12. _Terms used._--The representation of numbers by spoken sounds is
called _numeration_; their representation by written signs is called
_notation_. The systems adopted for numeration and for notation do not
always agree with one another; nor do they always correspond with the
idea which the numbers subjectively present. This latter presentation
may, in the absence of any accepted term, be called _number-ideation_;
this word covering not only the perception or recognition of particular
numbers, but also the formation of a number-concept.
13. _Notation of Numbers._--The system which is now almost universally
in use amongst civilized nations for representing cardinal numbers is
the Hindu, sometimes incorrectly called the Arabic, system. The
essential features which distinguish this from other systems are (1) the
limitation of the number of different symbols, only ten being used,
however large the number to be represented may be; (2) the use of the
_zero_ to indicate the absence of number; and (3) the principle of local
value, by which a symbol in effect represents different numbers,
according to its position. The symbols denoting a number are called its
_digits_.
A brief account of the development of the system will be found under
NUMERAL. Here we are concerned with the principle, the explanation of
which is different according as we proceed on the grouping or the
counting system.
(i) On the grouping system we may in the first instance consider that we
have separate symbols for numbers from "one" to "nine," but that when we
reach ten objects we put them in a group and denote this group by the
symbol used for "one," but printed in a different type or written of a
different size or (in teaching) of a different colour. Similarly when we
get to ten tens we denote them by a new representation of the figure
denoting one. Thus we may have:
ones 1 2 3 4 5 6 7 8 9
tens 1 2 3 4 5 6 7 8 9
hundreds,\ 1 2 3 4 5 6 7 8 9
&c. / &c. &c.
On this principle 24 would represent twenty-four, 24 two hundred and
forty, and 24 two hundred and four. To prevent confusion the _zero_ or
"nought" is introduced, so that the successive figures, beginning from
the right, may represent ones, tens, hundreds, ... We then have, e.g.,
240 to denote two hundreds and four tens; and we may now adopt a uniform
type for all the figures, writing this 240.
1 2 3 .. .. .. .. 8 9 10 1 2 3 .. .. .. ..
o o o .. .. .. .. o o o o o o .. .. .. ..
(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 (S 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 (S 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
1, 2, ..., 8, 9, 0, 1, 2, ... or 1, 2, ..., 8, 9, 0, 1, 2, ...
1
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
(6) (5) (4) (3) (2) (1) (0)
5 4 6 2 1 9 8
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 (SS
26, 93).
(iv) In graphic representation measurements are usually made upwards;
the adoption of this direction resting on certain deeply rooted ideas (S
23).
200
50
3
---
253
===
This question of direction is of importance in reference to the
development of useful number-forms (S 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 (S 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 (S 24 (iv)), and the difficulty of perceiving the
number of a group of objects beyond about three (S 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
(S 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 (S 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 two halfpennies). There are, therefore, four denominations, the
bases for conversion of one denomination into the next being
successively four (or two), twelve and twenty. Within each denomination,
however, the denary notation is employed exclusively, e.g. "twelve
shillings" is denoted by 12s.
The diversity of scales appears to be due mainly to four causes: (i) the
tendency to group into scores (S 20); (ii) the tendency to subdivide
into twelve; (in) the tendency to subdivide into two or four, with
repetitions, making subdivision into sixteen or sixty-four; and (iv) the
independent adoption of different units for measuring the same kind of
magnitude.
Where there is a division into sixteen parts, a binary scale may be
formed by dividing into groups of two, four or eight. Thus the weights
ordinarily in use for measuring from 1/4 oz. up to 2 lb. give the basis
for a binary scale up to not more than eight figures, only 0 and 1 being
used. The points of the compass might similarly be expressed by numbers
in a binary scale; but the numbers would be ordinal, and the expressions
would be analogous to those of decimals rather than to those of whole
numbers.
In order to apply arithmetical processes to a quantity expressed in two
or more denominations, we must first express it in terms of a single
denomination by means of a varying scale of notation. Thus L254, 13S.
6d. may be written
(20) (12)
L254 " 13s. " 6d.;
each of the numbers in brackets indicating the number of units in one
denomination that go to form a unit in the next higher denomination. To
express the quantity in terms of L, it ought to be written
(20) (12)
L254 " 13 " 6;
this would mean L254 13(6/12)/20 or L(254 + 13/20 + 6/(20.12)), and
therefore would involve a fractional number.
A quantity expressed in two or more denominations is usually called a
_compound number_ or _compound quantity_. The former term is obviously
incorrect, since a quantity is not a number; and the latter is not very
suggestive. For agreement with the terminology of fractional numbers (S
62) we shall describe such a quantity as a _mixed quantity_. The letters
or symbols descriptive of each denomination are visually placed after or
(in actual calculations) above the figures denoting the numbers of the
corresponding units; but in a few cases, e.g. in the case of L, the
symbol is placed before the figures. There would be great convenience in
a general adoption of this latter method; the combination of the two
methods in such an expression as L123, 16s. 4-1/2d. is especially
awkward.
18. _Numeration._--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. The words _eleven_ and _twelve_ have been supposed
to suggest etymologically a denary basis (see, however, NUMERAL).
Two exceptions, however, may be noted.
(i) The use of _dozen, gross_ (= dozen dozen), and _great gross_ (=
dozen gross) indicates an attempt at a duodenary basis. But the system
has never spread; and the word "dozen" itself is based on the denary
scale.
(ii) The _score_ (twenty) has been used as a basis, but to an even more
limited extent. There is no essential difference, however, between this
and the denary basis. As the latter is due to finger-reckoning, so the
use of the fingers and the toes produced a vigesimal scale. Examples of
this are given in S 20; it is worthy of notice that the vigesimal (or,
rather, quinary-quaternary) system was used by the Mayas of Yucatan, and
also, in a more perfect form, by the Nahuatl (Aztecs) of Mexico.
The number ten having been taken as the basis of numeration, there are
various methods that might consistently be adopted for naming large
numbers.
(i) We might merely name the figures contained in the number. This
method is often adopted in practical life, even as regards mixed
quantities; thus L57,593, 16s. 4d. would be read as _five seven, five
nine three, sixteen and four pence._
(ii) The word _ten_ might be introduced, e.g. 593 would be _five ten ten
ninety_ (= nine ten) _and three._
(iii) Names might be given to the successive powers of ten, up to the
point to which numeration of ones is likely to go. Partial applications
of this method are found in many languages.
(iv) A compromise between the last two methods would be to have names
for the series of numbers, beginning with ten, each of which is the
"square" of the preceding one. This would in effect be analysing numbers
into components of the form a. 10^b where a is less than 10, and the
index b is expressed in the binary scale, e.g. 7,000,000 would be
7 . 10^4 . 10^2, and 700,000 would be 7 . 10^4 . 10^1.
The British method is a mixture of the last two, but with an index-scale
which is partly ternary and partly binary. There are separate names for
ten, ten times ten (= _hundred_), and ten times ten times ten
(= _thousand_); but the next single name is _million_, representing a
thousand times a thousand. The next name is _billion_, which in Great
Britain properly means a million million, and in the United States (as
in France) a thousand million.
19. _Discrepancies between Numeration and Notation._--Although
numeration and notation are both ostensibly on the denary system, they
are not always exactly parallel. The following are a few of the
discrepancies.
(i) A set of written symbols is sometimes read in more than one way,
while on the other hand two different sets of symbols (at any rate if
denoting numerical quantities) may be read in the same way. Thus 1820
might be read as _one thousand eight hundred and twenty_ if it
represented a number of men, but it would be read as _eighteen hundred
and twenty_ if it represented a year of the Christian era; while 1s. 6d.
and 18d. might both be read as _eighteenpence_. As regards the first of
these two examples, however, it would be more correct to write 1,820 for
the former of the two meanings (cf. S 13).
(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 (S
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.
On the island of Ceylon there still exists, or existed till recently, a
system which combines some of the characteristics of the later Greek (or
Semitic) and the modern European notation; and it is conjectured that
this was the original Hindu system.
For a further account of the above systems see NUMERAL, and the
authorities quoted at the end of the present article.
21. _The Number-Concept._--It is probable that very few people have any
definite mental presentation of individual numbers (i.e. numbers
proceeding by differences of _one_) beyond 100, or at any rate beyond
144. Larger numbers are grasped by forming numbers into groups or by
treating some large number as a unit. A person would appreciate the
difference between 93,000,000 m. and 94,000,000 m. as the distance of
the centre of the sun from the centre of the earth at a particular
moment; but he certainly would not appreciate the relative difference
between 93,000,000 m. and 93,000,001 m. In order to get an idea of
93,000,000, he must take a million as his unit. Similarly, in the metric
system he cannot mentally compare two units, one of which is 1000 times
the other. The metre and the kilometre, for instance, or the metre and
the millimetre, are not directly comparable; but the metre can be
conceived as containing 100 centimetres.
On the other hand, it would seem that, for most educated people, sixteen
and seventeen or twenty-six and twenty-seven, and even eighty-six and
eighty-seven, are single numbers, just as six and seven are, and are not
made up of groups of tens and ones. 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 100.
Again, in the use of decimals, it is unusual to give less than two
figures. Thus 3.142 or 3.14 would be quite intelligible; but 3.1 does
not convey such a good idea to most people as either 3(1/10) or 3.10,
i.e. as an expression denoting a fraction or a percentage.
There appears therefore to be a tendency to use some larger number than
ten as a basis for grouping into new units or for subdivision into
parts. The Babylonians adopted 60 for both these purposes, thus giving
us the sexagesimal division of angles and of time.
This view is supported, not only by the intelligibility of percentages
to ordinary persons, but also by the tendency, noted above (S 19), to
group years into centuries, and to avoid the use of thousands. Thus 1876
is not 1 thousand, 8 hundred, 7 tens and 6, but 18 hundred and 76, each
of the numbers 18 and 76 being named as if it were a single number. It
is also in accordance with what is so far known about number-forms (S
23).
If there is this tendency to adopt 100 as a basis instead of 10, the
teaching of decimals might sometimes be simplified by proceeding from
percentages to percentages of percentages, i.e. by commencing with
_centesimals_ instead of with _decimals._
22. _Perception of Number._--In using material objects as a basis for
developing the number-concept, it must be remembered that it is only
when there are a few objects that their number can be perceived without
either counting or the performance of some arithmetical process such as
addition. If four coins are laid on a table, close together, they can
(by most adults) be seen to be four, without counting; but seven coins
have to be separated mentally into two groups, the numbers of which are
added, or one group has to be seen and the remaining objects counted,
before the number is known to be seven.
The actual limit of the number that can be "seen"--i.e. seen without
counting or adding--depends for any individual on the shape and
arrangement of the objects, but under similar conditions it is not the
same for all individuals. It has been suggested that as many as six
objects can be seen at once; but this is probably only the case with few
people, and with them only when the objects have a certain geometrical
arrangement. The limit for most adults, under favourable conditions, is
about four. Under certain conditions it is less; thus IIII, the old
Roman notation for _four_, is difficult to distinguish from III, and
this may have been the main reason for replacing it by IV (S 15).
In the case of young children the limit is probably two. That this was
also the limit in the case of primitive races, and that the
classification of things was into one, two and many, before any definite
process of counting (e.g. by the fingers) came to be adopted, is clear
from the use of the "dual number" in language, and from the way in which
the names for three and four are often based on those for one and two.
With the individual, as with the race, the limit of the number that can
be seen gradually increases up to four or five.
The statement that a number of objects can be seen to be three or four
is not to be taken as implying that there is a simultaneous perception
of all the objects. The attention may be directed in succession to the
different objects, so that the perception is rhythmical; the distinctive
rhythm thus aiding the perception of the particular number.
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. If a quinary-binary system (such as would naturally fit
in with counting on the fingers) is not adopted, teachers unconsciously
resort to a binary-quinary system. This is commonly done where cubes are
used; thus seven is represented by three pairs of cubes, with a single
cube at the top.
23. _Visualization of the Series._--A striking fact, in reference to
ideas of number, is the existence of number-forms, i.e. of definite
arrangements, on an imagined plane or in space, of the mental
representations of the successive numbers from 1 onwards. The proportion
of persons in whom number-forms exist has been variously estimated; but
there is reason to believe that the forms arise at a very early stage of
childhood, and that they did at some time exist in many individuals who
have afterwards forgotten them. Those persons who possess them are also
apt to make spatial arrangements of days of the week or the month,
months of the year, the letters of the alphabet, &c.; and it is
practically certain that only children would make such arrangements of
letters of the alphabet. The forms seem to result from a general
tendency to visualization as an aid to memory; the letter-forms may in
the first instance be quite as frequent as the number-forms, but they
vanish in early childhood, being of no practical value, while the
number-forms continue as an aid to arithmetical work.
The forms are varied, and have few points in common; but the following
tendencies are indicated.
(i) In the majority of cases the numbers lie on a continuous (but
possibly zigzag) line.
(ii) There is nearly always (at any rate in English cases) a break in
direction at 12. From 1 to 12 the numbers sometimes lie in the
circumference of a circle, an arrangement obviously suggested by a
clock-face; in these cases the series usually mounts upwards from 12. In
a large number of cases, however, the direction is steadily upwards from
1 to 12, then changing. In some cases the initial direction is from
right to left or from left to right; but there are very few in which it
is downwards.
(iii) The multiples of 10 are usually strongly marked; but special
stress is also laid on other important numbers, e.g. the multiples of
12.
(iv) The series sometimes goes up to very high numbers, but sometimes
stops at 100, or even earlier. It is not stated, in most cases, whether
all the numbers within the limits of the series have definite positions,
or whether there are only certain numbers which form an essential part
of the figure, while others only exist potentially. Probably the latter
is almost universally the case.
These forms are developed spontaneously, without suggestion from
outside. The possibility of replacing them by a standard form, which
could be utilized for performing arithmetical operations, is worthy of
consideration; some of the difficulties in the way of standardization
have already been indicated (S 14). The general tendency to prefer an
upward direction is important; and our current phraseology suggests that
this is the direction which increase is naturally regarded as taking.
Thus we speak of counting _up_ to a certain number; and similarly
mathematicians speak of _high_ and _ascending_ powers, while engineers
speak of high pressure, high speed, high power, &c. This tendency is
probably aided by the use of bricks or cubes in elementary
number-teaching.
24. _Primitive Ideas of Number._--The names of numbers give an idea of
the way in which the idea of number has developed. Where civilization is
at all advanced, there are usually certain names, the origin of which
cannot be traced; but, as we go farther back, these become fewer, and
the names are found to be composed on certain systems. The systems are
varied, and it is impossible to lay down any absolute laws, but the
following seem to be the main conclusions.
(i) Amongst some of the lowest tribes, as (with a few exceptions)
amongst animals, the only differentiation is between one and many, or
between one, two and many, or between one, two, three and many. As it
becomes necessary to use higher but still small numbers, they are formed
by combinations of one and two, or perhaps of three with one or two.
Thus many of the Australasian and South American tribes use only one and
two; seven, for instance, would be two two two one.
(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.
_Five_ may be signified by the word for hand; and either _ten_ or
_twenty_ by the word for _man_. Or the words signifying these numbers
may have reference to the completion of some act of counting. Between
five and ten; or beyond ten, the names may be due to combinations, e.g.
16 may be 10 + 5 + 1; or they may be the actual names of the fingers
last counted.
(iii) There are a few, but only a few, cases in which the number 6 or 8
is named as twice 3 or twice 4; and there are also a few cases in which
7, 8 and 9 are named as 6 + 1, 6 + 2 and 6 + 3. In the large majority of
cases the numbers 6, 7, 8 and 9 are 5 + 1, 5 + 2, 5 + 3 and 5 + 4, being
named either directly from their composition in this way or as the
fingers on the second hand.
(iv) There is a certain tendency to name 4, 9, 14 and 19 as being one
short of 5, 10, 15 and 20 respectively; the principle being thus the
same as that of the Roman IV, IX, &c. It is possible that at an early
stage the number of the fingers on one hand or on the two hands together
was only thought of vaguely as a large number in comparison with 2 or 3,
and that the number did not attain definiteness until it was linked up
with the smaller by insertion of the intermediate ones; and the linking
up might take place in both directions.
(v) In a few cases the names of certain small numbers are the names of
objects which present these numbers in some conspicuous way. Thus the
word used by the Abipones to denote 5 was the name of a certain hide of
five colours. It has been suggested that names of this kind may have
been the origin of the numeral words of different races; but it is
improbable that direct visual perception would lead to a name for a
number unless a name based on a process of counting had previously been
given to it.
25. _Growth of the Number-Concept._--The general principle that the
development of the individual follows the development of the race holds
good to a certain extent in the case of the number-concept, but it is
modified by the existence of language dealing with concepts which are
beyond the reach of the child, and also, of course, by the direct
attempts at instruction. One result is the formation of a number-series
as a mere succession of names without any corresponding ideas of number;
the series not being necessarily correct.
When numbering begins, the names of the successive numbers are attached
to the individual objects; thus the numbers are originally ordinal, not
cardinal.
The conception of number as cardinal, i.e. as something belonging to a
group of objects as a whole, is a comparatively late one, and does not
arise until the idea of a whole consisting of its parts has been formed.
This is the _quantitative_ aspect of number.
The development from the name-series to the quantitative conception is
aided by the numbering of material objects and the performance of
elementary processes of comparison, addition, &c., with them. It may
also be aided, to a certain extent, by the tendency to find rhythms in
sequences of sounds. This tendency is common in adults as well as in
children; the strokes of a clock may, for instance, be grouped into
fours, and thus eleven is represented as two fours and three.
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.
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