[Illustration: SIGNS OF THE MONTHS, FROM THE BOOK OF CHILAN BALAM OF CHUMAYEL.] The chronological portions of the "Books of Chilan Balam" re[TN-1] partly written with the ancient signs of the days, months and epochs, and they furnish us, also, delineations of the "wheels" which the natives used for computing time. The former are so important to the student of Maya hieroglyphics, that I have added photographic reproductions of them to this paper, giving also representations of those of Landa for comparison. It will be observed that the signs of the days are distinctly similar in the majority of cases, but that those of the months are hardly alike. [Illustration: SIGNS OF THE MONTHS, AS GIVEN BY BISHOP LANDA.] The hieroglyphs of the days taken from the "_Codex Troano_," an ancient Maya book written before the Conquest, probably about 1400, are also added to illustrate the variations which occurred in the hands of different scribes. Those from the "Books of Chilan Balam" are copied from a manuscript known to Maya scholars as the "_Codice Perez_," of undoubted authenticity and antiquity.[14-*] The result of the comparison I thus institute is a triumphant refutation of the doubts and slurs which have been cast on Bishop Landa's work and vindicate for it a very high degree of accuracy. The hieroglyphics for the months are quite complicated, and in the "Books of Chilan Balam" are rudely drawn; but, for all that, two or three of them are evidently identical with those in the calendar preserved by Landa. Some years ago, Professor de Rosny expressed himself in great doubt as to the fidelity in the tracing of these hierogylphs[TN-2] of the months, principally because he could not find them in the two codices at his command.[14-+] As he observes, they are _composite_ signs, and this goes to explain the discrepancy; for it may be regarded as established that the Maya script permitted the use of several signs for the same sound, and the sculptor or scribe was not obliged to represent the same word always by the same figure. In close relation to chronology is the system of numeration and the arithmetical signs. These are discussed with considerable fulness, especially in the "Book of Chilan Balam of Kaua." The numerals are represented by exactly the same figures as we find in the Maya manuscripts of the libraries of Dresden, Pesth, Paris and Madrid; that is, by points or dots up to five, and the fives by single straight lines, which may be indiscriminately drawn vertically or horizontally. The same book contains a table of multiplication in Spanish and Maya which settles some disputed points in the use of the vigesimal system by the Mayas.