It does not seem to have been generally recognized that the alternate contraposition of upright and reversed taus produces the best known and most widely spread primitive border-design, usually known as the Greek fret (fig. 40, no. 6). A plain demonstration of this is, oddly enough, visible on the two side-projections of the Scandinavian brooch (fig. 13) all symbols on which, I venture to assert, would have been perfectly intelligible and full of meaning to an ancient Mexican. The evolution of the fret, on the American continent, can be studied on the beautiful wooden clubs from Brazil and British Guiana, figured in Dr. Hjalmar Stolpes’ valuable work already referred to. As striking instances his fig. 8, pl. 1, figs. 3_a_ and 3_c_, pl. XIII, and figs. 1_a_ and 1_b_, pl. V, should be examined. The latter instance is extremely instructive as it not only exhibits single taus of two forms, but the same in different positions, as well as two double-headed figures joined in one, which illustrate the native association already discussed, of duality and of the curved lines as the opposite of the rectangular and both respectively figuring the Above and Below. It is impossible to study the decorations on these South American clubs without becoming convinced that their makers shared the same ideas as the ancient Mexicans. They offer, indeed, a whole set of variations on the native theme and idea of Heaven and Earth. Two instances (fig. 5_a_, pl. IX, and 6_a_, pl. XI) in which the union of two figures produces a third, or a single one produces two, elucidate the meaning sometimes expressed by the designs. In the round or spiral forms, which are most frequently accompanied by a zigzag border, I am inclined to see a presentation of air and water, corresponding to the Mexican symbols of the Above. As lack of space forbids my making here a more extended comparison of the native symbols, I shall but point out how the tau, in juxtaposition and contraposition painted in two colors, produces fig. 40, no. 3. The picture from the Codex Mendoza of a native tlachtli, the form of which is represented by two taus in contraposition, is partly painted black. The same division of a single tau into two parts, colored differently, transforms no. 3 into no. 4 and shows that a single tau could have been employed cursively to symbolize union. 2 and 7 are but variants of 3 and 4. If, instead of angles, curved lines be given to the taus, the first half of fig. 5 is the result. When spaces between the incurving hooks and the border are filled out with color, the familiar design on the second half of 5 results. With exception of the latter, the South American clubs exhibit each of the above forms, as well as no. 8. It will be shown later that these also occur in ancient Peru.