- SYMBOLS 11
the end, the left hand contains either five or nine sticks. After a sequence of such
procedures, a final step begins with 32 or 36 or 40 sticks, and as a result the number
of remaining sticks will be 24, 28, 32, or 36. This number is divided by four and the
quotient determines the bottom row of the symbol to be used for divination. Six
is called lesser yang, seven greater ying, eight lesser ying, and nine greater yang.
The ying and yang are respectively female and male principles. The greater cases
correspond to flux (tending to their opposites) and the lesser to stability. When this
entire procedure has been carried out six times, the result is a stack of six symbols
that can be interpreted according to the principles of divination. There are 64, that
is, 26 , different possible stackings of ying and yang, all discussed in the / Ching,
and the duality between stability and flux makes for 4096 possible symbols. One
must beware of attaching too much significance to numerical coincidences, but it
is intriguing that both Malagasy and Chinese forms of divination are based on the
number four.^1
Divination seems to fulfill a nearly universal human desire to feel in control
of the powerful forces that threaten human happiness and prosperity. It manifests
itself in a variety of ways, as just shown by the examples of the Malagasy and the /
Ching. We could also cite large parts of the Jewish Kabbalah, the mysticism of the
Pythagoreans, and many others, down to the geometric logic of Ramon Lull (1232-
1316), who was himself steeped in the Kabbalah. The variety of oracles that people
have consulted for advice about the conduct of their lives—tarot cards, crystal balls,
astrology, the entrails of animals and birds, palmistry, and the like—seems endless.
For the purposes of this book, however, we shall be interested only in those aspects
of divination that involve mathematics. Magic squares, for example, occur in both
the Kabbalah and the / Ching. Although the author puts no stock whatsoever in
the theories behind all this mysticism, it remains an important fact about human
behavior over the centuries and deserves to be studied for that reason alone. But
for now it is time to return to more prosaic matters.
Aids to computation, either tabular or mechanical, must be used to perform
computations in some of the more cumbersome notational systems. Just imagine
trying to multiply XLI by CCCIV! (However, Detlefsen and co-authors (1975)
demonstrate that this task is not so difficult as it might seem.) Even to use the
28 x 19 table of dates of Easter discussed in Problem 6.26, the Slavic calculators
had to introduce simplifications to accommodate the fact that dividing a four-digit
number by a two-digit number was beyond the skill of many of the users of the
table.
The earliest mathematical texts discuss arithmetical operations using everyday
words that were probably emptied of their usual meaning. Students had to learn
to generalize from a particular example to the abstract case, and many problems
that refer to specific objects probably became archetypes for completely abstract
reasoning, just as we use such expressions as "putting the cart before the horse" and
"comparing apples and oranges" to refer to situations having no connection at all
with horse-and-buggy travel or the harvesting of fruit. For example, problems of the
(^1) Like all numbers, the number four is bound to occur in many contexts. One website devoted to
spreading the lore found in the / Ching notes the coincidental fact that DNA code is written with
four amino acids as its alphabet and rhapsodizes that "The sophistication of this method has not
escaped modern interpretation, and the four-valued logic has been compared to the biochemistry
of DNA amino acids. How a Neolithic shaman's divination technique presaged the basic logic of
the human genome is one of the ageless mysteries."