826 PRIMITIVE NUMBERS [ETH. ANN. 19 
step, taken amid the mists of unwritten antiquity, was marked by the 
beginning of mathematics. In the absence of records, the rise of 
mathematics may be traced partly (like the growth of the next younger 
sciences) by vestigial features and functions; and these vestiges indi- 
vate that, just as scientific chemistry came out of mystical alchemy 
and as scientific astronomy sprang from mystical astrology, so rational 
mathematics grew out of a mystical system which long dominated the 
minds of men and slowly waned under the light of natural experience 
concentrated among the Arabs of past millenniums. In Arabia this 
mystical system preceded the simple and essentially natural, though 
happily conventional, system of enumeration and notation long known 
as algorithm (or algorism)—i. e., that inchoate form of arithmetic 
which permitted numerical treatment of quantities, and thus gave a 
foundation for science. The mystical system is even more clearly rep- 
resented in algebra, in which the conventional symbols now used to 
express natural values were originally employed as indices of magical 
potencies, like the characters inscribed on amulets and talismans; 
indeed the literature of science yields definite records of that long- 
abandoned side of algebra known as almacabala (sometimes written 
almachabel) from the Arabic word for learning and the Hebraic (or 
older) term for mystical or magical attainment of purpose,’ the whole 
constituting a jumble of occult or semi-occult redintegration such 
as appeals strongly to the ill-developed mind. Accordingly the step- 
ping stones to modern science may be enumerated as (1) almacabala, 
(2) astrology, (8) alchemy, leading respectively to mathematics and 
astronomy and chemistry, the oldest branches of definite knowledge. 
While the transition from almacabala to mathematics is indicated 
somewhat vaguely by the records and more clearly by vestiges among 
the peoples influenced by Arabic culture (including all the Aryans and 
their associates, who make up the intellectual world), the sequence is 
established by parallel developments displayed by other lines of cul- 
ture. The import of these parallelisms becomes clear in the light of 
principles pertaining both to science in general and to anthropology in 
particular; and some of these principles are worthy of enumeration: 
1. In all science it is necessarily (albeit often implicitly) postulated 
that knowledge grows by successive increments through experience 
and its assimilation, through observation and comparison (or general- 
ization), through discovery and invention, or’ in short, through natural 
processes. In the natural (or chiefly inductive) sciences and in recent 
decades this postulate is commonly made consciously and deliberately; 
in the more abstract (or chiefly deductive) sciences the postulate is less 
frequently made consciously, though a notable example of recognition 
1“ Cabala, or ‘practical cabala,’ as described by Hebraic authors, is the art of Snpihine eRnowe 
edge of the hidden world in order to attain one’s purpose in accordance with the mysticism expounded 
in the ‘Sefer Yezirah’ (Book of Creation), in which the creation of the world is ascribed to a com- 
bination and permutation of letters of the alphabet.’’—The Jewish Encyclopedia, Vol. 1, 1891, p. 548. 
