MORLBY] INTRODUCTION TO STUDY OF MAYA HIEEOGLYPHS 143 



Since the foregoing is true, it follows, that this number 18,980 or 

 any multiple thereof, may be deducted from the number which is to 

 be counted without affecting in any way the terminal date which the 

 number will reach when counted from the starting point. It is 

 obvious that this modification applies only to numbers which are 

 above 18,980, all others being divided by 13, 20, and 365 directly, as 

 indicated in rules 1, 2, and 3, respectively. This enables us to 

 formulate another rule, which should be applied to the number to 

 be counted before proceeding with rules 1,2, and 3 above, if that 

 number is above 18,980. 



Rule. If the number to be counted is above 18,980, fii"st deduct 

 from it the highest multiple of 18.980 which it contains. 



This rule should be applied whenever possible, since it reduces the 

 size of the number to be handled, and consequently involves fewer 

 calculations. 



In Table XVI are given 80 Calendar Rounds, that is, 80 multiples 

 of 18,980, in terms of both the Maya notation and our own. These 

 will be found sufficient to cover most numbers. 



Applying the above rule to the number 31,741, which was selected 

 for our first example, it is seen by Table XVI that 1 Calendar Round, 

 or 18,980 days, may be deducted from it; 31,741-18,980 = 12,761. 

 In other words, we can count the number 12,761 forward (or back- 

 ward had the count been backward in our example) from the starting 

 point 4 Ahau 8 Cumhu, and reach exactly the same terminal date as 

 though we had coimted forward 31,741, as in the first case. 



Mathematical proof of this point follows : 



12,761^ 13 =981^^ 12,761 -^20 = 638^0 12,761 -365 = 34f|i 



The numerators of the fractions in these three quotients are 8, 1, 

 and 351; these are identical with the numerators of the fractions in 

 the quotients obtained by dividing 31,741 by the same divisors, those 

 indicated in rules 1, 2, and 3, respectively. Consequently, if these 

 three numerators be counted forward from the corresponding parts 

 of the starting point, 4 Ahau 8 Cumliu, the resulting terms together 

 will form the corresponding parts of the same terminal date, 12 Imix 

 14 Kayab. 



Similarly it could be sho-vra that 50,721 or 69,701 counted forward 

 or backward from any starting point would both reach this same ter- 

 minal date, since subtracting 2 Calendar Rounds, 37,960 (see Table 

 XVI), from the first, and 3 Calendar Rounds, 56,940 (see Table XVI), 

 from the second, there would remain in each case 12,761 . The student 

 will find his calculations greatly facilitated if he will apply this rule 

 whenever possible. To familiarize the student with the working of 

 these rules, it is thought best to give several additional examples 

 involving their use. 



