(ii) On the counting system we may consider that we have a series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, repeating this series indefinitely. There is as yet no distinction between the first object marked 1 and the second object marked 1. We can, however, attach to the 0’s the same symbols, 1, 2, ... 0 in succession, in a separate column, repeating the series indefinitely; then do the same with every 0 of this new series; and so on. Any particular object is then defined completely by the combination of the symbols last written down in each series; and this combination of symbols can equally be used to denote the number of objects up to and including the last one (§ 10).
In writing down a number in excess of 1000 it is (except where the number represents a particular year) usual in England and America to group the figures in sets of three, starting from the right, and to mark off the sets by commas. On the continent of Europe the figures are taken in sets of three, but are merely spaced, the comma being used at the end of a number to denote the commencement of a decimal.
The zero, called “nought,” is of course a different thing from the letter O of the alphabet, but there may be a historical connexion between them (§ 79). It is perhaps interesting to note that the latter-day telephone operator calls 1907 “nineteen O seven” instead of “nineteen nought seven.”
14. Direction of the Number-Series.—There is no settled convention as to the direction in which the series of symbols denoting the successive numbers one, two, three, ... is to be written.
(i) If the numbers were written down in succession, they would naturally proceed from left to right, thus:—1, 2, 3, ... This system, however, would require that in passing to “double figures” the figure denoting tens should be written either above or below the figure denoting ones, e.g.
|1, 2, ... 8, 9,||0, 1, 2, ... or 1, 2, ... , 8, 9,||0, 1, 2, ...|
The placing of the tens-figure to the left of the ones-figure will not seem natural unless the number-series runs either up or down.
(ii) In writing down any particular number, the successive powers of ten are written from right to left, e.g. 5,462,198 is
the small figures in brackets indicating the successive powers. On the other hand, in writing decimals, the sequence (of negative powers) is from left to right.
(iii) In making out lists, schedules, mathematical tables (e.g. a multiplication-table), statistical tables, &c., the numbers are written vertically downwards. In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals, though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93).
(iv) In graphic representation measurements are usually made upwards; the adoption of this direction resting on certain deeply rooted ideas (§ 23).
This question of direction is of importance in reference to the development of useful number-forms (§ 23); and the existence of the two methods mentioned under (iii) and (iv) above produces confusion in comparing numerical tabulation with graphical representation. It is generally accepted that the horizontal direction of increase, where a horizontal direction is necessary, should be from left to right; but uniformity as regards vertical direction could only be attained either by printing mathematical tables upwards or by taking “downwards,” instead of “upwards,” as the “positive” direction for graphical purposes. The downwards direction will be taken in this article as the normal one for succession of numbers (e.g. in multiplication), and, where the arrangement is horizontal, it is to be understood that this is for convenience of printing. It should be noticed that, in writing the components of a number 253 as 200, 50 and 3, each component beneath the next larger one, we are really adopting the downwards principle, since the figures which make up 253 will on this principle be successively 2, 5 and 3 (§ 13 (ii)).
15. Roman Numerals.—Although the Roman numerals are no longer in use for representing cardinal numbers, except in certain special cases (e.g. clock-faces, milestones and chemists’ prescriptions), they are still used for ordinals.
The system differs completely from the Hindu system. There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz. I, V, X, L, C, D, M, proceeding by multiples of five and two alternately. Thus 1878 is MDCCCLXXVIII, i.e. thousand five-hundred hundred hundred hundred fifty ten ten five one one one.
The system is therefore essentially a cardinal and grouping one, i.e. it represents a number as the sum of sets of other numbers. It is therefore remarkable that it should now only be used for ordinal purposes, while the Hindu system, which is ordinal in its nature, since a single series is constantly repeated, is used almost exclusively for cardinal numbers. This fact seems to illustrate the truth that the counting principle is the fundamental one, to which the interpretation of grouped numbers must ultimately be referred.
The normal process of writing the larger numbers on the left is in certain cases modified in the Roman system by writing a number in front of a larger one to denote subtraction. Thus four, originally written IIII, was later written IV. This may have been due to one or both of two causes; a primitive tendency to refer numbers, in numeration, to the nearest large number (§ 24 (iv)), and the difficulty of perceiving the number of a group of objects beyond about three (§ 22). Similarly IX, XL and XC were written for nine, forty and ninety respectively. These, however, were later developments.
16. Scales of Notation.—In the Hindu system the numbering proceeds by tens, tens of tens, &c.; thus the figure in the fifth place, counting from the right, denotes the product of the corresponding number by four tens in succession. The notation is then said to be in the scale of which ten is the base, or in the denary scale. 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 introduction of these other symbols produces a compound scale, which may be called a quinary-binary, or, less correctly, a quinary-denary scale.
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. Thus 1878 in the quinary-binary scale would be 1131213, and 1828 would be 1130213; the meaning of these is seen at once by comparison with MDCCCLXXVIII and MDCCCXXVIII. Similarly the number which in the denary scale is 215 would in the quaternary scale (base 4) be 3113, being equal to 3·4·4·4 + 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. If mankind had had six fingers on each hand and six toes on each foot, we should be using a duodenary scale (base twelve), which would have been far more convenient.
17. Notation of Numerical Quantities.—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. In Great Britain and her colonies, however, and in the United States, other systems of notation still survive, though there is none which is consistently in one scale, other than the denary. The method is to form quantities into groups, and these again into larger groups; but the number of groups making one of the next largest groups varies as we proceed along the scale. The successive groups or units thus formed are called denominations. Thus twelve pennies make a shilling, and twenty shillings a pound, while the penny is itself divided into four farthings (or