Part 13

1 2 3 4 5 6 7 8 9 10 11 12.

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:--

1 2 3 4 5 6 7 8.

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:--

1 5 Chuen 360 2 18,980.

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:--

1,412,848 = XII Lamat I Muan (6 Muluc). 1,412,863 = I Akbal 16 Muan (6 Muluc). 1,412,878 = III Ezanab 11 Pax (6 Muluc). 1,434,748 = VII Lamat I Muan (1 Muluc).

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 2/3 (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 29-2/3 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:--

1742 = 8 × 177 + 148 + 178 1034 = 4 × 177 + 148 + 178 1210 = 6 × 177 + 148 ---------------------------- 3986 = 18 × 177 + 3 × 148 + 2 × 178.

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.

The _whole_ period of 11,958 days was divided as follows:--

3 × 1742 = 24 × 177 + 3 × 148 + 3 × 178 3 × 1034 = 12 × 177 + 3 × 148 + 3 × 178 3 × 1210 = 18 × 177 + 3 × 148 --------------------------------------- 3 × 3986 = 54 × 177 + 9 × 148 + 6 × 178.

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.