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One of these series simply adopts Italian numerals decimally. But the other, when it has reached 6, having had enough of novelty, makes 7 by 'six-one,' and so continues. It is for no abstract reason that 6 is thus made the turning-point, but simply because the costermonger is adding pence up to the silver sixpence, and then adding pence again up to the shilling. Thus our duodecimal coinage has led to the practice of counting by sixes, and produced a philological curiosity, a real senary notation.

On evidence such as has been brought forward in this essay, the apparent relations of savage to civilized culture, as regards the Art of Counting, may now be briefly stated in conclusion. The principal methods to which the development of the higher arithmetic are due, lie outside the problem. They are mostly ingenious plans of expressing numerical relation by written symbols. Among them are the Semitic scheme, and the Greek derived from it, of using the alphabet as a series of numerical symbols, a plan not quite discarded by ourselves, at least for ordinals, as in schedules A, B, &c.; the use of initials of numeral words as figures for the numbers themselves, as in Greek ΙΙ and Δ for 5 and 10, Roman C and M for 100 and 1,000; the device of expressing fractions, shown in a rudimentary stage in Greek γʹ, δʹ,, for ⅓, ¼, γ$δ$ for ¾; the introduction of the cipher or zero, by means of which the Arabic or Indian numerals have their value according to their position in a decimal order corresponding to the succession of the rows of the abacus; and lastly, the modern notation of decimal fractions by carrying down below the unit the proportional