,,papers ,,of ,,the ,,peabody ,,museum ,,of ,,american ,,archaeology ,,and ,,ethnology1 ,,harvard ,,university ,,vol4 ,,vi4--,,no4 #b ,a ,,possible ,,solution ,,of ,,the ,,number ,,series ,,on ,,pages #ea ,,to #eh ,,of ,,the ,,dresden ,,codex ,,by ,,carl ,e4 ,,guthe ,,cambridge1 ,,mass4 ,,published ,,by ,,the ,,museum #aiba ,,copyright1 #aiba ,,by ,,the ,,peabody ,,museum ,,of ,,american ,,archaeology ,,and ,,ethnology1 ,,harvard ,,university ,,note ,the solution set forth in this paper formed a part of a thesis for the degree of ,doctor of ,philosophy1 taken in ,anthropology at ,harvard ,university4 ,thanks are due ,dr4 ,a4 ,m4 ,tozzer of ,harvard ,university1 and ,mr4 ,s4 ,g4 ,morley of ,the ,carnegie ,institution of ,washington1 for the interest they have taken in my work and the valuable aid they have given4 ,i must thank especially ,professor ,r4 ,w4 ,willson of the ,astronomical ,department of ,harvard for the inestimable help and unfailing patience he has contributed to my work4 ,there are many statements throughout this paper which are the direct result of his teachings4 ,i wish also to thank ,mr4 ,charles ,p4 ,bowditch1 without whose inspiration and aid this work would not have been accomplished4 ,,carl ,e4 ,,guthe ,,andover1 ,,massachusetts ,july1 #aiai ,a ,,possible ,,solution ,,of ,,the ,,number ,,series ,,on ,,pages #ea ,,to #eh ,,of ,,the ,,dresden ,,codex ,,description ,in the ,dresden ,codex1 one of the three ,maya manuscripts in existence1 there is found a series of numbers covering eight pages1 #ea to #eh 7plate ,i74 ,as early as #ahhf1 ,dr4 ,frstemann recognized this series as an important one1 and one which probably referred to the moon in some way4 ,each page is divided into an upper and a lower half designated1 respectively1 8a0 and 8b40 ,pages #ea;a and #eb;a form a unit in themselves1 but are clearly associated with the remaining pages4 ,the probable meaning of this group is still so doubtful that it has been deemed best to omit entirely a discussion of it at the present time4 ,the remaining sections of these pages form one long series of numbers which should be read from left to right1 beginning at #ec;a1 reading to #eh;a1 continuing on #ea;b1 and ending the series at #eh;b4 ,each half-page is divided1 horizontally1 into four sections4 ,the upper section consists of two rows of hieroglyphs4 ,the section just below it contains a series of black numbers which increase in value from left to right4 ,the third section consists of three rows of day glyphs with red numbers attached to them4 ,the interval between the glyphs in successive rows can1 of course1 be mathematically obtained4 ,the last1 and bottom1 division of the page is filled with a series of black numbers which are of three values only1 namely1 #agg1 #adh1 and #agh1 of which the first is the most frequent4 ,at more or less regular intervals a vertical strip is run from the top of the half-page to the bottom4 ,this strip contains1 in the upper part1 eight or ten glyphs4 ,below them in all but the first strip is a constellation band1 and below that a figure of some kind4 ,these strips divide the number series into groups1 and are called 8pictures10 occurring on ten of the fourteen half-pages4 ,considered vertically the pages are composed of columns4 ,each column contains1 beginning at the top1 two hieroglyphs1 a long number1 three day glyphs1 and their numbers1 and finally1 at the bottom1 a short number4 ,the pictures occur between these columns4 ,the series covers a period of #aa1ifj days1 although the last number recorded in the upper series is only #aa1ieh4 ,by means of the columns this period of #aa1ifj days is divided into #fi unequal parts4 ,let columns #b1 #c1 and #d on page #ed;b be taken as examples4 ,then each column in the series should be read in the following manner3#a ,the lower number of column #c is #h4ag or #agg4 ,add this number to the upper number of column #b1 which is #a4 #b4 #aa4 #i or #hadi4 ,the result is #hcbf which is expressed correctly as #a4 #c4 #b4 #f in the upper number of column #c4 ,the lower number should also be added to the upper day glyph of column #b1 which is #aj ,caban1 giving #e ,ix1 which is the day glyph and number appearing as the first in column #c4 ,the second day glyph and number is that of the day following #e ,ix1 namely #f ,men4 ,similarly1 #g ,cib is the day after #f ,men4 ,going through the same process for column #d1 #adh1 that is1 the lower number #g4 #h of that column1 should be added to #hcbf to obtain #hdgd1 which is expressed in the upper number of column #d as #a4 #c4 #i4 #ad4 ,likewise1 #adh days after #e ,ix comes #aj ,ik1 which is the upper day glyph of column #d1 and below which are found the two days immediately following1 namely1 #aa ,akbal and #ab ,kan4 #a ,for those unacquainted with ,maya arithmetic the following points will explain matters3 the ,mayas used the vigesimal system of enumeration2 they counted by twenties instead of tens4 ,a bar represented five1 and a dot stood for one4 ,they represented numbers larger than twenty by position1 just as we do4 ,however1 instead of having the smallest denomination at the right and the largest at the left of a horizontal series of figures1 they had the smallest at the bottom and the largest at the top of a column of numbers4 ,instead of each unit in a given position representing ten times the value of that of the preceding position1 it represented twenty times the value1 except in the third position where it was only eighteen times as great4 ,thus each unit of the bottom number represented one 7,kin71 that of the number above it twenty 7,uinal71 that of the third number #bj #ah or #cfj 7,tun71 that of the fourth position #bj #cfj or #gbjj 7,katun71 etc4 ,for ease in handling1 these numbers are written in our script with arabic numerals1 the bottom number on the right1 and separated by periods4 ,thus in column three1 page #ec;a1 the upper number is #a4 #g4 #b1 which means that the kin of this group is #b1 the ,uinal #g1 7#g #bj7 and the ,tun #a 7#a #cfj71 making in all #b #adj #cfj or #ejb4 ,the ,maya calendar1 like ours1 consisted of a series of numbers and a series of names for each day1 each series repeating itself constantly1 irrespective of the other4 ,there were twenty different day names1 which remained in an unchangeable order1 and thirteen numbers4 ,in the pages under discussion these day names appear as glyphs preceded by the necessary number4 ,for further details consult ,s4 ,g4 ,morley1 ,an ,introduction to ,maya ,hieroglyphs1 ,bulletin #eg1 ,bureau of ,american ,ethnology1 ,washington1 ,d4 ,c41 #aiae1 and ,c4 ,p4 ,bowditch1 #aiaj4 ,in short1 then1 the ideal arrangement of the series is as follows3 ,each upper number is the sum of all the lower numbers of the preceding columns and its own column4 ,each lower number expresses the difference between the upper number of its own column and that of the column immediately preceding it4 ,the day names and numbers are three horizontal series1 each starting a day later than the one above it1 and recording three sets of day names and numbers which would fit the series formed by the upper numbers4 ,it should be noticed that the mathematical interpretation of the series does not appear to depend in any way upon the hieroglyphs appearing at the top of the columns1 or upon the pictures4 ,this series deals quite clearly with synodical revolutions of the moon4 ,the entire series records #aa1ifj days1 although the last number in the upper series is only #aa1ieh1 a condition that will be explained later4 ,four hundred and five synodical revolutions of the moon consume1 according to modern astronomy1 #aa1iei4hhi days1 or about 4#aa of a day less than the length of the series4 ,moreover1 the difference groups #adh1 #agg1 and #agh which separate the upper numbers1 also record synodical months1 for five months consume #adg4fe days1 and six months #agg4ah days4 ,in fact the correspondence between the numbers in the series and the synodical months is so exact1 that nowhere does an error of more than one day exist4#b #b ,bowditch1 #aiaj1 pp4 #bbb1 #bbc4 ,unfortunately the ideal arrangement given above is not followed exactly4 ,the actual series as it occurs in the manuscript appears to be full of errors1 a list of which will be found in ,table ,i1 p4 #d4 ,most of these errors have been pointed out and discussed repeatedly4#c ,there still exists some doubt as to which numbers should be considered errors of the original writer and which should be taken at their face value4 ,for this reason the errors are here discussed in some detail1 for in some cases the errors1 or supposed errors1 affect theories in regard to the series4 #c ,by ,dr4 ,frstemann1 ,dr4 ,thomas1 and ,mr4 ,bowditch4 ,,table ,i ,,apparent ,,irregularities ,lower number series3 ,absence of all #agh's that occur in upper series4 ,column #bc4 ,presence of #agh4 ,column #bf4 #agg instead of #adh4 ,column #ej4 #aeg 8 8 #agg4 ,upper number series3 ,column #a4 #aeg instead of #agg4 ,column #b4 #cec 8 8 #ced4 ,column #d4 #fgd 8 8 #fgi4 ,column #aj4 #agdh 8 8 #agdb4 ,column #ab4 #bjaf 8 8 #bjif4 ,column #ad4 #cadb4 8 8 #bdbb4 ,column #ae4 #beih 8 8 #beii4 ,column #bd4 #dafd 8 8 #dafc4 ,day series3 ,column #e4 #d ,chicchan instead of #aa ,chicchan4 ,column #aa4 ,omission of #a_/b tonalamatl4 ,column #ag4 #a ,ik instead of #b ,ik4 ,column #cf4 #d ,ben instead of #d ,ahau4 ,column #dg4 #aj ,eznab instead of #aa ,eznab4 ,column #di4 #aa ,kan instead of #ab ,kan4 ,columns3 ,columns #f and #g are reversed4 ,columns #eh and #ei are reversed4 ,totals3 ,upper number series totals #aa1ieh instead of #aa1ifj4 ,day series totals #aa1iei instead of #aa1ifj4 ,in ,table ,,ii1 pp4 #f1 #g1 both the corrected and the uncorrected series are given4 ,in the centre of the table are three columns containing the actual table4 ,the third column contains the uncorrected upper number2 the fourth the lower number2 and the fifth the first day sign and its number4 ,since the other two day series agree1 except in a very few cases which will be mentioned later1 with the first series1 they have been omitted from the table4 ,the sixth column contains the day signs as they probably should occur1 and the second contains the corrected upper number4 ,the first column gives the pages of the manuscript and the number of the columns on each in order to facilitate reference to the manuscript4 ,each column of ,table ,,ii1 with the exception of the first and fourth1 is composed of two series of numbers1 since each interval between the numbers of the manuscript has been placed in parentheses after the last of the pair of numbers it deals with1 in order to facilitate comparison with the lower numbers4 ,the names and numbers in the fifth column which have parentheses have been obliterated in the manuscript1 but are easily inferred from the other two rows of day signs and numbers4 ,the most prominent irregularity is the absence of the number #agh in the lower numbers when the differences in both the day series and the upper numbers show that #agh should be the difference4 ,this occurs in columns #g1 #ad1 #bi1 #cg1 #eb and #fj of the manuscript4 ,the only place in which #agh does occur in the lower number is in column #bc1 when it agrees with the difference in the day series1 but not with that of the upper number4 ,in other words1 the six occurrences of the #agh-;day group in the upper numbers are neglected in the lower numbers1 and the only occurrence of #agh in the lower numbers does not agree with the upper numbers4 ,this implies that it is of deeper significance than a mere error4 ,there is another disagreement between the upper and lower numbers which could very well be the result of carelessness4 ,in column #bf1 the lower number is #agg1 while both the upper number and the day series give a difference of #adh4 ,this is the only case in which the differences of #adh are not found at the same place in all series1 and1 consequently1 is probably an error of the scribe4 ,again in the lower number of column #ej1 the careless omission of one dot in the ,uinal place has resulted in the record of #aeg instead of the correct number1 #agg4 ,with one exception all of the errors in the upper numbers occur in the first third of the series4 ,that exception1 i4e41 the writing of #dafd for #dafc in the column #bd1 may be explained by the fact that the writer of the series had just added in column #bc the extra day to the day series1 which threw it out of agreement with the upper numbers4 ,for the moment this fact slipped his mind1 but he corrected the mistake by subtracting one day from the difference between the upper numbers of columns #bd and #be4 ,,table ,,ii 7#a7 7#b7 7#c7 7#d7 7#e7 7#f7 ,page ,corrected ,uncorrected and upper upper ,lower ,uncorrected ,corrected ,column number number number day signs day signs #j #aa ,manik #ec;a #a #agg 7#agg7 #aeg #agg #f ,kan #f ,kan 7#agg7 #b #ced 7#agg7 #cec 7#aif7 #agg #a ,imix 7#agg7 #a ,imix 7#agg7 #c #ejb 7#adh7 #ejb 7#adi7 #adh #f ,muluc 7#adh7 #f ,muluc 7#adh7 ,,picture #d #fgi 7#agg7 #fgd 7#agb7 #agg #a ,cimi 7#agg7 #a ,cimi 7#agg7 #e #hef 7#agg7 #hef 7#ahb7 #agg #i ,akbal 7#agg7 #i ,akbal 7#agg7 #f #ajcd 7#agh7 #ajcc 7#agg7 #agg #d 7,ahau7 7#agg7 #e ,imix 7#agh7 #ed;a #g #abaa 7#agg7 #abaa 7#agh7 #agg 7#ac7 ,enzab 7#agh7 #ac ,enzab 7#agg7 #h #achh 7#agg7 #achh 7#agg7 #agg #h ,men 7#agg7 #h ,men 7#agg7 #i #aefe 7#agg7 #aefe 7#agg7 #agg #c ,eb 7#agg7 #c ,eb 7#agg7 #aj #agdb 7#agg7 #agdh 7#ahc7 #agg #aa ,muluc 7#agg7 #aa ,muluc 7#agg7 #aa #aiai 7#agg7 #aiai 7#aga7 #agg #f ,cib 7 #dg7 #f ,cimi 7#agg7 #ab #bjif 7#agg7 #bjaf 7 #ig7 #agg #a ,akbal 7#cjg7 #a ,akbal 7#agg7 #ac #bbdd 7#adh7 #bbdd 7#bhh7 #adh #f ,chuen 7#adh7 #f ,chuen 7#adh7 #ee;a ,,picture #ad #bdbb 7#agh7 #cadb 7#hih7 #agg #b ,muluc 7#agh7 #b ,muluc 7#agh7 #ae #beii 7#agg7 #beih 7-#edd7 #agg #aj ,cimi 7#agg7 #aj ,cimi 7#agg7 #af #bggf 7#agg7 #bggf 7#agh7 #agg #e ,akbal 7#agg7 #e ,akbal 7#agg7 #ag #biec 7#agg7 #biec 7#agg7 #agg #ac ,ahau 7#agg7 #ac ,ahau 7#agg7 #ah #cacj 7#agg7 #cacj 7#agg7 #agg #h 8 8 #h ,caban 7#agg7 #ef;a #ai #cbgh 7#adh7 #cbgh 7#adh7 #adh 8 ,chicchan 8 #ac ,chicchan 7#adh7 ,,picture #bj #cdee 7#agg7 #cdee 7#agg7 #agg #h ,ik 7#agg7 #h ,ik 7#agg7 #ba #cfcb 7#agg7 #cfcb 7#agg7 #agg #c ,cauac 7#agg7 #c ,cauac 7#agg7 #bb #chji 7#agg7 #chji 7#agg7 #agg #aa ,cib 7#agg7 #aa ,cib 7#agg7 #eg;a #bc #cihf 7#agg7 #cihf 7#agg7 #agh #g ,ix 7#agh7 #g ,ix 7#agh7 #bd #dafc 7#agg7 #dafd 7#agh7 #agg #b ,chuen 7#agg7 #b ,chuen 7#agg7 #be #dcdj 7#agg7 #dcdj 7#agf7 #agg #aj ,lamat 7#agg7 #aj ,lamat 7#agg7 #bf #ddhh 7#adh7 #ddhh 7#adh7 #agg #b ,cib 7#adh7 #b ,cib 7#adh7 ,,picture #eh;a #bg #dffe 7#agg7 8 8 #agg #aj ,ben 7#agg7 #aj ,ben 7#agg7 #bh #dhdb 7#agg7 #dhdb 8 #agg #e ,oc 7#agg7 #e ,oc 7#agg7 #bi #ejbj 7#agh7 #ejbj 7#agh7 #agg #a ,lamat 7#agh7 #a ,lamat 7#agh7 #cj #eaig 7#agg7 #eaig 7#agg7 #agg #i ,chicchan7#agg7 #i ,chicchan 7#agg7 #ea;b #ca #ecgd 7#agg7 #ecgd 7#agg7 #agg #d ,ik 7#agg7 #d ,ik 7#agg7 #cb #eeea 7#agg7 #eeea 7#agg7 #agg #ab ,cauac 7#agg7 #ab ,cauac 7#agg7 #cc #egbh 7#agg7 #egbh 7#agg7 #agg #g ,cib 7#agg7 #g ,cib 7#agg7 #cd #eije 7#agg7 #eije 7#agg7 #agg #b ,ben 7#agg7 #b ,ben 7#agg7 #ce #fjhb 7#agg7 #fjhb 7#agg7 #agg #aj ,oc 7#agg7 #aj ,oc 7#agg7 #cf #fbcj 7#adh7 #fbcj 7#adh7 #adh #b ,enzab 7#adh7 #b ,enzab 7#adh7 #eb;b ,,picture #cg #fdjh 7#agh7 #fdjh 7#agh7 #agg #aa ,cib 7#agh7 #aa ,cib 7#agh7 #ch #fehe 7#agg7 #fehe 7#agg7 #agg #f ,ben 7#agg7 #f ,ben 7#agg7 #ci #fgfb 7#agg7 #fgfb 7#agg7 #agg #a ,oc 7#agg7 #a ,oc 7#agg7 #dj #fici 7#agg7 #fici 7#agg7 #agg #i ,manik 7#agg7 #i ,manik 7#agg7 #ec;b #da #gaaf 7#agg7 #gaaf 7#agg7 #agg #d ,kan 7#agg7 #d ,kan 7#agg7 #db #gbfd 7#adh7 #gbfd 7#adh7 #adh #i ,eb 7#adh7 #i ,eb 7#adh7 ,,picture #dc #gdda 7#agg7 #gdda 7#agg7 #agg #d ,muluc 7#agg7 #d ,muluc 7#agg7 #dd #gfah 7#agg7 #gfah 7#agg7 #agg #ab ,cimi 7#agg7 #ab ,cimi 7#agg7 #de #ggie 7#agg7 #ggie 7#agg7 #agg #g ,akbal 7#agg7 #g ,akbal 7#agg7 #ed;b #df #gigb 7#agg7 #gigb 7#agg7 #agg #b ,ahau 7#agg7 #b ,ahau 7#agg7 #dg #hadi 7#agg7 #hadi 7#agg7 #agg #aj ,caban 7#agg7 #aj ,caban 7#agg7 #dh #hcbf 7#agg7 #hcbf 7#agg7 #agg #e ,ix 7#agg7 #e ,ix 7#agg7 #di #hdgd 7#adh7 #hdgd 7#adh7 #adh #aj ,ik 7#adh7 #aj ,ik 7#adh7 ,,picture #ej #hfea 7#agg7 #hfea 7#agg7 #aeg #e ,cauac 7#agg7 #e ,cauac 7#agg7 #ee;b #ea #hhbh 7#agg7 #hhbh 7#agg7 #agg #ac ,cib 7#agg7 #ac ,cib 7#agg7 #eb #ijjf 7#agh7 #ijjf 7#agh7 #agg #i ,ix 7#agh7 #i ,ix 7#agh7 #ec #iahc 7#agg7 #iahc 7#agg7 #agg #d ,chuen 7#agg7 #d ,chuen 7#agg7 #ed #icfj 7#agg7 #icfj 7#agg7 #agg #ab ,lamat 7#agg7 #ab ,lamat 7#agg7 #ee #iecg 7#agg7 #iecg 7#agg7 #agg #g ,chicchan 7#agg7 #g ,chicchan 7#agg7 #ef #igad 7#agg7 #igad 7#agg7 #agg #b ,ik 7#agg7 #b ,ik 7#agg7 #eg #ihia 7#agg7 #ihia 7#agg7 #agg #aj ,cauac 7#agg7 #aj ,cauac 7#agg7 #eh #ajjci 7#adh7 #ajjci 7#adh7 #adh #b ,manik 7#adh7 #b ,manik 7#adh7 #ef;b ,,picture #ei #ajbaf 7#agg7 #ajbaf 7#agg7 #agg #aj ,kan 7#agg7 #aj ,kan 7#agg7 #fj #ajcid 7#agh7 #ajcid 7#agh7 #agg #f ,ik 7#agh7 #f ,ik 7#agh7 #fa #ajega 7#agg7 #ajega 7#agg7 #agg #a ,cauac 7#agg7 #a ,cauac 7#agg7 #fb #ajgdh 7#agg7 #ajgdh 7#agg7 #agg #i ,cib 7#agg7 #i ,cib 7#agg7 #eg;b #fc #ajibe 7#agg7 #ajibe 7#agg7 #agg #d ,ben 7#agg7 #d ,ben 7#agg7 #fd #aaajb 7#agg7 #aaajb 7#agg7 #agg #ab ,oc 7#agg7 #ab ,oc 7#agg7 #fe #aabej 7#adh7 #aabej 7#adh7 #adh #d ,eznab 7#adh7 #d ,eznab 7#adh7 ,,picture #ff #aadbg 7#agg7 #aadbg 7#agg7 #agg #ab ,men 7#agg7 #ab ,men 7#agg7 #fg #aafjd 7#agg7 #aafjd 7#agg7 #agg #g ,eb 7#agg7 #g ,eb 7#agg7 #eh;b #fh #aagha 7#agg7 #aagha 7#agg7 #agg #b ,muluc 7#agg7 #b ,muluc 7#agg7 #fi #aaieh 7#agg7 #aaieh 7#agg7 #agg #aj ,cimi 7#agg7 #aj ,cimi 7#agg7 ,,picture ,the apparent error due to the addition of two dots in the ,tun place in the upper number of column #ad is more the result of an error than an error in itself4 ,this number shows a very clear case of erasure4 ,the writer of this section of the manuscript in copying from the older source1 at first overlooked column #ad1 and placed #g4c4ah1 the upper number in column #ae in this place4 ,realizing his mistake he erased the three dots in the ,uinal place1 but utilized two of the bars and the three dots in the ,kin place as the #ac needed in the ,uinal in column #ad1 and erased the lower bar of the original #ah4 ,this procedure of the writer's threw the upper number of column #ad out of alignment1 for the two dots of the ,kin appear below the #ac1 somewhat below the line of ,kins of the other columns4 ,the seven in the ,tun place should have been a six1 so the scribe inserted an extra dot between the two of the original #g1 neglecting1 however1 to erase the other two dots4 ,as a result the upper number of column #ad records the number #cadb1 which is #gbj greater than it should be1 namely1 #bdbb4 ,in column #aj1 #agdh was recorded instead of #agdb1 for a bar and three dots were written in the ,kin place instead of only two dots4 ,this is a very peculiar and unexpected form of carelessness1 which is1 however1 corrected in the next column4 ,the remaining irregularities in the upper numbers are all due to the omission of a part of the number4 ,in columns #b and #ae1 one dot was omitted in the ,kin place1 thus recording #cec instead of #ced in the former1 and #beih instead of #beii in the latter4 ,in column #d one bar was omitted in the ,kin place1 making the number #fgd1 five less than it should be1 namely1 #fgi4 ,in column #a1 one dot was omitted in the ,uinal place1 and in column #ab1 four dots of the same denomination1 recording1 respectively1 #aeg and #bjaf instead of #agg and #bjif4 ,there is only one decided error in the day series4 ,in column #aa1 #f ,cib1 #g ,caban1 #h ,eznab were written instead of #f ,cimi1 #g ,manik1 #h ,lamat4 ,it should be noticed that the number of the day was right4 ,in fact just one-half a tonalamatl1 or #acj days1 was dropped before the day series of column #aa1 and added on again immediately afterwards4 ,this is an extremely curious error to make in calculating and may shed some light on the way in which the ,mayas reckoned4 ,the five remaining irregularities in the day series are of two kinds4 ,in column #e1 the number preceding the third day1 ,chicchan1 is #d instead of #aa4 ,apparently the writer of the manuscript forgot for the moment that the day was added to the one above it and not the one to the left1 and wrote #d because the number associated with the third day sign of column #d was #c4 ,the same mistake was made in the third day of column #cf1 except that in this case it was the day sign and not the number which was confused4 ,here1 instead of writing ,ahau1 which followed the ,cauac in the second series1 ,ben was recorded because the sign to the left was ,eb4 ,the other three irregularities are all due to carelessness in placing sufficient dots in the number associated with the day sign1 for in columns #ag1 #dg and #di1 #a ,ik1 #aj ,eznab and #aa ,kan were recorded1 respectively1 instead of the necessary #b ,ik1 #aa ,eznab and #ab ,kan4 ,there are two places in which columns seem to be misplaced1 although the mathematics of the series at these points is correct as it stands4 ,for the sake of uniformity in the arrangement of the difference groups1 the #agh-;day group of column #g should occur in column #f1 and for the same reason1 the #adh-;day group of column #eh should occur in column #ei4 ,professor ,frstemann calls both of these variations errors1 and arranges his version of the table so that each part is just like the other two4 ,he gives no reason for his opinion other than the phrase 8for the author of the manuscript had confused the differences #agh and #adh44440d ,mr4 ,bowditch1 on the other hand1 allows both of the variations to stand as they appear in the manuscript1 and quite rightly holds the opinion that1 8,it may possibly be that these numbers thus placed are errors of the scribe1 but the mere plea for uniformity is not sufficient to lead us to make these changes40#e #d ,frstemann1 #aija1 p4 #abc4 #e ,bowditch1 #aiaj1 p4 #bag4 ,in ,table ,,ii the apparent mistake in columns #eh and #ei remains as it occurs in the manuscript1 for the reason which ,mr4 ,bowditch gives4 ,in the case of columns #f and #g there seems to be some evidence that there actually was an error made4 ,the last column on page #ec;a1 which is the one under discussion1 contains no day glyph in the first day series4 ,the glyph should have been that of ,ahau4 ,there is distinct evidence1 altho very faint1 that a glyph was once there4 ,moreover1 the smooth coating which covered the material of the manuscript page is not broken4 ,there are other obliterated glyphs in these pages of the manuscript1 but few in which the surface1 although unbroken1 still contains a faint1 almost continuous outline of a glyph4 ,the glyph1 then1 was probably erased4 ,the writer of the manuscript had probably completely finished column #f and started column #g before he detected the error4 ,he began to erase the part that was wrong1 then realized what an amount of alteration would be necessary1 and finally compromised by making the difference come between columns #f and #g instead of columns #e and #f4 ,this hypothesis in regard to the manner in which the erasure was done may be wrong1 but the erasure still stands as a strong evidence to show that the #agh should have occurred in column #f rather than after it4 ,finally there appears to be an error in the totals of the series1 for the upper number series records as a total #aa1ieh days and the day series #aa1iei days1 although there is strong reason for believing that the series should record #aa1ifj days4 ,this discrepancy in the totals will be referred to again4 ,in general1 then1 the apparent irregularities in the manuscript fall into two great classes1 those which are corrected in the next column or are easily detected because of their disagreement with the record in the other two series1 and those which are not obviously due to carelessness4 ,the latter will be considered under the solutions4 ,the former may be dismissed as clerical errors not affecting the solution4 ,in this group are two of the irregularities in the lower numbers 7columns #bf and #ej71 and all eight in the upper numbers1 seven of which occur in the first third of the manuscript4 ,the six errors in the day series1 and the transposition of columns #f and #g1 also belong in this class4 ,by referring to ,table ,,ii it will be noticed that the pictures occur after the #adh-;day groups in each case4 ,the upper numbers immediately preceding the pictures are given in ,table ,,iii 7p4 #aa71 together with the differences between them4 ,by grouping these differences1 it becomes apparent that the pictures may be divided into three large groups of #cihf days2 two out of the three containing the same difference numbers1 #agdb1 #ajcd1 and #abaj4 ,if1 in the last group1 the number #aj1jci were changed to #aj1baf by adding #agg1 the differences for this group would also read as the others1 when the end of the series and the beginning of the series are added together 7#gjh #ejb #abaj71 for the #ajth picture is1 in a sense1 out of the grouping since it occurs after the last number in the series4 ,the #adh-;day groups are arranged in the same order for they occur in the same columns as the numbers used above4 ,by applying the same process to the #agh-;day groups1 it is found that they also can be divided into groups which contain #cihf days4 ,in this case the second and third groups contain the same numbers1 #beih and #achh 7,table ,,iv74 ,if the number #abaa in the first group is changed to #ajcd by subtracting #agg1 the last number of this group would be #achh2 and the first number #beih could be formed by adding the remainder at both ends of the series 7#aefd #ajcd #beih74 ,it should be remembered at this point that the only column in which the lower numbers contained #agh is column #bc1 of which the upper number is #cihf4 ,this gives further grounds for dividing the series as it stands into three parts of #cihf days1 each containing #bc columns4 ,,table ,,iii ,,upper ,,numbers ,,of #adh-;,,day ,,groups ,number ,difference ,group 7#j7 #ejb #ejb #agdb #bbdd #ajcd #cihf #cbgh #abaj _/ #ddhh #agdb #fbcj #ajcd #cihf #gbfd #abaj _/ #hdgd #aefe #ajjci #abaa #aabej #cihf #gjh 7#aaieh7 7#ejb7 _/ ,,table ,,iv ,,upper ,,numbers ,,of #agh-;,,day ,,groups ,number ,difference ,group 7#aefd7 7#j7 #abaa #cihf #abaa #abaa _/ #bdbb #beih #ejbj #cihf #achh _/ #fdjh #beih #ijjf #cihf #achh _/ #ajcid #aefd 7#aaieh7 ,the three parts are not exactly alike1 however1 as has already been pointed out in considering the probable errors4 ,if the upper numbers and day numbers in column #f should be altered1 so that the difference #agh might occur in that column instead of column #g1 and if1 by the same process1 the difference of #adh could occur in column #ei instead of #eh1 then the three parts of the series would be entirely alike4 ,the three facts mentioned are1 however1 very strong evidence for supposing that the people who used this table considered it as consisting of three equal parts4 ,this series in the ,dresden is very similar to other pages of the ,dresden and other manuscripts1 two examples of which are given as illustrations4 ,one of the most interesting parallels is the series on pages #df-ej of this same manuscript4 ,this series covers a period of #bibj days which is divided into #bj unequal subdivisions4 ,on page #bd1 which just precedes page #df1 this number is used as a unit in multiplication1 that is1 the numbers occurring on page #bd are separated from each other by #bibj or multiples of this number4 ,on pages #dd;b and #de;b the number #gh is divided into four unequal parts1 and on pages #dc;b and #dd;b it is used as a unit in a series which finally reaches the number #aidj #gh4 ,,solutions ,the first references to these pages in the manuscript were concerned chiefly with the reading of the numbers without any theories in regard to the probable meaning of the series4 ,dr4 ,frstemann1 in #ahhf1 was probably the first to mention these pages specifically4 ,at this time he corrected many of the errors in the series1 and related the rows of days to the number series4#f ,he had already recognized a close relation between the difference between the #ast and #ith pictures1 i4e41 #aj1gdh1 and the ,saturn sidereal period of #aj1gec days4 ,of course1 in order to do this he had also identified the various signs in the 8constellation bands10 assigning them to various planets4#g ,these identifications are based on little more than the wish he had that they might be those planets1 and for that reason they are seriously open to doubt4 #f ,frstemann1 #ahhf1 p4 #cd4 #g ,ibid41 pp4 #fh-ga4 ,cyrus ,thomas1 two years later1 also discussed this series at some length1 but confined his considerations entirely to the mathematical side of the work4 ,he also pointed out most of the errors1 agreeing in the main with ,frstemann4 ,he considered that the series contained #aa1ifj days4 ,in his conclusion he said 8the sum of the series as shown by the numbers over the second column of ,plate #eh;b is #cc years1 #c months1 and #ah days4 ,as this includes only the top day of this column 7#aj ,cimi71 we must add two days to complete the series1 which ends with #ab ,lamat40#h #h ,thomas1 #ahhh1 p1 #cbe4 ,during the following years1 ,dr4 ,frstemann repeatedly referred to these pages in his publications and1 in #ahih1 published an article devoted to these pages alone4#i ,the most detailed as well as the final discussion of these pages is that given in his book on the ,dresden ,codex4#aj ,in pages #ec-eh1 and #ea;b and #eb;b he recognizes the similarity to pages #df-ej1 and remarks that the ,mayas not only combined the tonalamatl and the ,mercury year1 but also attempted to bring the lunar revolution into accord with these two4 ,in other words1 ,frstemann seems to imply that the primary purpose of the series was the counting of the ,mercury years1 and that the lunar part of the problem was secondary4 #i ,frstemann1 #ahih4 #aj ,ibid41 #aija1 pp4 #aah-acc4 ,he explains the number #aa1ieh as the result of attempts to make the lunar count agree with #aa1ifj4 8,they the ,mayas found that #dje lunar revolutions amounted approximately to #aa1ieh days1 which is1 in fact1 the largest number on the second half page of page #eh40aa ,this will not stand at all as the reason for the #aa1ieh since #dje lunar revolutions come to #aa1iei4hhi days1 and if the ,mayas knew the revolutions accurately enough to know when to intercalate a day1 they most certainly would not have intentionally formed the number #aa1ieh1 when they were perfectly well aware of the fact that the time was more than #aa1iei days4 ,he recognizes in the numbers #agg1 #adh and #agh multiples of lunar months of #bi and #cj days4 #aa ,frstemann1 #aija1 p4 #aba4 ,dr4 ,frstemann at this time divides the series into the three equal divisions in which it has since been considered4 ,these are of #cihf days1 thus causing the intercalated days to come at the same time in all three4#ab ,he also divided each of these three divisions into three unequal groups of #agdb1 #ajcd1 and #abaj days each4 ,he advances theories1 based on the positions of the pictures in the series1 to show that the series also referred to the siderial periods of ,saturn and ,jupiter1 and discusses the meaning of the glyphs found on these pages4 #ab ,ibid41 p4 #abc4 ,this detailed discussion by ,dr4 ,frstemann of pages #ea-eh of the ,dresden has been used as a foundation by many in further studies of these pages4 ,it is highly probable1 however1 that a careful study of his interpretations will have to be made1 in which the proved assumptions must be clearly differentiated from those in which the 8wish is father to the thought40 ,mr4 ,bowditch1 in #aiaj1ac discussed these pages and their relation to the astronomical knowledge of the ,mayas4 ,he divided the series into the same groups as ,dr4 ,frstemann1 basing his division upon the pictures which occur in every case immediately after the number #adh4ad ,mr4 ,bowditch brought out very clearly that this series is a lunar series1 by means of a table which compares the numbers recorded in the manuscript and the multiples of true lunations4#ae ,there can be no question on this point1 for the difference between the recorded days and the true lunations is never more than 4#i of a day4 ,he also pointed out a way in which this series could be used over and over again in the form of a cycle1#af and then discussed the relation of this series to the ,saturn and ,mercury periods1 disagreeing with ,frstemann on several points4 #ac ,bowditch1 #aiaj1 pp4 #baa-bca4 #ad ,ibid41 p4 #bah4 #ae ,bowditch1 #aiaj1 pp4 #bbb1 #bbc4 #af ,ibid41 p4 #bbd4 ,mr4 ,bowditch also pointed out a peculiar coincidence between the synodical revolutions of ,jupiter and the numbers in the series1 but based his argument on quite different material from the similar theory of ,dr4 ,frstemann's4 ,the important fact brought out is that the three parts of the series under discussion are almost exactly equal to #aj revolutions of ,jupiter1 for one revolution of ,jupiter consumes #cih4hfg days4#ag 8,this would give a reason for the selection of #aa1ieh to #aa1ifj days or #dje revolutions1 and for the division of this number into three sections of #cihf days each40#ah #ag ,ibid41 pp4 #bbi1 #bcj4 #ah ,ibid41 p4 #bca4 ,dr4 ,frstemann and ,mr4 ,bowditch differ in regard to some of the corrections which should be made in the manuscript1 but on the whole the two discussions of these pages supplement one another4 ,the general conclusion to be drawn from them is that these pages of the ,dresden are closely associated with the synodical lunar month1 and possibly1 with the synodical revolution of ,jupiter4 ,three years after ,mr4 ,bowditch's discussion1 ,mr4 ,meinshausen published an article in which the relation of this series to eclipses was first brought out4#ai ,he compared1 by means of two tables1 recorded eclipses of the #ahth and #aith centuries with the numbers in the ,dresden ,codex4 ,out of the #fi dates in the manuscript all but #ae dates agreed with the first case1 and1 in the second1 all but #ac1 due to the fact that all the eclipses are not visible at one place on the earth's surface4 8,another indication that the numbers in the codex have arisen from the observation of eclipses lies in the fact that the exact grouping of the numbers which is induced by the insertion of pictures in the number periods is also possible in lunar eclipses which are visible at one particular point40#bj ,in the table given to uphold this statement1 the numbers1 to be sure1 can be grouped in the manner which he suggests2 but they can also be grouped in other series4 ,in his opinion the reason for the grouping 8lies in the close proximity of a solar eclipse to a lunar eclipse10#ba that is1 that at the date at which the pictures are inserted a solar eclipse occurred #ae days either before or after a lunar eclipse4 ,there are two facts which tend to uphold this theory4 ,one is the occurrence of the sun and the moon in shields over nearly all pictures1 which he interprets as 8signs of solar and lunar eclipses02 the other is the series of dates on pages #ea;a and #eb;a1 which are #ae days apart4 ,in a table of recorded eclipses proof is given that such double eclipses can occur at the intervals which separate the pictures in the manuscript4 ,since these intervals vary a great deal1 ,meinshausen believes that they will form the means of identifying the specific eclipses recorded in the manuscript4 #ai ,meinshausen1 #aiac1 pp4 #bba-bbg4 #bj ,ibid41 p4 #bbe4 #ba ,meinshausen1 #aiac1 p4 #bbe4 ,his general conclusion is that 8the material advanced will prove sufficiently that these numbers are associated in some way with solar and lunar eclipses1 and this explanation must remain standing at least until other numbers1 corresponding equally remarkably1 are found40#bb #bb ,ibid41 pp4 #bbf1 #bbg4 ,professor ,r4 ,w4 ,willson of the ,astronomical ,department of ,harvard ,university1 working on a similar theory at about the same time1 had found1 however1 that no series of solar eclipses corresponding to the intervals of the pictures in the text was visible in ,yucatan between the ,christian era and the time of the ,spanish conquest4#bc ,this apparently invalidates ,meinshausen's theory4 #bc ,professor ,willson's work on the ,dresden manuscript has not yet been published4 ,it is referred to here only through his kind permission4 ,professor ,willson believes that the table in the manuscript indicates the days of ecliptic conjunction 7that is1 ,new ,moon occurring so near the moon's node that eclipses may occur7 and1 as ,mr4 ,bowditch has shown1 with a high degree of accuracy4 ,sufficient proof of this1 in ,professor ,willson's opinion1 is the close correspondence of the intervals of the codex with the intervals of ,schram's lunar table4#bd #bd ,schram1 #aijh1 pp4 #ceh1 #cei4 ,the similarity between the numbers in the ,dresden and ,schram's table is so remarkable that it seems advisable to point out some of the most outstanding features4 ,in addition to giving the days of multiples of the lunar synodic months1 this table also gives the time of possible occurrences of both solar and lunar eclipses4 ,eclipses occur in cycles1 the best known of which is the ,saros1 although there are also smaller cycles which are not so accurate4 ,table ,v 7p4 #ag7 gives the occurrences of central solar eclipses according to ,schram4 ,it should be noticed that they occur in groups of threes and fours1 each set being separated from the preceding one by #bi synodical months4 ,the numbers in each group are only six months apart4 ,table ,,vi 7p4 #ag7 is a corresponding series of lunar eclipses1 which also occur in a grouping similar to that of the solar eclipses4 ,it should be noticed in passing that the first numbers of these groups1 in both the solar and lunar eclipses are separated by #dg and #da lunations1 the latter occurring after every third group in ,table ,v4 ,table ,,vii 7p4 #ag7 contains the numbers which are in the same columns as the #agh-;day groups in the ,dresden4 ,by comparing ,table ,v and ,table ,,vii1 it will be found that the numbers in the ,dresden are the same as the first numbers in groups #a1 #b1 #d1 #e1 #g and #h of the solar eclipses4 ,in the last two numbers there is a difference of one day1 which is explained by recalling the addition of an extra day in the day series but not in the upper numbers of the ,dresden4 ,if #fgi days are added to each number in ,table ,,vii1 which amounts to the same thing as advancing the ,dresden table #fgi days with respect to ,schram's table1 it will be found that these numbers will also agree with the first numbers in groups #b1 #c1 #e1 #f and #h and with the second number in group #i of the lunar eclipses1 in ,table ,,vi4 ,a similar agreement may be observed for the #adh-;day groups 7see ,table ,,iii74 ,this remarkable agreement between the #agh-;day groups in the ,dresden and the occurrences of eclipses may have several meanings4 7#a7 ,one possibility1 and one which should always be kept in mind1 is that this agreement is simply another coincidence1 of which there are always many in chronological work4 7#b7 ,it may be that the numbers refer to dates of prophesied eclipses which the ,mayas had learned occurred at more or less regular intervals4 7#c7 ,since this table has a place in the calendar of the ,mayas 7for a date probably occurs on page #eb;a71 it may be that these numbers refer to definite historical eclipses4 ,if they do1 they will afford a means by which an absolute correlation between the ,maya and the ,julian calendars may be obtained4 ,professor ,willson is at present working on this problem4 ,,table ,v ,,solar ,,eclipses ,group ,eclipse ,month _/ #ajcd #ce #a #abaa #da #achh #dg #aefe #ec _/ #bdbb #hb #b #beii #hh #bggf #id #biec #ajj _/ #cfcb #abc #c #chji #abi #cihg #ace #dafd #ada _/ #ejbj #agj #d #eaig #agf #ecge #ahb #eeeb #ahh _/ #fdjh #bag #e #fehe #bbc #fgfb #bbi _/ #gfai #beh #f #ggif #bfd #gigc #bgj #haej #bgf _/ #ijjg #cje #g #iahd #caa #icfa #cag #iech #cbc _/ #ajcie #ceb #h #ajegb #ceh #ajgej #cfd _/ #aafjf #cic #i #aaghc #cii #aaifj #dje ,,table ,,vi ,,lunar ,,eclipses ,group ,eclipse ,month _/ #ejb #ag #a #fgi #bc #hef #bi _/ #agac #eh #b #ahij #fd #bjfg #gj #bbdd #gf _/ #caja #aje #c #cbgh #aaa #cdee #aag _/ #dcaa #adf #d #ddhi #aeb #dfff #aeh #dhdc #afd _/ #efii #aic #e #ehgg #aii #fjed #bje #fbca #baa _/ #gjhg #bdj #f #gbfd #bdf #gddb #beb _/ #hbih #bha #g #hdge #bhg #hfeb #bic #hhcj #bii _/ #ifhf #cbh #h #ihfc #ccd #ajjdj #cdj _/ #ajhif #cfi #i #aajgd #cge #aabea #cha #aadbh #chg ,,table ,,vii #agh-;,,day ,,groups ,number ,month #ajcd #ce #bdbb #hb #ejbj #agj #fdjh #bag #ijjf #cje #ajcid #ceb ,in order to determine the exact extent to which the eclipse seasons affect these pages in the ,dresden ,codex it is necessary to work out in as great detail as possible the calendar represented4 ,modern astronomy shows that the synodical revolution of the moon consumes #bi4ecjei days1 about 4#jc days more than #bi-a_/b days4 ,since a calendar must be based on whole days the natural method of combining the months would be to alternate one of #bi days with one of #cj days4 ,at the end of two months or #ei days the true synodical month would be in advance of the calendrical month by 4#jfaah days4 ,every two months this error is doubled so that at the end of #cd months the calendar would have completed #ajjc days and the synodical month #ajjd4jd days4 7,see ,table ,,viii1 p4 #ai47 ,one method of correcting this would be to make the last month a #cj-;day month instead of one of #bi days as it would be by simple alternation4 ,this #cd-month period could then be repeated as a cycle with an accumulating error of 4#jd days at every repetition4 ,such a series utterly disregards1 however1 all other phenomena such as eclipses1 seasons1 etc4 ,as soon as eclipses are considered the arrangement of the months must be altered in order to use the periodicity of eclipses in the calendar4 ,eclipses occur at regular seasons1 approximately six months apart4 ,the average interval between eclipse seasons is #agc4caj days1 #c4hgd days less than six synodical lunar months4 ,in ,table ,,ix 7p4 #bj7 the eclipse season is compared with the nearest synodical lunar month4 ,it will be noticed that the difference increases between the two series until it is necessary to use five synodical months for one interval instead of six to keep the difference less than half a month4 ,it is necessary to do this three times in #ace synodical months1 or #cihf4fcj days1 which exceed #bc eclipse seasons1 or #cihf4aca days1 by practically one half-day4 ,it would be most logical to drop these extra months out of the set of six1 during that group in which the difference tends to become most nearly half a month4 ,that would be just before the #bc;d1 #gjth1 and #aagth month1 that is1 #dg months apart1 requiring #da months to complete the #ace-month period4 ,,table ,,viii ,number of ,number of ,elapsed days ,elapsed days month days in month calendar month synodical month ,error #a #cj #cj #bi4ec -4#dg #b #bi #ei #ei4jf 4#jf #c #cj #hi #hh4ei -4#da #d #bi #aah #aah4ab 4#ab #e #cj #adh #adg4fe -4#ce #f #bi #agg #agg4ah 4#ah #g #cj #bjg #bjf4ga -4#bi #h #bi #bcf #bcf4bd 4#bd #i #cj #bff #bfe4gh -4#bb #aj #bi #bie #bie4ca 4#ca #aa #cj #cbe #cbd4hd -4#af #ab #bi #ced #ced4cg 4#cg #ac #cj #chd #chc4ij -4#aj #ad #bi #dac #dac4dc 4#dc #ae #cj #ddc #ddb4if -4#jd #af #bi #dgb #dgb4di 4#di #ag #cj #ejb #ejb4jb 4#jb #ah #bi #eca #eca4ee 4#ee #ai #cj #efa #efa4jh 4#jh #bj #bi #eij #eij4fa 4#fa #ba #cj #fbj #fbj4ad 4#ad #bb #bi #fdi #fdi4fg 4#fg #bc #cj #fgi #fgi4bj 4#bj #bd #bi #gjh #gjh4gc 4#gc #be #cj #gch #gch4bf 4#bf #bf #bi #gfg #gfg4hj 4#hj #bg #cj #gig #gig4cc 4#cc #bh #bi #hbf #hbf4hf 4#hf #bi #cj #hef #hef4ci 4#ci #cj #bi #hhe #hhe4ib 4#ib #ca #cj #iae #iae4de 4#de #cb #bi #idd #idd4ih 4#ih #cc #cj #igd #igd4ea 4#ea #cd #bi #ajjc #ajjd4jd #a4jd ,this series of #ace lunar months1 or #bc eclipse seasons1 can be repeated almost indefinitely1 alternating #cihf and #cihg days to the series and still keep the synodical month in accord with the eclipse season4 ,but another factor must also be considered4 ,months of #bi and #cj days cannot be simply alternated and either conform with the true synodical month or complete the ecliptic series mentioned1 for #cihf contains three more days than sixty-eight #cj-;day months1 and sixty-seven #bi-;day months4 ,therefore in the #cihf series three of the #bi-;day months must be changed to #cj-;day months1 and in the #cihg series four must be changed4 ,the position of these changes is arbitrary4 ,they can1 for example1 be the #cdth1 #fhth1 and #ajb;d months1 and when necessary1 the #acdth4 ,,table ,,ix ,,comparison ,,of ,,synodic ,,months ,,and ,,eclipses ,eclipse season ,synodic month ,number ,days ,number ,days ,difference #a #agc4caj #f #agg4ahd #c4hgd #b #cdf4fbj #ab #ced4cfg #g4gdg #c #eai4icj #ah #eca4eea #aa4fba #d #fic4bdj #bc #fgi4bjd -#ad4jcf #e #hff4eej #bi #hef4chg -#aj4afc #f #ajci4hfj #ce #ajcc4ega -#f4bhi #g #abac4agj #da #abaj4ged -#b4daf #h #achf4dhj #dg #achg4ich #a4deh #i #aeei4gij #ec #aefe4aba #e4cca #aj #agcc4ajj #ei #agdb4cje #i4bje #aa #aijf4daa #fe #aiai4dhi #ac4jgh #ab #bjgi4gba #gj #bjfg4ada -#ab4ehj #ac #bbec4jca #gf #bbdd4cbe -#h4gjf #ad #bdbf4cda #hb #bdba4ejh -#d4hcc #ae #beii4fea #hh #beih4fib -4#iei #af #bggb4ifa #id #bgge4hge #b4iad #ag #bidf4bga #ajj #biec4jei #f4ghh #ah #caai4eha #ajf #cacj4bdc #aj4ffb #ai #cbib4hia #aab #ccjg4dbf #ad4ece #bj #cdff4bja #aag #cdee4jgi -#aa4abb #ba #cfci4eaa #abc #cfcb4bfc -#g4bdh #bb #chab4hba #abi #chji4ddf -#c4cge #bc #cihf4aca #ace #cihf4fcj 4#dii ,,table ,x #adh-;,,day ,,groups ,upper ,month number number ,interval ,groups #ejb #ag #ei #bbdd #gf #id7dg #dg7 #ce _/ #cbgh #aaa #da #da #ddhh #aeb #ei #fbcj #baa #id7dg #dg7 #ce _/ #gbfd #bdf #da #da #hdgd #bhg #ec #ajjci #cdj #id7dg #dg7 #da _/ #aabej #cha ,the next logical step is a comparison between the theoretical calendars just described and the manuscript4 ,a study of the manuscript reveals that3 7#a7 the series recorded represents #dje lunar months or three times #ace months1 and that the series naturally falls into three great subdivisions of #cihf days each2 7#b7 each third consists of #bc columns or unequal subdivisions2 7#c7 the intervals between the #agh-;day groups are #dg and #hh months2 7#d7 the #adh-;day groups fall approximately at #dg and #da month intervals 7see ,table ,x72 7#e7 the first #agh-;day group in each third occurs between the #cjth and #ceth month inclusive1 and the other #agh-;day group of the third comes #dg months later4 ,since the number #agh is composed of four #cj-;and two #bi-;day months1 an extra day must have been added1 that is1 a #cj-;day month was substituted for one of the #bi-;day months1 if the manuscript represents a regular alternating series4 ,the obvious conclusions to be drawn from these facts are3 7#a7 that the series was divided into three groups of #cihf days each in order to associate the lunar calendar closely with the ecliptic cycle of the same length2 7#b7 that the #bc columns in each third may represent the twenty-three eclipse seasons in each eclipse period of #cihf days2 7#c7 that groups of #dg and #da months were used in some way in the series1 for the #agh-;day groups are separated by #dg and #hh months and #hh is composed of #dg and #da1 the two periods so closely associated with the recurrence of eclipses2 7#d7 that the six months period was changed to one of five months of #adh days approximately every #dg and #da months1 which is the method already advanced in the theoretical ecliptic lunar series for keeping the synodical months and ecliptic seasons together2 7#e7 that one extra day was added to the alternating #bi-;and #cj-;day months1 between the #cjth and #ceth month inclusive of each third1 in accordance with the theoretical necessity for so doing already brought out1 and that another of the three extra days was added #dg months later4 ,when the difference groups#be are divided into months it is found that it is an easy matter to arrange the months in an alternating series4 ,the group of #agg days is composed of three #cj-;and three #bi-;day months1 either of which when alternated can begin the group1 which then ends with the other1 i4e41 #bi1 #cj1 #bi1 #cj1 #bi1 #cj1 or #cj1 #bi1 #cj1 #bi1 #cj1 #bi4 ,the group of #adh days consists of three #cj-;and two #bi-;day months1 necessitating that it begin and end with a #cj-;day month when alternated1 thus1 #cj1 #bi1 #cj1 #bi1 #cj4 ,in the #agh-;day group one of the #bi-;day months is replaced by a #cj-;day month1 otherwise the group is exactly like that of #agg days1 which it exceeds by one day4 ,it is evident that there will always be three #cj-;day months in succession in the #agh-;day group1 and that care must be taken in choosing the right sequence of the #agg-;day groups which fall near those of #adh days in order to avoid having two #cj-;day months in succession4 #be ,that is1 the #agg-;day1 #adh-;day and #agh-;day groups4 ,there remains simply the substitution of the six or five months1 as the case may be1 in place of the difference groups in the manuscript4 ,however1 if the ,mayas considered each third of the table as a unit1 it is reasonable to assume that the sequence of the months in each third is identical4 ,therefore it is necessary to arrange a sequence for only one-third1 that is1 #ace months1 and then1 if the assumption is correct1 this sequence will fit the other two-thirds of the series4 ,each third of the table consists of #ace months covering three more days than would be covered by a simple alternation of #cj-;and #bi-;day months4 ,these three intercalary days were inserted at definite intervals4 ,a clue to the position of two of them is given by the #agh-;day groups4 ,one was inserted between the #cjth and #ceth months1 another #dg months later1 between the #ggth and #hb;d months4 ,theoretically the extra day should be inserted in the #cdth month after the beginning of a series of alternating #bi-;and #cj-;day months1 for then the error between the synodical revolution of the moon and the calendrical months becomes more than one day4 ,in the #bi-;day month preceding the #cdth1 namely the #cb;d month1 the error at the end is also practically one day1 i4e41 4#ih days4 ,the #bi-;day month most nearly the centre of the first #agh-;day group is the #cb;d month of the series1 the third in the group4 ,the ,mayas may have chosen this month because of its position in the #agh-;day group1 making the sequence of the months #bi1 #cj1 #cj1 #cj1 #bi1 #cj1 if the #cjth month was a #bi-;day month as it would be by simple alternation4 ,the second time this intercalary day occurs in each third is #dg months later4 ,obviously1 this may be the recurrence of this intercalation in a repetition of a smaller group of months than the #ace-month group4 ,if #dg months are subtracted from the #gith month which is the third in the second #agh-;day group the result is #cb1 which implies that the smaller division is #dg months4 ,two #dg-month periods complete all but #da of the #ace months in each third4 ,then1 of necessity1 if each third of the manuscript is a unit1 a #da-month group follows two #dg-month groups1 an arrangement which also agrees with the eclipse groups in ,tables ,v4 ,the two #agh-;day groups account for only two of the three intercalated days1 and since no #agh-;day group occurs in the #da-month division1 the addition of this day must have been accomplished in some more obscure manner4 ,since both #dg and #da are odd numbers1 each group must contain at least one more month of one kind than the other4 ,since two synodical revolutions of the moon are slightly longer than two calendrical months it is wisest to start and end each group with a #cj-;day month4 ,if this is done1 the #dg-month group will contain twenty-five #cj-;day and twenty-two #bi-;day months1 and the #da-month group twenty-one #cj-;day and twenty #bi-;day months1 making for the composition of the #ace months1 seventy-one #cj-;day and sixty-four #bi-;day months1 that is1 seven more of the #cj-;day months than of those of #bi days1 showing that actually three of the sixty-seven #bi-;day months expected in a normal repetition have become #cj-;day months4 ,this is caused by the occurrence of two #cj-;day months in succession at the end of one series and the beginning of the next4 ,if the #ace months in each third are numbered in succession it will be seen that in the first #dg-month group and in the #da-month group1 the #cj-;day months are the odd numbers4 ,in the second #dg-month group they are the even numbers1 of which there is one more in this division than odd numbers1 thus accounting for the additional one of the three days4 ,if the period of #cihf days were considered by itself1 the arrangement given would be sufficient4 ,as soon1 however1 as this period is repeated a number of times an error develops1 since #ace synodical revolutions of the moon are completed in #cihf4fc days4 ,twice this number gives #gigc4bf1 or #a4bf days more than twice #cihf4 ,in order to keep the sequence of months in the arrangement given above in accordance with the moon1 it becomes necessary to intercalate one more day every two repetitions of the #cihf period4 ,this may be done by changing the last #bi-;day month in the #da-month group to a #cj-;day month1 making the last #agg-;day group in the third one of #agh days4 ,the ,mayas certainly did this in the first third of the series given and arranged for it in the last third in a manner which will be demonstrated later4 ,tabulating the solution here advanced will form ,table ,,xi 7p4 #be71 in which the #cj-;and #bi-;day months in one-third of the manuscript have been arranged in three columns1 the first two of which represent the #dg-month groups and the last one the #da-month group4 ,before the first column of months are numbers to facilitate locating any given month in the group4 ,the two kinds of months occur in direct alternation in each group1 with three exceptions4 ,the #cb;d month in both of the #dg-month groups is one of #cj days instead of #bi1 because of the addition of the intercalary day4 ,the #djth month in the #da-month group is given as one of #bi days with a #cj in parentheses before it1 representing the fact that every other third an extra intercalary day should be inserted in this month4 ,to the right of the month columns are three columns giving the difference groups as found in the manuscript 7see ,table ,,ii71 each column giving those numbers found in each third of the manuscript1 the first third being the left one of the three4 ,it should be noticed that the misplaced 787 #adh-;day group in the last third does not interfere with the sequence of the months4 ,finally it only remains to review the irregularities of the manuscript in the light of the solution just advanced4 ,those irregularities which are corrected immediately afterwards1 or are at variance with the rest of the column in which they occur1 are1 in all probability1 errors on the part of the writer of the series1 such as might have been caused by careless transcription from another copy of the table1 and correction in the following column to avoid the task of erasure4 ,eliminating these irregularities1 there remain three to be investigated1 namely1 7#a7 the absence of the #agh numbers in the lower number series1 with one conspicuous exception2 7#b7 the occurrence of #agh in column #g instead of #f1 and of #adh in column #eh instead of #ei2 and 7#c7 the discrepancies in the totals of the series4 ,,table ,,xi4--,,the ,,arrangement ,,of ,,lunar ,,months ,,in ,,the ,,dresden ,,table ,no4 ,days ,days ,days of in ,dresden groups in ,dresden groups in ,dresden groups month month #a #b #c month #a #b #c month #a #b #c #a #cj #cj #cj #b #bi #bi #bi #c #cj #cj #cj #d #bi #agg #agg #agg #bi #agg #agg #agg #bi #agg #agg #agg #e #cj #cj #cj #f #bi _/ #bi _/ #bi _/ #g #cj #cj #cj #h #bi #bi #bi #i #cj #cj #cj #aj #bi #agg #agg #agg #bi #agg #agg #agg #bi #agg #agg #agg #aa #cj #cj #cj #ab #bi _/ #bi _/ #bi _/ #ac #cj #cj #cj #ad #bi #bi #bi #ae #cj #adh #adh #adh #cj #cj #adh #adh #adh #af #bi #bi #agg #agg #agg #bi #ag #cj _/ #cj #cj _/ #ah #bi #bi _/ #bi #ai #cj #cj #cj #bj #bi #bi #adh #bi #ba #cj #agg #agg #agg #cj #agg #agg #cj #agg #agg #agg #bb #bi #bi #bi #bc #cj _/ #cj _/ #cj _/ #bd #bi #bi _/ #bi #be #cj #cj #cj #bf #bi #bi #agg #bi #bg #cj #agg #agg #agg #cj #adh #adh #cj #agg #agg #agg #bh #bi #bi #bi #bi #cj _/ #cj _/ _/ #cj #cj #bi #bi #bi _/ #ca #cj #cj #cj 99#cb #cj #cj #bi #cc #cj #agh #agh #agh #cj #agh #agh #agh #cj #agg #agg #agg #cd #bi #bi #bi #ce #cj _/ #cj _/ #cj _/ #cf #bi #bi #bi #cg #cj #cj #cj #ch #bi #agg #agg #agg #bi #agg #agg #agg #bi #agh #agg #agg #ci #cj #cj #cj 7#agh7 #dj #bi #bi 7#cj7bi #da #cj _/ #cj _/ -#cj _/ #db #bi #bi #dc #cj #cj #dd #bi #bi #de #cj #agg #agg #agg #cj #agg #agg #agg #df #bi #bi #dg #cj _/ #cj _/ ,the great bulk of the difference groups as expressed by the lower number series are #agg1 the only departure from these being the designation of the #adh-;day groups and the extra #agh-;day group at the end of the first third4 ,the complete disregard of all of the six normal #agh-;day groups by the lower numbers seems to imply that no attempt was made to have the latter agree with the actual differences in the upper numbers1 a conclusion which is strengthened by the fact that none of the lower numbers shows evidence of the clerical errors in the differences of the upper numbers4 ,it seems most probable that the lower number series was intended merely as a guide to indicate the position of the five month periods and to place emphasis on the extra intercalary day added in the #bc;d column1 without attempting to have this series accurate4 ,the presence of the #agh-;day group in column #g instead of #f has been discussed at some length under the description of the errors4 ,the scribe1 realizing that in neglecting to put in this #agh-;day group1 the first one of the series1 a serious error had been committed1 may have attempted to erase the incorrect record in column #f2 then1 realizing that four numbers and three glyphs would have to be altered1 decided to correct this mistake--although it was of more importance than the other two errors--as he had the former ones1 i4e41 by making the correction in the next column4 ,the very similar irregularity in the last third of the manuscript1 the placing of the #adh-;day group one column ahead of its expected position1 cannot be explained in the same manner4 ,it is very evident that this column has been deliberately placed where it is4 ,that it does not have to do with the month sequence is evident1 since it does not affect it4 ,it must then affect the ecliptic part of the series1 for it causes a short season to occur six months earlier than expected4 ,upon comparison of ,tables ,,vi and ,x1 it will be seen that all of the dates of the #adh-;day group occur during one of the eclipse groups given in ,schram's table4 ,however1 had the #adh-;day group under discussion been placed in the #eith column1 as uniformity demands1 this number1 #aj1baf1 would not have fallen in one of the eclipse groups given in ,table ,,vi4 ,this tends to show that there was some reason other than regulating the difference groups to agree with the eclipse seasons1 for the position of the #adh-;day groups4 ,this reason1 as yet undetermined1 is possibly associated with the pictures1 which immediately follow the #adh-;day groups4 ,finally there remain only the totals of the series to be considered4 ,the total of the upper number series records #aa1ieh days4 ,sixty-nine eclipse seasons complete #aa1ieh4ci days1 less than half a day more than the recorded number4 ,this close agreement and the failure to add the extra intercalary day to the upper number series at the end of the first third1 give rise to the belief that the upper number series is a calendar in itself1 and records a means by which dates of probable eclipses may be reckoned4 ,the units of the count were eclipse seasons expressed as lunar months1 #fi of which are represented in the calendar recorded on these pages4 ,the ,mayas undoubtedly knew the relation of the eclipses to the moon1 at least in a vague way1 and felt that it was necessary to associate this eclipse calendar in some way with the lunar calendar1 composed of #bi-;and #cj-;day months4 ,therefore the day series is found immediately below the upper number series4 ,this series of days constitutes a lunar calendar which coincides as closely as possible with the eclipse calendar4 ,it may be the formal lunar calendar of the ,mayas1 but it may also be an adaptation of the formal calendar to the eclipse periods4 ,the day series varies from the eclipse series in two places only4 ,at the end of the first third of the series1 it was necessary to add one day to the lunar calendar1 an addition strongly pointed out in the record1 but not to the eclipse calendar1 because of the increasing error between the revolutions of the moon and the calendrical lunar months4 ,therefore1 throughout the remaining two-thirds of the series1 the lunar calendar was one day in advance of the eclipse calendar4 ,at the end of the series1 since #dje of the moon's revolutions complete #aa1iei4hi days1 and the day series only #aa1iei1 one more day should be added1 in order to keep the error as small as possible4 ,this was accomplished by changing from the middle to the lower line of days4 ,on page #eb;a1 immediately preceding the calendar1 are four day signs with numbers4 ,one of these1 #ab ,lamat1 is the zero day of the day series1 but is associated with the middle line of day glyphs and not the upper line1 as might be expected4 ,the series of days which come1 calendrically speaking1 just before and after the actual series1 may have been placed in the record to show that slight variations from the average were to be expected4 ,the entire record is based on the middle line of days until the end of the series4 ,here the day just below the last day of the middle line is #ab ,lamat1 the end of #df tonalamatls 7#bfj-;day cycle71 and the zero day of the recorded series4 ,the tonalamatl was probably as easily used by the ,mayas as 8#fj days0 and 8#ij days0 are used now4 ,the entire calendrical system of the ,mayas is based on the cycle principle4 ,the series recorded in these pages was probably also a cycle1 and in order to repeat it1 #ab ,lamat must again be used as the zero date4 ,if to these arguments is added the fact that an additional day is necessary to keep the calendar in accord with the synodic revolution of the moon1 there remains little doubt but that the users of this calendar added the extra day by going from the middle to the lower line of day glyphs1 thereby keeping the error between the moon and the calendar as low as possible1 completing the #dfth tonalamatl1 and at the same time making it possible to repeat the recorded series as a cycle4 ,if the series is repeated once1 at the end of #haj months1 or about #ff-a_/b years1 the eclipse calendar will be behind the average eclipse season 4#gh days1 and the lunar calendar will be in advance of the synodical revolutions of the moon only 4#bb days4 ,in general1 then1 the irregularities in the calendar recorded on these pages fall into two groups1 those which are clerical errors of the scribe and do not therefore affect the solution advanced1 and those which do not appear to be of the clerical type4 ,in the light of the solution advanced1 it has been shown that there are perfectly logical reasons for the latter group of apparent irregularities4 ,,conclusion ,on pages #ea to #eh of the ,dresden ,codex occurs a series of numbers1 running continuously through all the pages except the upper halves of the first two4 ,this series records a period of #aa1ifj days1 divided by means of columns into sixty-nine unequal subdivisions1 of #agg1 #adh1 and #agh days1 of which the first is the most frequent4 ,there are three distinct series4 ,one series of numbers is in the upper part of the record1 and consists of totals increased step by step until the final total reached records #aa1ieh days4 ,just below this series are three series of day signs and numbers1 the middle one of which is the actual series4 ,these dates are separated by the same number of days as the upper number series1 except in the #bc;d group1 at which place one extra day is added to the day series and not to the upper number series1 causing the former to be in advance of the latter one day throughout the remainder of the record4 ,at the end of the day series another extra day is added by counting in the last day in the lower row of days1 thus completing the #aa1ifj-;day period4 ,below this day series is another number series no term of which exceeds #agh4 ,in a general way it records the differences between the dates appearing above each of its numbers4 ,the agreement is however so inaccurate that this lower number series could1 at best1 have been used only as a general guide to the user of the manuscript1 in that it calls attention to the intervals of unusual length4 ,the series recorded is composed of three equal parts1 each composed of #bc subdivisions and covering #cihf days4 ,the number series on these pages record an eclipse calendar1 that is1 a series of dates by means of which the occurrence of eclipses was foretold4 ,this calendar is composed of three identical parts1 with the exception of one #adh-;day group which occurs six months earlier in the last third than in the other two4 ,each third is composed of #bc unequal subdivisions which represent the twenty-three eclipse seasons1 expressed in lunar months1 in #cihf days4 ,the upper number series records this calendar1 and its total of #aa1ieh days is only 4#ci days less than #fi eclipse seasons4 ,in order to make it more intelligible this eclipse calendar is accompanied by a probably more generally known lunar calendar1 which may have been altered slightly to conform to the requirements of the eclipse calendar it accompanies4 ,this lunar calendar is contained in the day series just below the eclipse calendar4 ,it also is recorded in three divisions agreeing closely with the eclipse calendar4 ,one hundred and thirty-five lunar months of #cj and #bi days complete #cihf days1 4#fc days less than #ace synodical revolutions of the moon4 ,this error which amounts to more than one day when repeated once1 necessitates the addition of an extra day in the lunar calendar every other third1 which was done in the manuscript in the first and last third1 making the total recorded by the lunar calendar #aa1ifj1 two days more than the eclipse calendar1 and 4#aa days more than #dje synodical revolutions of the moon4 ,this period of #aa1ifj days may have been used as a cycle1 the zero day of which is #ab ,lamat4 ,each third of the lunar calendar consists of #cj-;and #bi-;day months arranged in alternating sequence1 with intercalary days added by the substitution of a #cj-;day for a #bi-;day month when the error arising from the nonconformity of the moon's revolution reaches more than one day4 ,in order that the lunar calendar might agree with the eclipse calendar more closely1 these months were recorded in groups of five and six4 ,the months in each third of the series were divided into three groups1 which are the same in each third4 ,the first two groups contained #dg months each1 and completed the first sixteen dates of the third4 ,the last group was one of #da months1 which was represented by the last seven dates of the third4 ,an intercalary day was added in the #cb;d month of each of the #dg-month groups to correct the accumulating error1 thereby causing the #fth and #adth subdivisions of the third to be of #agh days4 ,in the first and last third the #djth month of the #da-month group also contained an intercalary day for the same reason1 making the #bc;d subdivision #agh days1 but in the last column of the record this extra day is added by going from the middle to the lower line of day signs4 ,each of the #dg-;and #da-month divisions began and ended with a month of thirty days4 ,the numerical series of these pages of the ,dresden record1 then1 an eclipse calendar which is referred to a lunar calendar4 ,this solution explains all the irregularities of the series except those which seem clearly to be clerical errors of the scribe4 ,only the numerical and calendrical series on these pages have been considered4 ,no attempt has been made to explain the hieroglyphs1 the pictures1 or the first two pages1 which1 although showing a close association to the long series1 are nevertheless a unit in themselves4 ,,bibliography ,,bowditch1 ,c4 ,p4 #aiaj4 ,the ,numeration1 ,calendar ,systems and ,astronomical ,knowledge of the ,mayas4 ,cambridge4 ,f,,rstemann1 ,e4 #ahhf4 ,erluterungen zur ,mayahandschrift der ,kniglichen ffentlichen ,bibliothek zu ,dresden4 ,dresden4 #ahih4 ,zur ,entzifferung der ,mayahandschrift1 ,,vii4 ,dresden4 7,trans4 in ,bull4 #bh1 ,bur4 ,am4 ,ethnol41 pp4 #dfc-gb47 #aija4 ,commentar zur ,mayahandschrift der ,kniglichen ffentlichen ,bibliothek zu ,dresden4 ,dresden4 7,trans4 in ,papers of the ,peabody ,museum1 ,vol4 ,,iv1 ,no4 #b47 ,,meinshausen1 ,,martin4 #aiac4 ber ,sonnen- und ,mondfinsternisse in der ,dresdener ,mayahandschrift4 ,zeit4 fr ,ethnol41 ,bd4 ,,xlv1 pp4 #bba-bbg4 ,,schram1 ,r4 #aijh4 ,kalendariographische und ,chronologische ,tafeln4 ,leipzig4 ,,thomas1 ,,cyrus4 #ahhh4 ,aids to the ,study of the ,maya ,codices4 #fth ,ann4 ,rep41 ,bur4 ,am4 ,ethnol41 pp4 #bec-cga4 ,,printed ,,at ,,the ,,harvard ,,university ,,press ,,cambridge1 ,,mass41 ,u4 ,s4 ,a4 ,,peabody ,,museum ,,papers4 ,,vol4 ,,vi1 ,,no4 #b1 ,,plate #a ,illustration3 ,,dresden ,,codex1 ,,pages #ea ,,to #eh4 ,,transcriber',s ,,notes 99 ,this text has been preserved as in the original1 including archaic and inconsistent spelling1 punctuation and grammar1 except as noted below4 99 ,where the original work uses text in italics1 this e-text uses text4 ,small capitals in the original work are represented here in all capitals4 99 ,footnotes have been renumbered and then moved to directly below the paragraph to which they belong4 99 ,obvious printer's errors have been silently corrected4 99 ,page #c3 ,table ,i was relocated from mid-paragraph 7page #d7 to immediately prior to the beginning of the paragraph4 99 ,page #e3 ,table ,,ii was relocated from mid-paragraph 7pages #f and #g7 to immediately prior to the beginning of the paragraph4 99 ,page #h3 uinal changed to ,uinal4 99 ,page #aa3 ,tables ,,iii and ,,iv were relocated from within the paragraph to immediately prior to the paragraph4 99 ,page #aa3 ,table ,,iii and ,table ,,iv were side-by-side4 ,table ,,iv was relocated to below ,table ,,iii4 99 ,page #ag3 ,tables ,v1 ,,vi and ,,vii were side-by-side and ,table ,,vii was relocated to below ,tables ,v and ,,vi4 99 ,page #ah3 ,table ,,viii was relocated from mid-paragraph 7page #ai7 to immediately prior to the beginning of the paragraph4 99 ,page #ai3 ,table ,,ix and ,table ,x were relocated from mid-paragraph 7page #bj7 to immediately prior to the beginning of the paragraph4 99 ,page #bd3 ,table ,,xi was relocated from mid-paragraph 7page #be7to immediately prior to the relevant paragraph on page #bd4 99 ,page #bd1 ,table ,,xi3 ,the reference for the asterisk at ,no4 of ,month #cb was not found4