Page:American Anthropologist NS vol. 1.djvu/730

 mc gee] THE BEGINNING OF MA THEM A TICS 66 1

and mechanism. Primarily they are devices for divination, or for binding the real world to the supernal, and it is only later or in ancillary way that they are prostituted to practical uses; yet by reason of the extra-natural potency imputed to them they dominate thought and action in the culture stages to which they belong, and profoundly affect the course of intellectual develop- ment. The three systems correspond in that each rests on an exoteric base in the form of an even number, and in that each is really governed by a half-apperceived unity, itself the reflection of the Ego ; they differ in the value of the exoteric base, itself a measure of the intellectual capacity normal to the culture stage to which it pertains. The two higher systems have graphic equiva- lents which intensify and measurably shape their mystical potency (for the mechanical conditions attending graphic representation always interact with primary concepts in primitive thought) ; but the lowest and presumptively primeval system is without known graphic symbol.

Naturally, number-systems resting on inconstant and largely subjective bases are not susceptible of treatment in accordance with rational arithmetic ; but the two higher systems (and prob- ably the lowest also) are susceptible of combination in accordance with a law of augmentation^ which is neither addition nor multipli- cation, but which tends to generate both ; and this law of aug- mentation is significant (i) as indicating the evolution of number systems both mystical and rational, and (2) as a source of those vestigial features of almacabala still persisting in Aryan culture.

The augmentation of the widely diffused quaternary-quin- ary system is made clear by aid of its mechanical symbolism, which combined with the egoistic concept to shape the system. The commonest (and nearly world-wide) symbol is the cruciform figure -|-, or the quincunx, \-\ Now, magnification of the pe- ripheral powers or objects is readily and intuitively represented by adding a line or dot to each of the four extremities of the symbol, whereby it is converted into the simple swastika £JJ, or ^\?.

�� �