i^,o 



SCIENCE. 



[Vol. XXI. No. 527 



to g:ive credit to the tbeory. Notice of this, however, will be re- 

 served for a subsequent paper. 



Attention is called again to the series on pp. 63-64, in order to 

 remark that, by counting back from 13 Ix 91 days, we find that 

 the series commences with the first day of the year 12 Kan. Then, 

 by tracing it through, according to the usual year of 365 days, 

 we find that it ends with 13 Akbal, the last day of the year 3 Kan, 

 omitting the five supplemental days at the end. Adding these 

 five days, the total — 1835 — is exactly divisible by 365. However, 

 it seems that the series should be extended 42 days more to in- 

 clude the other days of the last column (see " Aids," p. 330); in 

 which case neither 365, 364, nor 360 would be an exact divisor of 

 the sum total. 



We refer nest to Dr. Forstemann's theory that the long series 

 on pp. 51-58 refers to the length of the lunar month. As he ad- 

 mits, the number of days, counting to the last, is 11960, though 

 the sum of the intervals between the columns, as shown by the 

 final numeral, is 11958. These intervals are generally 177 days, 

 but 9 of 148 days occur at nearly equal steps, and 6 of 178 at 

 irregular steps. He finds that by multiplying 29 by 3 and 30 by 

 3 and adding together the products he obtains 177; that the sum 

 of the products of 29 by 3 and 30 by 3 is 148. To obtain the 178, 

 he finds it necessary to arbitrarily add 1 to the products of 29 by 

 3 and 30 by 3. Next, he finds that by multiplying 177 by 54 — 

 the number of times this interval occurs in the series — 148 by 9 

 and 178 by 6, and adding thereto 6, he obtains as the sum of the 

 products 11958. He ascertains in this way that 29 occurs 198 

 times and that 30 occurs 207 times, making together 405, and that 

 11958 divided by this sum gives 29.526 days, which falls short of 

 the lunar month but one four- thousandth part of a day. As he 

 adds 6 days to his several products to obtain the number 11958, 

 would it not be as well to add 8 days, making 11960, the true 

 length of the series, which, divided by 405, gives as the quotient 

 29.53 days, precisely the desired figures? 



Notwithstanding my high appreciation of Dr. Forstemann's 

 ability as an investigator, and of his great caution in presenting 

 conclusions, I cannot help thinking that his love for numerical 

 coincidences, created by his long study of the time series of the 

 Dresden Codex, has, in this instance, led him to accept as satis- 

 factory what he would have hesitated to approve had it been pre- 

 sented by any one else. Now, 11960, the true length of the series, 

 embraces precisely 46 periods, or sacred years, of 260 days, so often 

 repeated in the Codices, the whole series and each of these 

 periods commencing with 12 Lamat and ending with 11 Manik, 

 initial and terminal year and month days, according to the 

 method of counting from the last day of the month, which I had 

 not discovered when my "Aids" was written. Is it not, there- 

 fore, more reasonable to conclude that the chief relation of the 

 series is to this sacred period? This inquiry is certainly perti- 

 nent in view of the fact that neither 29 nor 30 appears singly or 

 in multiple at any point in the series, that the total is first lessened 

 by subtracting 2 and the products increased by the addition of 6. 

 It is proper, however, to admit here that the interval 178, which 

 is an increase by 1 of the usual period of 177 days, is difficult to 

 account for, but such difiiculties occur at many points in this 

 Codex, and Dr. Forstemann's attempt at explanation involves so 

 many assumptions as to cause us to hesitate before accepting it. 



In order to show the uncertainty of the method adopted in re- 

 gard to the last mentioned series, we will apply it to one not 

 referred to by Dr. Forstemann, running through the lower divi 

 sion of pp. 30-33. In this case the total sum is 3340 days, and 

 the uniform interval 117. Now if we multiply this interval by 5 

 we obtain 585, but one day more than the time of the apparent 

 revolution of Venus. Or, if we multiply 584 by 4 and add 4, we 

 obtain 2340, the number of days in the series; and the result is 

 obtained by a less addition than that made by Dr. Forstemann in 

 obtaining the lunar period. Now let us try another experiment 

 in order to find the lunar period, thus : 39 x 3 -t- 30 x 1 = 117 

 and 3340 divided by 117 = 30. This will give us 60 periods of 29 

 days and 20 of 30 days, and dividing 3340 by 80, the sum total of 

 these, we obtain 29.25 days, lacking only about one-fourth of a 

 day of the correct time. Finally, we observe that 2340 days equal 

 9 of the sacred years of 260 days each, probably the real basis of 



the series, as 13 and 30, from which the latter is formed, are both 

 factors here — 9 x 13 = 117, 13 x 20 = 260, and 360 x 9 = 2340. 



If we turn to the series on pp. 46-50, in which Dr. Forstemann 

 thinks he finds the Venus period, and apply the method of figur- 

 ing above alluded to, we shall obtain some curious results. As 

 we have seen, the intervals which together make the 584 days 

 are 236, 90, 250, and 8 days. Are these intervals arbitrary, de- 

 pending upon arrangements by the priests or by the scribe, or 

 should we infer that they always depend upon the periodicity of 

 certain natural phenomena, and hence form factors or multiples 

 of time-periods? Although the latter may he generally true, 

 the proof of which seems to be the chief object Dr. Forstemann 

 has in view in his mathematical search, yet there are many of 

 the intervals and periods which apparently defy all efforts to fit 

 them into place. That 13, 30, and 18 will most frequently appear 

 is to be expected, as they are always factors, but the coincidences 

 in regard to other supposed time-periods (aside from the ordinary 

 and sacred years) are to be regarded with doubt unless there is 

 something more found than the occasional appearance thereof as 

 factors. For instance, if we take 336, one of the intervals men- 

 tioned above, we find that it can readily be made to coincide with 

 the lunar period ; thus : 39 X 4 -f 30 X 4 = 236. This will give as 

 the time of a revolution 29.5 days, which varies less than an hour 

 from the ti-ue period. Yet for all this shall we conclude that 

 here we find allusion to the moon's period? By no means, for 

 this is only a recurring interval; and the others, which go to 

 make up the 584, — the 90, 250, 8, — do not coincide with the moon's 

 revolution or any other known time-period ; 90 and 8 are factors 

 of 360, but this number, as we have found, is one of the counters 

 in these series. 



The supposition that the revolution of Mercury is indicated by 

 the numerals on p. 24 is certainly based on very slender data. 

 This is found only in the fact that 115, the time of a revolution, 

 is a divisor of the large number 11960, which is a multiple of 260, 

 on which it is doubtless based. Why he has referred in this con- 

 nection to p. 24 is not apparent. I do not find any relation here 

 between a 1 Ahau and 4 Ahau (the latter is found but once on the 

 page) ; nor do I find the number alluded to (11960) as the terminus 

 of a series or an interval. There axe two series on the page, or 

 one series in which the interval varies. That which occupies the 

 lower three-fifths of the right, commencing at the bottom, run- 

 ning to the left and up, has 2920 as its interval, of which 115 is 

 not a factor. The interval of the other, the terminal columns 

 of which are found at the left below, is 3300. This is not divisi- 

 ble by 115. Therefore, so far as I can see, Dr. Forstemann's only 

 basis for the supposition that the Mayas had ascertained the 

 period of the revolution of Mercury is found in the fact that the 

 large number 11960, which is found several times in the Codex, is 

 divisible by it. Can it be said that a conclusion based on no other 

 evidence than this " amounts to a demonstration ? " 



That Dr. Forstemann has made progress in the study of the 

 Codices by calling attention to the relations of these numerals to 

 one another is cheerfully admitted, and that he has thrown light 

 upon their meaning and suggested lines of investigation regarding 

 them is undoubtedly true. Yet his discussion in the paper alluded 

 to cannot be considered a "demonstration," when the same data 

 may be used legitimately to lead to quite different results from 

 those he obtains. The explanation which accords with the known 

 Maya Calendar should be accepted in preference to that which 

 requires a radical change, especially when that change is so radical 

 as to wipe out the chief land-marks by which the Mayas were ac- 

 customed to reckon time. 



Allusion has been made to the method of counting from the 

 last day of the preceding month, — or, as Dr. Seler holds, com- 

 mencing the months (and hence the years) vrith the days usually 

 counted the last. Although not essential to the present discussion, 

 we may say in reply to the suggestion which wi II arise in the mind 

 of the reader, that the first method would necessitate beginning 

 the count of the days from the last day of the preceding year, 

 that this may furnish an explanation of what has hitherto been 

 an unsolved problem — the numbering of the Ahaus. By count- 

 ing in this way we can readily see why the first Ahau of a Grand 

 Cycle cr Ahau-Katun would be numbered 13. 



