The five nemontemi, or “useless” days, were evidently a later interpolation, introduced at a period when it was discovered that an original time-count of 360 days did not fulfil the solar round. They were counted and distinguished, however, in precisely the same manner as the other days, that is the numerals and hieroglyphs of the tonalamatl were adjusted to them as well as to the rest, except that they had no “lords” or rulers of day or night. They were regarded as most unlucky and no business of any kind was transacted upon them, only the most necessary offices of life being undertaken whilst they lasted. They are in no sense to be regarded as intercalary days, for, despite classical statements to the contrary, the Mexicans were ignorant of the methods of chronological intercalation, and a study of the tonalamatl will show that the introduction of any intercalary period would render it nugatory and destroy that ability to return into itself which is one of its chief characteristics. These nemontemi did not always fall in the same period of the solar year, but were sometimes placed before Quaitleloa, now before Tititl, now before Atemoztli, or elsewhere, as the priestly authorities decided. For the Mexican year of 365 days was short of the true solar year by six hours and some minutes, therefore in the course of years the festivals became displaced and their chronological revision and balance became necessary and could be effected by the shifting of the nemontemi. THE VENUS PERIOD To Förstemann and Seler is due the discovery that the Mexicans possessed a system of computing time based upon the synodic revolution of the planet Venus. The Venus period or “year” comprised 584 days. It would seem as if the Maya and Mexicans had striven to discover a common measure for the numbers 584, 365, and 260. Five synodical revolutions of Venus are equivalent to eight solar years (5 × 584) = 2,920 = (8 × 365), but the number 2,920 is not divisible by 260, the number of days in the tonalamatl. Any accord between the two periods is not possible until the sum of 104 years is reached, that is to say, 65 Venus periods are equal to 146 tonalamatl periods both of which contain 37,960 days. Like the tonalamatl, the Venus period was productive of sacerdotal speculation, commencing with the day cipactli. At the end of six periods the Venus “year” recommenced with the same sign affected by a different figure. At the end of thirteen periods the sign differed, but the figure was the same. The question has been learnedly discussed in its entirety by Seler, to whose work the reader is referred. [344] SHORT BIBLIOGRAPHY OF WORKS RELATING TO THE TONALAMATL Bulletin 28 of the Bureau of American Ethnology has several papers by Seler and Förstemann on the tonalamatl. Morley, “An Introduction to the Study of Maya Hieroglyphs” (Bulletin 57 of the Bureau of American Ethnology). (Washington, 1915.)