3. Consequently, every fifth year any particular day-name returned to the same position in the divisions of the year. 4. Consequently, any particular day-name could occupy only one of four positions in the divisions of the year, each of which it held in successive years, returning to the same position every fifth year. 5. Consequently, the twenty day-names were divided into five groups of four day-names each, any day-name of any group being five days distant from the day-name of the same group next preceding it. 6. Finally, in any given year any particular day-name occupied the same relative position throughout the divisions of that year. Up to this point, however, as above stated, we have not been dealing with the complete designations of the Maya days, but only their _name parts_ or name glyphs, the positions of which in the several divisions of the year we have ascertained. It now remains to join the tonalamatl, which gives the complete names of the 260 Maya days, to the haab, which gives the positions of the days in the divisions of the year, in such a way that any one of the days whose name-part is Ik, Manik, Eb, or Caban shall occupy the first position of the first division of the year; that is, 0 Pop, or, as we should write it, the first day of Pop. It matters little which one of these four name parts we choose first, since in four years each one of them in succession will have appeared in the position 0 Pop. Perhaps the easiest way to visualize the combination of the tonalamatl and the haab is to conceive these two periods as two cogwheels revolving in contact with each other. Let us imagine that the first of these, A (fig. 21), has 260 teeth, or cogs, each one of which is named after one of the 260 days of the tonalamatl and follows the sequence shown in plate 5. The second wheel, B (fig. 21), is somewhat larger, having 365 cogs. Each of the spaces or sockets between these represents one of the 365 positions of the days in the divisions of the year, beginning with 0 Pop and ending with 4 Uayeb. See Table IV for the positions of the days at the end of one year and the commencement of the next. Finally, let us imagine that these two wheels are brought into contact with each other in such a way that the tooth or cog named 2 Ik in A shall fit into the socket named {57} 0 Pop in B, after which both wheels start to revolve in the directions indicated by the arrows. [Illustration: FIG. 21. Diagram showing engagement of tonalamatl wheel of 260 days (A), and haab wheel of 365 positions (B); the combination of the two giving the Calendar Round, or 52-year period.]