SCORPION

In two other ways, the production of numerals from
merely descriptive words may be observed both among
lower and higher races. The Gallas have no numerical
fractional terms, but they make an equivalent set of terms
from the division of the cakes of salt which they use as
money. Thus tchabnana, ' a broken piece ' (from tchaba,
4 to break/ as we say ' a fraction '), receives the meaning
of one-half ; a term which we may compare with Latin
dimidium, French demi. Ordinal numbers are generally
derived from cardinal numbers, as third, fourth, fifth, from

1 See Pott, ' Zahlmethode,' pp. 78, 99, 124, 161 ; Grimm, 'Deutsche
Rechtsalterthiimer,' ch. v.

258 , THE ART OF COUNTING.

three, four, five. But among the very low ones there is to
be seen evidence of independent formation quite uncon-
nected with a conventional system of numerals already
existing. Thus the Greenlander did not use his ' one ' to
make ' first,' but calls it sujugdlek, ' foremost/ nor ' two '
to make ' second/ which he calls aipd, ' his companion ; *
it is only at ' third ' that he takes to his cardinals, and
forms pingajuat in connexion with pingasut, 3. So, in
Indo-European languages, the ordinal prathamas, Trpwro?,
primus, first, has nothing to do with a numerical ' one/
but with the preposition pra, ' before/ as meaning simply
' foremost ; ' and although Greeks and Germans call the
next ordinal fovrepos, zweite, from Svo, zwei, we call it
second, Latin secundus, ' the following ' (sequi), which is
again a descriptive sense-word.

If we allow ourselves to mix for a moment what is with
what might be, we can see how unlimited is the field of
possible growth of numerals by mere adoption of the names
of familiar things. Following the example of the Sleswigers
we might make shilling a numeral for 12, and go on to ex-
press 4 by groat ; week would provide us with a name for 7,
and clover for 3. But this simple method of description
is not the only available one for the purpose of making
numerals. The moment any series of names is arranged in
regular order in our minds, it becomes a counting-machine.
I have read of a little girl who was set to count cards, and
she counted them accordingly, January, February, March,
April. She might, of course, have reckoned them as
Monday, Tuesday, Wednesday. It is interesting to find a
case coming under the same class in the language of grown
people. We know that the numerical value of the Hebrew
letters is given with reference to their place in the alphabet,
which was arranged for reasons tnat can hardly have had
anything to do with arithmetic. The Greek'alphabet is modi-
fied from a Semitic one, but instead of letting the numeral
value of their letters follow throughout their newly-arranged
alphabet, they reckon a, /3, y, 8, , properly, as i, 2, 3, 4, 5,

VARIOUS NUMERAL TERMS. 259

then put in r for 6, and so manage to let i stand for 10,
as does in Hebrew, where it is really the loth letter. Now,
having this conventional arrangement of letters made, it is
evident that a Greek who had to give up the regular I, 2, 3,
?<?, Svo, rpets, could supply their places at once by
adopting the names of the letters which had been settled to
stand for them, thus calling i alpha, 2 beta, 3 gamma, and
so onward. The thing has actually happened ; a remarkable
slang dialect of Albania, which is Greek in structure,
though full of borrowed and mystified words and metaphors
; and epithets understood only by the initiated, has, as its
equivalent for ' four ' and ' ten,' the words 8t\ra and

> 1

toxra.

While insisting on the value of such evidence as this in
making out the general principles of the formation of
numerals, I have not found it profitable to undertake the
task of etymologizing the actual numerals of the languages
of the world, outside the safe limits of the systems of digit-
numerals among the lower races, already discussed. There
may be in the languages of the lower races other relics of
the etymology of numerals, giving the clue to the ideas
according to which they were selected for an arithmetical
purpose, but such relics seem scanty and indistinct. 2 There
may even exist vestiges of a growth of numerals from de-
scriptive words in our Indo-European languages, in Hebrew
and Arabic, in Chinese. Such etymologies have been

1 Francisque-Michel, ' Argot,' p. 483.

2 Of evidence of this class, the following deserves attention : Dobrizhoffer
1 Abipones,' vol. ii. p. 169, gives geyenknate, 'ostrich-toes,' as the numeral
for 4, their ostrich having three toes before and one behind, and necnbalek,
1 a five-coloured spotted hide,' as the numeral ;. D'Orbigny, 'L'Homme
Amricain,' vol. ii. p. 163, remarks : ' Les Chiquitos ne savent compter que
jusqu'i un (tamo), n'ayant plus ensuite que des termes de comparaison.'
Kofie, ' Gr. of Vei Lang.,' notices that fera means both ' with ' and 2, and
thinks the former meaning original (compare the Tah. piti, ' together/
thence 2). Quichua cbuncu, ' heap,' cbunca, 10, may be connected. Aztec,
ce t i, cen-tli, 'grain,' may be connected. On possible derivations of 2 from
hand, &c., especially Hottentot, t'koam, ' hand, 2,' see Pott, ' Zahlmethode/
p. 29.

260 THE ART OF COUNTING.

brought forward, 1 and they are consistent with what is I
known of the principles on which numerals or quasi- 1
numerals are really formed. But so far as I have been able, 1
to examine the evidence, the cases all seem so philologically;!
doubtful, that I cannot bring them forward in aid of the
theory before us, and, indeed, think that if they succeed inp
establishing themselves, it will be by the theory supporting,
them, rather than by their supporting the theory. This;
state of things, indeed, fits perfectly with the view here
adopted, that when a word has once been taken up toi
serve as a numeral, and is thenceforth wanted as a mere,
symbol, it becomes the interest of language to allow it toj
break down into an apparent nonsense-word, from which
all traces of original etymology have disappeared.

Etymological research into the derivation of numeral
words thus hardly goes with safety beyond showing in the
languages of the lower culture frequent instances of digit-;
numerals, words taken from direct description of the ges- !
lures of counting on fingers and toes. Beyond this,
another strong argument is available, which indeed covers
almost the whole range of the problem. The numerical
systems of the world, by the actual schemes of their arrange-
ment, extend and confirm the opinion that counting on
fingers and toes was man's original method of reckoning,
taken up and represented in language. To count the
fingers on one hand up to 5, and then go on with a second

1 See Farrar, ' Chapters on Language,' p. 223. Benloew, ' Recherches sur
1'Origine des Noms de Nombre ;' Pictet, ' Origines Indo-Europ.' part ii. ch.
ii. ; Pott, ' Zahlmethode,' p. 128, &c. ; A. v. Humboldt's plausible compari-l
son between Skr. pancka, 5, and Pers. penjeh, ' the palm of the hand with the
fingers spread out j the outspread foot of a bird,' as though 5 were called
pancha from being like a hand, is erroneous. The Persian penjeh is itself
derived from the numeral 5, as in Skr. the hand is called pancbafakba, ' the
five-branched.' The same formation is found in English ; slang describes a
man's hand as his ' fives,' or ' bunch of fives,' thence the name of the game
of fives, played by striking the ball with the open hand, a term which
has made its way out of slang into accepted language. Burton describes
the polite Arab at a meal, calling his companion's attention to a grain of
rice fallen into his beard. ' The gazelle is in the garden,' he says, with a
smile. ' We will hunt her with the five, is the reply.

QUINARY, DECIMAL, AND VIGESIMAL. 26l

, is a notation by fives, or as it is called, a quinary nota-
tion. To count by the use of both hands to 10, and thence
* to reckon by tens, is a decimal notation. To go on by
: hands and feet to 20, and thence to reckon by twenties, is a
1 vigesimal notation. Now though in the larger proportion of
: known languages, no distinct mention of fingers and toes,
I hands and feet, is observable in the numerals themselves,
yet the very schemes of quinary, decimal, and vigesimal no-
tation remain to vouch for such hand-and-foot-counting
having been the original method on which they were
founded. There seems no doubt that the number of the
fingers led to the adoption of the not especially suitable
number 10 as a period in reckoning, so that decimal
arithmetic is based on human anatomy. This is so obvious,
that it is curious to see Ovid in his well-known lines putting
the two facts close together, without seeing that the second
was the consequence of the first.

' Annus erat, decimum cum luna receperat orbem.

Hie numerus magno tune in honore fuit.
Seu quia tot digiti, per quos numerare solemus :

Seu quia bis quino femina mense parit :
Seu quod adusque decem numero crescente venitur,

Principium spatiis sumitur inde novis.' l

In surveying the languages of the world at large, it is
found that among tribes or nations far enough advanced in
i arithmetic to count up to five in words, there prevails, with
scarcely an exception, a method founded on hand-counting,
quinary, decimal, vigesimal, or combined of these. For
perfect examples of the quinary method, we may take a
Polynesian series which runs i, 2, 3, 4, 5, 5-1, 5-2, &c. ; or
a Melanesian series which may be rendered as i, 2, 3, 4, 5,
2nd i, 2nd 2, &c. Quinary leading into decimal is well
shown in the Fellata series i ... 5, 5-1 ... 10, 10-1 . . .
10-5, io-5'i ... 20, ... 30, ... 40, &c. Pure decimal
may be instanced from Hebrew i, 2 . . . 10, 10-1 . . . 20,
20-1 ... &c. Pure vigesimal is not usual, for the obvious

1 Ovid, Fast. iii. 121.

262 ' THE ART OF COUNTING.

reason that a set of independent numerals to 20 would be
inconvenient; but it takes on from quinary, as in Aztec,
which may be analyzed as I, 2 ... 5, 5-1 ... 10, 10-1 . . j

IO'5, IO'5'I ... 20, 2O'I . . . 20* IO, 20-IO-I ... 40, &C. J

or from decimal, as in Basque, i . . . 10, 10-1 . . . 20, 20-1
. . . 20- 10, .20-10- 1 ... 40 &C. 1 It seems unnecessary to!
bring forward here the mass of linguistic details required for
any general demonstration of these principles of numeration
among the races of the world. Prof. Pott, of Halle, has treated
the subject on elaborate philological evidence, in a special
monograph, 2 which is incidentally the most extensive collec-
tion of details relating to numerals, indispensable to students
occupied with such enquiries. For the present purpose the
following rough generalization may suffice, that the quinary
system is frequent among the lower races, among whom also
the vigesimal system is considerably developed, but the ten-
dency of the higher nations has been to avoid the one as
too scanty, and the other as too cumbrous, and to use the in-'
termediate decimal system. These differences in the usage oi 1
various tribes and nations do not interfere with, but rather
confirm, the general principle which is their common cause,
that man originally learnt to reckon from his fingers and
toes, and in various ways stereotyped in language the result
of this primitive method.

Some curious points as to the relation of these systems
may be noticed in Europe. It was observed of a certain
deaf-and-dumb boy, Oliver Caswell, that he learnt to count
as high as 50 on his fingers, but always ' fived/ reckoning,
for instance, 18 objects as ' both hands, one hand, three i
fingers.' 3 The suggestion has been made that the Greek use

1 The actual word-numerals of the two quinary series are given as ex-
amples. Triton's Bay, i, samosi ; 2, ro'eeti ; 3, touwroe ; 4, faat ; 5, rimi; 6,
rim-samos ; 7, rim-roeeti; 8, rim-touwroe ; 9, rim-faat ; 10, woetsja, Lifu, i,
pacha; 2, lo; 3, kun; 4, tback; 5, tbabumb; 6, lo-acba; 7, lo-a-lo; 8, lo-kunn;
9, lo-tback; 10, te-bennete.

* A. F. Pott, ' Die Quinare und Vigesimale Zahlmethode bei Volkern
aller Welttheile,' Halle, 1847; supplemented in ' Festgabe zur xxv.
Versammlung Deutscher Philologen, &c., in Halle ' (1867).

8 ' Account of Laura Bridgman,' London, 1845, P- ! 59-

QUINARY, DECIMAL, AND VIGESIMAL. 263

of - /u7ra&iv, ' to five/ as an expression for counting, is a trace
of rude old quinary numeration (compare Finnish lokket ' to
count,' from lokke ' ten '). Certainly, the Roman numerals
I, II, ... V, VI ... X, XI ... XV, XVI, &c., form a
remarkably well-defined written quinary system. Remains
of vigesimal counting are still more instructive. Counting
by twenties is a strongly marked Keltic characteristic. The
cumbrous vigesimal notation could hardly be brought more
strongly into view in any savage race than in such examples
as Gaelic aon deug is da fhichead ' one, ten, arid two
twenties/ i.e., 51 ; or Welsh unarbymtheg ar ugain ' one
and fifteen over twenty/ i.e., 36 ; or Breton unnek ha tri-
ugent ' eleven and three twenties/ i.e., 71. Now French,
being a Romance language, has a regular system of Latin
tens up to 100 ; cinquante, soixante, septante, huitante,
nonante, which are to be found still in use in districts
within the limits of the French language, as in Belgium.
Nevertheless, the clumsy system of reckoning by twenties
has broken out through the decimal system in France.
The septante is to a great extent suppressed, soixante-
quatorze, for instance, standing for 74 ; quatre-vingts has
fairly established itself for 80, and its use continues into
the nineties, quatre-vingt-treize for 93 ; in numbers above
100 we find six-vingts, sept-vingts, huit-vingts, for 120, 140,
160, and a certain hospital has its name of Les Quinze-
vingts from its 300 inmates. It is, perhaps, the most
reasonable explanation of this curious phenomenon, to
suppose the earlier Keltic system of France to have held its
ground, modelling the later French into its own ruder
shape. In England, the Anglo-Saxon numeration is
decimal, hund-seofontig, 70 ; hund-eahtatig, 80 ; hund-ni-
gontig, 90 ; hund-teontig, 100 ; hund-enlufontig, no ; hund-
twelftig, 1 20. It may be here also by Keltic survival that
the vigesimal reckoning by the ' score/ threescore and ten,
fourscore and thirteen, &c., gained a position in English
which it has not yet totally lost. 1

1 Compare the Rajmahali tribes adopting Hindi numerals, yet reckoning

264 THE ART OF COUNTING.

From some minor details in numeration, ethnological
hints may be gained. Among rude tribes with scanty
series of numerals, combination to make out new numbers
is very soon resorted to. Among Australian tribes addition
makes ' two-one,' ' two-two,' express 3 and 4 ; in Guachi
* two-two' is 4 ; in San Antonio 'four and two-one' is 7.
The plan of making numerals by subtraction is known in
North America, and is well shown in the Aino language of
Yesso, where the words for 8 and 9 obviously mean ' two
from ten,' ' one from ten.' Multiplication appears, as in
San Antonio, ' two-and-one-two,' and in a Tupi dialect
4 two-three,' to express 6. Division seems not known for
such purposes among the lower races, and quite exceptional
among the higher. Facts of this class show variety in the
inventive devices of mankind, and independence in their
formation of language. They are consistent at the same
time with the general principles of hand-counting. The
traces of what might be called binary, ternary, quaternary,
senary reckoning, which turn on 2, 3, 4, 6, are mere
varieties, leading up to, or lapsing into, quinary and decimal
methods.

The contrast is a striking one between the educated
European, with his easy use of his boundless numeral series,
and the Tasmanian, who reckons 3, or anything beyond 2,
as ' many/ and makes shift by his whole hand to reach the
limit of ' man,' that is to say, 5. This contrast is due to
arrest of development in the savage, whose mind remains in
the childish state which the beginning of one of our nur-
sery number-rhymes illustrates curiously. It runs

' One's none,
Two's some,
Three's a many,
Four's a penny,
Five's a little hundred.'

by twenties. Shaw, I.e. The use of a ' score ' as an indefinite number in
England, and similarly of 20 in France, of 40 in the Hebrew of the Old
Testament and the Arabic of the Thousand and One Nights, may be among
other traces of vigesimal reckoning.

COMBINED NUMERALS. 265

To notice this state of things among savages and chil-
dren raises interesting points as to the early history of
grammar. W. von Humboldt suggested the analogy be-
tween the savage notion of 3 as many ' and the gram-

I matical use of 3 to form a kind of superlative, in forms
of which ' trismegistus,' ' ter felix,' ' thrice blest,' are

I familiar instances. The relation of single, dual, and plural
is well shown pictorially in the Egyptian hieroglyphics,
where the picture of an object, a horse for instance,
is marked by a single line | if but one is meant, by two
lines | | if two are meant, by three lines | | | if three or
an indefinite plural number are meant. The scheme of
grammatical number in some of the most ancient and im-
portant languages of the world is laid down on the same

I savage principle. Egyptian, Arabic, Hebrew, Sanskrit,
Greek, Gothic, are examples of languages using singular,

! dual, and plural number ; but the tendency of higher intel-
lectual culture has been to discard the plan as inconvenient

! and unprofitable, and only to distinguish singular and
plural. No doubt the dual held its place by inheritance
from an early period of culture, and Dr. D. Wilson seems
justified in his opinion that it ' preserves to us the
memorial of that stage of thought when all beyond two
was an idea of indefinite number/ 1

When two races at different levels of culture come into
contact, the ruder people adopt new art and knowledge, but
at the same time their own special culture usually comes to
a standstill, and even falls off. It is thus with the art of
counting. We may be able to prove that the lower race
had actually been making great and independent progress
in it, but when the higher race comes with a convenient
and unlimited means of not only naming all imaginable
numbers, but of writing them down and reckoning with
them by means of a few simple figures, what likelihood is
there that the barbarian's clumsy methods should be farther
worked out ? As to the ways in which the numerals of the

*D. Wilson, 'Prehistoric Man,' p. 616.

266 THE ART OF COUNTING.

superior race are grafted on the language of the inferior,
Captain Grant describes the native slaves of Equatorial
Africa occupying their lounging hours in learning the
numerals of their Arab masters. 1 Father Dobrizhoffer's
account of the arithmetical relations between the native
Brazilians and the Jesuits is a good description of the
intellectual contact between savages and missionaries.
The Guaranis, it appears, counted up to 4 with their native
numerals, and when they got beyond, they would say
' innumerable/ ' But as counting is both of manifold use
in common life, and in the confessional absolutely indis-
pensable in making a complete confession, the Indians were
daily taught at the public catechising in the church to
count in Spanish. On Sundays the whole people used to
count with a loud voice in Spanish, from i to 1,000.' The
missionary, it is true, did not find the natives use the
numbers thus learnt very accurately ' We were washing
at a blackamoor,' he says. 2 If, however, we examine the
modern vocabularies of savage or low barbarian tribes, they
will be found to afford interesting evidence how really
effective the influence of higher on lower civilization has
been in this matter. So far as the ruder system is com-
plete and moderately convenient, it may stand, but where
it ceases or grows cumbrous, and sometimes at a lower
limit than this, we can see the cleverer foreigner taking it
into his own hands, supplementing or supplanting the
scanty numerals of the lower race by his own. The higher
race, though advanced enough to act thus on the lower,
need not be itself at an extremely high level. Markham
observes that the Jivaras of the Maranon, with native
numerals up to 5, adopt for higher numbers those of the
Quichua, the language of the Peruvian Incas. 8 The cases
or the indigenes of India are instructive. The Khonds
reckon i and 2 in native words, and then take to borrowed

1 Grant in ' Tr. Eth. Soc.' vol. iii. p. 90.

2 Dobrizhoffer, ' Gesch. der Abiponer,' p. 205 ; Eng. Trans, vol. ii. p. 171.
a Markham in ' Tr. Eth. Soc.' vol. iii. p. 166.

ADOPTED FOREIGN NUMERALS. 267

Hindi numerals. The Oraon tribes, while belonging to a
race of the Dravidian stock, and having had a series of
native numerals accordingly, appear to have given up their
use beyond 4, or sometimes even 2, and adopted Hindi
numerals in their place. 1 The South American Conibos
were observed to count i and 2 with their own words, and
then to borrow Spanish numerals, much as a Brazilian
dialect of the Tupi family is noticed in the last century as
having lost the native 5, and settled down into using the
old native numerals up to 3, and then continuing in Portu-
guese. 2 In Melanesia, the Annatom language can only
count in its own numerals to 5, and then borrows English
siks, seven, eet, nain, &c. In some Polynesian islands,
though the native numerals are extensive enough, the
confusion arising from reckoning by pairs and fours as well
as units, has induced the natives to escape from perplexity
by adopting huneri and tausani* And though the Esqui-
maux counting by hands, feet, and whole men, is capable of
expressing high numbers, it becomes practically clumsy
even when it gets among the scores, and the Greenlander
has done well to adopt untrtte and tminte from his Danish
teachers. Similarity of numerals in two languages is a
point to which philologists attach great and deserved
importance in the question whether they are to be con-
sidered as sprung from a common stock. But it is clear
that so far as one race may have borrowed numerals from
another, this evidence breaks down. The fact that this
borrowing extends as low as 3, and may even go still lower
for all we know, is a reason for using the argument from
connected numerals cautiously, as tending rather to prove
intercourse than kinship.

At the other end of the scale of civilization, the adoption

1 Latham, ' Comp. Phil.' p. 186; Shaw in 'As. Res.' vol. iv. p. 96;
4 Journ. As. Soc. Bengal,' 1866, part ii. pp. 27, 204, 251.

a St. Cricq in ' Bulletin de la Soc. de Ge"og.' 1853, p. 286 ; Pott, ' Zahlme-
thode,' p. 7.

8 Gabelentz, p. 89 ; Hale, I.e.

268

THE ART OF COUNTING.

of numerals from nation to nation still .presents interest-
ing philological points. Our own language gives curious
instances, as second and million. The manner in which
English, in common with German, Dutch, Danish, and
even Russian, has adopted Mediaeval Latin dozena (from
duodecim) shows how convenient an arrangement it was
found to buy and sell by the dozen, and how necessary it
was to have a special word for it. But the borrowing
process has gone farther than this. If it were asked how
many sets of numerals are now in use among English-
speaking people in England, the probable reply would be
one set, the regular one, two, three, &c. There exist, however,
two borrowed sets as well. One is the well-known dicing-
set, ace, deuce, tray, cater, cinque, size ; thus size-ace is ' 6
and one/ cinques or sinks, ' double five.' These came to
us from France, and correspond with the common French
numerals, except ace, which is Latin as, a word of great
philological interest, meaning ' one.' The other borrowed
set is to be found in the Slang Dictionary. It appears
that the English street-folk have adopted as a means of
secret communication a set of Italian numerals from the
organ-grinders and image-sellers, or by other ways through
which Italian or Lingua Franca is brought into the low
neighbourhoods of London. In so doing, they have per-
formed a philological operation not only curious, but in-
structive. By copying such expressions as Italian due soldi,
ire soldi, as equivalent to ' twopence,' ' threepence,' the
word saltee became a recognized slang term for ' penny/
and pence are reckoned as follows :

Oney saltee .

Dooe saltee .

Tray saltee .

Quarterer saltee

Cbinker saltee

Say saltee .

Say oney saltee or setter saltee

Say dooe saltee or otter saltee .

Say tray saltee or nobba saltee

id. uno soldo.
^d. due soldi.
$d. tre soldi.
\d. quattro soldi.
$d. cinque soldi.
6d. sei soldi.
jd. sette soldi.
%d. otto soldi.
<. nove soldi.

DEVELOPMENT OF ARITHMETIC. 269

Say quarter er saltee or dacha sal tee . . io<l. dieci soldi.
Say cbinker saltee or dacha oney saltee . lid. undici soldi.
Oney beong ...... is.

A beong say saltee . . . . is. 6d.

Dooe beong say saltee or madza caroon . 25. 6d. (half crown,

mezza corona.) 1

One of these series simply adopts Italian numerals deci-
mally. 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 coster-
monger is adding pence up to the silver sixpence, and
then adding pence again up to the shilling. Thus our duo-
decimal 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 express-
ing 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 II and
A 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 / 8', for J, J, y s for f ; 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

1 J. C. Hotten, ' Slang Dictionary,' p. 218.

270 . THE ART OF COUNTING.

order which for ages had been in use above it. The ancient
Egyptian and the still-used Roman and Chinese numeration
are indeed founded on savage picture- writ ing, 1 while the
abacus and the swan-pan, the one still a valuable school-
instrument, and the other in full practical use, have their
germ in the savage counting by groups of objects, as when
South Sea Islanders count with coco-nut stalks, putting a
little one aside every time they come to 10, and a large one
when they come to 100, or when African negroes reckon
with pebbles or nuts, and every time they come to 5 put
them aside in a little heap.*

We are here especially concerned with gesture-counting
on the fingers, as an absolutely savage art still in use among
children and peasants, and with the system of numeral
words, as known to all mankind, appearing scantily among
the lowest tribes, and reaching within savage limits to deve-
lopments which the highest civilization has only improved in
detail. These two methods of computation by gesture and
word tell the story of primitive arithmetic in a way that can
be hardly perverted or misunderstood. We see the savage
who can only count to 2 or 3 or 4 in words, but can go
farther in dumb show. He has words for hands and fingers,
feet and toes, and the idea strikes him that the words which
describe the gesture will serve also to express its meaning,
and they become his numerals accordingly. This did not
happen only once, it happened among different races in
distant regions, for such terms as ' hand ' for 5, ' hand-
one ' for 6, ' hands ' for 10, ' two on the foot ' for 12,
' hands and feet ' or ' man ' for 20, ' two men ' for 40, &c.,
show such uniformity as is due to common principle, but
also such variety as is due to independent working-out.
These are ' pointer-facts ' which have their place and
explanation in a development-theory of culture, while a
degeneration-theory totally fails to take them in. They are
distinct records of development, and of independent deve-

1 ' Early History of Mankind,' p. 106.

z Ellis, ' Polyn. Res.' vol. i. p. 91 ; Klemm, C. G. vol. iii. p. 383.

DEVELOPMENT OF ARITHMETIC. 27!

lopment, among savage tribes to whom some writers on
civilization have rashly denied the very faculty of self-
improvement. The original meaning of a great part of the
stock of numerals of the lower races, especially of those from
fc I to 4, not suited to be named as hand-numerals, is obscure.
They may have been named from comparison with objects,
in a way which is shown actually to happen in such forms
as ' together ' for 2, ' throw ' for 3, ' knot ' for 4 ; but
any concrete meaning we may guess them to have once had
seems now by modification and mutilation to have passed
out of knowledge.

Remembering how ordinary words change and lose their
traces of original meaning in the course of ages, and that in
numerals such breaking down of meaning is actually
desirable, to make them fit for pure arithmetical symbols,
we cannot wonder that so large a proportion of existing
numerals should have no discernible etymology. This is
especially true of the i, 2, 3, 4, among low and high races
alike, the earliest to be made, and therefore the earliest to
lose their primary significance. Beyond these low numbers
the languages of the higher and lower races show a remark-
able difference. The hand-and-foot numerals, so prevalent
and unmistakable in savage tongues like Esquimaux and
Zulu, are scarcely if at all traceable in the great languages
of civilization, such as Sanskrit and Greek, Hebrew and
Arabic. This state of things is quite conformable to the
development-theory of language. We may argue that
it was in comparatively recent times that savages arrived
at the invention of hand-numerals, and that therefore
the etymology of such numerals remains obvious. But
it by no means follows from the non-appearance of such
primitive forms in cultured Asia and Europe, that they did
not exist there in remote ages ; they may since have been
rolled and battered like pebbles by the stream of .time, till
their original shapes can no longer be made out. Lastly,
among savage and civilized races alike, the general frame-
work of numeration stands throughout the world as an

272 ' THE ART OF COUNTING.

abiding monument of primaeval culture. This framework,
the all but universal scheme of reckoning by fives, tens, and
twenties, shows that the childish and savage practice of
counting on fingers and toes lies at the foundation of our
arithmetical science. Ten seems the most convenient
arithmetical basis offered by systems founded on hand-
counting, but twelve would have been better, and duodecimal
arithmetic is in fact a protest against the less convenient
decimal arithmetic in ordinary use. The case is the not
uncommon one of high civilization bearing evident traces of
the rudeness of its origin in ancient barbaric life.