As colors are not used in these figures the red numerals are indi cated 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 Oc273-1 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: 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.

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.

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 characters277-1 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 43b and Plate 1b, 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 work278-1 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:

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 35b and extending through Plates 36b and 37b 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:

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 work282-1 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:

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:

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:

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.

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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

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

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 30c are the four day columns here given, with the numeral eleven over each:

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 29c and running into 30c is a series which presents a difficulty I am unable to explain. The days and numerals in this case are as follows:

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 indicates that the variation from the usual rule must be accounted for in some other way.

In reference to this series, Dr. Förstemann292-1 remarks: