower.