(ii) The symbols 11 and 12 are read as eleven and twelve, not (except in elementary teaching) as ten-one and ten-two.
(iii) The names of the numbers next following these, up to 19 inclusive, only faintly suggest a ten. This difficulty is not always recognized by teachers, who forget that they themselves had to be told that eighteen means eight-and-ten.
(iv) Even beyond twenty, up to a hundred, the word ten is not used in numeration, e.g. we say thirty-four, not three ten four.
(v) The rule that the greater number comes first is not universally observed in numeration. It is not observed, for instance, in the names of numbers from 13 to 19; nor was it in the names from which eleven and twelve are derived. Beyond twenty it is usually, but not always, observed; we sometimes instead of twenty-four say four and twenty. (This latter is the universal system in German, up to 100, and for any portion of 100 in numbers beyond 100.)
20. Other Methods of Numeration and Notation.—It is only possible here to make a brief mention of systems other than those now ordinarily in use.
(i) Vigesimal Scale.—The system of counting by twenties instead of by tens has existed in many countries; and, though there is no corresponding notation, it still exhibits itself in the names of numbers. This is the case, for instance, in the Celtic languages; and the Breton or Gaulish names have affected the Latin system, so that the French names for some numbers are on the vigesimal system. This system also appears in the Danish numerals. In English the use of the word score to represent twenty—e.g. in “threescore and ten” for seventy—is superimposed on the denary system, and has never formed an essential part of the language. The word, like dozen and couple, is still in use, but rather in a vague than in a precise sense.
(ii) Roman System.—The Roman notation has been explained above (§ 15). Though convenient for exhibiting the composition of any particular number, it was inconvenient for purposes of calculation; and in fact calculation was entirely (or almost entirely) performed by means of the abacus (q.v.). The numeration was in the denary scale, so that it did not agree absolutely with the notation. The principle of subtraction from a higher number, which appeared in notation, also appeared in numeration, but not for exactly the same numbers or in exactly the same way; thus XVIII was two-from-twenty, and the next number was one-from-twenty, but it was written XIX, not IXX.
(iii) Other Systems of Antiquity.—The Egyptian notation was purely denary, the only separate signs being those for 1, 10, 100, &c. The ordinary notation of the Babylonians was denary, but they also used a sexagesimal scale, i.e. a scale whose base was 60. The Hebrews had a notation containing separate signs (the letters of the alphabet) for numbers from 1 to 10, then for multiples of 10 up to 100, and then for multiples of 100 up to 400, and later up to 1000.
The earliest Greek system of notation was similar to the Roman, except that the symbols for 50, 500, &c., were more complicated. Later, a system similar to the Hebrew was adopted, and extended by reproducing the first nine symbols of the series, preceded by accents, to denote multiplication by 1000.