To the combination of the tonalamatl and the solar calendar the tonalamatl contributed the names of the days, and the solar calendar the divisions of the year in which the days found positions. The tonalamatl and the solar year thus went side by side, each recommencing whenever it reached its own limits. The days in the solar year were known by the names of the days in the tonalamatl which were affixed to them. Thus it is plain that 105 of the 260 tonalamatl day-names had to be repeated in the solar year of 365 days. NAMES OF THE YEARS The year was known by the tonalamatl sign of the day with which it began. As there were 20 day-signs, and 5, the least common multiple of 365 and 20, goes into 20 exactly 4 times, the year could begin with one of the four signs only. These were Acatl, Tecpatl, Calli, Tochtli. Each month of a given year began with the same tonalamatl day-sign. The 20 day-signs always occupied the same position in all the months of a given year, as there were 20 days in a Mexican month. But since the last month was followed by the 5 nemontemi, or “unlucky” days, it follows that each year began with a day-sign five days later than the last. Also, since 365, the number of days in a year, is divisible by 13 with 1 as remainder, it follows that each year began with a day-number one in advance of the last. The commencement of the year coincided with the commencement of the tonalamatl once in four years. THE CALENDAR ROUND Fifty-two years made up what has been called by modern students the Calendar Round, and by the Mexicans was known as xiuhmolpilli or toxiuhmolpio, “year bundle” or “our years will be bound.” The Mexicans, differing in this from the Maya, never progressed beyond the Calendar Round in the development of their chronological system, as is proved by the fact that dates of precisely the same designation occurred at intervals of every 52 years.