850 PRIMITIVE NUMBERS [ETH. ANN. 19 
the same time connects the Book of Changes with the nearly world- 
wide Cult of the Quarters and its mystical Middle. The numbers 63 
and 65 are also mystical in Chinese philosophy, though their potency 
would seem to be dwarfed by the mechanical-arithmetical structure of 
the octonal square to which they have been adjusted evidently during 
recent centuries. Among the Hindu more or less mystical numbers 
abound, and many of these are found on analysis to correspond with 
conventional almacabalic augmentals and coincidentals; while the Budd- 
histic rituals and series of aphorisms often run in measures of fives, 
with an initial or final supernumerary—the feature being apparently 
fixed by a mnemonic finger-count superposed on the almacabalic sys- 
tem, much as the octonal count is superposed on the mystical figures 
in the Chinese hexagram. 
Suggestive vestiges of the mystical number-groups persist widely 
in the form of irrational and functionless supernumeraries, such as the 
thirteenth loaf in the baker’s dozen, the twenty-first skerret in the 
coster’s score, the thousand-and-first night of Arabian tale, and the 
conventional oyerplus in the legal ‘‘ year and a day.” It is possible 
that the supernumerary habit was crystallized in some cases by sim- 
ple object-counting so conducted as to include an additional object as 
a tally; but there are many indications that the habit originally sprang 
from almacabalic augmentation, in which the sum is always one more 
than the multiple of the even-number basis. Moreover, the super- 
numerary habit is especially characteristic of countries and culture- 
stages in which mystical number-jumbles are rife. 
Certain of the graphic vestiges of the quaternary-quinary system 
are of special significance; for just as the hexagrams of the senary- 
septenary system bridged the way from mystical almacabala to rational 
geometry, so the mechanical development of symbols exoterically 
quatern but esoterically quinary carried intelligence across the chasm 
dividing the morass of almacabala from the algorithmic forelands 
rising into the firm ground of arithmetic. True, the passage was 
made easier by the coincidental structure of the hand, that natural 
abacus which undoubtedly served to fix the quinary system in all 
minds trained up to the contemplation of fives; yet the way was 
apparently so long from the habitual perception of lowly twos and 
fours to the ready grasp and combination of fives that mechanical struc- 
ture was even more efficient than organic structure in guiding progress. 
The graphic number symbols of the Mexican codices illustrated and 
discussed by Dr Thomas and others epitomized the growth of a vigesi- 
mal system crystallized by the coincidence of manual and pedal strue- 
tures, while both the terms and the gestures of the Zuni finger-count 
analyzed by Cushing point the way in which binary prepossessions 
passed into quinary practices despite the obstruction of the senary 
