Aids to the Study of the Maya Codices Sixth Annual Report of the Bureau of Ethnology to the Secretary of the Smithsonian Institution, 1884-85, Government Printing Office, Washington, 1888, pages 253-372

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Transcriber's Note:

This book was originally published as a part of:

Powell, J. W. 1888 _Sixth Annual Report of the Bureau of Ethnology to the Secretary of the Smithsonian Institution, 1884-'85._ pp. 253-372. Government Printing Office, Washington, D.C.

The index included in this version of the book was extracted from the overall volume index.

A number of typographical errors found in the original text have been maintained in this version. They are marked in the text with a [TN-#]. A description of each error is found in the complete list at the end of the text.

Tables XX, XXI, and XXII were too wide to fit within the character limits of the text file for this ebook. They have been broken into two parts.

Special characters:

The following characters used in the original publication are not available in the character set used for this version of the book. They have been replaced with the following codes.

[=h] Small h with stroke [(1)] Circled 1 [(2)] Circled 2 [(3)] Circled 3 [(4)] Circled 4 [(5)] Circled 5 [(6)] Circled 6 [(7)] Circled 7 [(8)] Circled 8 [(9)] Circled 9 [(10)] Circled 10 [(11)] Circled 11 [(12)] Circled 12 [(13)] Circled 13 [(I)] Circled I

Smithsonian Institution--Bureau Of Ethnology.

Aids To The Study

Of

The Maya Codices.

By

Prof. Cyrus Thomas.

Contents.

Introduction 259 CHAP. I. The numerals in the Dresden Codex 261 II. Conclusions 339 III. The writing 345 Signification of the characters 347 Symbols of animals &c 348 Symbols of deities 358 Discussion as to phonetic features of the characters 365

Illustrations.

FIG. 359. Line of day and numeral symbols from Plates 36_c_ and 37_c_, Dresden Codex 272 360. Line of day and numeral characters from Plates 33-39, Dresden Codex 276 361. Unusual symbol for Akbal from Plate 8 of the Dresden Codex 284 362. Copy of Plate 50, Dresden Codex 297 363. Copy of Plate 51, Dresden Codex 306 364. Copy of Plate 52, Dresden Codex 307 365. Copy of Plate 53, Dresden Codex 308 366. Copy of Plate 54, Dresden Codex 309 367. Copy of Plate 55, Dresden Codex 310 368. Copy of Plate 56, Dresden Codex 311 369. Copy of Plate 57, Dresden Codex 312 370. Copy of Plate 58, Dresden Codex 313 371. Specimens of ornamental loops from page 72, Dresden Codex 337 372. Numeral character from the lower division of Plate XV, Manuscript Troano 343 373. Turtle from the Cortesian Codex, Plate 17 348 374. Jar from the Cortesian Codex, Plate 27 349 375. Worm and plant from Manuscript Troano, Plate XXIX 351 376. Figure of a woman from the Dresden Codex 351 377. Copy of middle and lower divisions of Plate XIX, Manuscript Troano 352 378. Copy of lower division of Plate 65, Dresden Codex 353 379. The moo or ara from Plate 16, Dresden Codex 355 380. The god Ekchuah, after the Troano and Cortesian Codices 358 381. The long nosed god (Kukulcan) or god with the snake-like tongue 359 382. Copy of head from the Borgian Codex (Quetzalcoatl?) 360 383. The supposed god of death from the Dresden Codex 361 384. The supposed god of death from the Troano Codex 361 385. The god with the banded face from the Troano Codex 362 386. The god with the old man's face 363 387. The god with face crossed by lines 364 388. Wooden idol in vessel with basket cover 371

Aids To The Study Of The Maya Codices.

By Cyrus Thomas.

Introduction.

The object of this paper is to present to students of American paleography a brief explanation of some discoveries, made in regard to certain Maya codices, which are not mentioned in my previous papers relating to these aboriginal manuscripts.

It is apparent to every one who has carefully studied these manuscripts that any attempt to decipher them on the supposition that they contain true alphabetic characters must end in failure. Although enough has been ascertained to render it more than probable that some of the characters are phonetic symbols, yet repeated trials have shown beyond any reasonable doubt that Landa's alphabet furnishes little or no aid in deciphering them, as it is evidently based on a misconception of the Maya graphic system. If the manuscripts are ever deciphered it must be by long and laborious comparisons and happy guesses, thus gaining point by point and proceeding slowly and cautiously step by step. Accepting this as true, it will be admitted that every real discovery in regard to the general signification or tenor of any of these codices, or of any of their symbols, characters, or figures, or even in reference to their proper order or relation to one another, will be one step gained toward the final interpretation. It is with this idea in view that the following pages have been written and are now presented to the students of American paleography.

It is impracticable to present fac simile copies of all the plates and figures referred to, but it is taken for granted that those sufficiently interested in this study to examine this paper have access to the published fac similes of these aboriginal documents.

Chapter I.

The Numerals In The Dresden Codex.

Before entering upon the discussion of the topic indicated it may be well to give a brief notice of the history and character of this aboriginal manuscript, quoting from Dr. Förstemann's introduction to the photolithographic copy of the codex,[261-1] he having had an opportunity to study the original for a number of years in the Royal Public Library of Dresden, of which he is chief librarian:

"Unfortunately, the history of the manuscript begins no further back than 1739. The man to whom we owe the discovery and perhaps the preservation of the codex was Johann Christian Götze, son of an evangelical pastor, born at Hohburg, near Wurzen, in the electorate of Saxony. He became a Catholic, and received his education first at Vienna, then in Rome; became first chaplain of the King of Poland and elector of Saxony; later on, papal prothonotary; presided over the Royal Library at Dresden from 1734, and died holding this position, greatly esteemed for learning and integrity, July 5, 1749. This sketch is taken from his obituary notice in Neue Zeitungen von gelehrten Sachen, Nr. 62, Leipzig, 1749. In his capacity as librarian he went to Italy four times, and brought thence rich collections of books and manuscripts for the Dresden library. One of these journeys took place in 1739, and concerning its literary results we have accurate information from a manuscript, in Götze's handwriting, which is found in the archives of the Royal Public Library, under A, Vol. II, No. 10, and bears the title: 'Books consigned to me for the Royal Library in January, 1740.' Under No. 300 we read: 'An invaluable Mexican book with hieroglyphic figures.' This is the same codex which we here reproduce.

"Götze also was the first to bring the existence of the manuscript to public notice. In 1744 he published at Dresden The Curiosities of the Royal Library at Dresden, First Collection. As showing what value Götze attributed to this manuscript, the very first page of the first volume of this work, which is of great merit and still highly useful, begins as follows: '1. A Mexican book with unknown characters and hieroglyphic figures, written on both sides and painted in all sorts of colors, in long octavo, laid orderly in folds of 39 leaves, which, when spread out lengthwise, make more than 6 yards.'

"Götze continues speaking of this book from page 1 to 5, adding, however, little of moment, but expatiating on Mexican painting and hieroglyphic writing in general. On page 4 he says:

"'Our royal library has this superiority over all others, that it possesses this rare treasure. It was obtained a few years ago at Vienna from a private person, for nothing, as being an unknown thing. It is doubtless from the personal effects of a Spaniard, who had either been in Mexico himself or whose ancestors had been there.'

"On page 5 Götze says:

"'In the Vatican library there are some leaves of similar Mexican writing, as stated by Mr. Joseph Simonius Asseman, who saw our copy four years ago at Rome.'

"Götze therefore received the manuscript as a present on his journey to Italy at Vienna and took it with him to Rome. Unfortunately we know nothing concerning its former possessor. A more accurate report of the journey does not seem to exist; at least the principal state archives at Dresden contain nothing concerning it, nor does the General Directory of the Royal Collections. As appears from the above note, Götze did not know that the Vatican Codex was of an entirely different nature from the Dresden Codex.

"In spite of the high value which Götze set upon the manuscript, it remained unnoticed and unmentioned far into our century. Even Johann Christoph Adelung, who as head librarian had it in his custody and who died in 1806, does not mention it in his Mithridates, of which that part which treats of American languages (III, 3) was published only in 1816, after Adelung's death, by J. S. Vater. This would have been a fitting occasion to mention the Dresden Codex, because in this volume (pp. 13 et seq.) the Maya language is largely treated of, and further on the other languages of Anahuac. Of course it was not possible at that time to know that our manuscript belongs to the former.

"After Götze, the first to mention our codex is C. A. Böttiger, in his Ideas on Archæology (Dresden, 1811, pp. 20, 21), without, however, saying anything that we did not already know from Götze. Still Böttiger rendered great and twofold service: first, as we shall see presently, because through him Alexander von Humboldt obtained some notice of the manuscript, and, second, because Böttiger's note, as he himself explains in the Dresden Anzeiger, No. 133, p. 5, 1832, induced Lord Kingsborough to have the manuscript copied in Dresden.

"We now come to A. von Humboldt. His Views of the Cordilleras and the Monuments of the Indigenous Peoples of America bears on the title page the year 1810, which certainly means only the year in which the printing was begun, the preface being dated 1813. To this work, which gave a mighty impulse to the study of Central American languages and literatures, belongs the Atlas pittoresque, and in this are found, on page 45, the reproductions of five pages of our manuscript. They are Nos. 47, 48, 50, 51, and 52 of Lord Kingsborough. In the volume of text belonging to this atlas Humboldt discusses our manuscript on pp. 266, 267. When he began his work he knew nothing as yet of the existence of the manuscript. It was brought to his knowledge by Böttiger, whose above named work he cites. Here we learn for the first time that the material of the manuscript consists of the plant metl (_Agave Mexicana_,) like other manuscripts that Humboldt had brought from New Spain. Furthermore, he correctly states the length of leaf as 0.295 and the breadth 0.085 meter. On the other hand, he commits two mistakes in saying that there are 40 leaves and that the whole folded table forming the codex has a length of almost 6 meters, for there are only 39 leaves and the length in question is only 3.5 meters, as calculation will approximately show, because the leaves are written on both sides. Humboldt's other remarks do not immediately concern our problem.

"In 1822 Fr. Ad. Ebert, then secretary and later head librarian, published his History and Description of the Royal Public Library at Dresden. Here we find, as well in the history (p. 66) as in the description (p. 161), some data concerning this 'treasure of highest value,' which indeed contain nothing new, but which certainly contributed to spread the knowledge of the subject among wider circles. We may remark right here that H. L. Fleischer, in his Catalogue of Oriental Manuscript Codices in the Royal Library of Dresden, p. 75, Leipzig, 1831, 4^o, makes but brief mention of our codex, as 'a Mexican book of wood, illustrated with pictures, which awaits its OEdipus;' whereupon he cites the writing of Böttiger. The signature of the manuscript here noted, E 451, is the one still in use.

"Between the above mentioned notices by Ebert and Fleischer falls the first and so far the only complete reproduction of the manuscript. Probably in 1826, there appeared at Dresden the Italian Augustino Aglio, a master of the art of making fac similes by means of tracing through transparent substances. He visited the European libraries, very probably even at that time under orders from Lord Kingsborough, to copy scattered manuscripts and pictures from Mexico or seemingly from Mexico.

"Now there arises the question, all important for interpretation, In which shape did the manuscript lie before Aglio? Was it a strip only 3.5 meters in length or did it consist of several pieces?

"To render clear the answer which we proceed to give, it is first necessary to remark that of the 39 leaves of the codex 35 are written on both sides and 4 on one side only, so that we can speak only of 74 pages of manuscript, not of 78. These 74 pages we shall in the following always designate by the numbers which they bear in Lord Kingsborough, and it is advisable to abide by these numbers, for the sake of avoiding all error, until the manuscript can be read with perfect certainty; the 4 empty pages I shall designate with 0 when there is need of mentioning them expressly.

"Furthermore it is necessary to state which of these pages so numbered belong together in such way that they are the front and back of the same leaf. This condition is as follows: One leaf is formed of pages 1 45, 2 44, 3 43, 4 42, 5 41, 6 40, 7 39, 8 38, 9 37, 10 36, 11 35, 12 34, 13 33, 14 32, 15 31, 16 30, 17 29, 18 0, 19 0, 20 0, 21 28, 22 27, 23 26, 24 25, 46 74, 47 73, 48 72, 49 71, 50 70, 51 69, 52 68, 53 67, 54 66, 55 65, 56 64, 57 63, 58 62, 59 61, 60 0. [That is to say, each pair of this series forms one leaf, one page on one side and the other on the reverse side of the leaf.]

"But now we are justified in the assumption, which at least is very probable, that neither did Aglio change arbitrarily the order of the original, nor Lord Kingsborough the order of Aglio. Consequently Aglio must already have had the manuscript before him in two pieces, be it that the thin pellicles by which the single leaves are connected were loosened in one place or that the whole was separated only then in order not to be obliged to manipulate the whole unwieldy strip in the operation of copying. A third possibility, to which we shall presently return, is that of assuming two separate pieces from the beginning; in this case Götze and the others must be supposed to have seen it in this condition, but to have omitted the mention of the circumstance, believing that the original unity had been destroyed by tearing.

"Of the two pieces one must have comprised 24, the other 15 leaves. But Aglio copied each of the two pieces in such way as to trace first the whole of one side and then the other of the entire piece, always progressing from left to right, in European style. Therefore Aglio's model was as follows:

"_First piece_:

"Front (from left to right): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.

"Back (from right to left): 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 0, 0, 0, 28, 27, 26, 25.

"_Second piece_:

"Front (from left to right): 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60.

"Back (from right to left): 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 0.

"In considering this, our attention is attracted by the position of the four blank pages, three of which are together, the fourth alone. It might be expected that the separate blank page began or concluded the second piece and was purposely left blank, because in the folding of the whole it would have lain outside and thus been exposed to injury; the other three would be expected at the end of the first piece. The former, as is easily seen, was quite possible, but the latter was not, unless we assume that even at the time Aglio took his copy the original order had been entirely disturbed by cutting and stitching together again. The four blank pages show no trace of ever having contained writing; the red brown spots which appear on them are to be found also on the sides that contain writing. Perhaps, therefore, those three continuous pages indicate a section in the representation; perhaps it was intended to fill them later on; in a similar way also page three has been left unfinished, because the lower half was only _begun_ by the writer.

"I do not wish to conceal my view that the two pieces which Aglio found were separated from the beginning; that they belong even to two different manuscripts, though written in the same form; but, since it is human to err, I will here and there follow custom in the succeeding pages in speaking of one codex.

"My conviction rests especially on the fact that the writer of manuscript A (pp. 1-45) endeavors to divide each page by two horizontal lines into three parts, which the writer of manuscript B (pp. 46-74) rarely does. The more precise statement is as follows: In A, pp. 1-23 and 29-43 always show two such lines in red color; pp. 25-28 have no red lines, but clearly show a division into three parts; p. 24 is the only one of this manuscript that has only writing and no pictures and where the greater continuity of the written speech forbids tripartition (here ends one side of the manuscript); finally, p. 45 seems to be marked as the real end of the whole by the fact that it contains three very light lines, dividing it into four parts; moreover, everything on this page is more crowded, and the figures are smaller than on the preceding pages, just as in some modern books the last page is printed more closely or in smaller type for want of space. In the same manner I suspect that p. 1 is the real beginning of the manuscript. This is indicated by the bad condition of leaf 2 44, which has lost one corner and whose page 44 has lost its writing altogether. For, if in folding the codex leaf 1 45 was turned from within outward, somewhat against the rule, leaf 2 44 was the outer one, and p. 44 lay above or below, and was thus most exposed to injury. I will not omit mentioning that my attention has been called by Dr. Carl Schultz-Sellack, of Berlin, to the possibility of leaves 1 45 and 2 44 having been fastened to the rest in a reversed position, so that 43, 1 and 2 and on the other side 44, 45, 3 were adjoining; then the gods would here be grouped together, which follow each other also on pages 29 and 30. It cannot be denied that this supposition explains the bad condition of leaf 2 44 still better, because then it must have been the outermost of the manuscript; 44 would be the real title page, so to say, and on p. 45 the writer began, not ended, his representation, with the closer writing of which I have spoken, and only afterward passed on to a more splendid style; and this assumption tallies very well with some other facts. But all this can only be cleared up after further progress has been made in deciphering the manuscript.

"In two places, moreover, this first manuscript shows an extension of the drawings from one page over to the neighboring one, namely, from 4 to 5 and from 30 to 31. This is not found on the second manuscript. From continuity of contents, if we are allowed to assume it from similarity of pictures and partition, we may suppose this manuscript to be divided into chapters in the following manner: pp. 1-2 (then follows the unfinished and disconnected page 3), 4-17, 18-23 (here follows p. 24, without pictures), 25-28, 29-33, 34-35, 36-41.

"Compared with this, manuscript B rarely shows a tripartition, but on pp. 65-68 and 51-57 a bipartition by one line. A further difference is this, that A out of 45 pages has only one (p. 24) without pictures, while B out of 29 pages has 9 without pictures (51, 52, 59, 63, 64, 70, 71, 72, 73), nothing but writing being found on them. Page 74, differing from all others, forms the closing tableau of the whole; and, similarly, p. 60, the last of the front, shows a peculiar character. A closer connection of contents may be suspected between pp. 46-50, 53-58, 61-62, 65-68.

"The two manuscripts also differ greatly in the employment of the sign, or rather signs, differing little from each other, which resemble a representation of the human eye and consist of two curves, one opening above and the other below and joined at their right and left ends. These signs occur only on 5 out of the 45 pages of Codex A (1, 2, 24, 31, 43), while they occur on 16 pages out of the 29 of Codex B (48, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 70, 71, 72, 73).

"I believe that the differences above mentioned, to which others will probably be added, are sufficient to justify my hypothesis of the original independence of the two codices. Whoever looks over the whole series of leaves without preconception cannot escape the feeling, on passing from leaf 45 to leaf 46, that something different begins here.

"Thus the copy of Aglio has made it possible to venture a hypothesis bordering on certainty concerning the original form of this monument. Five years after Aglio had finished the copying there appeared, in 1831, the first volumes of Lord Kingsborough's Mexican Antiquities. The work in the trade cost 175_l_.; the expense of publication had been over 30,000_l_. The eighth and ninth volumes followed only in 1848. The ponderous work has undoubtedly great value from its many illustrations of old monuments of Central American art and literature, which in great part had never been published. As regards the Spanish and English text, it is of much less value. We may pass in silence over the notes added by Lord Kingsborough himself, in which he tries to give support to his favorite hypothesis that the Jews were the first settlers of America. Whoever wishes to obtain exact information concerning the character and contents of the whole work and dreads the labor of lifting and opening the volumes, may find a comprehensive review of it in the Foreign Quarterly Review, No. 17, pp. 90-124, 8vo, London, January, 1832, where he will also find a lucid exposition of the history of the literature of Mexican antiquarian studies.

"In the middle of the third volume of the Mexican Antiquities (side numbers are here absent) there is found the title 'Fac simile of an original Mexican painting preserved in the Royal Library at Dresden, 74 pages.' These 74 pages are here arranged on 27 leaves in the following manner:

Codex A. Codex B.

1, 2, 3, 46, 47, 48, 4, 5, 6, 49, 50, 51, 7, 8, 9, 52, 53, 54, 10, 11, 55, 56, 57, 12, 13, 14, 58, 59, 60, 15, 16, 17, 61, 62, 63, 18, 19, 64, 65, 66, 20, 67, 68, 69, 21, 22, 23, 70, 71, 72, 24, 25, 73, 74. 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45.

"On the whole, therefore, each leaf in Kingsborough comprises three pages of our manuscript. Why the publisher joined only two pages in the case of 10 and 11, 18 and 19, 24 and 25, and left page 20 entirely separate, I cannot say; but when he failed to add 46 to 44 and 45 it was due to the fact that here there is indication of a different manuscript.

"On January 27, 1832, Lord Kingsborough wrote a letter from Mitchellstown, near Cork, in Ireland, to Fr. Ad. Ebert, then head librarian at Dresden, thanking him again for the permission to have the manuscript copied and telling him that he had ordered his publisher in London to send to the Royal Public Library at Dresden one of the ten copies of the work in folio. The original of the letter is in Ebert's manuscript correspondence in the Dresden library.

"On April 27, 1832, when the copy had not yet arrived at Dresden, an anonymous writer, in No. 101 of the Leipziger Zeitung, gave a notice of this donation, being unfortunate enough to confound Humboldt's copy with that of Lord Kingsborough, not having seen the work himself. Ebert, in the Dresden Anzeiger, May 5, made an angry rejoinder to this "hasty and obtrusive notice."[TN-1] Böttiger, whom we mentioned above and who till then was a close friend of Ebert, on May 12, in the last named journal, defended the anonymous writer (who perhaps was himself) in an extremely violent tone. Ebert's replies in the same journal became more and more ferocious, till Böttiger, in an article of May 25 (No. 150 of the same journal), broke off the dispute at this point. Thus the great bibliographer and the great archæologist were made enemies for a long time by means of our codex.

"From Kingsborough's work various specimens of the manuscript passed into other books; thus we find some in Silvestre, Paléographie universelle, Paris, 1839-'41, fol.; in Rosny, Les écritures figuratives et hiéroglyphiques des peuples anciens et modernes, Paris, 1860, 4to; and also in Madier de Montjou, Archives de la société américaine de France, 2^de série, tome I, table V.

"In 1834 Ebert died, and was followed as head librarian by K. C. Falkenstein. He, unlike his predecessor, strove especially to make the library as much as possible accessible to the public. Visits and examinations of the library became much more frequent, and our manuscript, being very liable to injury, on account of its material, had to be withdrawn from the hands of visitors, if it was desired to make it accessible to their sight. It was therefore laid between glass plates and thus hung up freely, so that both sides were visible. In this position it still hangs in the hall of the library, protected from rude hands, it is true, but at the same time exposed to another enemy, daylight, against which it has been protected only in recent time by green screens. Still it does not seem to have suffered much from light during these four decades; at least two former officers of the library, who were appointed one in 1828 and the other in 1834, affirm that at that time the colors were not notably fresher than now. This remark is important, because the coloring in Humboldt, as well as in Lord Kingsborough, by its freshness gives a wrong impression of the coloring of the original, which in fact is but feeble; it may have resembled these copies some 300 years ago.

"In 1836, when the manuscript was being preserved in the manner indicated, the two unequal parts, which were considered as a whole and which no one seems to have thought susceptible of being deciphered, were divided into two approximately equal parts from considerations of space and for esthetic reasons.

"The first five leaves of Codex A, that is, pp. 1-5, with the backs containing pp. 41-45, were cut off and prefixed to Codex B in such way as to have p. 46 and p. 5 adjoining; when I examined the codex more closely I found that between 5 and 46, and therefore also between 41 and 74, there was no such pellicle as generally connects the other leaves. By this change one part was made to contain 20 leaves, the other 19.

"At the same time another change was made. The three blank pages between pp. 28 and 29 had a marring effect, and they were put at the end by cutting through between leaves 18 0 and 17 29 and turning the severed leaves around, so that p. 24 joined on to p. 29 and 17 to 25. The pellicle loosened on this occasion was fastened again.

"I must expressly state that I have no written or oral account of these two manipulations, but conclude they have taken place merely from a comparison of the present arrangement with that which Aglio must have had before him.

"Thus the arrangement in which I found the manuscript, which it may be best to preserve until my views are recognized, is the following:

"(1) _The diminished Codex A (19 leaves):_

Front: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 25, 26, 27, 28, 0, 0, 0.

Back: 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40.

"Or, if we enumerate the numbers on the back from right to left, so that the back of each leaf stands beneath its front:

6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | 25, 26, 27, 28, 0, 0, 0. 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29 | 24, 23, 22, 21, 20, 19, 18.

"(2) _The enlarged Codex B (20 leaves):_

Front: 1, 2, 3, 4, 5, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60.

Back: 0, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 41, 43, 43, 44, 45.

"Or, reversing, as in the preceding case, the numbers on the back:

1, 2, 3, 4, 5 | 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60. 45, 44, 43, 42, 41 | 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 0."

One of the most difficult things to account for in regard to this codex is the immense number of numeral characters it contains, many of which appear to have no reference to day or other time symbols.

Although it is not claimed that the key which will fully unlock this mystery has been found, it is believed that the discoveries made will throw considerable light on this difficult subject and limit the field of investigation relating to the signification of the Maya codices.

Before proceeding with the discussion of the subject proposed, it will not be amiss to state, for the benefit of those readers not familiar with these ancient American manuscripts, that the Maya method of designating numbers was by means of dots and lines, thus: . (one dot) signifying one; .. (two dots) two, and so on up to four; five was indicated by a single short straight line, thus, ----; ten, by two similar lines, [Illustration: Two horizontal lines, stacked]; and fifteen, by three such lines: [Illustration: Three horizontal lines, stacked]. According to this system, a straight line and a dot, thus, [Illustration: Dot above horizontal line], would denote 6; two straight lines and two dots, [Illustration: Two dots above two stacked horizontal lines], 12; and three straight lines and four dots, [Illustration: Four dots in a line above 3 stacked horizontal lines], 19. But these symbols do not appear to have been used for any greater number than nineteen. They are found of two colors in all the Maya codices, one class black, the other red, though the latter (except in a few instances, where the reason for the variation from the rule is not apparent) are never used to denote a greater number than thirteen, and refer chiefly to the numbers of the days of the Maya week and the numbers of the years of the "Indication" or "week of years." On the other hand, the black numerals appear to be used in all other cases where numbers not exceeding nineteen are introduced. As will appear in the course of this discussion, there are satisfactory reasons for believing that other symbols, quite different from these dots and lines, are used for certain other numbers, at least for 20 and for 0.

In order that the reader may understand what follows, it is necessary to explain the methods of counting the days, months, and years in the order in which they succeed one another. Much relating to this will be found in a previous work,[269-1] but a particular point needs further explanation.

According to the older and also the more recent authorities, the Maya years--there being 20 names for days and 365 days in a year--commenced alternately on the first, sixth, eleventh, and sixteenth of the series, that is to say, on the days Kan, Muluc, Ix, and Cauac, following one another in the order here given; hence they are spoken of as Kan years, Muluc years, Ix years, and Cauac years.

Writing out in the form of an ordinary counting house calendar the 365 days of the year, commencing with 1 Kan and numbering them according to the Maya custom (that is, up to thirteen to form their week and then commencing again with one) they would be as shown in Table I.

TABLE I.--_Names and numbers of the months and days of the Maya system._

_______________________________________________________________________ | | | | | | | | | | | | | | | | | | | |N t| | | | | | | | | | | | | | | | | | | | |u h| | | | | | | | |Y | | | | | | |K | | | | |m e| | | | | | | | |a | | | | | | |a | | |K |C |b | | | | | |T |T | |x | |C | | | | |n |M | |a |u |e d| | |P | |Z |z |z |X |k |M |h |Y |Z |C |M |k |u |P |y |m |r a| | |o |U |i |o |e |u |i |o |e |a |a |e |a |i |a |a |e |h |s y| | |p |o |p |z |c |l |n |l |n |x |c |h |c |n |n |x |b |u | s| | |--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+o | | | 1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14|15|16|17|18|f | |-------------+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+---| |_Names of the| | | | | | | | | | | | | | | | | | | | | days._ | | | | | | | | | | | | | | | | | | | | |Kan | 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3| 1| |Chicchan | 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4| 2| |Cimi | 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5| 3| |Manik | 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6| 4| |Lamat | 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 5| |Muluc | 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 6| |Oc | 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 7| |Chuen | 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 8| |Eb | 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 9| |Been |10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 10| |Ix |11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 11| |Men |12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 12| |Cib |13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 13| |Caban | 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3| 14| |Ezanab | 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4| 15| |Cauac | 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5| 16| |Ahau | 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6| 17| |Ymix | 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 18| |Ik | 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 19| |Akbal | 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 20| |----------------------------------------------------------------+--+---| | _Intercalated days._ | | | |Kan |10| | |Chicchan |11| | |Cimi |12| | |Manik |13| | |Lamat | 1| | -----------------------------------------------------------------------

Each of these eighteen columns forms one month, and the whole taken together, with the 5 days added at the end of the eighteenth month, form one continuous series, the second column following the first as though placed at the end of it, the third following the second, and so on to the end of the eighteenth. Whether or not it was the ancient custom to include the 5 added days in the year, as asserted by the old Spanish writers, is somewhat doubtful, at least in studying the Dresden Codex, we shall find but few occasions, if any, to use them, for there are few if any positive indications in this codex that they were added.

As stated, each column of the table forms a month, though the numbering is carried to thirteen only; but at present the chief object in view in presenting it is to use it in explaining the method of counting the days and the intervals of time. The table is in truth a continuous series, and it is to be understood as though the 365 days were written in one column, thus:

1. Kan. 2. Chicchan. 3. Cimi. 4. Manik. 5. Lamat. 6. Muluc. 7. Oc. 8. Chuen. 9. Eb. 10. Been. 11. Ix. 12. Men. 13. Cib. 1. Caban. 2. Ezanab, &c.,

the 20 days being repeated over and over in the order in which they stand in the table. This order is never changed; we may commence at whatever point in the series occasion may require, but the order here given must always be maintained, just as in our calendar the order of our days is always Sunday, Monday, Tuesday, &c. In other words, Chicchan must always follow Kan, Cimi must always follow Chicchan, &c.

The method of counting intervals in the Maya calendar is very simple, if these explanations are borne in mind, and may be illustrated thus: Counting 14 days from 1 Kan--the first day of the year given in Table I--brings us to 2 Ezanab (the day we count from being excluded); 12 days more bring us to 1 Oc, in the second column of our table; 17 days more to 5 Manik, in the third column; and 17 days more, to 9 Kan, in the fourth column.

The number of the day required is readily ascertained by adding together the number of the day counted from and the number of days to be counted, casting out the thirteens when the sum exceeds this number (excepting where the remainder is thirteen); thus: 1 + 14 - 13 = 2, the number of the day Ezanab given above. So 1 + 14 + 12 - 13 - 13 = 1, the number of the day Oc, second column, Table I; and 1 + 14 + 12 + 17 + 17 - 13 - 13 - 13 - 13 = 9, the number of the day Kan, fourth column. The reason for this is so apparent that it is unnecessary to state it.

Suppose the day counted from is 11 Muluc of the eleventh month, and the number of days to be counted (or the interval) is 19; by adding together the numbers and casting out the thirteens the following result is obtained: 11 + 19 - 13 - 13 = 4. Counting forward on the table 19 days from 11 Muluc (the sixth number in the eleventh figure column), we reach 4 Lamat (the fourth day of the twelfth month). When the sum of the numbers is a multiple of 13 the number obtained is 13, as there can be no blanks, that is to say, no day without a number.

As the plates of the codices are usually divided into two or three compartments by transverse lines, it is necessary to adopt some method of referring to these in order to avoid the constant repetition of "upper," "middle," and "lower" division. On the plan proposed by Dr. Förstemann, in his late work on the Dresden Codex (Erläuterungen zur Mayahandschrift der Königlichen öffentlichen Bibliothek zu Dresden), these divisions are designated by the letters _a_, _b_, and _c_; this plan will be adopted in this paper. The letter _a_ joined to the number of a plate, therefore, will signify that the division referred to is the upper one, as Plate 12_a_; the letter _b_ signifies the middle one where there are three divisions or the lower one where there are but two; and the letter _c_ signifies the lowest or bottom division where there are three.

Where reference is made to the fac simile of the Dresden Codex, Kingsborough's colored edition is always to be understood, except where another is specially mentioned.

Running through Plates 36_c_ and 37_c_ is a continuous line of day symbols and red and black numeral characters as follows, the numbers and names below the characters being explanatory and of course not on the original:

[Illustration: FIG. 359. Lines of day and numeral symbols. Pl. 36 10 XI Men | 15 XII Oc | 9 IX Cauac

Pl. 37 11 VII Oc | 20 I Oc | 10 XI Ahau]

As colors are not used in these figures the red numerals are indi cated[TN-2] by hollow or outline dots and lines and the black numerals by solid lines and dots.[272-1]

In order further to assist those unacquainted with the symbols the same line is here given in another form, in which the names of the days are substituted for the symbols, Roman numerals for the red numbers, and Arabic for the black: 10, XI Men; 15, XIII Oc; 9, IX Cauac; 11, VII Oc; S, I Oc; 10, XI Ahau.

The S is introduced to represent a numeral symbol different from the lines and dots and will be explained when reached in the course of the illustration.

Starting from 11 Men, found in the twelfth figure column of Table I, and counting forward fifteen days, we come to 13 Oc of the thirteenth figure column, the second day of the above quoted line. Counting nine days from 13 Oc[273-1] brings us to 9 Cauac, the third day of the line; eleven days more, to 7 Oc, the fourth day of the line. Following this day in the line, instead of a black numeral of the usual form, is this symbol: [Illustration: Hieroglyph] represented by S in the second form, where the names and numbers are substituted for the symbols. Taking for granted, from the position it occupies in the line, that it is a numeral character, it must represent 20, as the day which follows is 1 Oc, and counting twenty days from 7 Oc brings us to 1 Oc. Counting ten days more we reach 11 Ahau, the last day of the line given above.

In this example the black numerals appear to have been used simply as counters, or as numbers indicating intervals; for example, 15 is the interval between 11 Men and 13 Oc.[273-2]

This furnishes a clew which, if followed up, may lead to important results. That it explains the signification of one symbol undetermined until this relation of the numerals to one another was discovered, is now admitted. In the work of Dr. Förstemann before alluded to the discovery of the symbol for 20 is announced. Although I was not aware of the signification of this symbol until after my second paper, "Notes on certain Maya and Mexican manuscripts," was written, I had made this discovery as early as 1884.[273-3]

As there will be occasion to refer to the days of the four different series of years (the Cauac, Kan, Muluc, and Ix years), a combined calendar, similar to an ordinary counting house calendar, is introduced here. For the Cauac years the left or Cauac column is to be used; for the Kan years, the Kan column, and so on.

TABLE II.--_Names and numbers of the four series of years of the Maya system._

__________________________________________________________________________________ | | | | {Numbers Cauac | Kan | Muluc | Ix | 1 2 3 4 5 6 7 8 9 10 11 12 13{of the column. |column. |column. |column. |14 15 16 17 18 {months. --------+--------+--------+--------+--+--+--+--+--+--+--+--+--+--+--+--+--+------- | | | | | | | | | | | | | | | | |Days of | | | | | | | | | | | | | | | | |month. Cauac |Kan |Muluc |Ix | 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1 Ahau |Chicchan|Oc |Men | 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2 Ymix |Cimi |Chuen |Cib | 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3 Ik |Manik |Eb |Caban | 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4 Akbal |Lamat |Been |Ezanab | 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5 Kan |Muluc |Ix |Cauac | 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6 Chicchan|Oc |Men |Ahau | 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7 Cimi |Chuen |Cib |Ymix | 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8 Manik |Eb |Caban |Ik | 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9 Lamat |Been |Ezanab |Akbal |10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3| 10 Muluc |Ix |Cauac |Kan |11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4| 11 Oc |Men |Ahau |Chicchan|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5| 12 Chuen |Cib |Ymix |Cimi |13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6| 13 Eb |Caban |Ik |Manik | 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 14 Been |Ezanab |Akbal |Lamat | 2| 9| 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 15 Ix |Cauac |Kan |Muluc | 3|10| 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 16 Men |Ahau |Chicchan|Oc | 4|11| 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 17 Cib |Ymix |Cimi |Chuen | 5|12| 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 18 Caban |Ik |Manik |Eb | 6|13| 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 19 Ezanab |Akbal |Lamat |Been | 7| 1| 8| 2| 9| 3|10| 4|11| 5|12| 6|13| 20 ----------------------------------------------------------------------------------

As this table has been explained in my previous papers it is only necessary to add here that the thirteen figure columns form a single series; therefore, when we reach the bottom of the thirteenth column we go back to the top of the first. The day reached will be the one directly opposite (that is, in the same horizontal line) in the day column for the given year.

For example, taking the fifth column of numbers (the one having 3 for the top figure) and counting down nine days from the top number we reach the number 12. This will be 12 Lamat if a Cauac year, 12 Been if a Kan year, 12 Ezanab if a Muluc year, and 12 Akbal if an Ix year. Therefore it is necessary in counting to refer always to the year (year column) with which the count begins. So long as the particular year referred to is unknown (as is Usually the case, the day series being apparently of general rather than of special application) it is immaterial which day column is selected, as the result will be the same with any. This will be apparent if we bear in mind that, when 260 days with their numbers attached have been written down in proper order as a series, we have therein all the possible combinations of days and numbers. This, it is true, does not give us all the months and years (to include these it is necessary to write out fifty-two entire years), but the same series of numerals will be applicable to each of the four year series (Kan, Muluc, Ix, and Cauac years). As any one of the thirteen figure columns of the table may be taken as the commencement of a year and any of the four day columns may be used, it is apparent that we have all the possible combinations (4 × 13 = 52).

I say above that "it is necessary in counting to refer always to the year (year column) which the count begins." This I admit does not agree with the generally received idea of the Maya calendar, upon which Table II is constructed, as, according to this theory (which I have accepted in my previous papers), after passing through a year of one series (corresponding with one of the day columns of the table), we should enter upon a year of the next series; for example, when the year 1 Kan is completed we should enter upon the year 2 Muluc.

Although this calendar system seems to have been in vogue at the time of the conquest and is indicated in one or two of the codices, and possibly in the one now under consideration, the chronological series of the latter, as will hereafter appear, do not seem to be based upon it or to agree with it.

These explanations, with the further statement that the lines in the codex are to be read from left to right and the columns from the top downward, except where variations from this rule are noted, will enable the reader to follow the discussion. Another reason for using a table with only thirteen columns (though it would be difficult to devise a combined calendar of any other form) is that the 260 days they contain form one complete cycle, which, as will appear in the course of this discussion, was one of the chief periods in Maya time computations.

Examining Plates 33 to 39 of the codex the reader will observe that the line already alluded to extends continuously through division _c_, commencing with the two characters over the figure (picture) in the lower right hand corner of Plate 33.

The first of these characters as given in Kingsborough's work is the symbol of the day Ezanab, with the red numeral 13 to the left of it and the black numeral 9 over it; but referring to Förstemann's photolithographic copy of the codex it is found to be the symbol of Ahau.

The entire line, with this correction (that is to say, as given by Förstemann), is represented in Fig. 360. In order to assist the reader, the names of the days and numbers of the symbols have been added immediately below the characters.

As the year to which the line relates is unknown, we select the Muluc series, designated "Muluc column" in Table II, and commence with 13 Ahau, the twelfth number of the third figure column. Counting 9 days from this brings us to 9 Muluc, the top number of the fourth figure column and also the second day of the line above given. (the symbol is a face in Kingsborough's copy, but is plainly the Muluc sign in Förstemann's photograph). Eleven days more bring us to 7 Ahau, the third day of the above line; 20 more to 1 Ahau, the fourth day of the line (the 20 here is the symbol represented by S); 10 more to 11 Oc, the fifth day of the line; 15 more to 13 Chicchan, the sixth day of the line; 9 more to 9 Ix, the seventh day of the line; 11 more to 7 Chicchan, the eighth day of the line; line; 20 (S) more to 1 Chicchan, the ninth day of the line; 10 more to 11 Men, the tenth day of the line, and so on to the end.

[Illustration: FIG. 360. Line of day and numeral characters. Pl. 33 XIII Ahau IX Muluc Pl. 34 11 VII Ahau | 20 I Ahau | 10 XI Oc | 15 XIII Chicchan Pl. 35 9 IX Ix | 11 VII Chicchan | 20 I Chicchan Pl. 36 10 XI Men | 15 XIII Oc | 9 IX Cauac Pl. 37 11 VII Oc | 20 I Oc | 10 XI Ahau Pl. 38 15 XIII Men | 9 IX Kan | 11 VII Men Pl. 39 20 I Men(?) | 10 XI Chicchan | 15 XIII Ahau]

That the order of the series may be clearly seen the numbers are given here as they stand in the line, omitting the days: XIII; 9, IX; 11, VII; 20, I; 10, XI; 15, XIII; 9, IX; 11, VII; 20, I; 10, XI; 15, XIII; 9, IX; 11, VII; 20, I; 10, XI; 15, XIII; 9, IX; 11, VII; 20, I; 10, XI; 15, XIII.

By adding together a black numeral and the preceding red one and casting out thirteen (or thirteens, as the case maybe), when the sum exceeds this number, we obtain the following red one, thus: XIII + 9 - 13 = IX; IX + 11 - 13 = VII; VII + 20 - 13 - 13 = I; I + 10 = XI, and so on through the entire series. Attention is also called to the fact that the sum of the black (Arabic) numbers 9, 11, 20, 10, 15, 9, 11, 20, 10, 15, 9, 11, 20, 10, 15, 9, 11, 20, 10, 15, is 260, a multiple of 13.

If this relation of days and numerals holds good as a general thing throughout the codex, it is apparent that where the break is not too extensive it will enable the student to restore the missing and defective numerals and day symbols, to detect the errors of both copyists and original artists, and to determine the proper relation of the plates to one another. By it he learns, as before stated, that the symbol (see page 273) denotes 20, and if phonetic probably stands for the Maya word _Kal_.

Comparing Plates 42 and 43 with Plates 1 and 2, the resemblance is found to be so strong as to lead to the belief that they belong together. It is apparent from the figures, numerals, and characters[277-1] in the middle division (_b_) of Plates 1 and 2 that they belong together, as they now stand in Kingsborough's work and Förstemann's copy; that Plates 42 and 43 are properly placed in regard to each other is also apparent from the figures and numerals in divisions _a_ and _b_.

Taking for granted that the lines are to be read from left to right and the plates to follow each other in the same order, our next step is to ascertain on which side of the pair (Plates 42 and 43) Plates 1 and 2 should be placed.

The series of days and of numbers in Plate 43_b_ and Plate 1_b_, which evidently belong together, can only be brought into proper relation by placing the latter to the right of the former. Yet, strange as it may appear, the days and numerals in this division are to be read from right to left, while all the other numeral series of these four plates are to be read as usual, from left to right. This change in the order of the pages also brings together the similar figures in the upper division of these plates. That Plate 42 properly follows Plate 41 is apparent from the line of alternate red and black numerals in division _b_. As shown in a previous work[278-1] and as will appear hereafter, these horizontal lines of alternate red and black numerals without day symbols interspersed are usually, if not always, connected at the left with a column of days over which there is a red numeral, as in the Codex Troano. Running back along the line of numerals in the middle division of Plates 42 and 41, the day column with which it is connected is found at the left margin of Plate 38. Unfortunately the red numeral over this column is obliterated, but can easily be restored. Starting with the first black numeral to the right of this, the entire line, which ends in the second column of the middle division of Plate 43 (representing the black numerals by Arabic numbers and the red by Roman numbers), is as follows: 16, IX; 8, IV; 11, II; 10, XII; 1, XIII; 12, XII; 6, VI(?); 12, IV; 11, II; 11, XIII; 6, VI; 12, V; 7, XII; 6, V; S + 1, XIII; 6, VI.

The number over the day column, Plate 38, must have been VI, as VI + 16 - 13 = 9, a conclusion which is sustained by Förstemann's copy, which shows here very plainly the red character for VI.

By adding the black (Arabic) numeral to the preceding red (Roman) one and casting out the thirteens, as heretofore explained, we obtain the following red (Roman) numerals, thus: VI + 16 - 13 = IX; IX + 8 - 13 = IV; IV + 11 - 13 = II; II + 10 = XII; XII + 1 = XIII; XIII + 12 - 13 = XII; XII + 6 - 13 = V.

Here the result differs from what is found at this point in the line, as we obtain V instead of VI. In this case the mistake, if one has been made, cannot be attributed to Lord Kingsborough's copyist; the Maya artist must have made a mistake or there must be an error in the theory here advanced. But let us continue according to our own figures: V + 12 - 13 = IV; IV + 11 - 13 = II; II + 11 = XIII; XIII + 6 - 13 = VI; VI + 12 - 13 = V; V + 7 = 12; XII + 6 - 13 = V; V + 20 + 1 - 13 = XIII; XIII + 6 - 13 = VI.

There is no doubt, therefore, that the line forms one continuous series, and if so it links together pages 38 and 43 as they are now numbered. It follows, then, that if Plates 1 and 2 and Plates 42 and 43 belong together, the former pair must be placed to the right of 43. This is conceded by Dr. Förstemann,[278-2] as he says that, Dr. Karl Schultz-Sellack having pointed out the error in his paging, he changed pages 1 and 2 to 44 and 45 and pages 44 and 45 to 1 and 2; that is to say, the two leaves containing these pages were loosened from the strip and reversed, so that page 1 would be 44 and page 2 would be 45.

Having brought together these plates so that 1 and 2 stand to the right of 43, attention is called to the lines of day symbols running through division _c_. Substituting names and numbers as heretofore, they are as follows:

Plate 42: IV Ahau; XII Lamat; VII Cib; II Kan; X Eb; V Ahau; XIII Lamat. 17 8 8 8 8 8 8

Plate 43: IV Chicchan; XII Been; VII Ymix; II Muluc; X Caban; V Chicchan; XIII Been. 17 8 8 8 8 8 8

Plate 1: IV Oc; XII Ezanab; VII Cimi; II Ix; X Ik; V Oc; (?) Ezanab. 17 8 8 8 8 8 8

Plate 2: IV Men; XIII Akbal; VII Chuen; II Cauac; X Manik; V Men; XIII Akbal. 17 8 8 8 8 8 8

The chief objects in view at present in selecting this series are, as before indicated, to prove the relation of the plates to one another and to determine the use of the black numerals which stand under the day symbols. These numerals consist of but two different numbers, the first on each page being 17, the rest 8's.

As the particular year or years to which the series refers is unknown we turn to our calendar--Table II--and select the Kan column, as we find that 4 Ahau, the first day of the series, is the seventeenth day of the year 1 Kan. This corresponds with the first black numeral. Counting 8 days from this we reach 12 Lamat, the second day of our series; 8 more bring us to 7 Cib, the third day of the series; 8 more to 2 Kan; 8 more to 10 Eb; 8 more to 5 Ahau; 8 more to 13 Lamat, and 17 more to 4 Chicchan. The red numeral at this point in some of the colored copies of Kingsborough's work is III, but a close inspection shows the missing dot which has not been colored. IV Chicchan is therefore correct.

Continuing our count, 8 days more bring us to 12 Been: 8 more to 7 Ymix; 8 more to 2 Muluc; 8 more to 10 Caban; 8 more to 5 Chicchan; 8 more to 13 Been; 17 more to 4 Oc; 8 more to 12 Ezanab; 8 more to 7 Cimi; 8 more to 2 Ix; 8 more to 10 Ik; 8 more to 5 Oc, and 8 more to 13 Ezanab. Here the red numeral is wanting, but a comparison of the numbers on the different plates and the order of the series make it evident that it should be XIII.

Continuing our count, 17 more bring us to 4 Men (here a dot is missing in Kingsborough's copy, but is present in the photograph); 8 more to 12 Akbal. Here there is one dot too many, which we may attribute to a mistake of the original artist. Assuming XII to be correct, 8 more bring us to 7 Chuen; 8 more to 2 Cauac; 8 more to 10 Manik; 8 more to 5 Men; 8 more to 13 Akbal, and to the end of our table; thus, if we include the first seventeen days, completing the series of thirteen months or 260 days.

These illustrations will probably satisfy any one that the black numerals in these lines denote the intervals between the days indicated by the symbols and that the series so far examined are to be read from left to right.

Although the succession of days and numbers in the lines of the last example would seem to furnish conclusive evidence that the whole is one continuous series, yet the peculiar combinations of numbers used by the Maya priests render these series very deceptive. There can be no doubt that the black numbers--8's--are used to indicate the intervals between the days specified; but there is another possible way of explaining the 17 with which the lines on the different plates begin.

Here are four plates, evidently closely related to one another; the lines of days and numbers in the lowest division of each are precisely alike, except as to the days indicated; in the left hand column of characters of each is one of the cardinal point symbols. It is possible, therefore, that these four plates relate to the four different years or series of years; that is to say, one to the Kan years, one to the Muluc years, and so on. This view is somewhat strengthened by the fact that 4 Ahau, first of the line on Plate 42, is the seventeenth day of the first month of the year 1 Kan; 4 Chicchan, first of the line on plate 43, the seventeenth day of the first month of the year 1 Muluc; 4 Oc, the seventeenth day of 1 Ix, and 4 Men the seventeenth day of 1 Cauac. The four figures in the middle division of Plates 1 and 2 seem also to favor this idea, not so much by the peculiar animals represented (of which we have no explanation to give) as by the double symbols from which they are suspended, which I am quite confident denote the union of years or the time at which two years meet--the close of one and the commencement of another--although fully aware that Dr. Förstemann has interpreted them as symbols of the heavenly bodies.[280-1]

In the text above these figures are seen two characters or symbols of this type, which in all probability, as will hereafter appear, denote or symbolize the "tying of the years." We may also add that the five days of each plate or group are the five assigned, as I have explained in "Notes on certain Maya and Mexican manuscripts," to the cardinal points. For example, those on Plate 42 are Ahau, Eb, Kan, Cib, Lamat.[280-2] Still it must be admitted, on the other hand, that as the four lines form precisely one complete cycle of 13 months or 260 days there is a very strong inference that they together form one continuous series and that the arrangement into four parts or divisions has reference to the four seasons or four cardinal points. The final decision on this point therefore still remains in doubt.

As it has been shown that Plates 33 to 39 and Plates 38 to 43 are properly placed as they stand in Kingsborough's copy and also in Förstemann's and that Plates 1 and 2 follow Plate 43, we have proof that the following plates succeed one another to the right, as here given: 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 1, 2.

A slight inspection is sufficient to show that Plates 29 to 33 follow one another in the same order, a conclusion which is easily verified by testing the lines of numerals in the manner explained. It is apparent, therefore, that the following plates form one unbroken series, running from left to right: 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 1, 2; a conclusion which Dr. Förstemann, who has had the opportunity of studying the original, has now reached.

Having ascertained the object and use of at least one class of black numerals and the relation they bear to the days and day numbers, it may be well to test further the discovery by other examples, in order to see how far it holds good and what new facts it may bring out. In doing this it will be necessary to repeat in part what has already been shown by Dr. Förstemann in his late work; but as these discoveries were made independently and before this work came to hand, and as our conclusions differ in some respects from those reached by him, the plan and scope of this paper would be incomplete without these illustrations.

Commencing with the day column in the middle of Plate 35_b_ and extending through Plates 36_b_ and 37_b_ to the right margin of the latter, is a line of alternate red and black numerals, which may be taken as an example of the most common series found in the Dresden and other codices. It is selected because it is short, complete, and has no doubtful symbols or numerals in it.

Using names and numbers in place of the symbols, it is as follows:

I. Caban, 11, XII; 6, V; 9, I; 4, V; 7, XII; 9, VIII; 6, I. Muluc. Ymix. Been. Chicchan.

In this case the red numeral over the day column is I. It is to be observed that the last number of the series is also I, a fact which it will be well to keep in mind, as it has an important bearing on what is now to be presented. But it is proper to show first that this series is continuous and is connected with the day column.

Adding the I over the column to the 11, the first black numeral; gives XII, the red numeral following the 11. That this holds good in all cases of this kind will become apparent from the examples which will be given in the course of this discussion. Adding together the remaining pairs, as follows: XII + 6 - 13 = V; V + 9 - 13 = 1; 1 + 4 = V; V + 7 = XII; XII + 9 - 13 = VIII; VIII + 6 - 13 = I, we obtain proof that the line is one unbroken series. It is apparent that if the black numerals are simply counters used to indicate intervals, as has been suggested, then, by adding them and the red numerals over the column together and casting out the thirteens, we should obtain the last red number of the series. In this case the sum of the numbers 1, 11, 6, 9, 4, 7, 9, 6, is 53; casting out the thirteens the remainder is 1, the last of the series. If we take the sum of the black numbers, which in this case is 52, and count the number of days on our calendar (Table II) from 1 Caban, the fourteenth day of the first month of the year 1 Kan, we shall find that it brings us to 1 Muluc, the sixth day of the fourth month; 52 days more to 1 Ymix; 52 more to 1 Been, and 52 more to 1 Chicchan, thus completing the day column in the example given. This proves, in this case at least, that the red numeral over the day column applies to all the days of the column and that the whole numeral series--that is to say, the sum of the counters--represents the interval between the successive days of the column. The total number of days from 1 Caban, first of the column, to 1 Chicchan, the last, is 208. Adding 52 more gives 260 and brings us back to 1 Caban, our starting point.

It will be observed that the sum of the black numbers--which denotes the interval between the days of the column--is 52, which is a multiple of 13, the number of days in a Maya week. It follows, therefore, that so far as this rule holds good the last red numeral of the series must be the same as that over the day column. In a former work[282-1] I explained the method of ascertaining the relations of the days of a column to one another by means of the intervals without reference to the numbers attached to them, a subject to which Charency had previously called attention;[282-2] by the explanation now given we ascertain the true intervals between the days _as numbered_. The two modes therefore form checks to each other and will aid very materially in restoring obliterated and doubtful days.

There is another point in regard to these series which may as well be illustrated by means of the example given as any other. What is the signification of the red numerals of the series? They are unnecessary if the only object in view was to indicate the intervals between the days of the column. Nor will the supposition that the Mayas had not discovered a means of representing higher numbers than 20 suffice, as the introduction of 13 would have lessened the labor and shortened the calculation. But one answer to this inquiry appears possible, viz, that these numbers are intended to denote certain intermediate days to which importance was for some reason attached. These intermediate days can readily be determined from the data given, and in the present example are as follows:

(1) Between 1 Caban and 1 Muluc they are 12 Lamat, 5 Ix, 1 Akbal, 5 Manik, 12 Ix, and 8 Akbal.

(2) Between 1 Muluc and 1 Ymix they are 12 Ahau, 5 Cimi, 1 Men, 5 Cauac, 12 Cimi, and 8 Men.

(3) Between 1 Ymix and 1 Been they are 12 Eb, 5 Ezanab, 1 Manik, 5 Chuen, 12 Ezanab, and 8 Manik.

(4) Between 1 Been and 1 Chicchan they are 12 Kan, 5 Oc, 1 Cauac, 5 Akbal, 12 Oc, and 8 Cauac.

These, as will be readily perceived, are found by counting on the calendar from 1 Caban, 1 Muluc, &c., as heretofore explained.[283-1]

Our interpretation of the series of this particular class is now complete, except as to their application or the object in view in forming them and the determination of the particular years to which they apply. Possibly they may be of general application, so far as consistent with the calendar system. The conclusion on this point depends largely upon the conclusion as regards the system, as it is evident their location in time--if the year of 365 days and the four series of years formed the basis of the system--would not correspond with their position in a system based upon the year of 360 days, in which the four year series does not play any necessary part.

Dr. Förstemann calls attention to the fact that the pairs of numerals representing the intermediate days are usually placed in separate compartments, each containing a figure or a picture generally symbolic or of a priest dressed to indicate some particular god. It is therefore very probable that these intermediate days are to be devoted to ceremonies relating to the divinities or subjects indicated by these figures.

In order to confirm the theory we are now discussing and at the same time show some of the different varieties of the series of the type now under consideration, the following additional examples are given.

In the middle division of Plate 5 is a day column and a numeral series, as follows:

I. Manik } Cauac } 16, IV; 9, XIII; S + 5, XII; 2, I. Chuen } Akbal Men

This series terminates with I, as it should according to the theory. The sum of the black numerals--16, 9, 20, 5, 2--is 52, a multiple of thirteen, and the interval between the successive days, reading downwards, is 52, agreeing in these particulars with the theory. It will also be observed that the symbol represented by S answers to the number 20.

In the lowest division of the same plate is another similar series, as follows:

XII Ezanab } Akbal } 20 + 9, II; 11, XIII; 18, V; 7, XII. Lamat } Been Ezanab

This terminates with XII, the number over the column; the sum of the black numbers is 65, a multiple of thirteen and precisely the interval between the successive days of the column, taking the week numbers into consideration, which is always to be understood in speaking of these intervals unless the contrary is expressly stated.

[Illustration: FIG. 361.]

In the middle division of Plate 8 is a short series connected with a day column containing the following days, reading downwards, as usual: Manik, Cauac, Chuen, Akbal, Men. The symbol for Akbal (Fig. 361), is a very unusual one, reminding us strongly of a skull, which may possibly have given origin to the symbol. The numerals of the series are as follows: 20 + 6, VIII; 20 + 6, VIII; the number over the column, VIII; and the interval between the days, 52.

In Plate 15, division _c_, is the following series, which differs from those given in having two day columns instead of one:

III III Lamat Ix Ahau Cimi } Eb Ezanab } 12, II; 14, III. Kan Oc Cib Ik

The final number is the same as that over the columns; the sum of the black numbers is 26, which is a multiple of 13; but in this case in counting the intervals the days are to be taken alternately from the two columns.

Commencing with 3 Lamat on our calendar and counting 26 days brings us to 3 Ix; 26 more to 3 Ahau; 26 more to 3 Cimi, and so on to the end.

In the lower division of Plate 9 is a series arranged as follows:

III III VI VIII Cauac Been 3 2 {XI II Chuen Chicchan { 3 4 {VI VII Akbal Caban { 4 1 Men Muluc I III Manik Ymix 7 2

The sum of the black numerals is 26 and the final red number is III, the same as that over the columns. The interval between the days, taken alternately from the two columns, as in the preceding example, is 26. The numbers are also to be taken alternately from the two number columns.

It is apparent that these examples sustain the theory advanced. This will also be found true in regard to all the series of this type in this and the other codices where the copy is correct. Brasseur's copy of the Manuscript Troano is so full of mistakes that no satisfactory examination of this codex can be made until a photographic copy is obtained; nevertheless a few examples are given as proof of the above statement.

In the third division of Plate XI* is the following series:

IV Ahau } Eb } 17, VIII; 13, VIII; 10 V; 12, IV. Kan } Cib Lamat

As will be readily seen, after the explanations given, this agrees with the theory advanced.

The last red number is the same as that over the day column, the sum of the black numbers is 52, and the interval between the days 52.

Commencing in the right margin of the lowest division of Plate XXIII* and running through Plates XXII* and XXI*, is the series here represented:

VII VII Cib Cimi } Ik Eb } 7, I; 7, VIII; 7, II; 5, VII. Lamat Ezanab } Ix Kan Ahau Oc

An examination of this shows it to be of the type of the double column series of the other codex, except that here the days of one column are to be taken in the order in which they stand before proceeding to the other column. The sum of the black numbers is 26 and the interval between 7 Cib and 7 Ik 26 days. The interval between 7 Ik and 7 Lamat, 7 Lamat and 7 Ix, and between 7 Ix and 7 Ahau is, in each case, 26 days. The interval between 7 Ahau, last day of the left hand column, and 7 Cimi, the first day of the right hand column, is also 26 days.

The order in which the days of these double column series of this manuscript follow one another is not uniform, as in some cases (see Plate XXV*, division _a_) they are to be taken alternately from the two columns, as in the examples heretofore given from the Dresden Codex.

In the middle division (Plate XXXIII*, same codex) is a series of the following form, but with the days so nearly obliterated that restoration is necessary:

{ VI I { 5 8 I { VI I Ymix (?) { 5 8 Cimi (?) { VI I Chuen { 5 8 Cib (?) { VI I (?) { 5 8 { VI I { 5 8

The symbol of the first day has only the upper circle of dots to indicate that it is Ymix, that of the second day is almost obliterated, the third is clearly Chuen, the lower half of the fourth is obliterated, and the interior of the fifth is a blank.

Fortunately there are sufficient data by which to make the restoration. Chuen, we observe, is the middle of the column; that is, two days are above it and two days below it; the sum of the black numerals is 65; hence the interval between the days, considering the week numbers as attached, is 65, and the simple interval in the month series, without regard to the week numbers, is 5. Counting back on our calendar (Table II) 65 days from 1 Chuen we reach 1 Cimi, and 65 more bring us to 1 Ymix. In like manner we find the fourth day to be 1 Cib and the fifth 1 Ymix. The numbers in the figure columns are to be taken alternately, thus: 5, VI; 8, I; 5, VI; 8, I, &c.

These examples are sufficient to show that the series of the Manuscript Troano are arranged upon the same plan and based upon the same system as those of the Dresden Codex. The following examples from the Codex Cortesianus prove the same thing to be true in reference to the series found in it.

The first is taken from the lower division of Plates 10 and 11, Rosny's reproduction:

XIII Ahau } 11, XI; 5, III; 5, VIII; 5, XIII; 9, IX; 3, XII; 6, V; Chicchan } 1, VI; X, XIII. Oc Men

The S in the line of numerals represents the usual symbol for 20. The sum of the black numbers is 65, the interval between the days 65, and the last red numeral the same as that over the day column, thus agreeing in plan with those in the other codices.

The following double column series is found in the middle division of Plate 30:

XI XI Ahau Ymix } Eb Been } 20 + 6, XI; 20 + 6, XI. Kan Caban } Cib Chicchan Lamat Manik

The number 20 is denoted by the usual symbol. The sum of the black numbers is 52 and the interval between the days in each column 52, but in this case there does not appear to be any connection between the columns, there being, in fact, two distinct series.

In the upper division of the same plate is this series:

XI Ezanab { VI XI { 8 5 Oc { VI XI { 8 5 Ik { VI XI { 8 5 Ix { VI XI { 8 5 Cimi

The order in which these numerals are to be read is as follows: 8, VI; 5, XI; 8, VI; 5, XI, &c., which gives, as the final red number of the series, XI, the same as that over the column. The sum of the black numbers is 52 and the interval between the days 52.

Taking for granted that the correctness of the theory advanced is conceded, some attempts at its further application, especially its use in making restorations and corrections in defective series and in settling doubtful questions relating thereto, will now be presented.

In the upper division of Plate 32, Dresden Codex, are the four day columns and lines of numerals over them here represented:

1 4 13 9 4 15 13 2 11 XIII XIII XIII XIII Manik Cib Chicchan Ix Chuen Ahau Muluc Ezanab Men Kan Been Ik Cauac Lamat Caban Cimi Akbal Eb Ymix Oc

Connected with these numbers is a line of alternate black and red numbers running along over the figures of Plates 32 to 39, division _a_. There are several breaks and some partially obliterated characters in it which must be restored in order to use it. It has been selected partly on this account, that the method of filling such breaks and making such restorations may be seen.

Representing the numerals and symbols as heretofore and substituting a cipher where the numbers are wanting or are too much obliterated to be determined by inspection, the series will be as follows: 11, XI; 8 + 20, 0; 12 (or 13), XIII; 6 + 20, XIII; 12, VII (?); 16 (?), V; 5, X; 1, XI; 20, V; 12, IV, 6, X; 0, V; 5, X; 7, IV; 12 (?), II; 5, VII; 8, II; 11, 0.

Commencing with the XIII over the day columns and counting as heretofore, we obtain the following result: XIII + 11 - 13 = XI; XI + 8 + 20 - 13 - 13 = XIII. The first blank should therefore be filled with XIII. Continuing, XIII + 13 - 13 = XIII; the black numeral in this case should be 13, although apparently 12 in the codex; XIII + 6 + 20 - 13 - 13 = XIII; XIII + 12 - 13 = XII. Here the result obtained differs from the red numeral in the codex, which is apparently one line and two dots, or VII; but, by carefully examining it or inspecting an uncolored copy, the two lines which have been covered in the colored copy by a single broad red line are readily detected. The next black numeral is partially obliterated, the remaining portion indicating 16, but it is apparent from the following red numeral that it should be 19. Making this correction we proceed with our count: XII + 19 - 13 - 13 = V; V + 5 = X; X + I = XI; XI + 20 - 13 - 13 = V; V + 12 - 13 = IV; IV + 6 = X. The next black numeral is obliterated, but is readily restored, as X + 8 - 13 = V; V + 5 = X; X + 7 - 13 = IV. The next step presents a difficulty which we are unable to explain satisfactorily. The black numeral to be counted here, which stands over the animal figure in the upper division of Plate 39, is 12, both in Kingsborough's copy and in Förstemann's photograph, and is clear and distinct in each, and the following red numeral is as distinctly II, whereas IV + 12 - 13 = III. Moreover it is evident from the remaining numbers in the line that this red numeral should be II. We may assume that the Maya artist has made a mistake and written 12 instead of 11, which is evidently the number to be used in the count; but this arbitrary correction should not be resorted to so long as any other explanation is possible. From the fact that immediately under these numbers there are certain symbols which appear to have some reference to the termination of one year or cycle and the commencement of another, it is possible that a supplemental, unnumbered, but not uncounted day has been added. The fact that this interval of twelve days includes the day Ymix lends some probability to this supposition. Using 11 instead of 12, we continue our count as follows: IV + 11 - 13 = II; II + 5 = VII; VII + 8 - 13 = II; II + 11 = XIII. Thirteen is, therefore, the last number of the series, which is wanting in the codex. The 8 and II next to the last pair of the series are not in line with the other numbers, but thrust into and near the bottom of the column of characters in the upper division of Plate 39. Adding together the black numbers as thus amended and restored, viz, 11, 8, 20, 13, 6, 20, 12, 19, 5, 1, 20, 12, 6, 8, 5, 7, 11, 5, 8, 11, the sum is found to be 208, which is a multiple of 13, and the final number of the series is 13. On the other hand, the sum of the series does not indicate the interval between the days of a column counting downwards, nor between two consecutive days or the corresponding days of two adjoining columns in any direction. The number of days from 13 Manik to 13 Chuen is 104, but counting 208 days from 13 Manik brings us to 13 Men, the third day of the first (left hand) column; 208 more to 13 Akbal, the fifth; 208 more to 13 Chuen, the second; and 208 more to 13 Cauac, the fourth, thus completing the column.

As these columns do not appear to form a continuous series it is possible they pertain to four different series of years, though the fact that each includes more than one year would seem to forbid this idea. It is more probable that they pertain to four different series, to each of which the line of numerals is to be considered as belonging.

The black numerals above the columns present a problem which I am unable to explain. The numbers stand in the original as follows:

1 4 13 9 4 15 13 2 11

If we suppose that the lowest line denotes days, the one next above, months, and the uppermost, in which there is but a single number, years, the series will appear to be ascending toward the left, with the difference 4 months and 11 days, as shown by addition, thus:

Y. M. D. 4 11 Numbers over the fourth column. 4 11 --------------- 9 2 Numbers over the third column. 4 11 --------------- 13 13 Numbers over the second column.

Doubling the difference and adding we obtain the numbers over the first column:

Y. M. D. 13 13 9 2 --------------- 1 4 15

What adds to the difficulty is the fact that if the columns are taken in reverse order the interval between the corresponding days is 4 months and 11 days; that is to say, counting from 13 Ix, first day of the fourth column, to 13 Chicchan, first day of the third column, we find the interval to be exactly 4 months and 11 days; and the same rule holds good throughout, so that reading across the upper line of days, from right to left, and following with the second line in the same way, ending with Akbal, the interval will be 4 months and 11 days between the consecutive days. Another significant fact is that by counting 4 months and 11 days from the first day of the year 1 Kan we reach 13 Ix; counting 9 months and 2 days from the same date brings us to 13 Chicchan; 13 months and 13 days, to 13 Cib; and 1 year and 4 days, to 13 Manik, which corresponds with the regular interval; it is therefore probable that there is an error in the numerals over the first or left hand column.

It is apparent from the illustrations given that in numeral series of the preceding type restorations can be made where not more than two numbers in succession are wanting. Even three can generally be restored if the numbers preceding and those following the break are distinct, but such restorations should be cautiously made.

In the middle division of Plate 9 is a short series where the number over the day column is wanting; moreover, there is uncertainty as to the number of days in the column and as to the signification of the red numerals, which are in pairs in Kingsborough's work instead of single as usual. Is it possible to explain these uncertainties and to reduce them to the usual simple form? Let us make the trial.

The days in the column are apparently the following: Ahau, Muluc, Ix, Cauac, Kan. The symbols, except that for Cauac, are too plain to admit of doubt, and there is no difficulty in reference to Cauac, the question of doubt being with regard to the Ahau, which is partially surrounded by other characters and may, apparently, be as correctly considered a part of the hieroglyphic inscription as of the day column.

Counting on the list of days in the calendar (Table II), as, for example, the Muluc column, we find the interval from Muluc to Ix is 5 days, from Ix to Cauac is 5 days, and from Cauac to Kan 5 days; but the interval from Ahau to Muluc is 9 days. From this fact we may reasonably infer that Ahau does not belong to the column. Moreover, the other 4 days are the four year bearers, and when they occur together the column usually consists of but 4 days, as, for example, in the lowest division of Plate 29 of this codex and Plate XXXII* of the Manuscript Troano. The numerals are 20; XIII, X; 20, XII, III; the number over the day column, as before stated, is wanting. The interval from 1 Muluc (or 2 or 3 Muluc) to Ix of the same number is 65 days. It is evident, therefore, that one of each pair of red numerals of the series given must be a counter and has been colored red by mistake. As the numbers in the last pair are III and XII, the number over the column must be 3 or 12. Suppose it is 12 and that XIII of the first pair is a counter, then XII + 20 + 13 - 13 - 13 - 13 = VI. As the number in the series is X this will not do. Supposing the X of the first pair of red numerals to be the counter, colored by mistake, the result is as follows: XII + 20 + 10 - 13 - 13 - 13 = III. This is also wrong, as the remainder should be XIII. Supposing the number over the column to be III and the XIII of the first pair and XII of the second to be the counters, the result agrees with the theory in every particular. Thus, III + 20 + 13 - 13 - 13 = X; X + 20 + 12 - 13 - 13 - 13 = III; and 20 + 13 + 20 + 12 = 65, the interval between 3 Muluc and 3 Ix. In Förstemann's copy the XIII and XII are black, thus verifying the conclusion here reached.

The series running through Plates 10_c_ and 11_c_ presents some difficulties which I have, so far, been unable to solve. The day columns and numerals are as follows:

I XIII Ymix Cimi } Been Ezanab } 1, I; 5, VI; 10, III; 13, III; 15, V; 9 (?), XIII. Chicchan Oc } Caban Ik Muluc Ix[290-1]

The numerals in this case are very distinct, especially in the photographic copy, and there can be no doubt as to the days. Here the last black number, 9, is wrong; it should be 8, a fact noticed by Förstemann.[290-2] Making this correction, the series is regular and consistent, so far as it relates to the right hand column, which has the red thirteen over it. But there is no series for the left hand column. Can it be that those who used the manuscript were expected to find the proper numbers by the line given? Possibly this is the reason the other series is not written out, as by adding one to each red number we obtain the proper result, which, if written out, would be as follows: 1, II; 5, VII; 10, IV; 13, IV; 15, VI; 3, I.

In Plate 30_c_ are the four day columns here given, with the numeral eleven over each:

XI XI XI XI Ahau Chicchan Oc Men Caban Ik Manik Eb Ix Cauac Kan Muluc Chuen Cib Ymix Cimi Lamat Been Ezanab Akbal.

Extending from the right of this group is a numeral series consisting of nine pairs of numbers, each pair the same, 13, XI. The sum of the black numbers (nine 13's) is 117 and the interval between the successive days of each column is 117; thus, from 11 Ahau to 11 Caban is 117 days, and so on down to Lamat, the last of the left hand column. From 11 Lamat to 11 Chicchan (first day of second column) is also 117, and so on to the end of the fourth column. These four columns, therefore, form one continuous series of 2,223 days, commencing with 11 Ahau and ending with 11 Akbal; but, by adding 117 days more, so as to bring us back to 11 Ahau--which appears to be in accordance with the plan of these series--the sum is 2,340 days, or nine cycles of 260 days each.[291-1]

The interval between the days, without reference to the numbers attached to them, is 17. It may be well to notice here the relation of the intervals between the days when counted in the two ways: (1) the apparent interval, or that which indicates their position in the month; (2) the true interval between the days, indicated by the symbols and numbers. When the first is 6 the latter, as we have found, is 20; when the first is 12 the latter is 52; when the first is 5 the latter is 65, and when it is 17 the latter is 117.

Particular attention is also called here to the fact that so far no indications of the use of the year period of 365 days have been observed; on the contrary the cycle of 260 days appears to be the period to which reference is chiefly made.

Attached to the day column in Plate 29_c_ and running into 30_c_ is a series which presents a difficulty I am unable to explain. The days and numerals in this case are as follows:

III Ix Cauac } 16, VI; 16, IX; 16, XII; 16, (?) Kan } Muluc

The red numeral over the day column is very distinctly III in Kingsborough's work, but is II, though somewhat blurred, in Förstemann's photograph. As III + 16 - 13 = VI, and the remaining numerals agree with this result, III must be correct. Adding together the pairs and casting out the thirteens, thus, III + 16 - 13 = VI; VI + 16 - 13 = IX; IX + 16 - 13 = XII; XII + 16 - 13 - 13 = II, we find the last red number, which is wanting in both copies of the codex, to be II, whereas, according to the theory advanced, it should be III. The sum of the black numerals (four 16's) is 64, while the interval between the days is 65. The only way of correcting the mistake, if one has been made, is by arbitrarily changing the last 16 to 17; but uniformity in the black numerals apparently forbids this change and and[TN-3] indicates that the variation from the usual rule must be accounted for in some other way.

In reference to this series, Dr. Förstemann[292-1] remarks:

The column of the days has the difference 5; the fifth sign (in this case really superfluous), that of the thirteenth day, appears in a remarkable form, apparently as an inscription on a vessel. The black figures ought to give the sum 65, but we get only 4 × 16, or 64. But this appears to be merely an oversight by the copyist, for although in the Codex Troano, also, we find 64 several times instead of 65, still this has always appeared to me merely as a sign of the great negligence of the copyist of that manuscript.

Turning to the Manuscript Troano, Plate XXVIII*_b_, we find a column consisting of the four terminal days of the year, Been, Ezanab, Akbal, and Lamat, which of course have the same relation to one another as the first days. It is evident from the space that only four were intended to be given. The numerals in Brasseur's fac simile are XI; 20, 12, IV; 9, XIII; 10, X; 13, XI.

The red numeral over the column is XI, as is also the last of the series, but the sum of the black numbers is only 64, which would give X as the final number, as is evident from the following operation: XI + 32 - 13 - 13 - 13 = IV; IV + 9 = XIII; XIII + 10 - 13 = X; X + 13 - 13 = X. The interval between the days is 65. We have, therefore, precisely the same difficulty in this instance as in the case from the Dresden Codex under consideration. Moreover, the only method of correcting the mistake, if there is one, is by adding _one_ to the last black number. It would be hazardous to assume that two mistakes, precisely the same in every respect, should have been made in regard to these exactly similar series. The probability that a mistake has been made is lessened by the fact that on Plate XXIX*_b_ of the manuscript is another four day column, the last days of the years, as the preceding. The numeral over the column is XIII and the series is as follows: 13, XIII; 20, 18, XII; 13, XIII. Adding these and casting out the thirteens, we have this result: XIII + 13 - 13 = XIII; XIII + 20 + 18 - 13 - 13 - 13 = XII; XII + 13 - 13 = XII. This gives XII as the last number when it should be XIII. If a mistake has been made the only method of correcting it is by increasing the last black number by one, as in the other two cases alluded to.

It is proper to state that on the other hand there is another four day column on Plate XXXII*_a_ of the last mentioned codex, the days of which are precisely the same as those on Plate 29_c_ of the Dresden Codex, to wit, Ix, Cauac, Kan, Muluc. The numeral over it is XII and the series is as follows: 13, XII; 13, XII; 13, XII; 13, XII; 13, XII. This presents no difficulty, as it conforms in every respect to the rules given, but only serves to deepen the mystery in the other cases.

Going back to the series on Plate 29_c_ of the Dresden Codex, we observe not only that the days of the column are the four year bearers, but also that one of the four cardinal symbols is found--in the superscription--in each of the four compartments through which the series extends. It is possible, therefore, that the series is intended to be applied separately to each of the four years. Supposing this to be the case, counting 64 days from 3 Ix would bring us to 2 Ezanab; 64 days from 3 Cauac to 2 Akbal; 64 days from 3 Kan to 2 Lamat; and 64 days from 3 Muluc to 2 Been. It is significant that in each case the day reached is that on which the given year terminates; for example, the Ix years (counting the five added days) terminate on Ezanab; the Cauac years on Akbal &c. If the intention was to have the series terminate with the end of the respective years, then these years must necessarily have been 2 Ix, 2 Cauac, 2 Kan, and 2 Muluc. I must confess that this explanation is not satisfactory; it is thrown out simply as a suggestion.

Running through the middle division of Plates 30 and 31 is this series:

3, VIII; 3, VIII; 3, VIII; 3, VIII 5, Oc 5, Men 5, Ahau 5, Chicchan.

Commencing with 8 Oc (omitting for the present the 3 and 5 to the left) and counting thence 3 months and 5 days we reach 8 Men; 3 months and 5 days more and we reach 8 Ahau; 3 months and 5 days more bring us to 8 Chicchan, and 3 months and 5 days more bring us again to 8 Oc, thus completing a cycle of 260 days (13 months) and also accounting for the first pair of numerals--3 and 5 in the series. It appears to be a pretty general rule to commence a series of this type with the difference between the numbers of the series. One reason for this is apparent: that is, to complete the cycle of 260 days, to which most, if not all, of these groups appear to refer.

Dr. Förstemann says in regard to this line:[293-1]

This is the place where I first discovered how numbers of several figures are to be read; here for the first time I understood that the figure 3 with 5 below it is nothing but 3 × 20 + 5, or 65, and that they mean nothing else than the interval between the days, such as we have frequently met with so far; 4 × 65 is again the well known period of 260 days.

Plate 3 appears to be isolated and unfinished; at least it presents nothing on its face by which it can be directly connected with any other plate of the codex, notwithstanding the change made by Dr. Förstemann, by which 45 was brought next to it. The day column in this case is in the middle compartment of the upper division and consists of the following days: Ahau, Eb, Kan, Cib, Lamat; the red numeral over it is I. The numerals and days are arranged as follows:

(?) (?) 4, V(?) 15, XIII

I Ahau 8, XIII Eb Kan Cib 14 (?) Lamat

As numerals belonging to two different series are never found in the same compartment it is fair to assume that those of the middle and right compartments pertain to one series. But what shall we say in reference to those in the left compartment, the upper pair of which is almost entirely obliterated? So far we have found no series extending to the left of the day column. Is this an exceptional case? I am inclined to believe it is, for the following reasons:

Taking the 4, V over the bird as the first pair of the series, we have I + 4 = V, which is so far correct; after this follows the pair in the lower left hand corner, 8, XIII, as V + 8 = XIII. It is probable that the obliterated pair in the upper left hand corner followed next, then the pair in the upper right hand corner, and last the partly obliterated one in the lower right hand corner. In this case the obliterated pair in the upper left hand corner should be 11, XI, as XIII + 11 - 13 = XI, and XI + 15 - 13 = XIII, and XIII + 14 - 13 - 13 = I, which makes the terminal red number of the series the same as that over the day column. This restoration requires no change of any of the numbers which can be distinctly read. By adding together the black numbers 4, 8, 11, 15, 14, the sum is found to be 52, precisely the interval between the days of the column. These facts are sufficient to render it more than probable that the restoration and the order as here given are correct. The series as thus given, including the number over the day column, is: I; 4, V; 8, XIII; 11, XI; 15, XIII; 14, I.

This is repeated, because on turning to Dr. Förstemann's comment on this series I find that he has restored and amended it so as to read thus: I; 10, XI; 4, V; 15, XIII; 9, XIII; 14, I; and he remarks that all would be plain sailing if, for the V before and the XIII after 15, we could read II and IV. This is true, but these numbers are too distinct to justify such change; moreover his "9" is not to be found on the page; it is true that the three dots over the line are not exactly spaced, but there are no indications of a fourth; the number is 8 and should, I think, be so read. His 10 is the obliterated black numeral; of course the value attributed to it depends upon the order given to the series. The fragments remaining of the red number of this pair I think warrant his making it XI.

Plates 46, 47, 48, 49, and 50 are peculiar and seemingly have no direct relation to any other part of the codex. In the upper left hand corner of each are four day columns, all more or less injured, but each column evidently contained, originally, thirteen days, or, more correctly speaking, the symbol for one day repeated thirteen times. In every case the day in the first (left hand) column and that in the third column are the same. As the numbers attached to them are absolutely unreadable in Kingsborough and much obliterated in the photograph, I give here restorations for the benefit of those studying this codex. This restoration is easily made by finding the order of the series, which can be obtained from Plates 49 and 50 of the photographic copy.

_Plate_ 46: III Cib. II Cimi. V Cib. XIII Kan. XI Cib. X Cimi. XIII Cib. VIII Kan. VI Cib. V Cimi. VIII Cib. III Kan. I Cib. XIII Cimi. III Cib. XI Kan. IX Cib. VIII Cimi. XI Cib. VI Kan. IV Cib. III Cimi. VI Cib. I Kan. XII Cib. XI Cimi. I Cib. IX Kan. VII Cib. VI Cimi. IX Cib. IV Kan. II Cib. I Cimi. IV Cib. XII Kan. X Cib. IX Cimi. XII Cib. VII Kan. V Cib. IV Cimi. VII Cib. II Kan. XIII Cib. XII Cimi. II Cib. X Kan. VIII Cib. VII Cimi. X Cib. V Kan.

_Plate_ 47: II Ahau. I Oc. IV Ahau. XII Lamat. X Ahau. IX Oc. XII Ahau. VII Lamat. V Ahau. IV Oc. VII Ahau. II Lamat. XIII Ahau. XII Oc. II Ahau. X Lamat. VIII Ahau. VII Oc. X Ahau. V Lamat. III Ahau. II Oc. V Ahau. XIII Lamat. XI Ahau. X Oc. XIII Ahau. VIII Lamat. VI Ahau. V Oc. VIII Ahau. III Lamat. I Ahau. XIII Oc. III Ahau. XI Lamat. IX Ahau. VIII Oc. XI Ahau. VI Lamat. IV Ahau. III Oc. VI Ahau. I Lamat. XII Ahau. XI Oc. I Ahau. IX Lamat. VII Ahau. VI Oc. IX Ahau. IV Lamat.

As the arrangement and the order of the series are readily seen from the two examples given, only the top and bottom lines of the remaining series will be presented.

_Plate_ 48: I Kan. XIII Ix. III Kan. XI Eb. * * * * * * * [TN-4] VI Kan. V Ix. VIII Kan. III Eb.

_Plate_ 49: XIII Lamat. XII Ezanab. II Lamat. X Cib. * * * * * * * * V Lamat. IV Ezanab. VII Lamat. II Cib.

_Plate_ 50: XII Eb. XI Ik. I Eb. IX Ahau. * * * * * * * * IV Eb. III Ik. VI Eb. I Ahau.

A careful examination of these groups will bring to light the following relations of the numbers, days, columns, and series to one another:

The numerals of any one column, counting downwards, differ from one another by 8; that is to say, by adding 8 to any one and casting out 13 when the sum exceeds that number, the next lower number will be obtained; or, reversing the operation and counting upward, the difference is found to be 5. The true interval between the days of the columns (counting downwards) is 3 months (60 days), a rule which holds good as to all the series and each column. Thus, from 3 Cib to 11 Cib is 3 months, or 60 days; from 11 Cib to 6 Cib, 3 months; from 2 Cimi to 10 Cimi, 3 months, and from 13 Kan to 8 Kan, 3 months.

Counting on the list of the days of the month, without reference to the week numbers attached to them, it will be found that from Cib to Cimi is an interval of 10 days, and from Cib to Kan is an interval of 8 days. This rule holds good as to all the series, showing that all are arranged upon precisely the same plan. The true interval between any day of the first column of either series (the week number attached being considered) and the opposite or corresponding day in the second column, is 4 months and 10 days, that between the corresponding days of the second and third columns is 12 months and 10 days, that between the days of the third and fourth columns is 8 days, and that between the corresponding days of the fourth or last column of one series or plate and the first column of the following series or plate (taking the plates in the order they are paged) is 11 months and 16 days.

In order to illustrate this we will run through the lowest line of each series, taking them in the order of the pages.[296-1]

These are as follows:

_Plate_ 46: VIII Cib. VII Cimi. X Cib. V Kan.

_Plate_ 47: VII Ahau. VI Oc. IX Ahau. IV Lamat.

_Plate_ 48: VI Kan. V Ix. VIII Kan. III Eb.

_Plate_ 49: V Lamat. IV Ezanab. VII Lamat. II Cib.

_Plate_ 50: IV Eb. III Ik. VI Eb. I Ahau.

[Illustration: FIG 362. Copy of Plate 50, Dresden Codex.[TN-5]]

By counting on the calendar (our Table II), as heretofore explained, the reader will observe that the interval from 8 Cib to 7 Cimi is 4 months and 10 days; from 7 Cimi to 10 Cib is 12 months and 10 days; from 10 Cib to 5 Kan is 8 days; from 5 Kan to 7 Ahau is 11 months and 16 days; from 7 Ahau to 6 Oc, 4 months and 10 days; from 6 Oc to 9 Ahau, 12 months and 10 days; from 9 Ahau to 4 Lamat, 8 days; from 4 Lamat to 6 Kan, 11 months and 16 days, and so on to the end of the series on Plate 50. Referring to the codex the reader will observe at the bottom of each plate and directly under--that is to say, in the same vertical lines as the day columns--two lines of red numerals. It is impossible to determine these in Kingsborough's copy (except on Plate 50), but they can readily be made out on the photographed plates. (See the copy of Plate 50, given in Fig. 362.) Those on a single plate are as follows:

{ XI, IV, XII, 0, { XVI, X, X, VIII.

The 0 here represents a red, diamond shaped symbol.

If the upper line represents months and the lower line days, these numbers will indicate the intervals between the columns and are properly placed. For example, the XI and XVI signify 11 months and 16 days, the interval between the last column of the preceding plate and the first column of the plate on which they stand; the IV and X, the interval of 4 months and 10 days between the first and second columns; XII and X, the interval of 12 months and 10 days between the second and third columns; and 0, VIII, the interval of 8 days between the third and fourth columns. It is apparent from this that the red, diamond shaped symbol represented by 0 over the VIII denotes a cipher or nought, a conclusion reached independently by Förstemann.

If this supposition as to the arrangement of the series and the signification of these numbers be correct, it is apparent that the plates are to be taken in the order in which they are paged, that is, from left to right, as the others so far noticed, an inference borne out by another fact now to be mentioned.

Immediately below each of these four column day series are four lines of characters (hieroglyphics), and immediately under the latter three horizontal lines of black numerals, with here and there a red, diamond shaped symbol inserted. As these numerals stand directly in the vertical lines of the day columns, it is possible the two have some connection with each other, a supposition somewhat strengthened by what has been observed in regard to the red numerals at the bottom of the plates. To test this and also for the reason that we propose to discuss their relations and their use, we give here the bottom line of days of each of the five series (or plates), together with their week numbers attached; also, the numbers of the three lines of black numerals mentioned, taking them in the order of the paging as here shown:

_Plate_ 46: VIII Cib. VII Cimi. X Cib. V Kan. 1 1 11 16 10 11 16 6 16 4

_Plate_ 47: VII Ahau. VI Oc. IX Ahau. IV Lamat. 2 2 3 3 5 9 4 4 0 10 0 8

_Plate_ 48: VI Kan. V Ix. VIII Kan. III Eb. 3 4 4 4 16 2 15 15 3(?) 14 4 12

_Plate_ 49: V Lamat. IV Ezanab. VII Lamat. II Cib. 5 5 6 6 9 13 8 8 8 18 8 16

_Plate_ 50: IV Eb. III Ik. VI Eb. 1 Ahau. 7 7 8 8 3 7 1 2 12 2 12 0

In considering these horizontal lines it is to be understood that the series runs through the five pages, 46-50.

Let us proceed upon the supposition that the figures of the lowest of the three lines denote days of the month, the numbers of the middle line months, and those of the upper line years. As already shown, the interval between 8 Cib and 7 Cimi is 4 months and 10 days; adding 4 months and 10 days to 11 months and 16 days (bearing in mind that 20 days make a month and 18 months a year), the sum is found to be 16 months and 6 days, precisely the figures under 7 Cimi. As already ascertained, the interval between 7 Cimi and 10 Cib is 12 months and 10 days; this added to 16 months and 6 days gives 1 year, 10 months, 16 days, precisely the figures under 10 Cib. The interval between 10 Cib and 5 Kan is 8 days; this added to the 1 year, 10 months, and 16 days gives 1 year, 11 months, and 4 days, the figures under 5 Kan. The interval between 5 Kan and 7 Ahau is 11 months, 16 days, which, added to the preceding, gives 2 years, 5 months, 0 day, agreeing with the figures under 7 Ahau, if the symbol represented by 0 signifies nought. That this rule holds good throughout the entire series, by making one correction, is shown by the following additions:

Years. Months. Days. 11 16 Under VIII Cib, Plate 46. 4 10 -- -- 16 6 Under VII Cimi, Plate 46. 12 10 -- -- 1 10 16 Under X Cib, Plate 46. 8 -- -- -- 1 11 4 Under V Kan, Plate 46. 11 16 -- -- -- 2 5 0 Under VII Ahau, Plate 47. 4 10 -- -- -- 2 9 10 Under VI Oc, Plate 47. 12 10 -- -- -- 3 4 0 Under IX Ahau, Plate 47. 8 -- -- -- 3 4 8 Under IV Lamat, Plate 47. 11 16 -- -- -- 3 16 4[300-1] Under VI Kan, Plate 48. 4 10 -- -- -- 4 2 14 Under V Ix, Plate 48. 12 10 -- -- -- 4 15 4 Under VIII Kan, Plate 48. 8 -- -- -- 4 15 12 Under III Eb, Plate 48. 11 16 -- -- -- 5 9 8 Under V Lamat, Plate 49. 4 10 -- -- -- 5 13 18 Under IV Ezanab, Plate 49. 12 10 -- -- -- 6 8 8 Under VII Lamat, Plate 49. 8 -- -- -- 6 8 16 Under II Cib, Plate 49. 11 16 -- -- -- 7 2 12 Under IV Eb, Plate 50. 4 10 -- -- -- 7 7 2 Under III Ik, Plate 50. 12 10 -- -- -- 8 1 12 Under VI Eb, Plate 50. 8 -- -- -- 8 2 0 Under I Ahau, Plate 50.

The proof of the correctness of the theory advanced may, therefore, be considered conclusive, as it amounts, in fact, to a mathematical demonstration.

Dr. Förstemann, who considers these lines of black numbers, standing one above another, as representing different grades of units--thus, the lowest, single units; the second, units twenty-fold the lower; the third, eighteen-fold the second; the fourth, twenty-fold the third, &c.--has found the correct intervals of the series, which he states are 236, 90, 250, and 8 days, agreeing with our 11 months, 16 days; 4 months, 10 days; 12 months, 10 days, and 8 days.

As all the discoveries mentioned herein were made previous to the receipt of Dr. Förstemann's work, I give them according to my own method, acknowledging any modification due to his work. Although I shall compare special results from time to time, an explanation of Dr. Förstemann's method is reserved for a future paper, as his work was not received until I was revising my notes for publication.

The foregoing explanation of the series shows it to be very simple and makes it clear that it relates to the day columns at the top of the pages. Still, there is one point somewhat difficult to understand. Are the numbers of the third or lowest line intended to denote the positions in the month of the days in the columns above? If so, the month must have commenced with Ymix, as can readily be shown in the following manner:

Table III.

1. Ymix. 2. Ik. 3. Akbal. 4. Kan. 5. Chicchan. 6. Cimi. 7. Manik. 8. Lamat. 9. Muluc. 10. Oc. 11. Chuen. 12. Eb. 13. Been. 14. Ix. 15. Men. 16. Cib. 17. Caban. 18. Ezanab. 19. Cauac. 20. Ahau.

If we write in a column in proper order the 20 days of the Maya month, commencing with Ymix, and number them consecutively, as in Table III, we shall find by comparison that the numbers in the lower line indicate the position, in this column, of the days directly over them. Take, for example, the lower line of black numerals on Plate 46, writing over them the respective days of the columns, thus:

Cib. Cimi. Cib. Kan. 16 6 16 4

Referring to Table III we see that Cib is the sixteenth day, Cimi the sixth, and Kan the fourth.

The days and numbers of Plate 47 are:

Ahau. Oc. Ahau. Lamat. 0 10 0 8

Ahau is the twentieth day--here is the diamond shaped symbol--Oc is the tenth, and Lamat the eighth, and so on to the end of the series on Plate 50.

It may be justly argued that such relation to some given day of the month would necessarily follow in any series of this kind made up by adding together intervals of days and months. Still it is not at all likely that these series were made up without reference to fitted and determinable dates. If so, the months given must be months of certain determinable years, and the days denoted must be days of particular months. In other words, if we had the proper starting point we should be able to determine the position in the calendar of any day or month mentioned in the series.

First. It is easily seen by reference to the calendar (Table II) that Cib is not the sixteenth day of the month of any of the four years, nor is Cimi the sixth nor Kan the fourth. The idea that the figures of this lower line represent the days of the month must, therefore, be given up unless we assume that the year commenced with Ymix. It may be worthy of notice at this point that the list of days on the so-called "title page" of the Manuscript Troano begins with Ymix. It is also true that the remarkable quadruple series in the Codex Cortesianus on Plates 13-18 commences with Ymix; as this is evidently some kind of a calendar table, its bearing on the question now before us is important.

Second. It can easily be shown that the months referred to in the series, if the numbers given denote specific months, are not those of the Kan years. The first, 8 Cib, if in the eleventh month, must be in the year 4 Kan; counting forward from this 4 months and 10 days to 7 Cimi brings us into the sixteenth month of the year 4 Kan; this agrees with our figures on Plate 46. Counting forward 12 months and 10 days to 10 Cib, we reach the tenth month of the next year; 8 days more carry us to the eleventh month, which still agrees with the figures in the codex. Counting 11 months and 16 days more to 7 Ahau, we reach but do not pass the fourth month of the next year; hence the result does not correspond with the series, which has at this point a 5 in the middle line. The same will be found true in regard to the other years as given in our calendar (Table II). This result, as a matter of course, must follow if the figures in the lower line of the series do not denote the month days of some one of the year series as usually given.

Another fact also becomes apparent here, viz, that the 5 supplemental days of the year are not brought into the count, the year consisting throughout of 360 days. There is, in fact, nothing here indicating the four year series as given in the authorities and as represented in our calendar table; yet this ought to appear wherever a series extends over more than one year.

Dr. Förstemann says that this entire series of black numerals covers 2,920 days, or 8 years of 365 days. This is true, but the concluding figures show that it is given by the writer of the codex as 8 years and 2 months, which would also be 2,920 days, counting the years at 360 days each and the months 20 days each; moreover, the members of the series are based throughout upon the year of 360 days. His theory that the intervals of the series relate to the movements of the planet Venus is, as yet, a mere hypothesis, which needs further proof before it can demand acceptance; but his discovery of the methods of identifying the month symbols on the five plates now under consideration is important. Although I had noticed that most of the characters which he mentions are month symbols, I did not succeed in identifying all of them.

According to his conclusion, which appears to be justified not only by the evidence he gives but by an additional fact that I shall, presently mention, there are four of these symbols in the upper row of the middle group of written characters on each plate and four in the upper and lower lines of the lower group on each plate (see, for example, Fig. 362). Each of these symbols (except three or four) has a black number attached to it which denotes the day of the month represented by the symbol.

These months and days as given by Dr. Förstemann are as follows, the positions of the lines as here given corresponding with those of the plates:

TABLE IV.--_Table showing months and days._

_____________________________________________________________________ | Month. Day. | Month. Day. | Month. Day. | Month. Day. |--------------+--------------+--------------+-------------- Plate 46 | 7 4 | 11 14 | 5 19 | 6 7 | 11 8 | 15 18 | 10 4 | 10 12 | 1 14 | 6 4 | 18 14 | 1 2 Plate 47 | 18 3 | 4 8 | 16 18 | 17 6 | 4 3 | 8 13 | 2 18 | 3(not 2) 6 | 10 10 | 15 3 | 9 8 | 9 16 Plate 48 | 10 17 | 15 7 | 9 12 | 10 20 | 15 2 | 1 7 | 13 17 | 14 5 | 3 7 | 7 17 | 2 2 | 2 10 Plate 49 | 3 11 | 8 1 | 2 6 | 2 14 | 7 16 | 12 6 | 6 11 | 6 19 | 14 6 | 18 16 | 13 1 | 13 9 Plate 50 | 14 10 | 18 20 | 13 5 | 13 13 | 18 15 | 5 20 | 17 10 | 17 18 | 6 20 | 11 10 | 5 15 | 6 3 ---------------------------------------------------------------------

An examination of the plates will show that Dr. Förstemann has filled out the following obliterated or wanting day numbers, to wit, the first of the upper line of Plate 46, the fourth of the upper line of Plate 47, and the second of the middle line and first of the lower line of Plate 50. He has also ventured to change the first day number of the lower line of Plate 46 from 16 to 14. Where the number 20 is found in his list there is no corresponding number in the codex, the month symbol only being given. It is evident he has proceeded in these cases upon the theory that the absence of a number indicated that the month was completed. Although probably correct in this conclusion, the question will arise, Does the symbol in such cases denote the _month completed_ or the _month reached?_

The intervals between these dates are as follows, the left hand column being those between the first and second columns of Förstemann's list (our Table IV), the second column those between the second and third columns of his list, the third column those between the third and fourth columns of his list, and the fourth column those between the last date of one plate and the first of the next:

TABLE V.--_Table showing intervals between dates._

_____________________________________________________________________ | Month. Day. | Month. Day. | Month. Day. | Month. Day. |--------------+--------------+--------------+-------------- | | | | Plate 46 | 4 10 | 12 5 | 0 8 | 11 16 | 4 10 | 12 6_b_| 0 8 | 11 11 | 4 10 | 12 10 | 0 8 | 9 8_d_ Plate 47 | 4 5 | 12 10 | 0 8 | 11 11 | 4 10 | 12 5 | 0 8_c_| 11 16_e_ | 4 13_a_| 12 5 | 0 8 | 11 11 Plate 48 | 4 10 | 12 5 | 8 | 11 11 | 4 5 | 12 10 | 0 8 | 11 11 | 4 10 | 12 5 | 0 8 | 11 16 Plate 49 | 4 10 | 12 5 | 0 8 | 11 16 | 4 10 | 12 5 | 0 8 | 11 16 | 4 10 | 12 5 | 0 8 | 11 11 Plate 50 | 4 10 | 12 5 | 0 8 | 11 11 | 4 5 | 12 10 | 0 8 | 11 10 | 4 10 | 12 5 | 0 8 | 12 11_g_ ---------------------------------------------------------------------

Although it is apparent that the variations from the intervals of the black numeral and day series above them are too numerous and too uniform to be considered mistakes, yet there is little reason to doubt that these month numbers are connected with and depend upon the day series given in the columns above.

That there are some errors is quite clear; for instance, the variation at _a_ arises from the fact that Dr. Förstemann gives the date here as 10 months, 10 days, whereas the codex has it 10 months, 13 days. Making this correction the interval will be 4 months, 10 days. The correction will make the interval at _d_ 9, 11, instead of 9, 8. Still there is a variation of two months from the usual interval, which, if corrected on the supposition that Dr. Förstemann has mistaken the month, would necessitate a change of the remainder of the series given in this line. The interval at _c_, according to the figure given by Dr. Förstemann, would be retrograde, that is, minus 12. This arises from the fact that he gives the last date in the middle line on Plate 47 as 2 months, 6 days, whereas the symbol is very distinctly that of the third month, and the eight day series is unbroken if this correction is made.

When these evident errors are corrected the series of intervals show very clearly a system and periodicity depending on the day column series in the upper part of the pages. In the first column (Table V) the interval is usually 4 months, 10 days, precisely the same as between the first and second day columns, but occasionally it is 4 months, 5 days, which will still bring it to one of the four day series, including the day indicated by the date--4 months, 10 days. This will be understood by examining our calendar (Table II). The corresponding days in the four year columns were, by the Maya system, necessarily brought together in the calendar; for example, they are arranged in the series pictured on Plates 13-18 of the Cortesian Codex precisely as given in our Table II. This skip of five days is also apparent in the second and fourth columns of differences (Table V). Whether Dr. Förstemann is correct in all his identifications of months among the symbols on the five plates now under consideration is a question I feel unqualified to answer without a much more careful comparison and study of these characters than I have given them.

Running through the upper division of Plates 53 to 58 and continued through the lower division of Plates 51 to 58--that is to say, commencing in the upper division of 53 and running into 58, then back to the lower division of 51 and ending in 58--is a remarkable compound series. It consists, first, of a three line series of black numerals standing above; second, a middle series of short, three day columns, or columns each of three day symbols, with red numerals attached; and, third, below, a two line series of numerals, those of the upper line red and of the lower black numbers.

As this series is a very important one in the study of the relations of the numerals to one another and to the days indicated, an exact copy of it is given in Figs. 363-370, each figure representing a page and the whole standing in the same order as in the original. The red numerals and red symbols are, as usual, given in outline as an indication of their color.

[Illustration: FIG. 363. Copy of Plate 51, Dresden Codex.]

[Illustration: FIG. 364. Copy of Plate 52, Dresden Codex.]

[Illustration: FIG. 365. Copy of Plate 55, Dresden Codex.]

[Illustration: FIG. 366. Copy of Plate 54, Dresden Codex.]

[Illustration: FIG. 367. Copy of Plate 55, Dresden Codex.]

[Illustration: FIG. 368. Copy of Plate 56, Dresden Codex.]

[Illustration: FIG. 369. Copy of Plate 57, Dresden Codex.]

[Illustration: FIG. 370. Copy of Plate 58, Dresden Codex.]

In order to assist those not familiar with the numeral and day symbols, the entire series is given in the following tables in names and Arabic and Roman numerals, as usual. The obliterated symbols and numbers are restored.

TABLE VI.--_Table of numeral and day symbols._ (Plate 51_b_.)

______________________________________________________________________ 14 | 15 | 15 | 16 | 16 | 17 16 | 7 | 16 | 7 | 16 | 5 14 | 11 | 8 | 5 | 2 | 10 IV Ik. | XII Cauac.| VII Cib. | II Been.| X Oc. | II Ezanab. V Akbal.|XIII Ahau. |VIII Caban. |III Ix | XI Chuen|III Cauac. VI Kan. | I Ymix. | IX Ezanab.| IV Men. |XII Eb. | IV Ahau.[VI-1] VIII | VIII | VIII | VIII | VIII | VII 17 | 17 | 17 | 17 | 17 | 8 ----------------------------------------------------------------------

[VI-1] The symbol in this case is that of Been, but this is a manifest error, as Ahau follows Cauac.

TABLE VII.--_Table of numeral and day symbols._ (Plate 52_b_.)

____________________________________________________________ | 17 | 18 | 18 | 19 | 14 | 5 | 14 | 4 | 8 | 5 | 2 | 19 | XI Cib. | VI Been.| I Oc. | IX Manik. [Picture.] | XII Caban. | VII Ix. | II Chuen.| X Lamat. | XIII Ezanab. |VIII Men. |III Eb. | XI Muluc. | VIII | VIII | VIII | VIII |17? (18)[VII-1]| 17 | 17 | 17 ------------------------------------------------------------

[VII-1] The variation from the rule found here is explained a little further on.

TABLE VIII.--_Table of numeral and day symbols._ (Plate 53_a_.)

______________________________________________________________________________________ | 1 | 1 | | 2 | 2 | 7 | 17 | 7 | | 15 | 6 | 15 17 | 18 | 2 | |14?(19)[VIII-1]| 16 | 13 VI Kan. | I Ymix. | VI Muluc.|[Picture.]| I Cimi. |IX Akbal. |IV Ahau. VII Chicchan.| II Ik. | VII Oc. | | II Manik. | X Kan. | V Ymix. VIII Cimi. |III Akbal.|VIII Chuen.| | III Lamat. |XI Chicchan.|VI Ik. VIII | VIII | VII | | VIII | VIII | VIII 17 | 17 | 8 | | 17 | 17 | 17 --------------------------------------------------------------------------------------

[VIII-1] The 14 here is manifestly an error, one of the lines in the number symbol having been omitted; it should be 19.

TABLE IX.--_Table of numeral and day symbols._ (Plate 53_b_.)

_____________________________________________________________________ | 1 | | 1 | 1 | 1 19 | 0 | | 0 | 1 | 1 13 | 3 | | 12 | 2 | 11 16 | 4 | | 1 | 18 | 15 IV Kan. |IX Eb. |[Picture.]|IV Muluc.| XII Cimi. | VII Akbal. V Chicchan.| X Been.| | V Oc. |XIII Manik.|VIII Kan. VI Cimi. |XI Ix. | |VI Chuen.| I Lamat.| IX Chicchan. VIII | VII | | VIII | VIII | VIII 17 | 8 | | 17 | 17 | 17 ---------------------------------------------------------------------

TABLE X.--_Table of numeral and day symbols._ (Plate 54_a_.)

____________________________________________________________________________________ 3 | 3 | 4 | 4 | 5 | 5 | 6 6 | 15 | 6 | 15 | 5 | 10 | 4 11 | 8 | 5 | 5 | 19 | 16 | 4 XIII Ezanab.|VIII Men. |III Eb. | XI Muluc.| VI Cib. | I Akbal. | VI Chuen. I Cauac. | IX Cib. | IV Been.| XII Oc. | VII Caban. | II Kan. | VII Eb. II Ahau. | X Caban.| V Ix. |XIII Chuen.|VIII Ezanab.|III Chicchan.|VIII Been. VIII | VIII | VIII | VIII | VIII | VIII | VII 17 | 17 | 17 | 17 | 17 | 17 | 8 ------------------------------------------------------------------------------------

TABLE XI.--_Table of numeral and day symbols._ (Plate 54_b_.)

_______________________________________________________________ 1 | 1 | 1 | 1 | | 1 2 | 2 | 3 | 3 | | 4 2 | 11 | 2 | 9 | | 0[XI-1] 12 | 9 | 6 | 14 | | 11 II Ahau.| X Caban. | V Ix. | X Ik. |[Picture]| V Cauac. III Ymix.| XI Ezanab.| VI Men.| XI Akbal.| | VI Ahau. IV Ik. |XII Cauac. |VII Cib.|XII Kan. | |VII Ymix. VIII | VIII | VIII | VII | | VII[XI-2] 17 | 17 | 17 | 8 | | 17 ---------------------------------------------------------------

[XI-1] The 0 inserted at various points in these tables denotes as usual the red, diamond shaped symbol, which apparently signifies "nought."

[XI-2] The numeral symbol in this case, both in Kingsborough's copy and in the photograph, is VII, one dot having been omitted by a mistake of the original artist.

TABLE XII.--_Table of numeral and day symbols._ (Plate 55_a_.)

____________________________________________________________________________ | 8 | 7 | 7 | 8 | 8 | 13 | 3 | 12 | 3 | 12 | 2 | 18 | 16 | 13 | 10 | II Muluc.[XII-1]| X Cimi. | V Akbal. |XIII Ahau.|VIII Caban. [Picture]|III Oc. | XI Manik.| VI Kan. | I Ymix.| IX Ezanab. | IV Chuen. |XII Lamat.|VII Chicchan.| II Ik. | X Cauac. | VIII | VIII | VIII | VIII | VIII | 17 | 17 | 17 | 17 | 17 ----------------------------------------------------------------------------

[XII-1] In Kingsborough's work the symbol in this case is that of Been, but should be Muluc, as it is in the photograph.

TABLE XIII.--_Table of numeral and day symbols._ (Plate 55_b_.)

__________________________________________________________________________________________ 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 4 | 5 | 5 | 6 | 6 | 6 | 7 | 7 9 | 0 | 9 | 0 | 8 | 17 | 8 | 15 8 | 6 | 3 | 0 | 17 | 14 | 11 | 19 XIII Cib. |IX Ix. |IV Chuen.| XII Lamat.| VII Chicchan.| II Ik. | X Cauac.| II Manik. I Caban. | X Men.| V Eb. |XIII Muluc.|VIII Cimi. |III Akbal.| XI Ahau. |III Lamat. II Ezanab.|XI Cib.|VI Been. | I Oc. | IX Manik. | IV Kan. |XII Ymix. | IV Muluc. VIII | VIII | VIII | VIII | VIII | VIII | VIII | VII 17 |17?(18)| 17 | 17 | 17 | 17 | 17 | 8 ------------------------------------------------------------------------------------------

TABLE XIV.--_Table of numeral and day symbols._ (Plate 56_a_.)

____________________________________________________________ 9 | | 9 | 10 | 10 1 | | 10 | 1 | 10 18 | | 15 | 12 | 9 XIII Chicchan.| |VIII Ik. |III Cauac.| XI Cib. I Cimi. |[Picture]| IX Akbal.| IV Ahau. | XII Caban. II Manik. | | X Kan. | V Ymix. |XIII Ezanab. VII | | VIII | VIII | VIII 8 | | 17 | 17 | 17 ------------------------------------------------------------

TABLE XV.--_Table of numeral and day symbols._ (Plate 56_b_.)

_________________________________________________________ | 1 | 1 | 1 | 1 | 8 | 8 | 9 | 9 | 6 | 15 | 6 | 15 | 16 | 14 | 11 | 8 [Picture]| X Kan. | VI Ik. | I Cauac.|IX Cib. | XI Chicchan.| VII Akbal.| II Ahau. | X Caban. |XII Cimi. |VIII Kan. |III Ymix. |XI Ezanab. | VIII | VIII | VIII | VIII | 17 | 17?(8) | 17 | 17 ---------------------------------------------------------

TABLE XVI.--_Table of numeral and day symbols._ (Plate 57_a_.)

______________________________________________________ 11 | 11 | 12 | 12 | 1 | 10 | 1 | 8 | 6 | 4 | 0 | 8 | VII Ix. | II Chuen.| X Lamat.| II Cib. | VIII Men.|III Eb. | XI Muluc.|III Caban. |[Picture] IX Cib.| IV Been. |XII Oc. | IV Ezanab. | VIII | VIII | VIII | VIII[XVI-1]| 17 | 17 | 17 | 17[XVI-2]| ------------------------------------------------------

[XVI-1] This should be VII.

[XVI-2] This should be 8.

TABLE XVII.--_Table of numeral and day symbols._ (Plate 57_b_.)

__________________________________________________________________ 1 | 1 | 1 | | 1 | 1 10 | 10 | 11 | | 11 | 12 6 | 15 | 4 | | 13 | 4 5 | 2 | 10 | | 7 | 4 IV Been.| XII Oc. |IV Ezanab.|[Picture]| XII Men. | VII Eb. V Ix. |XIII Chuen.| V Cauac. | |XIII Cib. |VIII Been. VI Men. | I Eb. |VI Ahau. | | I Caban.| IX Ix. VIII | VIII | VII | | VIII | VIII 17 | 17 | 8 | | 17 | 17 ------------------------------------------------------------------

TABLE XVIII.--_Table of numeral and day symbols._ (Plate 58_a_.)

____________________________________________ 12 | 13 | 13 | 14 17 | 8 | 17 | 7 5 | 2 | 0 | 17 X Been.| V Oc. | I Lamat.|II Chicchan. XI Ix. | VI Chuen.| II Muluc.| X Cimi. XII Men. |VII Eb. |III Oc. |XI Manik. VIII | VIII | VIII | VIII 17 | 17 | 17 | 17 --------------------------------------------

TABLE XIX.--_Table of numeral and day symbols._ (Plate 58_b_.)

________________________________ 1 | 1 | 12 | 13 | 13 | 3 | 1 | 18 | II Muluc.| X Cimi. |[Picture.] III Oc. | XI Manik.| IV Chuen.|XII Lamat.| VIII | VIII | 17 | 17 | --------------------------------

The spaces in the lists indicate the positions of the pictures of persons and curtain-like ornaments inserted here and there, as seen in Figs. 363-370.

In order to explain this series, we commence with that portion of it found in the lower division of Plate 51 (Fig. 363).

Omitting any reference for the present to the black numbers over the day columns, we call attention first to the days and to the red numerals attached to them. Those in the division selected as an illustration are as follows:

IV Ik. XII Cauac. VII Cib. II Been. X Oc. II Ezanab. V Akbal. XIII Ahau. VIII Caban. III Ix. XI Chuen. III Cauac. VI Kan. I Ymix. IX Ezanab. IV Men. XII Eb. IV Ahau.[317-1]

It will be observed that the week numbers of the days in each single column follow one another in regular arithmetical order, thus: in the first column, 4, 5, 6; in the second, 12, 13, 1; in the third, 7, 8, 9; and so on throughout the entire series. The interval, therefore, between the successive days of a column is 1; or, in other words, the days follow one another in regular order, as in the month series, so that having the first day of a column given we know at once the other two. It is apparent, therefore, that the intervals between the three correspondingly opposite days of any two associate columns are the same; that is to say, the interval between 5 Akbal and 13 Ahau, in the first two columns given above is the same as that between 4 Ik and 12 Cauac, and also as that between 6 Kan and 1 Ymix. This is also true if the attached week numbers are omitted; for instance, the interval between Ik and Cauac, counting on the list of days forming the month, is 17 days, and it is the same between Kan and Ymix. Taking the second and third columns we find here the same interval. This holds good in that part of the series above given until we reach the last two columns; here the interval between Oc and Ezanab is 8 days and it is the same between the other days of these two columns.

This being ascertained, the next step is to determine the true interval between the first days of these columns, taking the numbers attached to them into consideration. Referring to our calendar (Table II) and (for reasons which will be given hereafter) using the Muluc column and counting from 4 Ik, as heretofore explained, we find the interval between this and 12 Cauac to be 8 months and 17 days; counting in the same way from 12 Cauac, 8 months and 17 days more bring us to 7 Cib; 8 months and 17 days more to 10 Oc. So far the intervals have been the same; but at this point we find a variation from the rule, as the interval between 10 Oc and 2 Ezanab (first of the next column) is 7 months and 8 days.

These intervals furnish the explanation of the red and black numerals below the day columns.

These numerals, as the reader will observe by reference to Fig. 363 or the written interpretation thereof in Table VI, are 8 and 17 under the first five columns, but 7 and 8 under the sixth column, the red (8 under the first five and 7 under the sixth) indicating the months and the black (17 under the first five and 8 under the sixth) the days of the intervals. This holds good throughout all that portion of the series running through the lower divisions of Plates 51 to 58, with three exceptions, which will now be pointed out.

In order to do this it will be necessary to repeat here a part of the series on Plate 51_b_ and part of that on Plate 52_b_; that is, the two right hand columns of the former and the two left hand columns of the latter, between which is the singular picture shown in the _lower left hand corner_ of our Fig. 364:

___________________________________________________________________ Plate 51_b_. || Plate 52_b_. --------------------------++--------------------------------------- X Oc. | II Ezanab. || | XI Cib. | VI Been. XI Chuen. | III Cauac. || | XII Caban. | VII Ix. XII Eb. | IV Ahau. || [Picture.] | XIII Ezanab. | VIII Men. VIII | VII || | VIII | VIII 17 | 8 || | 17 | 17 -------------------------------------------------------------------

As before stated, the interval between 10 Oc and 2 Ezanab is 7 months and 8 days, as indicated by the red and black numerals under the latter. According to the red and black numbers under the column commencing with 11 Cib, the interval between 2 Ezanab and 11 Cib should be 8 months and 17 days, the usual difference, when, in fact, as we see by counting on the calendar, it is 8 months and 18 days. That this variation cannot be attributed to a mistake on the part of the author or of the artist is evident from the fact that the interval between 11 Cib and 6 Been (first of the next column) is 8 months and 17 days and that the difference throughout the rest of the series follows the rule given; that is to say, each is 8 months and 17 days, except at two other points where this variation is found and at the regular intervals where the difference of 7 months and 8 days occurs.[319-1] Precisely the same variation occurs on Plate 55_b_ in passing from the first to the second column and on Plate 56_b_ between columns 1 and 2.

Why these singular exceptions? It is difficult, if not impossible, for us, with our still imperfect knowledge of the calendar system formerly in vogue among the Mayas, to give a satisfactory answer to this question. But we reserve further notice of it until other parts of the series have been explained.

Reference will now be made to the three lines of black numerals immediately above the day columns. Still confining our examinations to the lower divisions, the reader's attention is directed to these lines, as given in Tables VI, VII, IX, XI, XIII, XV, XVII, and XIX. As there are three numbers in each short column we take for granted, judging by what has been shown in regard to the series on Plates 46-50, that the lowest of the three denotes days, the middle months, and the upper years, and that the intervals are the same between these columns as between the day columns under them. The correctness of this supposition is shown by the following additions: Starting with the first or left hand column on Plate 51_b_, we add successively the differences indicated by the corresponding red and black numbers under the day columns. If this gives in each case (save the two or three exceptions heretofore referred to) the numbers in the next column to the right throughout the series, the demonstration will be complete.

Years. Months. Days. 14 16 14 First column on Plate 51_b_. 8 17 -- -- -- 15 7 11 Second column on Plate 51_b_. 8 17 -- -- -- 15 16 8 Third column on Plate 51_b_. 8 17 -- -- -- 16 7 5 Fourth column on Plate 51_b_. 8 17 -- -- -- 16 16 2 Fifth column on Plate 51_b_. 7 8 -- -- -- 17 5 10 Sixth column on Plate 51_b_. 8 18[319-1] -- -- -- 17 14 8 First column on Plate 52_b_. 8 17 -- -- -- 18 5 5 Second column on Plate 52_b_. 8 17 -- -- -- 18 14 2 Third column on Plate 52_b_. 8 17 -- -- -- 19 4 19 Fourth column on Plate 52_b_. 8 17 -- -- -- 19 13 16 First column on Plate 53_b_. 7 8 -- -- -- 20 3 4 Second column on Plate 53_b_.

At this point in the original, instead of 20 in the year series, we find a diamond shaped symbol, represented by 0 in our tables, with one black dot over it. From this it would seem that when this codex was written the Maya method of counting years was by periods of 20 each, as in the case of the month days. Whether there is any reference here to the ahaues is uncertain. I am inclined to think with Dr. Förstemann that it was rather in consequence of the use of the vigesimal system in representing numbers. It would have been very inconvenient and cumbersome to represent high numbers by means of dots and lines; hence a more practicable method was devised. It is evident, from the picture inserted at this point in the series, that some important chronological event is indicated. Here also in the written characters over this picture is the symbol for 20. The last number given in the above addition may therefore, in order to correspond with the method of the codex, be written as follows:

Twenty year periods. Years. Months. Days. 1 0 3 4

Continuing the addition in this way the result is as follows:

Twenty year periods. Years. Months. Days. 1 0 3 4 8 17 -- -- -- -- 1 0 12 1 Third column on Plate 53_b_. 8 17 -- -- -- -- 1 1 2 18 Fourth column on Plate 53_b_. 8 17 -- -- -- -- 1 1 11 15 Fifth column on Plate 53_b_. 8 17 -- -- -- -- 1 2 2 12 First column on Plate 54_b_. 8 17 -- -- -- -- 1 2 11 9 Second column on Plate 54_b_. 8 17 -- -- -- -- 1 3 2 6 Third column on Plate 54_b_. 7 8 -- -- -- -- 1 3 9 14 Fourth column on Plate 54_b_. 8 17 -- -- -- -- 1 4 0 11 Fifth column on Plate 54_b_. 8 17 -- -- -- -- 1 4 0 8 First column on Plate 55_b_. 8 18[321-1] -- -- -- -- 1 5 0 6 Second column on Plate 55_b_. 8 17 -- -- -- -- 1 5 9 3 Third column on Plate 55_b_. 8 17 -- -- -- -- 1 6 0 0 Fourth column on Plate 55_b_. 8 17 -- -- -- -- 1 6 8 17 Fifth column on Plate 55_b_. 8 17 -- -- -- -- 1 6 17 14 Sixth column on Plate 55_b_. 8 17 -- -- -- -- 1 7 8 11 Seventh column on Plate 55_b_. 7 8 -- -- -- -- 1 7 15 19 Eighth column on Plate 55_b_. 8 17 -- -- -- -- 1 8 6 16 First column on Plate 56_b_. 8 18[321-2] -- -- -- -- 1 8 15 14 Second column on Plate 56_b_. 8 17 -- -- -- -- 1 6 6 11 Third column on Plate 56_b_.