Commentary on the Maya Manuscript in the Royal Public Library of Dresden/4

Pages 46—74.

The first glance at the form and contents of the second part of the Manuscript shows that it is very different from the first. The pages are no longer divided into the usual three parts and there are fewer pictures. The Tonalamatls, which form the principal contents of the first part, disappear wholly, and with them both the vertical columns of day-signs and the horizontal lines of numerals alternating between red and black. On the other hand, the large number series as well as the high numbers significantly increase and we note the appearance of the large vertical columns of hieroglyphs, which were impossible in the triple division of the earlier pages. We also find a large number of hieroglyphs which did not occur in the first part. The contents are essentially astronomical.

And yet the two parts are so closely connected with one another that the idea of two independent Manuscripts must be dismissed. Especially the front side of the second part as far as page 60 is nothing more than an amplification of page 24. The contents of pages 61-74 are of a more independent nature, but special attention should be called to the relation of 31a-32a to 62-63.

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 left side 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 the division of the Venus revolution.

3. The Akbal sign 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 the close of 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 repeated in 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 37,960 which 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 = 2$3$ × 5 × 7 × 13$2$, and between the second and third 49,920 days = 2$8$ × 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, owing to 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 the right half 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 sign c, 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 head q with the Ben-Ik superfix, then the sign a, 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 sign c again 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 again c and the last is a half-effaced sign, of which only a Muluc is distinguishable.

Our knowledge of the middle section of these pages is somewhat 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 the beginning of 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 as the 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 a prefix; 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 cross b, 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 sign p as 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 (hieroglyph c and the lower part of d) 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 basis of 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 year accord, 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 complete normal 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 fractions having 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 what proportion these 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:—

The whole period 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 confused the 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 not before but after group 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 is doubtless 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:—



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 are 3986, 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

The tenth picture has the cross b as 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, occur several 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 cross b appears 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 eighth picture 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 but one point in favor of the first possibility, viz:—the Venus sign in group 4b (I will designate the upper hieroglyphs by a and the lower by b). 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 found only on these pages and here not less than eleven times, possibly with the addition of the effaced 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? Accident alone is 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 repeated at 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 Cumhu i.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.