This linguistic poverty proves that the Australian tongue has no affinity to the Polynesian group of languages, where denary enumeration prevails: the nearest Polynesians, the Maoris, counting in thousands.
Thus, on the denary system (§ 16) we can give independent definitions to the numbers up to ten, and then regard (e.g.) fifty-three as a composite number made up of five tens and three ones.
The Roman system, except for the use of symbols for five, fifty, &c., is also in the denary scale, though expressed in a different way.
The figures used in the Hindu notation might be used to express numbers in any other scale than the denary, provided new symbols were introduced if the base of the scale exceeded ten.
Similarly the number which in the denary scale is 215 would in the quaternary scale (base 4) be 3113, being equal to 126.96.36.199+ 1.4.4+1.4+3.
The use of the denary scale in notation is due to its use in numeration (§ 18); this again being due (as exemplified by the use of the word digit) to the primitive use of the fingers for counting.
Over a large part of the civilized world the introduction of the metric system (§ 118) has caused the notation of all numerical quantities to be in the denary scale.
Within each denomination, however, the denary notation is employed exclusively, e.g.
- The names of numbers are almost wholly based on the denary scale; thus eighteen means eight and ten, and twenty-four means twice ten and four.
But the system has never spread; and the word " dozen " itself is based on the denary scale.
There is no essential difference, however, between this and the denary basis.
- Although numeration and notation are both ostensibly on the denary system, they are not always exactly parallel.
In " threescore and ten " for seventy - is superimposed on the denary system, and has never formed an essential part of the language.
The numeration was in the denary scale, so that it did not agree absolutely with the notation.
- The Egyptian notation was purely denary, the only separate signs being those for 1, io, too, &c. The ordinary notation of the Babylonians was denary, but they also used a sexagesimal scale, i.e.
In other words, the denary scale, though adopted in notation and in numeration, does not arise in the corresponding mental concept until we get beyond too.
In consequence of this limitation of the power of perception of number, it is practically impossible to use a pure denary scale in elementary number-teaching.
(ii) Beyond ten, and in many cases beyond five, the names have reference to the use of the fingers, and sometimes of the toes, for counting; and the scale may be quinary, denary or vigesimal, according as one hand, the pair of hands, or the hands and feet, are taken as the new unit.
Finger-counting is of course natural to children, and leads to grouping into fives, and ultimately to an understanding of the denary system of notation.
They only apply accurately to divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of fitting in with the denary scale of notation, and they can be extended to other divisions by using a mixed number as numerator.
- Instead of regarding the 153 in 27.153 as meaning o h, we may regard the different figures in the expression as denoting numbers in the successive orders of submultiples of i on a denary scale.