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 mc gee] THE BEGINNING OF MA THEM A TICS 647

for science; it even antedated conventional algebra, in which symbols are used to represent natural values, and seems to have dropped into the background of thought with that long- abandoned side of algebra, almacabala (or almachabel) — a jumble of occult or semi-occult redintegration which appeals strongly to the half-developed mind. So the stepping-stones to science may be enumerated as almacabala, astrology, alchemy, leading respec- tively to mathematics and astronomy and chemistry, the oldest branches of definite knowledge.

While the transition from almacabala to mathematics is indi- cated by vestigial characteristics among the peoples influenced by Arabic culture (the Aryans and their associates, who make up the intellectual world), the sequence is established by parallel phenomena displayed by other lines of culture. The import of these parallel phenomena becomes clear in the light of cardinal principles, pertaining both to science in general and to an- thropology in particular: (1) In all science it is necessarily postulated that knowledge grows by successive increments, through experience and its assimilation, through observa- tion and comparison (or generalization), through discovery and invention — i. e., through natural processes ; (2) in all branches of definite knowledge, but especially in anthropology, it is implicitly if not explicitly postulated that knowledge is diffused and its acquisition stimulated through association and interchange among individuals and peoples ; (3) in anthropology, as in other sciences, it is necessary to recognize a body or volume of knowledge proper to each people, made up of the combined possessions of all the individuals, increasing with successive experiences, decreasing only through neglect or disuse, and in greater part perpetuated by record and tradition if not by direct heritage ; (4) in anthropology, as measurably in other sciences, it is desirable to assume that (a) mental capacity and (b) the sum of knowledge, either in the individual or in the group, are in the long run practically equivalent ; and (5) in anthropology it is

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