We have then a little cycle of five revolutions of Venus equal to eight solar years, which are 2,920 days. Each of the frames inclosing five dots, each of the little pentagons, expresses this chronological value. Supposing the planet at the beginning of its matutinal apparition or its heliacal rising, it will have recovered the identical position, with respect to the star of day, at the termination of the cycle. The fact is a phenomenon of astronomical observation which could not pass unobserved to scrutinizers of the heavens like the Indians: an in order to commemorate it, they celebrated the festival atamalqualiztli of which Sahagún speaks. But we have seen that this unity is repeated thirteen times. There are two reasons for this: one, to equal with Venus years the great period of 104 solar years, a cycle equivalent to 65 synodical movements of the planet; the other, to equalize the two calendars, because when five Venus years have passed, the sixth commences anew with Cipactli, but this character goes this second time not accompanied by the numeral 1, but by 9, necessitating that the five years shall be repeated thirteen times, in order that Cipactli should return to be accompanied by 1, as at the beginning of the period, and the commencement of the one and the other calendar.
The cause of this phenomenon is known. The tonalámatl, that is to say, the series of twenty thirteens, run through the book of the planet the same as through that of the sun, calendars, one and the other, which are made up by the combination of thirteen numbers in order with the twenty day characters, so that these may not be confounded on being repeated. As the number 584 does not contain an exact number of thirteens, there are twelve units over in the first Venus year, eleven in the next, ten in the next, and so on successively, so that Cipactli comes to be accompanied by different numerals, the thirteen times that it begins the year, until the 65 counts of the planet's calendar are complete. Although all archaeologists know this, we copy anew the distribution of the thirteens and of the day characters in the computation which we are considering.
Figures which accompany the initial signs of the Venus year, in a series of 65 years:
| Cipactli | 1 | — | 9 | — | 4 | — | 12 | — | 7 | — | 2 | — | 10 | — | 5 | — | 13 | — | 8 | — | 3 | — | 11 | — | 6 |
| Cóatl | 13 | — | 8 | — | 3 | — | 11 | — | 6 | — | 1 | — | 9 | — | 4 | — | 12 | — | 7 | — | 2 | — | 10 | — | 5 |
| Atl | 12 | — | 7 | — | 2 | — | 10 | — | 5 | — | 13 | — | 8 | — | 3 | — | 11 | — | 6 | — | 1 | — | 9 | — | 4 |
| Acatl | 11 | — | 6 | — | 1 | — | 9 | — | 4 | — | 12 | — | 7 | — | 2 | — | 10 | — | 5 | — | 13 | — | 8 | — | 3 |
| Ollin | 10 | — | 5 | — | 13 | — | 8 | — | 3 | — | 11 | — | 6 | — | 1 | — | 9 | — | 4 | — | 12 | — | 7 | — | 2 |
14
