REAL NATURE OF SO-CALLED TIME PERIODS 259 As ii means of illustratiug the use of uumbers by the Mayas, iu

relation totime, the following

example — ^which

^{ispart of a series on}

plate LIX ofthe

Dresden

codex ^(figure 159)

—

^{is presented:}

13Caban 13Cauac 13Imix 13Akbal 13Chicclian 13Manik

As

^{this series} ascends towardthe left

hand

the forward count will

be ⁱⁿthatdirection. Starting withthe

column

at the right hand,

we

subtract it(3-lS)

from

thenext one totheleft,

and

^thisone fromthat immediately to theleftof it,

and

so

on

to the ^last.

The

difference in each case is

found

to be 3-18; thatis, 3 twenties (3x20) plus18 equal 78daA's, the

day

being theunit. Counting for-

• •••

• ••

• •

^•

^••i

cc ®

Fig,159. Lower^divisionofplate Lix,Dresdencodex.

ward

78days

from

13

Manik

of

any

year (say 13

Manik

²⁰ Zotz, 3'ear 12

Lamat) we

reach 13 Chiechau (in this case 18 Mol,

same

year).

Counting

forward 78 days

from

the last date

we

reach 13

Akbal

^l(j Ceh,

same

year; 78

more

(always counting

from

the last date), 13

Imix

14Pax,

same

year; 78 more, 13

Cauac

⁷ Uo, year 13 Ben. If

we

count Ijack 78 days from 13

Manik

20 Zotz (first

column

at the right hand),

we

reach 13 3Iuluc 2 Pop, year 12Akbal,

which

is the initial

day

^of the whole series, the

month and

yearof the first given

day

beingas

assumed

above.

Attentioniscalled to this seriesnotbecauseitpresents

any

peculiar feature, but to

show

that considering the numerals merelj^ as

num-

bersinrespective ordersof iinits will furnish a full

aud

.satisfactory explanation of their object

and

use. I take for granted that'the simplest explanation, if it meets every requirement

and

presents nothing im^onsisteut with the

known

facts regarding the

Maja

time

260 MAYAN CALENDAR SYSTEMS

^{[eth. axx.22}

ami numeral

systems, should Ix' accepted ratlier than a tlieory

which

introduces

new and

hitherto

unknown

featui'es.

If

we

use ordinary

numbers

in place of tlie

numeral

syml)ols,

and keep them

in therelative pctsitions givenabove, the result will be as follows

:

3rd orderof units

THOMAS]

REAL NATl'RE

OB^

SO-CALLED TIME PERIODS 261

If, iiisteail of adding the written names, simiily the figure should begiven, the rehxtivepositions being maintained

and

understood,

we would have

the ^Maya method,

and

the value

would

be

known

as well as

by

our ordinaiy

method

^ofwriting

numbers

horizontally.

I

have

given these details of elementary rules

and

principles iu ordertolead

up

to this point, viz, that symbols

may

be ixsed to indi- cate orders of units instead of position. In the last

example

given above, a .symbol

may

be adopted for the

"hundred

thousands,"

another for

"ten

thousands," another for "thousands," ^etc.

They

maj-then be

grouped

in

any

regular order ^inost convenient,

and

yet be as coi-rectly read as

by

position. This is precisely

what

has be(>n

done

in theinscriptions. Sj-mbols

have

beeii adopted to indicate the orders of units, asit

was

inconvenient to do ^this

by means

of relative position alone? with the dots

and

short lines

—

^at

^any

^{rateitisapparent}

that the latter

method

is uotsowell adapted ^{to the}glyph form iu the inscriptions; but even here

we

see a strong tendency^to maintain the relative position

which

almost universally obtains

and

^is often the onlj-

means

^of determination. If

we

take

Goodnmn's work and

go through ^it

from

beginning to

end and

^{substitute in} everyseries

where

they occur "unitsof the 2nd order" for hischueus, "unitsofthe 3rd order" for his ahaus, "units ofthe -tthorder" for his katuns, "units of the othorder" for his cycles,

and

"tinits of the 0th order" for his great cycles, the result will be correct ⁱⁿ every instance. I

am

fully

aware

thatthis will be true

whether we

call

them

^realtime peri- ods or ordersof units.

The

point,however, for

which

I

am

contending

is,that as theINIaj'as

had

a .sy.stemof

numeration and must have

used

it ill expressing

numbers

in the codices

and

inscriptions,

and

this

numeral

sj'stem corresponds exactly with (xoodman's supposed time periods so far as these are given numerically correct

by

him, there

is

no

necessity or reason for the theoryof a separate

Maya

chrono- logical .system (identical sofaras correctlygivenw-itli the

Maya num-

eral system as used in counting time), differing

from

their calendar sj'stem.

From

the evidence given intheearlier part ^{of this} paper

and what

has been presente(l in mj'preceding pajier, the following conclusions appearto be clearlj-justified:

That Mr Goodman

has dl.scovered independentlythe sigtiification

and

numeral

valuesofthesymbols found in the inscrii)tious which he designates

by

the

names

^cycle, katuu, ahau, chuen,

and

calendar round, thoughthis

had been

already

done

in part

by

others.

That

he has discovered that certain faceand other characters are

number

symbols,

and

has ascertained theii-values.

That

he has determined the object

and

useof the

numeral

series,

aud

the

method

of counting

by

^the

same

^.series from the preceding

and

followingdates, as well as tothem.

'J62

MAYAN CALENDAR SYSTEMS

^[eth.ann.22

It isalso cqTially apparoutthat liis theoryofa Maj'a chronological system, ^distinci,

from

theMaj^a calendar system

—

^{the Maj'aii}

^method

of niimeration in counting time

— ^and

^his

^method

^{of counting 13}

so-called cj'cles onlj- to the so-called great cycle

and

73 great cycles to his so-called grand era are not justified

by

the facts, nor is his

method

of

numbering

the cycles, katuns, etc., beginning with 73, 13,

and

20, satisfactorilyproved;

and

alsothathis selection of Ik,

Manik,

Eb,

and Caban

as the dominicaldaysis erroneous, the truedominical daj'Sbeing Akbal, Lamat, Ben,

and

Ezanab, both inthe inscriptions

and Dresden

codex.

TjCt us turn next to liis

method

of

numbering

the so-called great cycles. According to his theory, as

we have

seen, 73 great cycles are counted to

what

he calls the grand era, the

common

multiple of

aU

the factors ^of the calendar sj'stem

and

supposed "chronological system."

The

reason

why

he adopted ^this theory isexplained in

my

previouspaper,

and

theexplanation need notberepeatedhere, except so far asmerely^{to state} that in order tofind a

common

multijile of the various time periods, one

must

include the Tiumber 30.5,

which

contains the i:)rime

number

^73.

That

there

was

in the

Maya

system a

number

or order of units corresponding with

Goodman's

great cycle is certainlytrue, but ^this pertained to their numeral,

and

not their time, system. It is also admittedthat the large quadruple glyph ^thatusuallyheadstheinitial seriesis the

symbol

used to rejiresent this

number

or orderof units.

But, ashas been shown, thereisnoreason

whatever

forbelieving that they were

numbered

otherwise than in accordance withthe vigesimal system^; that is to say, 20 cyclestothe greatcycle,

and

20greatcj^cles tothenext higherunit. Itisnecessary,therefore, for

Goodman,

before his theory can beaccepted,to

show

bj'satisfactorj' evidencethat, on reachingthe cycles

and

greatcycles,the ordinary

method

ofproceed- ing

by

the vigesimal system

was abandoned and

other multipleswere introduced.

That

there

was

a

change

from this rule in passing

from

the 2ndorderof units, orchuens, to the 3rd oi-der, orahaus,

where

18

was made

the multiple, is jiroved

by

incontrovertible evidence

and

hence

must

be admitted, even though

we

maj'not be able to

show by

absolute demonstration

why

the cliange

was

made. Nevertheless,

we

arejustified in believingthat, in this instance, the

method

of

numera-

tion

was made

to correspontlwith the

number

of

months

in theyear.

But

no such reason appears for

Goodman's

proposed

change

in the higher orders of units;

we

are, therefore, justified in rejecting the idea until other proof, besides its necessity to support a theory, is

shown. It

must

be

made

evident

by

proof that theseries can notbe otherwise explained, which

we have shown

^isnotthecase, or it

must

be

shown

that the great cycle symbols present,

by

their forms, the nuntbersassigned them.

THOMAS]

NUMBERING OF SO-CALLED GREAT CYCLES 263