mc gee] THE BEGINNING OF MA THEM A TICS 67 1
of trigrams — triangles, triskelions, hearts, etc. — of mystic or sym- bolic character.
The quaternary-quinary system survives conspicuously in the form of graphic devices, especially the world-wide cruciform symbol, which has taken on meanings of constantly increasing nobility and refinement with the growth of intelligence. Hardly less conspicuous are the classic and later literary survivals in the Four Elements (Air, Earth, Fire, Water) of alchemistic philoso- phy, the Four Winds of astrology and medieval cartography, the Four Iddhis of Buddha, and the Four Beasts of Revelation and their reflections in the ecclesiastic writing of two millenniums ; while the survivals in lighter lore are innumerable. The system persists significantly also in its augmentals, especially nine, thirteen, twenty-five, forty-nine, and sixty-one. The numerical vestiges are naturally for the most part quaternary, since the quinary aspect is merged and largely lost in algorism.
The senary-septenary system survives as the bridge connecting almacabala and mathematics. In the graphic form it became Pythagoras' hexagram of two superposed triangles, the equally mystical hexagram of Brianchon with which Paracelsus wrought his marvels, and the sub-rational hexagram of Pascal, while the current hexagram of the Chinese is apparently a composite of this and the binary as well as algorismic systems. In the numerical form, six and (more especially) seven play large roles in lore and in the classical and sacred literature revived during the Elizabethan period ; even so recently as the middle of the century the hold of the astrologic seven was so strong as to retard general acceptance of the double discovery of the eighth planet, Neptune ; and equally strong is the hold on the average mind of certain senary-septenary augmentals, particularly those coinciding with the augmentals of the lower systems.
In tracing vestiges in the form of augmentals, it is clearly to be borne in mind that their significance, like that of the primary numbers, is mystical rather, than quantitative, so that certain
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