These examples are sufficient to show that the series of the Ma.n.u.script 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 Cortesia.n.u.s 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 subst.i.tuting 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 Forstemann"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 a.s.sume 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 ill.u.s.trations 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 x.x.xII* of the Ma.n.u.script 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 Forstemann"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 Forstemann.[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 ma.n.u.script 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 Forstemann"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. Forstemann[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 ma.n.u.script.
Turning to the Ma.n.u.script 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 s.p.a.ce that only four were intended to be given. The numerals in Bra.s.seur"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 a.s.sume 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 ma.n.u.script 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 x.x.xII*_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.