The most primitive mathematical activity of man is counting, but here
his first efforts are lost in the obscurity of the past. The lower
races, however, yield us evidence that is not without value. Although
the savage mind is not identical with the mind of primitive man, there
is much in the activities of undeveloped races that can throw light upon
the behavior of peoples more advanced. We must be careful in our
inferences, however. Among the Australians and South Americans there are
peoples whose numerical systems go little, or not at all, beyond the
first two or three numbers. "It has been inferred from this," writes
Professor Boas (_Mind of Primitive Man_, pp. 152-53), "that the people
speaking these languages are not capable of forming the concept of
higher numbers.... People like the South American Indians, ... or like
the Esquimo ... are presumably not in need of higher numerical
expressions, because there are not many objects that they have to count.
On the other hand, just as soon as these same people find themselves in
contact with civilization, and when they acquire standards of value that
have to be counted, they adopt with perfect ease higher numerals from
other languages, and develop a more or less perfect system of
counting.... It must be borne in mind that counting does not become
necessary until objects are considered in such generalized form that
their individualities are entirely lost sight of. For this reason it is
possible that even a person who owns a herd of domesticated animals may
know them by name and by their characteristics, without even desiring to
count them."
And there is one other false interpretation to be avoided. Man does not
feel the need of counting and then develop a system of numerals to meet
the need. Such an assumption is as ridiculous as to assume prehistoric
man thinking to himself: "I must speak," and then inventing voice
culture and grammar to make speaking pleasant and possible. Rather, when
powers of communication are once attained, presumably in their
beginnings also without forethought, man being still more animal than
man, there were gradually dissociated communications of a kind
approaching what numbers mean to us. But the number is not yet a symbol
apart from that of the things numbered. Picture writing, re-representing
the things meant, preceded developmentally any kind of symbolization
representing the number by mere one-one correspondence with
non-particularized symbols. It is plausible, although I have no
anthropological authority for the statement, that the prevalence of
finger words as number symbols (cf. infra) is originally a consequence
of the fact that our organization makes the hand the natural instrument
of pointing.
The difficulty of passing from concrete representations to abstract
symbols has been keenly stated by Conant (_The Number Concept_, pp.
72-73), although his terminology is that of an old psychology and the
limitations implied for the primitive mind are limitations of practice
rather than of capacity as Mr. Conant seems to believe. "An abstract
conception is something quite foreign to the essentially primitive mind,
as missionaries and explorers have found to their chagrin. The savage
can form no mental concept of what civilized man means by such a word as
_soul_; nor would his idea of the abstract number 5 be much clearer.
When he says _five_, he uses, in many cases at least, the same word that
serves him when he wishes to say _hand_; and his mental concept when he
says _five_ is a hand. The concrete idea of a closed fist, of an open
hand with outstretched fingers, is what is uppermost in his mind. He
knows no more and cares no more about the pure number 5 than he does
about the law of conservation of energy. He sees in his mental picture
only the real, material image, and his only comprehension of the number
is, "these objects are as many as the fingers on my hand." Then, in the
lapse of the long interval of centuries which intervene between lowest
barbarism and highest civilization, the abstract and concrete become
slowly dissociated, the one from the other. First the actual hand
picture fades away, and the number is recognized without the original
assistance furnished by the derivation of the word. But the number is
still for a long time a certain number _of objects_, and not an
independent concept."
An excellent fur trader's story, reported to me by Mr. Dewey, suggests a
further impulse to count besides that given by the need of keeping a
tally, namely, the need of making one thing correspond to another in a
business transaction. The Indian laid down one skin and the trader two
dollars; if he proposed to count several skins at once and pay for all
together, the former replied "too much cheatem." The result, however,
demanded a tally either by the fingers, a pebble, or a mark made in the
sand, and as the magnitude of such transactions grows the need of a
specific number symbol becomes ever more acute.
The first obstacle, then, to overcome--and it has already been
successfully passed by many primitive peoples--is the need of fortuitous
attainment of a numerical symbol, which is not the mere repeated symbol
of the things numbered. Significantly, this symbol is usually derived
from the hand, suggesting gestures of tallying, and not from the words
of already developed language. Consequently, number words relate
themselves for the most part to the hand, and written number symbols,
which are among the earliest writings of most peoples, tend to depict it
as soon as they have passed beyond the stage mentioned above of merely
repeating the symbol of the things numbered. W. C. Eells, in writing of
the Number Systems of the North American Indians (_Am. Math. Mo._, Nov.,
1913; pp. 263-72), finds clear linguistic evidence for a digital origin
in about 40% of the languages examined. Of the non-digital instances, 1
was sometimes connected with the first personal pronoun, 2 with roots
meaning separation, 3, rarely, meaning more, or plural as distinguished
from the dual, just as the Greek uses a plural as well as a dual in
nouns and verbs, 4 is often the perfect, complete right. It is often a
sacred number and the base of a quarternary system. Conant (_loc. cit._
p. 98) also gives a classification of the meanings of simple number
words for more advanced languages; and even in them the hand is
constantly in evidence, as in 5, the hand; 10, two hands, half a man,
when fingers and toes are both considered, or a man, when the hands
alone are considered; 20, one man, two feet. The other meanings hang
upon the ideas of existence, piece, group, beginning, for 1; and
repetition, division, and collection for higher numerals.
A peculiar difficulty lies in the fact that when once numbering has
become a self-conscious effort, the collection of things to be numbered
frequently tends to exceed the number of names that have become
available. Sometimes the difficulty is met by using a second man when
the fingers and toes of the first are used up, sometimes by a method of
repetition with the record of the number of the repetition itself added
to the numerical significance of the whole process. Hence arise the
various systems of bases that occur in developed mathematics. But the
inertia to be overcome in the recognition of the base idea is nowhere
more obvious than in the retention by the comparatively developed
Babylonian system of a second base of 60 to supplement the decimal one
for smaller numbers. Among the American Indians (Eells, _loc. cit._) the
system of bases used varies from the cumbersome binary scale, that
exercised such a fascination over Leibniz (_Opera_, _III_, p. 346),
through the rare ternary, and the more common quarternary to the
"natural" quinary, decimal, and vigesimal systems derived from the
use of the fingers and toes in counting. The achievement of a number
base and number words, however, does not always open the way to
further mathematical development. Only too often a complexity of
expression is involved that almost immediately cuts off further
progress. Thus the Youcos of the Amazon cannot get beyond the number
three, for the simplest expression for the idea in their language is
"pzettarrarorincoaroac" (Conant, _loc. cit._, pp. 145, 83, 53). Such
names as "99, tongo solo manani nun solo manani" (i.e., 10, understood,
5 plus 4 times, and 5 plus 4) of the Soussous of Sierra Leone; "399,
caxtolli onnauh poalli ipan caxtolli onnaui" (15 plus 4 times 20 plus 15
plus 4) of the Aztec; "29, wick a chimen ne nompah sam pah nep e chu
wink a" (Sioux), make it easy to understand the proverb of the Yorubas
of Abeokuta, "You may be very clever, but you can't tell 9 times 9."
Almost contemporaneously with the beginnings of counting various
auxiliary devices were introduced to help out the difficult task. In
place of many men, notched sticks, knotted strings, pebbles, or finger
pantomime were used. In the best form, these devices resulted in the
abacus; indeed, it was not until after the introduction of arabic
numerals and well into the Renaissance period that instrumental
arithmetic gave way to graphical in Europe (D. E. Smith, _Rara
Arithmetica_, under "Counters"). "In eastern Europe," say Smith and
Mikami (_Japanese Mathematics_, pp. 18-19), "it"--the abacus--"has never
been replaced, for the tschotue is used everywhere in Russia to-day, and
when one passes over into Persia the same type of abacus is common in
all the bazaars. In China the swan-pan is universally used for the
purposes of computation, and in Japan the soroban is as strongly
entrenched as it was before the invasion of western ideas."
Given, then, the idea of counting, and a mechanical device to aid
computation, it still remains necessary to obtain some notation in which
to record results. At the early dawn of history the Egyptians seem to
have been already possessed of number signs (cf. Cantor, _Gesch. de.
Math._, p. 44) and the Phoenicians either wrote out their number words
or used a few simple signs, vertical, horizontal, and oblique lines, a
process which the Arabians perpetuated up to the beginning of the
eleventh century (Fink, p. 15); the Greeks, as early as 600 B. C., used
the initial letters of words for numbers. But speaking generally,
historical beginnings of European number signs are too obscure to
furnish us good material.
Our Indians have few number symbols other than words, but when they
occur (cf. Eells, _loc. cit._) they usually take the form of pictorial
presentation of some counting device such as strokes, lines dotted to
suggest a knotted cord, etc. Indeed, the smaller Roman numerals were
probably but a pictorial representation of finger symbols. However, a
beautiful concrete instance is furnished us in the Japanese mathematics
(cf. Smith and Mikami, Ch. III). The earliest instrument of reckoning in
Japan seems to have been the rod, Ch'eou, adapted from the Chinese under
the name of Chikusaku (bamboo rods) about 600 A. D. At first relatively
large (measuring rods?), they became reduced to about 12 cm., but from
their tendency to roll were quickly replaced by the sangi (square
prisms, about 7 mm. thick and 5 cm. long) and the number symbols were
evidently derived from the use of these rods:
_ __ ___ ____
|, ||, |||, ||||, |||||, |, ||, |||, ||||.
For the sake of clearness, tens, hundreds, etc., were expressed in the
even place by horizontal instead of vertical lines and vice versa; thus
1267 would be formed
__
- || | ||.
-
The rods were arranged on a sort of chessboard called the swan-pan. Much
later the lines were transferred to paper, and a circle used to denote
the vacant square. The use of squares, however, rendered it unnecessary
to arrange the even places differently from the odd, so numbers like
38057 came to be written
+-------+-------+-------+-------+------+
| | ___ | | | __ |
| ||| | ||| | | ||||| | || |
| | | | | |
+-------+-------+-------+-------+------+
instead of
+-------+-------+-------+-------+------+
| | | | | |
| | | | - | |
| | | | | - | __ |
| ||| | - | | - | || |
| | - | | - | |
| | - | | - | |
| | | | | |
+-------+-------+-------+-------+------+
as in the earlier notation.
Somewhere in the course of these early mathematical activities the
process has changed from the more or less spontaneous operating that led
primitive man to the first enunciation of arithmetical ideas, and has
become a self-conscious striving for the solution of problems. This
change had already taken place before the historical origins of
arithmetic are met. Thus, the treatise of Ahmes (2000 B. C.) contains
the curious problem: 7 persons each have 7 cats; each cat eats 7 mice;
each mouse eats 7 ears of barley; from each ear 7 measures of corn may
grow; how much grain has been saved? Such problems are, however, half
play, as appears in a Leonardo of Pisa version some 3000 years later: 7
old women go to Rome; each woman has 7 mules; each mule, 7 sacks; each
sack contains 7 loaves; with each loaf are 7 knives; each knife is in 7
sheaths. Similarly in Diophantus' epitaph (330 A. D.): "Diophantus
passed 1/6 of his life in childhood, 1/12 in youth, and 1/7 more as a
bachelor; 5 years after his marriage, was born a son who died 4 years
before his father at 1/2 his age." Often among peoples such puzzles were
a favorite social amusement. Thus Braymagupta (628 A. D.) reads, "These
problems are proposed simply for pleasure; the wise man can invent a
thousand others, or he can solve the problems of others by the rules
given here. As the sun eclipses the stars by its brilliancy, so the man
of knowledge will eclipse the fame of others in assemblies of the people
if he proposes algebraic problems, and still more if he solves them"
(Cajori, _Hist. of Math._, p. 92).
The limitation of these early methods is that the notation merely
records and does not aid computation. And this is true even of such a
highly developed system as was in use among the Romans. If the reader is
unconvinced, let him attempt some such problem as the multiplication of
CCCXVI by CCCCLXVIII, expressing it and carrying it through in Roman
numerals, and he will long for the abacus to assist his labors. It was
the positional arithmetic of the Arabians, of which the origins are
obscure, that made possible the development of modern technique. Of this
discovery, or rediscovery from the Hindoos, together with the zero
symbol, Cajori (_Hist. of Math._, p. 11) has said "of all mathematical
discoveries, no one has contributed more to the general progress of
intelligence than this." The notation no longer merely records results,
but now assists in performing operations.
The origins of geometry are even more obscure than those of arithmetic.
Not only is geometry as highly developed as arithmetic when it first
appears in occidental civilization, but, in addition, the problems of
primitive peoples seem to have been such that they have developed no
geometrical formulae striking enough to be recorded by investigators, so
far as I have been able to discover. But just as the commercial life of
the Phoenicians early forced them self-consciously to develop
arithmetical calculation, so environmental conditions seem to have
forced upon the Egyptians a need for geometrical considerations.
It is almost platitudinous to quote Herodotus' remark that the invention
of geometry was necessary because of the floods of the Nile, which
washed away the boundaries and changed the contours of the fields. And
as Proclus Diadochus adds (_Procli Diadochi, in primum Euclidis
elementorum librum commentarii_--quoted Cantor, I, p. 125): "It is not
surprising that the discovery of this as well as other sciences has
sprung from need, because everything in the process of beginning
proceeds from the incomplete to the complete. There takes place a
suitable transition from sensible perception to thoughtful consideration
and rational knowledge. Just as with the Phoenicians, for the sake of
business and commerce, an exact knowledge of numbers had its beginning,
so with the Egyptians, for the above-mentioned reasons, was geometry
contrived."
The earliest Egyptian mathematical writing that we know is that of Ahmes
(2000 B. C.), but long before this the mural decorations of the temple
wall involved many figures, the construction of which involved a certain
amount of working knowledge of such operations as may be performed with
the aid of a ruler and compass. The fact that these operations did not
earlier lead to geometry, as ruler and compass work seems to have
done in Japan in the nineteenth century (Smith and Mikami, index,
"Geometry"), is probably due to the stage at which the development of
Egyptian intelligence had arrived, feebly advanced on the road to higher
abstract thinking. It is everywhere characteristic of Egyptian genius
that little purely intellectual curiosity is shown. Even astronomical
knowledge was limited to those determinations which had religious or
magically practical significance, and its arithmetic and geometry never
escaped these bounds as with the more imaginative Pythagoreans, where
mystical interpretation seems to have been a consequence of rather than
a stimulus to investigation. An old Egyptian treatise reads (Cantor, p.
63): "I hold the wooden pin (Nebi) and the handle of the mallet (semes),
I hold the line in concurrence with the Goddess S[a.]fech. My glance
follows the course of the stars. When my eye comes to the constellation
of the great bear and the time of the number of the hour determined by
me is fulfilled, I place the corner of the temple." This incantation
method could hardly advance intelligence; but the methods of practical
measuring were more effective. Here the rather happy device of using
knotted cords, carried about by the Harpedonapts, or cord stretchers,
was of some moment. Especially, the fact that the lengths 3, 4, and 5,
brought into triangular form, served for an interesting connection
between arithmetic and the right triangle, was not a little gain, later
making possible the discovery of the Pythagorean theorem, although in
Egypt the theoretical properties of the triangle were never developed.
The triangle obviously must have been practically considered by the
decorators of the temple and its builders, but the cord stretchers
rendered clear its arithmetical significance. However, Ahmes' "Rules for
attaining the knowledge of all dark things ... all secrets that are
contained in objects" (Cantor, _loc. cit._, p. 22) contains merely a
mixture of all sorts of mathematical information of a practical
nature,--"rules for making a round fruit house," "rules for measuring
fields," "rules for making an ornament," etc., but hardly a word of
arithmetical and geometrical processes in themselves, unless it be
certain devices for writing fractions and the like.