I think we need no more examples to show that people—­in almost all countries—­reckon by fives, tens, or twenties, merely because they began to count upon their fingers and toes.  If the strong man who had six fingers on each hand, and six toes on each foot, had invented a system of numeration, it would have gone in twelves, nearly like the duodecimals which our carpenters use; unless, indeed, he had been stupid after the manner of very strong men, and not gone beyond sixes.  We see how the Romans, though they inherited from their Eastern ancestors a numeration by tens up to decem, and then beginning again undecim, &c., yet when they began to write a notation could get no farther than five—­I., II., III., IV., V.; and then on again, VI., VII., up to ten, from ten to fifteen, and so on.

There is a very curious vulgar error which prevails, even among people who have a good practical acquaintance with arithmetic.  It is that the number ten has some special virtue which fits it for counting up to.  The fact is that ten is not the best number for the purpose; you can halve it, it is true, but that is about all you can do with it, for its being divisible by five is of hardly any use for practical purposes. Eight would be a much better number, for you can halve it three times in succession; and twelve is perhaps the most convenient number possible, as it will divide by two, three, and four.  It is this convenient property that leads tradesmen to sell by dozens, and grosses, rather than by tens and hundreds.  If we used eights or twelves instead of tens for numeration, we might of course preserve all the advantages of the Indian or Arabic numerals; in the first case, we should discard the ciphers 8 and 9, and reckon 5, 6, 7, 10; and in the second case, we should want two new ciphers for ten and eleven; and 10 would stand for twelve, and 11 for thirteen.  Our happening to have ten fingers has really led us into a rather inconvenient numerical system.

[Illustration:  AZTEC HEAD, IN TERRA COTTA. (PROBABLY EITHER A HOUSEHOLD-GOD OR A VOTIVE OFFERING).]

* * * * *

NOTE.

The unique Knife figured at page 101 and two masks incrusted with a similar mosaic work (of turquoise and obsidian) are in Mr. Christy’s collection; and a mask and head of similar workmanship are in the collection at Copenhagen.  These are the only known examples of this advanced style of Aztec art.

The whole once belonged probably to one set, brought to Europe soon after the Conquest of Mexico.  The two at Copenhagen were obtained at a convent in Rome; and, of the other three, two were for a long period in a collection at Florence, and the other was obtained at Bruges, where it was most probably brought by the Spaniards during their rule in the Low Countries.

CHAPTER V.

MEXICO.  GUADALUPE.