6 8 8 Under VII Lamat, Plate 49. 8 -- -- -- 6 8 16 Under II Cib, Plate 49. 11 16 -- -- -- 7 2 12 Under IV Eb, Plate 50. 4 10 -- -- -- 7 7 2 Under III Ik, Plate 50. 12 10 -- -- -- 8 1 12 Under VI Eb, Plate 50. 8 -- -- -- 8 2 0 Under I Ahau, Plate 50. The proof of the correctness of the theory advanced may, therefore, be considered conclusive, as it amounts, in fact, to a mathematical demonstration. Dr. Förstemann, who considers these lines of black numbers, standing one above another, as representing different grades of units--thus, the lowest, single units; the second, units twenty-fold the lower; the third, eighteen-fold the second; the fourth, twenty-fold the third, &c.--has found the correct intervals of the series, which he states are 236, 90, 250, and 8 days, agreeing with our 11 months, 16 days; 4 months, 10 days; 12 months, 10 days, and 8 days. As all the discoveries mentioned herein were made previous to the receipt of Dr. Förstemann's work, I give them according to my own method, acknowledging any modification due to his work. Although I shall compare special results from time to time, an explanation of Dr. Förstemann's method is reserved for a future paper, as his work was not received until I was revising my notes for publication. The foregoing explanation of the series shows it to be very simple and makes it clear that it relates to the day columns at the top of the pages. Still, there is one point somewhat difficult to understand. Are the numbers of the third or lowest line intended to denote the positions in the month of the days in the columns above? If so, the month must have commenced with Ymix, as can readily be shown in the following manner: TABLE III. 1. Ymix. 2. Ik. 3. Akbal. 4. Kan. 5. Chicchan. 6. Cimi. 7. Manik. 8. Lamat. 9. Muluc. 10. Oc. 11. Chuen. 12. Eb. 13. Been. 14. Ix. 15. Men. 16. Cib. 17. Caban. 18. Ezanab. 19. Cauac. 20. Ahau. If we write in a column in proper order the 20 days of the Maya month, commencing with Ymix, and number them consecutively, as in Table III, we shall find by comparison that the numbers in the lower line indicate the position, in this column, of the days directly over them. Take, for example, the lower line of black numerals on Plate 46, writing over them the respective days of the columns, thus: Cib. Cimi. Cib. Kan. 16 6 16 4 Referring to Table III we see that Cib is the sixteenth day, Cimi the sixth, and Kan the fourth. The days and numbers of Plate 47 are: Ahau. Oc. Ahau. Lamat. 0 10 0 8