These figures actually occur in all the nine pictures with the exception of the first, which has no rectangle at all, and where in true Maya fashion, the zero-point is concealed.
I go still further in my bold hypothesis. The time of the apparent revolution of Jupiter has been placed at 397 days. The Mayas, I think, computed it at 398 days. In the passage alluded to I regarded the following as the sign for Jupiter:—
We find these signs in pictures 4, 6, 7 and 9. The corresponding numbers reduced for the revolution of Saturn are3986, 6762, 7972 and 10,748. I assume that the third picture,i.e., the number 2776, is another zero-point, in consequence of which the sign is here suppressed, and that still another is the tenth picture with the number 11,958, which has no relation to the revolution of Saturn.
If we compare these numbers with the 398,i.e., the apparent revolution of Jupiter, we have the following:—
The differences 10, 6, 4, 12, 2 and 18 are so small in comparison with 398, that the numbers 2776, etc., might very well have been regarded as approximate multiples of the revolution of Jupiter. And the remainders in the seventh and tenth pictures could be still further reduced. In the seventh picture, the first sign is very unusual and one which I do not remember having met with elsewhere. If it should be possible to regard it as the number of the thirteen week days, then it would follow (the Saturn sign being regarded as unimportant) that the contents of the rectangle meant:— 13 + a multiple of 398, by which this remainder would be reduced to-1.
The tenth picture has the crossbas the beginning of the rectangle. This is the sign for union, very often denoting especially the union of all the twenty days. Thus we have here (aside from the middle sign to be discussed later) the formula:—20 + 30 × 398-2 = 11,958, or even 20 + 30 × 398 = 11,960.
The regular progression from the 7th multiple to the 10th, 17th, 20th, 27th, and 30th multiples in the above six equations is also somewhat in favor of my theory, while the four rectangles without the Jupiter sign are by no means multiples of the Jupiter revolution:—
Let us now try to interpret the meaning of the remaining rectangles (always omitting the Saturn sign as a matter of course.)
In pictures 2 and 8 the rectangle also contains the sign of the moon or of the twenty days. Beside it in picture 2 is the sign, which in my "Erläuterungen," page 16, I regarded as the sign for Mercury. Hence we have here 20 + 15 × 115 = 1745,i.e., only 3 units more than the required 1742.
The rectangle with the eighth picture contains in addition to the moon a sign which looks as if it were intended for a whole divided into four parts. Until something better (perhaps the the sign of Venus) is proposed, I will assume that it is the quarter of the Tonalamatl,i.e., 65, and I take the required number to be 9714 in the form of 20 + 149 × 65 + 9.
Above the third picture I see a Mercury and a Venus sign and I read 584 + 19 × 115 = 2769, which is only 7 units less than the required 2776.
The fifth picture still remains to be discussed, but I do not know how to unite the Mercury revolution here with the 5728. For the present, however, I am inclined to believe that there is a mistake in this passage.
We pass now from the obscure contents of the rectangles to the equally mysterious pictures themselves.
Aside from the tenth picture, I find human forms in four pictures.
Picture 1, page 53a, is the death-god (A) seated and pointing upward, an appropriate representation for the zero-point of the Saturn series,i.e., for the end of the preceding revolution.
Picture 2, page 55a, contains the head of a deity, probably D's with the suggestion of a beard and the sun-sign on his forehead. The head is surrounded by a ring striped black and white.
Picture 3, page 56a, is the head of B, again with a beard and with the sign Kin (sun) above. The head is surrounded by a design, the left part of which is black and the right white.
Picture 6, page 53b, represents a hanged woman, which Schellhas, "Göttergestalten," page 11, takes to be the Maya goddess Ixtab, the goddess of the halter,i.e., of the hanged.
The centre of picture 4 on page 57a, contains the suggestion of a face, perhaps in place of the Ahau sign, and on either side of it is a black and white surface.
It is further important to note that four times in this section Kin (sun) forms the centre of the picture, viz:—pictures 5, 7, 8 and 9, pages 52b, 54b, 56b and 57b. In all four cases there is on either side of Kin a black and white surface, such as we have already seen in picture 4 and similar to that in picture 3. Pictures 8 and 9 are vomited up, as it were, by a serpent placed below them, in the same way as B is represented on pages 34b and 35b. In pictures 5 and 8, four objects suggesting arrows extend from the Kin in four directions and probably denote the four cardinal points or the four Bacabs, of which we shall have more to say presently. Two of these arrow-like signs also appear in picture 7, page 54b, but only on the black and not on the white surface.
I will postpone discussing picture 10 until later and pass on to the hieroglyphs above the first nine pictures, about which it is true I have nothing satisfactory to say. There are always properly speaking ten of these hieroglyphs, among them the two signs for the sun and moon. But the scribe introduced the latter only in pictures 1-4, and also with the more elaborate last picture 10. With pictures 5 and 9 he omitted these signs in order to represent the other eight larger and with greater distinctness of detail. Among these hieroglyphs are several of gods, especially that of A with pictures 1, 5 and 9, and H with picture 5, and with pictures 1, 3, 5, 7, 8 and 9 there are other heads, some of them bird-heads, regarding which I am uncertain.
The Ben-Ik sign, to which I have assigned the meaning of a lunar month, belongs with pictures 4, 8 and 9 and occurs twice each with pictures 1 and 10.
I am inclined to see the sign for Mercury in the crouching figure belonging to pictures 9 and 10, which is drawn upside down and combined with the half Venus sign (11958 = 104 × 115-2).
Hands grasping a hieroglyph (a sign for 20 days?) are represented in pictures 1, 7, 8 and 10.
The enigmatical numbers, prefixed to the hieroglyphs, occurseveral times, thus a 1 with pictures 1 and 10, and a 4 twice with picture 8 and a 6 with picture 3.
Now let us examine picture 10 somewhat in detail and also the signs standing above it, since both are of special significance here. This representation treats of the period of 11,960 days in which the Mercury and lunar revolutions meet. And this is proved by the ten hieroglyphs, which I will number as follows:—
I can omit Signs 3 and 8, sun and moon, since they refer to a period of time only in a general way. Sign 1 seems to me, as I have already stated, to have reference to the revolutions of Mercury. Then follows sign 2, the upper part of which is a mat and the lower the Muluc sign. I believe this sign is intended to denote that the beginning of this period is in a Muluc year. Indeed, our examination of pages 51-52 showed that it was the year 6 Muluc. The mat (Pop) is very properly the symbol of beginning, since the first month of the year was likewise called Pop. Sign 7, it seems to me, indicates that this period should be divided into lunar months (denoted by Ben-Ik), and, as I have already demonstrated in my examination of page 24, the length of the period is stated here by Signs 4, 5, 6 and 9, but the dot before the fifth should be placed before the fourth, as is actually the case on page 24. Therefore:—
It is perhaps not accidental that the ninth sign is that of the fourteenth month, which signifies the expiration of the preceding lunar month, for here the month begins with the first day of the fifteenth month.
Sign 10 is doubtless Xul = end, as it so often is, for example, on pages 61-62 below. But I have not solved the meaning of the two prefixes. The end would be XII Lamat 16 Yax (13 Ix).
The picture represents a human form, which has in place of a head a design somewhat resembling the head of a lance. It is sitting with legs spread apart, and in this respect may be compared with god B of Cort. 9, who is represented in the same way. In the picture before us, the figure holds in its upraised hands the sun and moon signs, which are constantly repeated throughout the series. The Venus sign is placed between the outspread legs. In the rectangle above the figure, this sign is repeated in a more concise form, while on the left the crossbappears as the sign of union or multiplication, and on the right that of Jupiter, whose period of revolution is here multiplied by 30 (30 × 398 = 11,940). And the two Venus signs can mean nothing more than that this period of 11,960 also serves the purpose of filling up the gap between the two large Venus-solar periods of 37,960 days, like the similar process which we saw on pages 46-50.
We have examined first the series and then the pictures with the hieroglyphs belonging to them. Let us pass now, as the third step, to the examination of the two rows of hieroglyphs extending above the numbers throughout the whole section. First of all, I will again set down here the position of each of the sixty-nine groups:—
Since each group contains two hieroglyphs, this makes 138. in all. Of these, however, about 24 on the upper halves of the pages, are wholly or almost wholly effaced which very materially hinders the trustworthy determination of the context.
Furthermore group 59 is entirely lacking or rather group 58, in the place of which the 59th has been set down. The eighthpicture was probably already drawn, when the artist saw that there was not room enough left for the 58th and 59th groups. Hence he omitted the 58th, setting down in place of it the 59th and in the place of the latter he set down the zero mentioned above.
The question now arises:—Are these hieroglyphs dependent upon the days and numbers of the series and upon the pictures, or are they entirely independent of them?
I find butonepoint in favor of the first possibility, viz:—the Venus sign in group 4b (I will designate the upper hieroglyphs byaand the lower byb). It is placed in the period indicated in which 502-679 days elapse, and in which, therefore, Venus has finished a revolution of 584 days. It may be, that by way of exception, this significant date was intentionally recorded.
On the other hand, there are many things, which favor an entirely different interpretation of these hieroglyphs. Thus I am of the opinion that the ritual year of 364 days with its four Bacab periods of 91 days each is referred to here, as we have already found it referred to on pages 31a-32a and on page 45a, and shall find it again on pages 65-69 and 71-73. In that case the single groups would be separated from one another by one Maya week = 13 days.
I will now arrange the sixty-nine groups in the following order (the reason for which will become clear directly):—
The groups in a horizontal row are separated from one another by 7 or a multiple of 7. If now a hieroglyph is repeated in those places, which are in the same horizontal row, then this is a confirmation of the supposition that Bacab periods are meant to be represented here. Hence I will examine each row in turn. These rows extend over the long period of 69 × 13 days probably merely for the purpose of filling up the space.
I. In 39b, 46b, 53b and 60b,i.e., after every seven groups, perhaps also in 18b, we find the following sign, which I identified as that of a Bacab, in Globus, Vol. LXXI:—
Hence this denotes the beginning of the Bacab period. In 4b the sign is replaced by that for Venus. In 11b, 25b, 32b and 67b we find other signs, it is true, nevertheless the regularity stated above cannot be accidental. The upper signs of groups 39a, 46a, 53a and 60a contain an Imix and corroborate the connection.
II. 5b and 26b (after 3 × 91 days) contain a head very like the preceding, which readily suggests the idea that it is merely a Bacab sign pushed one group ahead, but it also appears in 13b, 50b and 52b.
Then 12b, 54b and 61b correspond,i.e., after six groups of 91 days and one more of the same length, but the same sign appears also in 34b, 48b and 56b.
III. 41b and 69a are Xul = end and are therefore separated by 28 × 13 = 4 × 91 days,i.e., the length of a year. It is singular that both signs of 41 are like those of 47; if we assume that 47 was set down one group too soon, it would be in excellent keeping with the rest. The Xul also appears in 11b and 28b. 34b and 48b correspond after 2 × 91 days, as already mentioned under II.
IV. 42a and 49b both contain the sign for the sun between clouds.
V. 36b and 57b agree after 3 × 91 days; the same sign appears again in 10b and 20b.
15a and 36a correspond after 3 × 91 days; we shall continue the examination of this sign under pages 71-73.
VI. 37a and 65a agree,i.e., after 4 × 91 days = a year. The sign contains a human figure stretching both arms aloft. The passing of a year was likewise indicated in III, but a year coming 52 days later than this.
VII. 10a and 31a agree,i.e., after 3 × 91 days. The sign is composed of the crouching figure prefixed to the cross, which we also find in 12b, 35a and 65b; it is prefixed to a different hieroglyph in 30a. In 38b, 52b and 59b (58 in the Manuscript) we see bird-like heads resembling the Bacab sign. We should expect to find a familiar sign in 45, which is drawn between these, but a Moan appears there instead. These signs seem to indicate the end of the Bacab period. Does the Moan sign here, too, suggest the end of the year?
In 38a, 52a and 59a we again see an Imix, and I consider it a corroboration of my theory that all the four signs of groups 38 and 39 are repeated in 52 and 53 after 2 × 91 days.
I believe a further corroboration is the fact that though many of these hieroglyphs have no connection with these periods of 7 × 13,i.e., with the divisions of the ritual year, they do correspond with the usual divisions of the Tonalamatl,i.e., 4 × 13 and 5 × 13 days.
After 4 × 13 or a multiple of it the signs recur in 20b, 24b, 40b, 44b-12b, 48b, 56b-16b, 32b, 64b-26b, 50b-10a, 30a-37a, 65a-15b, 51b-11b and 47b.
As examples of 5 × 13 I would mention 3b, 63b-10a, 20a, 30a-5b, 50b-24b, 29b-35b, 65b-15b, 20b, 40b.
Finally, I must mention two more hieroglyphs, which are limited almost entirely to these pages:—
In the first sign, which occurred on page 10a, I thought I recognized the lunar month of 28 days. It occurs in this section in connection with the third picture on page 56, and besides in the following groups of hieroglyphs:—16b, 32b and 64b, always combined with a Yax. The regularity of the intervals is striking, but as yet I can neither explain that, nor the crouching personage (Mercury?) in the 10th, 20th and 30th groups and again in the next, the 31st.
The second sign is foundonlyon these pages and here not less than eleven times, possibly with the addition of theeffaced sign in 6b and 27b which may have been the same hieroglyph. The eleven places in which it occurs are as follows:—3b, 15b, 17b, 23b, 24b, 29b, 40b, 44b, 49a, 51b and 63b. Two different prefixes are added to it; one in the first two and the last two places and also in the last but two joined with Kin, and the other in the six middle places. Of the eleven groups, 17 and 24, 44 and 51 are 7 groups apart, 3 and 17, 15 and 29, 49 and 63 are 14 groups apart, 23 and 44 are 21 groups apart, and hence 23 and 51 are separated by 28 groups or 1 year. Group 40 alone is not concerned with these intervals of seven or multiples of seven.
Now, how far may all these periods of time be due to accident and how far to design? Accidentaloneis quite out of the question. The frequent repetition of the sun-sign in groups 49, 50 and 51 on pages 54b and 55b, seems to me to refer to the conjunctions of the sun with certain stars, which occur at intervals of thirteen days.
Pages 58—59.
This section is also based on a series occupying the whole of page 59, which contains nothing but number and day signs. This series has the difference 78, which we found once before on page 44. There the starting-point was III Lamat, here it is the day XIII Muluc, probably coming in the year XIII Muluc, as in Cort. 40b, as I shall have occasion to suggest later. The series extends, with the usual errors and variations, in four divisions from top to bottom. The days, which are always two days behindhand, owing to the number 78, in 780 again reach the day XIII Muluc, at which point the succeeding members remain stationary, since from here on the difference is always 780 or a multiple of it. 780 days are, however, the apparent time of the revolution of Mars, which is the only planet now left to be discussed, the subject of pages 46-50 having been Venus and the sun, and of pages 51-58, Mercury and the moon with incidental treatment of Saturn and Jupiter. With 780 as its difference, the series ascends to 19 × 780 = 14,820, and then continues with this large number as its difference until the series is lost in the effaced passages.
Curiously enough, however, directly under the line containing the 14,820, there is a new series composed of nine members, or ten counting the suppressed starting-point. But this starting-point is again the day IX Ik, the difference, as proved by the annexed days, is again 78 and the series ends with 780. Thus the starting-point is the only difference between the two series. The principal series contains all the even and the secondary series all the uneven days. Can the starting-point of the revolution of Mars have been determined according to different principles? Is it possible that in one case the beginning of the planet's retrogression was adopted as the starting-point, and in the other case the date on which the planet, after completing its retrograde course, again reached the degree of right ascension at which it had begun its retrogression? This is a difficult matter to decide, since the period of the retrogression of Mars fluctuates between 62 and 81 days. The interval from IX Ik to XIII Muluc is 147 and in reversed order 113 days.
It can hardly be assumed that the 19 of the IX 19 or IX Ik is connected with the 19 × 780 mentioned above or with the 19 + 19 + 19 + 21 into which the 78 is divided on pages 44-45, or finally with the 19, which four times forms the principal part of the sub-divisions of 65 on pages 33-34.
Numbers amounting to millions accompany this series in the usual way. Two of these are on page 58, viz:—1,426,360 and 1,386,580; but with the sign of the sixth day, which is important here, between them. Below these numbers, however, are two month dates:—first the normal date IV Ahau 8 Cumhu and, if I have correctly restored the effaced number before the month sign, which in its turn is indistinct, the second is XIII Muluc 2 Zac, which would fall in the year VIII Muluc. The encircled numbers also occur here. They are set down beside the lower number of seven figures. We find here a red 12 with a black 1 inserted, below this a black 7 and below this again, enclosed in a red band, a black 11, which I regard as also representing the value of a red number. We shall find a similar instance among the serpent numerals. Then we have here 1. 7. 11. = 511 and 12. 11. = 251. But 511 = 260 + 251 and 251 is the interval between XIII Muluc and IV Ahau.
With the day XIII Muluc and the interval 9 between IV Ahau and XIII Muluc, numbers for XIII Muluc have been formed amounting to millions, which, however, have been suppressed in the Manuscript, just as they were on page 31 where, in like manner, numbers were first formed for day XIII Akbal.
I assume that to begin with, 76 Tonalamatls (= 19,760) were added to this 9 and then 228 Tonalamatls (= 59,280), the 228 being = 3 × 76 and the 59,280 including 76 revolutions of Mars.
The result in one case was 19,769 and in the other 59,289. If the 12 Ahau-Katuns, which are specified as 1,366,560 on page 24b be added here, we have the following numbers:—
and if the two encircled numbers of the Manuscript:—251 and 511 be added, the sums are 1,386,580 and 1,426,360,i.e., the two large numbers of the Manuscript.
The dates corresponding to these numbers are as follows:—
If we compare the two numbers with the normal date, the curious result follows that:—
1) 1,386,580 - 1,366,560 = 20,020.
This number is equal to 55 × 364, including therefore the ritual year of 364 days.
2) 1,426,360 - 1,366,560 = 59,800.
This number is five times 11,960 days, which is assumed to be the time in which the lunar and Mercury revolutions accord. This 59,800 was found once before on page 24 as the suppressed difference between 68,900 and 9,100.
Thus the separate sections (of the book) are very closely connected.
If the two large numbers be compared with one another their difference will be found to be 39,780. This is equal first to 51 Mars revolutions of 780 days, and second to 4420 × 9,i.e., a multiple of the interval between IV Ahau and XIII Muluc.
Now we must direct our attention to the seventeen hieroglyphs, which we find in the two columns on page 58, apart from the matter-of-course calendar date at the top, which is repeatedat the bottom. One column contains 11 hieroglyphs and the other 6. I will here advance the following theory in regard to these hieroglyphs, which may serve until a better is found:—
Since, as a rule, the Tonalamatl is divided into 5 × 52 days, I believe that each group of three Tonalamatls treated of on page 59, is divided into 15 of these parts; that each hieroglyph, therefore, denotes 52 days and that the first three parts are separated from the others by the signs of beginning and end in the first and fifth places, so that three of these parts, which equal 156 days, always form a separate group. 156 is the 5th part of 780. With the omission of the first and fifth signs, the passage, I think, stands thus:—
If we adopt this arrangement for the present we cannot fail to see that the author had an aim in view, when we consider the following:—
1. The zero-point lies 15,609 days later than the normal date IV Ahau 8 Cumhu (9 Ix). This is equal to 20 × 780 or 60 × 260 increased by the interval between IV Ahau and XIII Muluc = 9. There are 86 days between 2 Kankin and 8 Cumhui.e., 15,609 = 43 × 365-86, and from 9 Ix to 13 Muluc it is 43 years.
2. The same zero-point, 13 Muluc, lies in the year with the same name, that is, the very point where a Tonalamatl of the year ends.
3. In this arrangement the first as well as the last day of the year 1 Ix is exactly reached in the second and ninth groups. While the meaning of the second is as yet unintelligible to me, the end of the year is appropriately indicated by the ninth with its compound of Kin and the year-sign, above which there may be an Ix as a superfix, but misshapen for want of room.
4. Also the fact that it is the first of the two columns, which closes with this year-end, seems to show a purpose.
5. Several instances of similarity appear among the hieroglyphs in these groups of three:—an Akbal sign in 1 and 4 suggests the god D, the superfix and prefix of 2 and 14 the god K and 5 and 11 the screech-owl and therefore A.
Little else is to be said of these hieroglyphs.
C might be denoted by 3 (13 Zip) and 10 (12 Zip). Group 8, the central point of the series, has on the left and right the signs for the north and south as if the time between the north (Muluc) years and the south (Cauac) years were meant to be indicated here.
I am inclined to consider the crouching personage in 12 as the revolution of Mercury, which requires 115 days:—573 is 5 × 114 + 3 or 5 × 115-2.
Is 7 a sign, as yet unknown, for the year of 364 days?
15 looks like two signs for the month Mac, placed back to back, which here designates the Tonalamatl as it does on page 24. The superfix of three parts might denote three Tonalamatls = 780 days. The familiar sign in the fifth place in connection with the expiration of the first Tonalamatl is striking; it is the one usually identified as that of the screech-owl or death-bird.
Page 60.
This is the last page of the front of the second part and is divided into four sections:—at the top we find hieroglyphs, below these a picture, then hieroglyphs again and in the lowest section another picture.
The upper picture contains first a rectangular elevation like a platform. Enclosed in this rectangle is the picture of the animal resembling a dog lying down, which we have often met with, the last time on page 47. In front of the dog is a hieroglyph, which, I regret to say, is still unknown and which occurred six times as a heading on page 23b. On the platform two personages are fighting; one is in war-dress holding in his left hand the throwing-stick or atlatl, and in his right probably arrows; the other figure, whose back is somewhat indistinct owing to obliteration, is apparently unarmed and is making a defensive gesture with one hand. Beside the platform, and therefore on a lower level, is a second person walking behind the armed person as if to help him. He too is in war-dress and likewise holds an atlatl. A black 3 is set down between the two combatants, and there may also have been a red 2, which is indistinct owing to the red background of the picture.
Let us next examine the lower picture. A blindfolded personage is kneeling on the left. A serpent's head rises from the ground in front of him. A second serpent rises in several coils on the shoulders of the blindfolded personage and on the serpent's neck sits enthroned another personage, who is rather indistinct, holding a spear and a shield. On the right, opposite this group and facing it, is a second. A personage with arms bound and bowed head is sitting on the ground. There is a black ring around his eye. Behind him stands the victor in war-dress and again equipped with spear and shield. There is a red 11 and a black 2 between the two groups.
We see that the reference here is to combat, just as it was on the right side of pages 46-50. And since the subject of these pages like that of 46-59 is confined to the revolutions of the planets, it is natural that the pursuit of one by the other, their periodical disappearance, the crossing of their orbits and the variation in the length of their revolutions should be looked upon as a contest. Therefore, since the sun, the moon and the five planets have hitherto been treated of on these pages, I look for these seven heavenly bodies in the seven personages pictured here on page 60. I will attempt to explain them, hoping that my interpretation may be replaced by a better one.
The sun and moon stand on the platform in the upper picture; their combat is equivalent to the eclipses to which they at times succumb. The moon is the assailant and the sun makes only a proud defensive gesture. The person behind the moon must be Mars. The animal under the two persons is the embodiment of the eclipses, which the Aztecs interpreted as the act of being devoured by the jaguar. The hieroglyph in front looks very much like the meeting of two circles. Does it refer to the day Lamat (Aztec tochtli = rabbit)?
At the left, bottom, the powerful Venus triumphs over the weak Mercury. The two planets are real chronometers by reason of the regular alternation of their appearance as morning and evening stars, and also by their disappearance twice in each revolution and finally even in the variation in the length of the two periods of invisibility. Hence they are each accompanied by a serpent as the usual symbol of time.
On the right, on the other hand, Jupiter as the stronger has vanquished Saturn, whose bound arms symbolize his slowness of motion and the fact that he is confined to the same region of the sky. Should not the ring around his eye have a very special meaning? But we must guard against an excess of imagination. Jupiter and Saturn are the last to be represented, as they were of but secondary importance, on pages 51-58 and perhaps also in the 2200 on page 24.
I will not deny that yet another interpretation of this page is possible. The top picture may be Venus and the moon opposing one another, and the bottom picture may represent the sun as victor over Mercury. There are some things in favor of this point of view.
The correct order of the twenty-four hieroglyphs is the following, in my opinion, which is borne out by the different colors of the four groups:—
These signs can have no relation to mythology. There is not a hieroglyph of a god among them, for if sign 6 could be taken for B's hieroglyph, the resemblance to the sign of the fist, familiar from the inscriptions, as well as the Imix and the cross-hatching as a prefix, makes this doubtful. The latter component would rather suggest the summer solstice. If sign 12 were intended to denote the Bacab, then it would refer to chronology rather than to mythology. Also the Cimi in 17 might equally well mean the day as the god. Indeed several things refer here to chronology and astronomy, among them the unmistakable union of numbers and month signs, which occur here repeatedly. Thus from what remains of the almost obliterated signs 1 and 2, they might denote the normal date IV Ahau 8 Cumhu, which always occupies the first place. Signs 7 and 20 are plainly the same, 9 Xul (sixth month) and sign 14 is 10 Yaxkin (seventh month). Sign 5 might be Caban combined with Uo (second month) and a ten. In sign 19 we again see Yaxkin without a number. Signs 9 and 23 are Zec (fifth month) and signs 21 and 22 may be Kankin (fourteenth month). The days occur in the same manner as the months. It is true that Kin is only a part of hieroglyph 10, the rest of which is effaced, but the familiar compound of Caban and Muluc appears in 18 and 24 and Cimi is in 17, as we have seen. In sign 13, Ahau is combined with a red number, which must lie between X and XV. But this should not be regarded as forming a calendar date with the 10 Yaxkin near by, for Ahau is never the tenth day of a month. Can 16 be the sign of the twelfth month, Ceh, combined with that for 7200? Hieroglyphs 3 and 8 are effaced and I do not understand 4, 11 and 15.
There are no parallels in the kindred passages 46-50, unless it be 7 Zec on the bottom of page 49 and here in signs 9 and 23, but without a number. Cf. my paper on this page 60 in the "Weltall," year 6, pages 251-257.
Page 61—64.
On examining the reverse of the second part of our Manuscript,i.e., pages 61-74, we find an empty page on the left, the back of which is occupied by page 60. This may be explained by assuming that the scribe wrote pages 61-64 and possiblyeven pages 61-74 from right to left, the great series having occasioned such a proceeding, and that his material came to an end when he had finished page 61. Nevertheless, it is advisable to continue with the original numbering in order to avoid confusion.
Aside from the concluding (or beginning) page 74, this whole section of pages 61-74 consists of three parts:—61-64, 65-69 and 70-73. Let us first consider the first section, which I have already discussed in my treatise "Zur Erläuterung der Mayahandschriften II."
The basis of this section is a series, the beginning of which is on the bottom, right, of page 64. Its primary difference is always that which we found on pages 31-32, viz:—the Bacab period of 91 days, the quarter of the ritual year of 364 days = 7 weeks of 13 days each. It ascends by 91 until it reaches 1820, which number is a multiple of both 364 and 260 and is also divisible by 28, the number of weeks in a year. Just as on page 32 the series continues with the new difference 1820 as far as 7280, its fourth multiple, which then becomes the third difference. Indeed, I believe that even the partially effaced numbers could be so restored as to carry the series to the number 36,400 = 400 × 91, which would then become the fourth difference and the series would close at the top of page 63 with 145,600 = 1600 × 91,i.e., with the numbers 1. 0. 4. 8. 0. of which the 1 is entirely and the 0 half effaced. The series on pages 31-32, however, closed with 29,120 = 320 × 91, but there is still room for a higher series.
Under this largest number (1600 × 91) there is on page 63 a large red number consisting of 19. 0. 4. 4. which is crowded into a very small space between the figures of 1820. I can only understand it by replacing the first 4 by a 3, for then it is 136,864 = 1504 × 91 or by addition of a zero. We shall return to this number in the examination of the serpent numerals.
The series is accompanied in the regular way by five days. At the beginning of this series, page 64, right bottom, are the days III Cib, III Men, III Chicchan, III Caban and XIII Ix; the III is set down only with the first of these days and is to besupplied with the next three. Hence the actual zero point is to be found 91 days back in the days III Chicchan, III Kan, III Ix, III Cimi and XIII Akbal, the last of which is also the beginning of the corresponding series on page 32. From 1820 on, these last-named days, of course unchanged, accompany the numbers. The most important of these days are the first and last, but we shall see later in connection with the serpent numbers that the other three, which are separated from one another by 39, 130 and 52,i.e., 3 × 13, 10 × 13 and 4 × 13, are likewise not set down here by mere accident.
We come now to the five columns, three on page 63 and two on page 62, which join this series on the left. They contain the large numbers, which invariably accompany these series. Here there are six numbers, four of which, in my opinion, refer to the past and two to the future. Two of these numbers, the two largest, are set down together in the third column on page 63, one with red numbers and the other with black. Of these black numbers, I take the second from the top to be not 8 but 13, assuming that a line is omitted. The normal date IV Ahau 8 Cumhu from which, as the starting-point, all these numbers are to be computed, is set down below at the end of each of the five columns.
I now give the six numbers, first the two highest, then the other four from right to left, adding in each case the calendar date and the year in which they should be situated:—
The first, third and fifth numbers are already known from page 31a, and hence they need no further discussion here.
As these three numbers depend on the day XIII Akbal, so the other three all proceed from the day III Chicchan in the following positions, which are again suppressed in the Manuscript:—
The second date in the manuscript is 13 Kankin and the third is 13 Zip; hence there is one line too many in the former number and one too few in the latter. While on page 31a the origin of the numbers belonging to the day XIII Akbal seems to be quite clear, here their relation to one another is entirely concealed. I must, therefore, refrain from expressing any conjecture in regard to them.
Now the numbers set down in the Manuscript are formed only by the addition of the encircled numbers also found there. The encircled number for the first expressed number is 51,419, which is the same number we found with the corresponding day XIII Akbal; the second has 235 and the third 456 = 260 + 196. The 51,419 was 197 × 260 + 199; but 199 is the interval from III Chicchan to VII Kan, just as it is from XIII Akbal to IV Ik. The 235 is the interval between III Chicchan and IV Ahau and the 196 that from III Chicchan and IV Imix.
By the addition of these differences, the numbers written out in the Manuscript are obtained:—
Keeping in mind what was said in reference to page 31a, let us now examine the six numbers and dates collectively.
The fact that the days IV Ahau and XIII Akbal occur here and consequently also III Chicchan is not surprising. Nor is the choice of VII Kan and IV Ik an accident, for the interval between these days is exactly the same as that between III Chicchan and XIII Akbal, viz:—218 days.
Hence the distance from III Chicchan to VII Kan is also exactly equal to that between XIII Akbal to IV Ik, viz:—199 days.
Finally, the distance from VII Kan to III Chicchan is exactly equal to that between IV Ik and XIII Akbal, viz:—61 days.
IV Imix and IV Kan are separated from the normal date IV Ahau by 3 × 13 = 39 and 8 × 13 = 104 days.
Regarding the encircled numbers, so far as they are independent of 260, I would note the following:—
In addition let me remark that 36 = VII Kan to IV Ahau, 39 = IV Imix to IV Ahau and 104 = IV Ahau to IV Kan.
The following arrangement will prove that these numbers were as usual also employed to form the large numbers by multiplication:—
But the highest number, 1,538,342, was formed in a different way; it = 59,167 × 26; but the interval from IV Ahau to IV Ik = 182 = 7 × 26, and from IV Ik to IV Ahau = 78 = 3 × 26.
If in conclusion, we now examine the twelve numbers of seven figures given in this section, we will clearly see that by twos and twos they plainly belong together in pairs:—
The three pairs of numbers found by computation are as follows:—
Difference 3338 = 12 × 260 + 218 (VII Kan to IV Ik, III Chicchan to XIII Akbal).
Difference 34,538 = 132 × 260 + 218 (as above).
Difference 42 (which is 260 - 218); 42 = IV Kan to IV Imix.
On the other hand the three pairs specified in the Manuscript are as follows:—
Difference 3338 = 12 × 260 + 218 as above, by reason of the encircled number 51,419 which is common to both numbers.
Difference 34,320 = 132 × 260, on account of the same day.
Difference 117 = IV Kan to IV Imix; strictly speaking 377 = 260 + 117.
The upper part of the five columns just now under discussion still remains to be examined. Here are five vertical rows of hieroglyphs, the first four each containing seven, and the fifth only six owing to lack of space.
The two rows at the top are as usual much obliterated, which is the more to be deplored since they consisted of five calendar dates, which would have contributed materially to the comprehension of the entire section. Fortunately, however, one of these dates is preserved complete, and we are able to see in what relation it stands to the rest. Thus we find in the third column of page 63 the date XIII Imix 9 Uo. It comes in the year 12 Ix and represents the number 1,523,921 (or a number separated from it by a multiple of 18,980). Now 1,523,921 = 4175 × 365 + 46 and = 5861 × 260 + 61. This agrees with the lower number inserted in red:—1,538,342 = IV Ik 15 Zac (12 Muluc), which comes later by 14,421 = 39 × 365 + 186 and = 55 × 260 + 121. 121, however, is the difference between both XIII Imix and IV Ik and the days XIII Akbal and IV Kan in the last column of page 62. If we set down with these two numbers, those of the normal date just preceding and the normal date next following, we have
This is a period of 37,960 = 2 × 18,980 days. It is possible that at some future time an indication of such a transition fromone Katun to the other will be found in the writings. Now these two top lines contain two dates; on page 62 we find 13 Ceh, and on page 63, 13 Xul, but nothing further is to be learned from this than that one or the other of the day-signs, 2, 7, 12, 17, must have been set down in the effaced indication of the position in the Tonalamatl. All else is obliterated. From the third to the seventh row of these five columns it is all extremely simple. The third row consists only of five signs for beginning, the fourth, of five for end, the sixth of B's sign five times and the seventh of the elongated headqfour times. But in the fifth, two deities alternate, one is apparently male and the other female; the god is in columns 1, 3 and 5 and the goddess in 2 and 4; the god probably belonging to the days III Chicchan and the goddess to XIII Akbal.
If we look upon this series as the first story of a structure and the large numbers just now discussed as the second, then we find the third story here, as we shall find it again on page 69. In the passage on page 31, which is so closely related to the present one, a timid attempt has already been made with the number 2,804,100 to erect a third story of this kind, which however barely attained to a quarter of the height of the one which now engages our attention. If the numbers hitherto examined refer to a time not very far from the present, we now come to numbers which lie in so remote a future that they can hardly suggest anything else than the destruction of the world or a sort of twilight of the gods. Nevertheless the starting-point of the whole, the series, which is built up with the number 91,i.e., the Bacab period or the quarter of a ritual year, continually comes to view. Indeed, the number of serpents is suggestive of this.
There are four large serpents, which fill most of the space on the left half of page 62 and the right of page 61. The two outer ones are bluish and the two inner ones white. They rise in several coils, their tails below and their heads above. A deity is represented above the gaping jaws of each of the four serpents, having apparently been vomited up. Above the first and third serpents B is represented in a fashion very similar to that which we have already seen on pages 33-35. Above the first serpent B has the pouch hanging from his neck and his hatchet is helddownward; above the third he wears the pouch and the gala mantle and his hatchet is raised. Above the second fourth serpents, on the other hand, there are four-footed animals, but of a species not represented elsewhere. They might suggest a (four-footed?) walrus and a bear. We have here a double contrast, apparently referring to the four cardinal points.
The veil enveloping this representation would be lifted to a considerable extent, if all the eight hieroglyphs written above each serpent, were still legible. But, unfortunately, the second group is wholly and the third almost wholly effaced, while the first is partially effaced and only the fourth is preserved in its entirety. I read these groups in the following order:—
Of these 7 and 8 in the first and fourth groups form the date IX Kan 12 Kayab, which is in the year 4 Ix; this same date probably occurred also in the other two groups. That it is of special importance here, is shown by the two columns of hieroglyphs on the left side of page 61, where this date occurs again in the lowest place. The last three large numbers are not computed from the normal date IV Ahau 8 Cumhu, but from this very date and the other five from a similar one. The sixth hieroglyph in the first group seems to correspond to the fifth in the fourth, since both contain the elongated headq, though with different accompaniments.
In the first group the fourth hieroglyph is the Bacab sign familiar to us from pages 51-58, suggesting that the series here is closely connected with the one which had the difference 91. The fifth sign of the same group is that forbeginning, probably to confirm the fact that this section begins here. The third sign of the first group is probably an Imix, as it is in the first and fourth of the fourth group, combined here with the woman's head, which we saw repeated on pages 62 and 63 at the top; and over it in the second place of the fourth group is B's hieroglyph, whichis also repeated on pages 62 and 63 at the top. The third place of the fourth group is occupied by a head, which may be C's and which is distinguished by the same kind of circle which on page 9b surrounded the Ahau.
Eight complete dates are set down below the serpents, among which are the XIII Akbal already found with the previous large numbers, and III Chicchan (repeated three times), and then III Kan (twice), forming the beginning and end of the series (page 64), and also III Cimi and III Ix. As we shall see directly these are the end dates of the large numbers, and Xul = end repeated eight times at the extreme bottom corresponds with this. On the other hand, the starting-points must be found by computation, with the exception of the date IX Kan 12 Kayab, which is actually written down and is the point of departure for three of the numbers.
I will designate the black numbers byaand the red byb. Seven of the eight numbers are undoubtedly absolutely correct; but I must alter the number 1b, the red number belonging to the first serpent. I assume that a line is wanting in the lowest figure,i.e., it should be 8 instead of 3, and that the conspicuously large 1 further down on the page serves also as the red number, which belongs here. Only one slight change is necessary in the dates on the bottom of the pages, which were mentioned above. To the 16 in the date 4b I add a dot, and read it 17.
I will now give a table of the numbers, the starting-points of the periods obtained by computation, and the ends of the latter which are indicated below the serpents:—
See my treatise, "Die Schlangenzahlen in der Dresdener Mayahandschrift" (Weltall, year 5, pages 199-203).
Several details show how this number-structure forms a definite, closely connected whole.
1. The beginning day in each case is the day Kan, which thereby indicates its position as the first.
2. The last three starting-points are the same; the first three end dates, at least, are the same in the Tonalamatl, though not in the year.
3. The two numbers 2b and 4b are exactly the same.
4. The first three numbers are each divisible without a remainder by 17, the interval from XIII Akbal to IV Ahau, which was true also of the 1,268,540 in the second column on page 63, although only this last number has anything to do with these important days, of which the other three numbers are independent.
On the other hand, a notable difference between the first serpent and the other three is, that the day XI Kan is the starting-point of the first and IX Kan of the others. There are, however, 80 days between IX Kan and XI Kan. Hence the numbers 2a and 1b are separated from each other by 66,640 = 256 × 260 + 80, although they have the same end III Chicchan.
Further it is to be noted that the largest of the eight numbers, 12,489,781, is separated from the lowest, 12,388,121,i.e., the black number from the red one of the first serpent, by only 101,660,i.e., by not a full one per cent of the entire magnitude. 101,660 = 5 × 18,980 + 26 × 260 or 391 × 260 or 7820 × 13.
It is to be noted also that the differences between the black and red numbers in the second and third serpents (60,021 and 28,132) are divisible by 13 (4617 × 13 and 2164 × 13). Theymustbe, since all six numbers refer to the day III.
Finally the question naturally arises, how did the computer obtain these values,i.e., how was the whole structure built up? On page 63 we found a 136,864 (not 136,884) set down instrikingly small characters and crowded between the other numbers, which would remain a mystery unless one assumed that it was reserved there for this structure; it is 91 × 1504. At first I thought it possible that this 136,864 had been again multiplied by 91, the real basal number of this section; for we had found a second power once before (on pages 46-50) by computation, viz:—2 × 260 × 260. The result of multiplication in this case would be 12,454,624, and the differences between the eight numbers in the serpents would be as follows:—1a + 35,157, 1b-66,503, 2a + 137, 2b and 4b-60,884, 3a-17,814, 3b + 12,318, 4a-165. But these differences are doubtful, inasmuch as they bear no relation to the dates beginning and ending the serpent numbers.
On the other hand, another number contains the desired properties. I refer to the 12,412,920,i.e., it is 109 times the so-called Ahau-Katun of 113,880 days, and I believe I have found that the Ahau-Katun and its multiples were mostly used in the formation of the large numbers. In the following table I have placed this number beside each of the serpent numbers, have then found the difference between the two and have added to it the interval between the first and last day of each serpent number:—
Where the serpent number is less than 12,412,920, I have had to place the last day before the initial day.
The work of the Indian computer was, therefore, as follows:—
He took the days for granted. First he determined the differences between them; then he added to each difference a multiple of 260; and the choice of the multiple seems to have been quite arbitrary. The number thus obtained he added to 12,412,920, unless it was the smaller, in which case he subtracted it from 12,412,920, and the result he wrote down in the serpents.
We shall find the same process, only somewhat amplified, with the serpent on page 69.
Are the seven numbers intended to denote the destruction of the seven planets? I hope this question will be answered in the near future.
There now remains of the contents of these pages only the two columns on the left of page 61, which we will now examine and at the same time compare them with the corresponding column of page 69, the upper part of which is exactly the same, and the lower very nearly so. Each column consists of 18 hieroglyphs, which I count from the top downward, designating those of the first column byaand those of the second byb.
At the first glance these double columns remind one of the inscriptions in the temples and on the stelae, especially of their beginnings, the so-called initial series. Here, in the second column, we find the statement of the usual periods:—144,000, 7200, 360, 20, 1, but in the first column we find faces belonging to them. In his work "The Archaic Maya Inscriptions," 1897, which, on the whole, contains more of imagination than of science, J. T. Goodman unqualifiedly declares these faces to be numbers by which the periods indicated beside them are to be multiplied, and this theory has already found considerable recognition; we will therefore try to follow where he leads.
1a and 1b are effaced on page 61; they probably contained a sort of superscription as on the inscriptions. 2a is effaced on page 61, but the sign may be recognized from page 69 as that with which on page 46 the series of the twenty deities begins after 236 (4 × 59) days. On pages 61 and 69 it takes the placeof a face, to which I am inclined to assign the numerical value 4. In 2b, which is C's head, I am inclined to look for the value 2,880,000 = 20 × 20 × 20 × 360 days, which is not at all inappropriate for C, as the sign of the north pole around which everything revolves. I therefore propose to read 2ab as 4 × 2,880,000 = 11,520,000. 3b, it seems to me, resembles the sign for 144,000, which I found in the inscriptions and which is repeated in 12a. It must, however, be left undecided by what this same number in 3a is to be multiplied; 3a is repeated besides in 8a and 13b. 4a contains the head of E, and 4b that of the Moan. 4a seems to refer to 5a, and 4b to 5b. But 5a and 5b are the same sign, which, inserted between the 144,000 and the 7200, can scarcely mean anything else than the so-called Ahau-Katun of 6 × 18,980 = 113,880 days. Have we two such periods here? Were they designated by consecutive numbers? Now comes the 7200 in 6a, and the number 8 with E's head and the inserted sign for 360 days in 6b (on page 69 without E's head), therefore 8 × 360 = 2880. Seler also thinks 7a has the numerical value 16 (Einiges mehr über die Monumente von Copan, etc., page 217); 7b belongs to 7a. 7b, a Kin with a I and a suffix and a leaf-shaped prefix, is inserted between the 360 and 20. What can it mean? Hardly the 260, for this is represented elsewhere (e.g., page 24) by the thirteenth month Mac. Or can it possibly refer to the month Yaxkin (days 120-140)?