Pages 46—50.
The entire contents of these pages must be represented as a unit, for what is in the main true of page 24 is also true of these pages, namely that they treat exclusively of the period of 2920 days, in which five Venus years of 584 days each are brought into accord with eight solar or terrestrial years of 365 days each. Each page is a direct continuation of the preceding one. Each period of 2920 days is taken 13 times, the result being 37,960 days, which are equal to 146 Tonalamatls.
I will give here first a reproduction, as it were, of the leftside of the five pages, omitting for greater clearness a few indifferent matters, which are intended only to fill the blank spaces, viz:—
1. The twenty hands pointing to the right, with a knife placed over them, in the middle of the pages, which mean nothing more than that these parts of the Venus year are to be read from left to right.
2. The Venus hieroglyph three lines below, repeated twenty times with the sign of the knife, to denote thedivisionof the Venus revolution.
3. TheAkbalsign occurring further down, four times on each page, except on pages 46 and 47. This is the last of the day-signs, again counting forward from the day Kan, and means only that henceforth thecloseof the four periods of the Venus year is indicated below, as the beginning is indicated above.
4. The sixteen Venus signs also occurring below, except on page 48. This sign likewise occurs in a very similar form on Altar R of Copan.
With these omissions, the left side of these pages presents the following appearance:—
Let us first examine the numbers which are regularly repeatedin the lowest line:—236, 90, 250, and 8, and we shall find that the 584 days of the apparent Venus revolution are divided into these four periods.
The number 236 denotes the time of the western elongation, when Venus is the morning star, 90 the time of the invisibility of the planet, during its superior conjunction, 250 that of its eastern elongation, when Venus is the evening star, and 8 the time of its invisibility during inferior conjunction. The disproportion between 236 and 250 is somewhat striking. These periods which need not of course be exactly equal are usually computed at 243 days. The short period of eight days is only calculated for very sharp eyes; we actually find in the Anales del Museo Nacional de Mexico II, 341 (Mex. 1882), that the Aztecs calculated only eight days for the invisibility of Venus, and this period is also mentioned in the Anales de Quauhtitlan. The repetition of the cardinal points in the 15th and 20th lines of the extract given above refer to these periods; in the upper line to their beginning and in the lower to their close. Hence in the lower line the cardinal points must advance one place and the gods belonging to them in the 16th and 19th lines must follow the same course.
The numbers in the 17th line indicate to which day of the period of 2920 days the position has advanced.
But now we see that the indication of days in the lines 1-13, the indication of months in lines 14, 18 and 21, and the numbers in line 17 are separated from those directly to the right of them by a number of days equal to the numbers given in the lowest line.
From this it follows that each day of the thirteen top lines is joined to each of the month dates placed just below them, forming a complete calendar date. Therefore from the III Cib on the left upper corner of page 46 a III Cib 4 Yaxkin, a III Cib 8 Zac, a III Cib 19 Kayab must be formed.
All the 4 × 13 × 5 = 260 day indications combined with three month indications each, show therefore that this whole passage is a huge abbreviation for 780 calendar dates and that the whole refers to 3 × 37,960 days = 113,880 days. But 37,960which we already found on page 24, is equal to 146 × 260, 104 × 365, 65 × 584, 13 × 2920. I am inclined to think that I also found 113,880 on page 24.
But the 3 × 37,960 = 113,880 days do not form the entire period treated of here. For the three periods begin and end with the days:—
Hence these three dates, the second of which was found on page 24, prove that the three periods of 37,960 are not consecutive, but that there is an interval between them. Now between the first and second of the three dates the interval is 19 years + 85 days = 7020 days, and between the second and third, the interval is 26 years + 130 days = 9620 days. If these two periods be added to the 113,880 days, the sum is the whole period treated of here, viz:— 130,520 = 502 × 260 days.
But a truly surprising result is obtained, if, as must often be the case with series, we begin not with the upper of the three dates, but with the lower.
From I Ahau 3 Xul (4 Cauac) to I Ahau 18 Kayab (3 Kan) there is a lapse of 9360 days or 12 apparent Mars years of 780 days, such as we shall find as the principal subject of page 59. 9360, however, equals 25 × 365 + 235 days. We shall meet with this 235 again as a difference on page 63.
But from I Ahau 18 Kayab (3 Kan) to I Ahau 13 Mac (10 Muluc) there are 11,960 days,i.e., the 104 Mercury years, which we found on page 24, and which we shall find again as the principal period on pages 51-58. But this is equal to 32 years + 280 or 33 years-85 days. Now if 113,880, 9360, 11,960 are added together, we have for the entire period under discussion here, 135,200 days, and this is equal to 2 × 260 × 260 days. Thus the Mayas seem actually to have had an idea of a second power.
Finally I would call attention to a singular double connection between the numbers occurring here:—
But if we subtract 2 × 11,960 = 23,920 from 37,960, the remainder is 14,040,i.e., an extraordinary number which often occurs and is equal to 54 × 260, 39 × 360 and 18 × 780.
In short, a Mars and a Mercury-lunar period are inserted in the two spaces between the three solar-Venus periods.
Now, let us try to gain a clearer understanding of this subject by approaching from another side.
As we have seen, the beginning of the middle one of the three equal periods of 37,960 days, is the date I Ahau 18 Kayab (3 Kan). Now, however, page 24 furnished us with a day number for this date, 1,364,360, and from this the beginnings of the other two periods may be computed in the following way:—
Between the first number and the second there are 47,320 days = 23× 5 × 7 × 132, and between the second and third 49,920 days = 28× 3 × 5 × 13 days.
But, according to what has been stated above, 47,320 = 37,960 + 9360, and 49,920 = 37,960 + 11,960.
The whole period is therefore divided as follows:—
It begins with a Venus-solar-Tonalamatl-period followed by 12 Mars years, then the great period again followed by 8 × 13 = 104 Mercury years, and lastly, apparently about the present time, comes the third great period, which, as already stated, ends 135,200 days after the first date.
The case assumes a different aspect, if we insert between the three dates the other two from page 24:—
Here we have again, as examination of page 24 showed, the difference 11,960 between the second and third numbers, while there is no longer any connection with the periods of 37,960 days.
Of the left halves of the pages we have now examined all except the twenty hieroglyphs of the gods. I shall mention them according to the upper place in line 16; the lower in line 19, where the hieroglyphs move forward only one place, is only referred to when the two signs differ. They offer many problems still unsolved.
The first sign on page 46 is an unknown sign, which, however, is repeated several times on the right side of the pages; the second is probably an Ahau (i.e., D) with a prefix suggesting the snail, the symbol of birth; the third is a head also occurring elsewhere, which I have not yet determined; the fourth is A; compare page 24, hieroglyph 25.
Page 47. The first sign is probably K; compare the third picture on page 7a with its hieroglyph; the second is C's hieroglyph with an Akbal appropriate to it; the third sign is Moan with the 13 belonging to it; the fourth sign is N's with a prefixed 4; the year-sign in the lower series is replaced by Zac, which agrees equally well; compare page 24, hieroglyph 21.
Page 48. The first sign is Kin with the Ben-Ik superfix, perhaps denoting G; the second is a figure similar to the year-sign with a prefixed 6. This same sign in the line below has a 6, but is very different in other respects; the third is an Akbal with superfix and prefix, perhaps denoting D; the fourth is a head which might easily be F's; compare page 24, hieroglyph 22.
Page 49. The first sign is B's; the second, A's; the 3d, K's; compare page 24, hieroglyph 38; the fourth is H's with a prefixed 1; compare page 24, hieroglyphs 23 and perhaps 37.
Page 50. The first sign is E's; compare page 24, hieroglyph 38; the second is L, the black deity; compare page 24, hieroglyph 32; the third is an unknown hieroglyph with a prefixed 7, which also occurs on page 5a and 19b; the fourth is the bat-god; compare page 24, hieroglyph 24.
I find it impossible to discover any relation between these hieroglyphs and the periods and I have as little success with the hieroglyphs apparently belonging to the same cardinal point. Perhaps we should follow Seler here (Quetzalcoatl and Kukulcan, p. 403), who thinks these passages suggest constellations with which Venus is in conjunction; this question, owingto the retrogression of the planet, raises increased difficulties. It is curious that the fourth of these signs on page 46 is like the fourteenth on page 49 (A), and perhaps the two following refer to the same god K; the first two are separated by 1494 days and the latter by 1508 days.
We come now to therighthalf of the pages. Interpretation is rendered impossible by the destruction of the top part. For we do not even know whether the upper hieroglyphs occupied three or four rows each, the latter being the case at least in part, and there may have been a superscription over the day signs in the left half.
These upper signs are always followed by a picture, then three rows of hieroglyphs, then a second picture and next two rows of hieroglyphs and lastly a third picture.
Let us first examine the pictures:—
At the top of pages 46-49 there is on each page a deity, who with his right arm extended is offering or receiving something. He is seated on astronomical signs; on page 46 B's head accompanies these signs. On pages 46 and 48 the deity is undoubtedly the old woman with tiger claws, who usually pours streams of water from a jug (compare pages 39, 43 and 74). I cannot identify the personage on pages 47 and 49. The object in the deity's hand seems invariably to be a cup of foaming pulque. On page 49 another object is placed above the cup, which I am unable to determine. The fifth, page 50, differs from the other four and forms the connecting link, as it were, between the upper row of pictures and the middle and lower ones. Here, too, a personage is represented sitting on astronomical signs and exhibiting symptoms of violent anger toward a second person opposite him holding the cup in his hands. Both personages are painted as warriors.
The middle pictures on all the pages represent a warrior in a half-kneeling, half-crouching posture, holding spears or a shield in the right hand and brandishing a hatchet in the left. The shield on page 46 is doubtless a representation of the sun-glyph; and on 47 the Venus sign is combined with the head ornament. The hieroglyphs of these deities occupy the first place in the middle line of the three lines above the pictures.
The five lower pictures represent a creature lying on the ground, pierced by arrows and spears. On page 47 it is a jaguar; at any rate it is the same animal found on pages 29a, 30c and 45c; a very similar creature pierced by arrows is given in the Cod. Vat. B.; compare also the pictures in Seler's "Venus-periode," page 371. On the other four pages this creature is in human guise. On page 50 where, differing from the other four pages, this figure is represented lying with the head to the right, it is plainly shedding tears. Seler takes this figure to be E on page 48 and the tortoise on page 49. The varying periods of time occupied by the revolution of the planets is plainly conceived of as contest. But who is the victor in this contest? The planet with the longer or with the shorter period of revolution? Owing to obliteration only a small part of the hieroglyphs of the top section is legible.
On page 46 we see the Venus sign and E's hieroglyph; on page 47 the signc, which occurs frequently on these pages, and is probably always connected with Moan (the Pleiades and thus with the year). The numeral 1, prefixed to an obliterated sign on page 47, is still legible, and we find it repeated on the lower part of the same page. There is rather more to be seen on page 48:—first the elongated headqwith the Ben-Ik superfix, then the signa, beside it that for the west with a prefix, in the line below an Ahau, next, an Akbal sign with the prefix of the north, and lastly a Moan sign.
On page 49 we see signcagain occupying the first place, then o with Ben-Ik, and in the lower line the year sign with that for 20 or the moon as a superfix, and to the right of it the head with the Akbal eye, probably denoting D.
This top part of the page is best preserved on page 50. In the third line from the bottom we see the Venus sign and beside it the Moan sign, below, a Cauac, then a Kin with the Ben-Ik superfix, then a Kan-Imix. Finally, in the first place in the lowest row there is a Kin sign and in the second place a sign resembling the year-sign, both having the same superfix, the next sign is againcand the last is a half-effaced sign, of which only a Muluc is distinguishable.
Our knowledge of the middle section of these pages issomewhat more definite. There are twelve hieroglyphs on each page, which I will number in the following order:—
Unquestionably these 5 × 12 signs refer to a Venus year, more exactly to thebeginningof it, the period of the east. The first sign, which is a hand pointing to the right, merely refers here, as on the left side, to the direction in which this is to be read; the second sign is always the sign for the east, and the sixth invariably that for Venus.
Notice should be taken of the fact that the signs of the Moan and screech-owl or death-bird are recurrent, that of the Moan appearing on page 46, sign 7; 48, 3; 49, 11; 50, 11; and that of the death-bird on page 47, sign 3; 48, 11, 49, 3, 50, 3 and 7,i.e., only in places 3, 7 and 11, which indicates that the 12 signs are divided each time into three times four.
It is further to be noted that the five gods, who are represented on page 24 by hieroglyphs 36-40, always recur in the ninth place in the order of the pages:—the god represented on page 24 by sign 36 is the 8th on page 49; the 38th on the same page is the 11th on page 46 and the 12th on page 50; the 39th is the 12th on page 47, and the 40th may be the 5th on page 49, though this is hardly possible. On page 49 the 9th hieroglyph seems to be the 39th on page 24 joined to the sign for the month Kayab.
Of the twenty gods on the left side of these pages, I have already remarked that E, who on page 24 occupies the 38th place, and the 11th on page 46, also occurs as the 9th on page 48 and the 12th on page 50.
It is doubtless of special significance that the sign of the first of the twenty deities on the left side of page 46 is repeated on the right as the tenth sign on all the pages (on page 47 also in the eleventh place where it has a prefixed 3). It seems as if this sign, which is otherwise quite unfamiliar, might be connected with the sun and regarded as a contrast to the Venus sign in the sixth place.
Also the 9th deity of the left side, the 1st of page 48,reappears in the 4th place on page 49; the 10th deity, the 2nd on page 48, in the 12th on page 49; the 15th deity, the 3d on page 49, in the 9th on page 46 and the 8th on page 49 (as already stated); the 18th, the 2nd on page 50, in the 5th on page 46.
The 2nd of these deities is suggested by the 8th on page 47, perhaps also by the 5th on page 50; the 3d and 13th seem to be A and to recur in the 3d place on page 46.
On the other hand C, the god who, as I believe I have proved, is connected with the day-sign Chuen, does not appear on the left side. Now the 4th sign on page 46 contains a Chuen, which in the 12th sign on page 48 is probably combined with a Muluc, in the 12th on page 49 with Yax and a prefixed 6, and in the 4th sign on page 50 with C's sign,i.e., as a rule Chuen stands in the 4th place in a line.
As the gods E and K already mentioned also appear on pages 25-28 in connection with the change of the year, so we find the tiger on the top of page 26, and I believe this animal occurs again in the 7th sign on page 47.
Of the day-signs I take the 4th on page 47 to be Kan, the 7th on page 48 to be Caban, and the next sign, the 8th on page 48, to be Muluc. Now if we take into consideration the fact, that of the three periods of the month signs on the left side of these pages, the 18th (the middle) line is the most important, owing to its ending, 18 Kayab, alone, if for no other reason; furthermore, that in this middle period the second Venus year always ends with a Kan year and the third with a Muluc year, one is naturally led to suppose that the illegible sign 12 on page 46 is an Ix (for thus the first Venus year ends) and that the days Cauac and Kan might have been found among the obliterated day-signs on pages 49 and 50.
I shall examine the remaining signs in the order of the pages.
Sign 8 on page 46 is the same compound of Yax and Kin having as a superfix the sign assumed by me to be the numeral 18, which occurs again in the lower group on page 50 and also on page 27.
In the number 11 prefixed to the fifth sign on page 47, the 1 seems to be indistinct and may not belong here. If we correctly assume that this number is 10, then the sign is the same asthe 34th on page 24, to the discussion of which I beg to refer my readers.
Sign 8 on page 47 is an indistinct compound, the first part of which I supposed above to be the sign of the second deity on page 46.
I cannot explain 4 and 5 on page 48.
As yet I do not understand sign 5 on page 49, which we seem to have met before on page 22c.
Sign 7 on page 49 is the moon, which is very curious here.
I would like to call special attention to signs 5 to 8 on page 50. I interpret the passage thus:—At the time of the summer solstice after the reappearance of the Pleiades, the change of the Venus year takes place (this time). I have already discussed the Venus sign in the sixth place and the screech-owl so closely connected with the Moan (Pleiades) in the seventh place. Sign 5 connects the sun (Kin) with the Ahau (lord) and the cross-hatching on the left of it, which I have assigned to the tortoise and thus to the summer solstice (Zur Entzifferung III, 3). Sign 8 is recognized as very appropriate to the change of year; compare the first sign of the middle section on pages 25-28. All this points to the day 18 Kayab, of one of the Kan years, if, as I stated above, we base our computation on the middle series of dates.
Now we have yet to examine the eight signs of the lower group, which we will do in the following order:—
Regarding the beginnings of these groups, I will venture a bold surmise, which will, I hope, be improved upon by some one else. It concerns the first sign of four of these five groups, which seem to me to refer to the end of the Venus year, as those above refer to the beginning. This sign has the following form:—
I see in this the term of 73 days, which is the fifth part of the 365 days of the solar year and the eighth part of the 584 days of the Venus year:—
It is combined with Chuen in all four cases (pages 46, 48, 49 and 50). But I attribute the meaning of eight days to this Chuen sign, as I did on pages 25-28 and 42c-45c, though I am doubtful in these as in other cases.
Page 46 contains the sign for 73 with a Chuen under it, and a 1 prefixed to each sign;i.e., 1 × 8 × 73 = expiration of the first Venus year.
On page 48 Chuen follows the sign for 73 and each sign has a 3 prefixed to it;i.e., 3 × 8 × 73, expiration of the third Venus year.
On page 49 the two signs again stand side by side, but the prefix is a 7 instead of the expected 4. By an error this 4 has been added to the 3 of the preceding page, but, for a wholly unintelligible reason, prefixed to the crouching person below the Chuen, as if to correct the 7.
Page 50 again has the sign for 73 above and the Chuen below. A prefixed 5 would seem to be in order; instead of it, there is a 10, one 5 for the 73 and another 5 for the 8 days. In this connection let me say that I believe I have found on page 27, top left, the year of 365 days divided into 5 × 73.
Page 47 differs from the others. Above two oval bodies appears the cross-hatched figure resembling a clamp, like the one in the middle group of page 50 in the fifth place, which I ventured to refer to the summer solstice. There is a 1 prefixed to it. Is this equivalent to a union of two Venus revolutions?
Next we repeatedly meet here, as we did in the middle groups, with the Moan sign and that of the screech-owl belonging with it; the former is the 6th sign on page 46 and page 50, and the latter is the 3d and 7th on page 47, the 7th on page 49 and finally the 2nd and 4th on page 50.
The moon is represented in the 5th sign on page 48 and in the 3d on page 49 and indistinctly in the 4th on page 48.
The cardinal points occur here several times. The 3d and 7th signs on page 46 have at least the superfix of the south as aprefix; the 8th on page 47 apparently has the east, but with the familiar cross-hatched sign prefixed; the 7th on page 48 plainly has the east, the 3d on page 50 the prefix of the north prefixed to the crossb, and the 8th on page 50 the west, thus approximating the usual order and distribution.
Of the gods I note the Akbal head, perhaps intended for D, in the 4th place on page 46, also in the 3d on page 48, and lastly in the 5th on page 49, the first two times with the Ben-Ik superfix, and in the 2nd place on page 47 the sign for A.
In the 4th place on page 47 we have the tortoise as the sign of the month Kayab or of the summer solstice, in the 6th on page 47 the lightning-beast or the month Kankin with a Ben-Ik superfix; the beast itself is pictured below, and the same hieroglyph also with the Ben-Ik superfix is the 8th sign on page 49.
It is hard to decide whether the sign 4 on page 49 represents the god F owing to the line through the eye, or a female by reason of the prefixed lock.
Sign 7 on page 50 represents the deity whose sign began the series of twenty gods on the left of page 46 and which we have already met with several times in the centre of the right side. We recognize the prefix as having occurred in the middle group of the same page.
Sign 6 on page 48 is a Kin combined with an unfamiliar sign. Sign 5 on page 50 contains a Kin with a Yax and probably with 18 as a superfix (as on pages 27 and 46 middle).
Sign 6 on page 49 contains a crouching person with a 4 which is probably out of place here and to be regarded as a correction of the 7 above it.
Sign 5 on page 46 contains a Mac denoting the thirteenth Uinal or a Tonalamatl, and having the signpas a superfix and a double Ik as a prefix.
Sign 3 on page 46 merits special attention, because it contains the duplication of the sign, which, at the end of the first part of the Manuscript, pages 29-41, always began the groups of hieroglyphs on the lower third of the pages.
I do not understand the second hieroglyph on page 46 and the 5th on page 47.
In conclusion I would call attention to the fact that the last hieroglyph on page 48 is very peculiar. As on pages 51, 52, 61 and 69 it has the meaning of 18,980 days and consists of an Imix with a comprehensive superfix; its prefix is a 7.
But what is the meaning here of 7 × 18,980 = 132,860? When we recall the statement made above that the whole section of pages 46-50 embraces 130,520 days, or, according to another calculation 135,200 days, it is a striking fact that 132,860 is exactly the mean of the two numbers, being separated from each by 2340 days = 9 × 260. Can it be an accident that on the next page (page 49) the fourth Venus revolution is reached, for 4 × 584 = 2336,i.e., almost 2340? The hieroglyph discussed here would not be so extraordinary on page 50. I will not venture to assert as to the 511 in 132,860 = 511 × 260, that it is connected with the 511 which will appear as the difference on page 58.
Before leaving these pages, I will give a brief survey of the two signs of the screech-owl and the Moan (hieroglyphcand the lower part ofd) which occur on these pages with such marked frequency.
In spite of obliteration, the first of these two signs is distinguishable in the top groups on pages 47, 49 and 50, in the middle groups on pages 47, 48, 49 and twice on page 50, in the lower groups on page 46, twice on page 47, once on 49, twice again on 50, making 14 times in all. A few additional cases might be added to these where the similar hieroglyph of the moon may have been set down instead of the one in question.
On the other hand the second sign, always provided with the same prefix and suffix as the first, occurs in the top groups on page 48 and 50, in the middle of pages 46, 48, 49 and 50, and in the lowest on pages 46 and 50, 8 times in all.
Since the subject here is astronomical, it is suggestive less of a deity or a sacrifice than of a period of time to which the allied page 24 has already referred (see page 110 of this book). The inner meaning of these pages is of course still enveloped in mystery.
Pages 51a—52a.
I shall begin the discussion of this very peculiar section with the remarkable fourth column on page 52, which, very possibly, the scribe ought to have placed at the beginning; for it looks like a repetition of the section on pages 46-50, while everything else on the left and right of it, apparently belongs together.
If we omit the two hieroglyphs at the top, which I regard as belonging to the two rows of hieroglyphs extending over these two pages, we shall have the following result, according to my point of view:—
Since, as is frequently the case, the Chuen will here have the value of 8 days and the 5 with the sign for 360 may be regarded as 365, this group might denote 8 × 365 = 2920, but actually be 2 × 18,980 = 37,960. Both numbers are the basis of the section included on pages 46-50. And in the same way the 13 repeated 13 times seems to me to refer to the 13 series of days on those pages, which begin with the 13th day of the Uinal.
The two rows of hieroglyphs are in the main destroyed. We can still recognize in the second and third columns of page 51 the signs for end and beginning, which we often find in the vicinity of numbers; in the second and third columns of page 52, the sun and moon; in the fourth column, the 8 days of such significance here and in the fifth and sixth, the normal date IV Ahau 8 Cumhu repeated twice.
As the problem on pages 46-50 was to bring into accord the solar year with the Venus year and consequently also the Tonalamatl,i.e., to combine 365, 584 and 260, so the aim here is first of all to bring the Tonalamatl into unison with the Mercury year (115). For this purpose the number 11,960 is employed. This is equal to 46 × 260 = 104 × 115, including, therefore just as many Mercury years as there were solar years in the preceding section. 11,960 is also 8 × 1495, and this 8 is significant here, for, as we shall see directly, the day forming the basisof this calculation is XII Lamat, which comes 8 days after the normal date IV Ahau.
The series given here is based, therefore, on 11,960 and consists entirely of multiples of this number, which, it is true, are recorded with the usual irregularity. The members of this series, representing the greatest values, which are set down in red numbers among the black, are the 31st and 39th multiples of 11,960, which are separated from each other by 8 × 11,960, viz:—370,760 and 466,440. All these numbers, of course, denote the day IV Ahau.
The day XII Lamat as the actual starting-point of the Mercury revolution is not introduced until we come to the dates placed below the series. Here we find the days XII Lamat, I Akbal, III Ezanab, V Ben and VII Lamat written one below the other, and repeated seven times. Each of these days is separated from the next by 15, and the last of one row and the first of the next on the left are 200 days apart, hence the whole is equal to 7 × 260 = 1820 days. From XII Lamat begins also the Peresianus, pages 21-22.
Now these dates are connected with the four large numbers, which we find on page 52, but between the third and fourth, one number corresponding to the day V Ben is omitted for lack of space.
These four numbers, to which I have added the corresponding dates, are as follows:—
It is curious that while the first three are separated from each other by 15, between the 3d and 4th, or rather between the missing 4th and 5th, 84 × 260 days are inserted in excess of the required 15,i.e., 21,855. This, however, is not accidental, but is due to the fact that between the first number and the last exactly 21,900 = 60 × 365 days have elapsed. This number is, however, = 18,980 + 2920, i. e., the sum of two very important numbers, in the first of which the Tonalamatl and the solar yearaccord, while both the solar and Venus years occur in the second.
I must here call attention to the fact that the four numbers are not obtained without slight corrections, since in the 20-place of the third, I have put a 11 instead of 10, while in the 360-place of the fourth, I have omitted the three dots,i.e., set down a 5 instead of the 8.
Of these four dates, which were doubtless not far removed from the time of the scribe, the three last are only the result of the first. Day XII Lamat is the most important. As the beginning of a Mercury period it should be regarded in the same way as I Ahau of the Venus period and IV Ahau of the solar period; and the very next day, XIII Muluc, will subsequently be seen to be the beginning day of the Mars period.
The four dates XII Lamat, I Akbal, III Ezanab and VII Lamat are set down in the Manuscript directly below the numbers.
Now in the first column on page 51 we again find a day XII Lamat, as is expressly stated beneath it. It has the number 1,578,988 and the corresponding date is XII Lamat 6 Cumhu (6 Kan). This day, however, is separated from the same day on page 52 (1,412,848 = XII Lamat I Muan 6 Muluc) by 166,140 days, that is by 8 × 18,980 + 14,300 = 639 × 260,i.e., by 8 so-called Katuns increased by 55 Tonalamatls. Here 58 × 260 = 15,080 seems to have been added to 252 (XII Lamat-IV Ahau) and the sum subtracted from 14 Ahau-Katuns = 1,594, 320. I could obtain this number only by substituting 1 for 0 in the 20-place.
In the Manuscript the sign XII Lamat is set down above and below this number. I must leave undetermined whether the 8 directly above the number and combined with Kin and the Katun sign refers only to the 8 Katuns or at the same time also to the 8 days from IV Ahau to XII Lamat.
It is also to be noted here that once before on page 24 of this Manuscript (which forms the basis of this section) 8 × 18,980 = 151,840 days was found to be the difference between 185,120 and 33,280, and that there, too, if my restoration is correct, it was the highest term of the series = 4 × 37,960.
Finally, in the first column of page 51, we have the completenormal date 4 Ahau 8 Cumhu (9 Ix). But below this, between red numerals denoting the 1,578,988 mentioned above, there is set down in black the number 1,268,800. This corresponds to the date IV Ahau 3 Zip (2 Cauac). It may have been formed by adding 16,120 = 62 × 260 to 11 Ahau-Katuns = 1,252,680. It is, however, not only equal to 4880 × 260, but also to 158,600 × 8, therefore also divisible by the interval between IV Ahau XII Lamat, as well as by 104 = 8 × 13, while on the contrary it is not as we should expect, divisible by 11,960. I have changed the 11, in the 20 × 11, to 8 by omitting one line and adding two dots, for otherwise the result would not be the one required.
The magnitude of the number recalls the one on page 31, which is only 260 less, and that on page 62.
Finally it should be noted that the two large numbers on page 51 are separated from one another by 310,188 days = 849 years and 303 days, which corresponds exactly to the dates given for each. One may be situated as far in the future as the other is in the past, but this does not necessarily mean that the present coincides exactly with 1,423,894.
Pages 51—58.
Thus far we have examined only the upper halves of pages 51 and 52 and have still to consider the lower, but not until we have finished the upper parts of pages 53-58 of which the former are the continuation. We have first to consider the series, then the pictures and lastly the hieroglyphs.
As on page 24 we found multiples of the number 2920 (= 8 × 365 = 5 × 584), while on pages 46-50 it was divided into four unequal parts, so on pages 51-52 we find multiples of the number 11,960 (104 × 115 = 46 × 260) while on pages 53-58 it is divided into 69 unequal parts. On pages 51-52 it was the aim to combine only the Mercury course with the Tonalamatl, but here we are confronted with the additional problem of bringing the lunar revolution into accord with these two.
The lunar revolution, which we assume to be 29.53 days, of course requires fractional computation, of which the Mayas either were ignorant or which they timorously avoided; like the ancient Egyptians, who were acquainted only with fractionshaving 1 as numerator, or beyond these at most with ⅔ (see Hultsch, "Die Elemente der ägyptischen Teilungsrechnung," 1895, page 16).
Now the Mayas had determined the lunar revolution so exactly that they perceived the incompatibility of the period of 11,960 days with a multiple of lunar revolutions. They found that 405 lunar revolutions amounted approximately to 11,958 days, which is, in fact, the largest number on the second half of page 58. In order not to drop the significant 11,960 altogether, they made use of a very shrewd artifice. They took as the starting-point the day XII Lamat, corresponding to the number 11,960, and set down XI Manik before it and XIII Muluc after it. Now if the count began with XIII Muluc and ended with XI Manik, it actually resulted in 11,958.
Therefore what the Manuscript presents here is, in the first place, the series, which is this time to be read from left to right. Below it are the three days belonging to each member of the series and then a number for each member stating the interval between it and the preceding one. The members, the days and the differences must correspond with one another. It is, therefore, no longer necessary to pay especial attention to the two latter. They will serve merely to control and to correct the manifold errors.
The entire period of 11,958 days was doubtless first divided into three equal periods of 3986 days. And in order still further to subdivide these shorter periods, the term of 177 days was employed as far as it would go; 177, however, is the half of a lunar year of 354 days, made up of 6 months of 30 days and 6 of 29 days, thus allowing 29.5 days in round numbers for each month.
Now 177 is = 3 × 29 + 3 × 30. The average, 29.5, however, is too short for the length of the lunar revolution. In order to raise it as nearly as possible to the exact time, two other numbers were introduced at certain points of the series, viz:—148 = 2 × 29 + 3 × 30, 178 = 2 × 29 + 4 × 30. 148 = 5 months of 29.6 days, while 178 = 6 months of 29⅔ days. Now let us see in whatproportionthese 148 and 178 days were distributed among the periods of 177.
First we see that the period of 3986 days (i.e., a third of the whole) was divided into 3 sections of 1742, 1034 and 1210 days, as follows:—
This is equal to 135 months of 29.526 days each. Now the question arises how did the Mayas express this fraction?
Perhaps some time in the future it will be found, that following their vigesimal system, they designated it approximately thus:—
29 + ½ +1⁄40+1⁄800.
Thewholeperiod of 11,958 days was divided as follows:—
Thus for every 6 parts of 177 days there was consequently 1 of 148 and to every 9 parts of 177, 1 of 178.
Since 177 and 178 include 6 months each, while 148 equals 5 months, the entire length of the period is 405 months, which are divided into 69 periods.
It was necessary to discuss all this before I could introduce the entire series itself. In the following table I have set down the numbers and added to them the differences between each number and the preceding one (to the first, the interval between it and the zero point), just as they are given in the Manuscript. An asterisk is added to show that the number has been corrected by me and is wrong in the Manuscript, owing to a mistake either in writing or in computation. The three columns correspond to the three divisions of 3986 days, and the two horizontal lines divide the periods of 1742, 1034 and 1210 days.
No one acquainted with the cursoriness of the Maya Manuscripts will be surprised that among 138 numbers I have declared 21 to be wrong. Furthermore the 21 errors are lessened by the fact that six of them are really only one, for in all 6 cases where the difference is 178, the scribe has overlooked this and written down the usual 177, although the numbers and the days of the series very correctly indicate 178. Again the three errors in groups 58 and 59 are also only one, for the author had confusedthe differences 177 and 148 and had, therefore, to write down 10,039 instead of 10,068. In group 4 the error is merely the omission of a line meaning 5. The scribe must have been at the same time the computer and therefore the actual author of the Manuscript.
Furthermore I must call attention to the regular position of the differences 178 and 148. In the three periods of 1742 days the 178 always occupies the 6th place and in the periods of 1034 it is always in the 4th place. This difference appears, therefore, in groups 6, 14, 29, 37, 52 and 60,i.e., 8, 15, 8, 15 and 8 groups apart; but it is entirely lacking in the periods of 1210 days. And in all nine sections the 148 occupies the third place,i.e., directly in front of the pictures, which will be discussed immediately, therefore in groups 3, 13, 19, 26, 36, 42, 49, 59, 65,i.e., at intervals of 10, 6, 7, 10, 6, 7, 10 and 6 groups. But I must point out an error fraught with consequences. Groups 22 and 23 quite correctly have the difference 177, but in this single instance the scribe has written down 178 and hence has computed the three days belonging to it as VII Ix, VIII Men and IX Cib instead of VI Ben, VII Ix and VIII Men, and from here on to the close he is always one day in advance, so that on page 58 group 69 ends with the days X Cimi, XI Manik and XII Lamat, while it ought to have ended with IX Chicchan, X Cimi and XI Manik.
So much for the series. Vid. on this series my paper "Zwei Hieroglyphenreihen in der Dresdener Mayahandschrift" (Zeitschrift für Ethnologie, 1905, numbers 2 and 3). Let us turn next to the ten pictures which are inserted in this series, three of which appear at the end of each period of 2920 days as on pages 46-50. Let us attempt to advance a step further in the darkness which still surrounds us here.
One of these pictures, the 8th, which is on page 56b, is in the wrong place, owing to the error in computation in Groups 58 and 59 to which I called attention above. It belongs notbeforebutaftergroup 59, the first on page 56b. This is indicated in the Manuscript itself. For in group 59 the two hieroglyphs, usually placed above each group, are missing and we find instead of them the sign resembling a snail, which isdoubtless a very much emphasized zero (compare my "Erläuterungen," page 29), which indicates that the section designated by a picture closes with this group.
Having corrected this error we see that the ten pictures are on the following pages and come after the following numbers of the series:—
From this it follows that a picture is assigned to each of the nine sections composing the series. They are placed, however, not at the beginning or end of the section, but always after the expiration of 502 (= 2 × 177 + 148) days. The pictures are thus separated from one another by 1742, 1034 and 1210 days, which intervals correspond exactly to the length of the nine sections. But the last picture is separated from the preceding one by 708 days, and as it has a character quite its own, it must be discussed separately. But these 708 days with the 502 days of the beginning quite regularly amount to 1210 days, and the series is therefore to be considered as a recurring one.
Now these nine pictures might very easily be regarded as forming a new series, which is inserted in the original one and which has the day 502 as its zero-point. In that case, we shall have to subtract 502 every time from the days set down in the Manuscript. This new series may be represented thus:—
It is certainly remarkable that the last number, 10748, corresponds so closely to the time of the revolution of Saturn, which is computed at 10753 days. For owing to the slowness of its progress, the Mayas may have known not only the apparent but also the actual revolution of Saturn. Besides the apparent revolution of Saturn (378 days from one superior conjunction to the next) could not be made to coincide very well with the length of the solar year. I will immediately present a further confirmation of my theory.
All these pictures have rectangles above them, of which I have spoken in my "Erläuterungen," page 16, and which always enclosed two or three hieroglyphs in which, with due hesitation, I assumed to be the signs of the sun, moon, and planets. This theory has as yet called forth no serious opposition.
Now in the passages just mentioned, I indicated the following figures as the signs of Saturn:—