EARLY HINDU FORMS WITH NO PLACE VALUE
While it is generally conceded that the scientific development of astronomy
among the Hindus towards the beginning of the Christian era rested upon
Greek[42] or Chinese[43] sources, yet their ancient literature testifies to
a high state of civilization, and to a considerable advance in sciences, in
philosophy, and along literary lines, long before the golden age of Greece.
From the earliest times even up to the present day the Hindu has been wont
to put his thought into rhythmic form. The first of this poetry--it well
deserves this name, being also worthy from a metaphysical point of
view[44]--consists of the Vedas, hymns of praise and poems of worship,
collected during the Vedic period which dates from approximately 2000 B.C.
to 1400 B.C.[45] Following this work, or possibly contemporary with it, is
the Brahmanic literature, which is partly ritualistic (the
Br[=a]hma[n.]as), and partly philosophical (the Upanishads). Our especial
interest is {13} in the S[=u]tras, versified abridgments of the ritual and
of ceremonial rules, which contain considerable geometric material used in
connection with altar construction, and also numerous examples of rational
numbers the sum of whose squares is also a square, i.e. "Pythagorean
numbers," although this was long before Pythagoras lived. Whitney[46]
places the whole of the Veda literature, including the Vedas, the
Br[=a]hma[n.]as, and the S[=u]tras, between 1500 B.C. and 800 B.C., thus
agreeing with Buerk[47] who holds that the knowledge of the Pythagorean
theorem revealed in the S[=u]tras goes back to the eighth century B.C.
The importance of the S[=u]tras as showing an independent origin of Hindu
geometry, contrary to the opinion long held by Cantor[48] of a Greek
origin, has been repeatedly emphasized in recent literature,[49] especially
since the appearance of the important work of Von Schroeder.[50] Further
fundamental mathematical notions such as the conception of irrationals and
the use of gnomons, as well as the philosophical doctrine of the
transmigration of souls,--all of these having long been attributed to the
Greeks,--are shown in these works to be native to India. Although this
discussion does not bear directly upon the {14} origin of our numerals, yet
it is highly pertinent as showing the aptitude of the Hindu for
mathematical and mental work, a fact further attested by the independent
development of the drama and of epic and lyric poetry.
It should be stated definitely at the outset, however, that we are not at
all sure that the most ancient forms of the numerals commonly known as
Arabic had their origin in India. As will presently be seen, their forms
may have been suggested by those used in Egypt, or in Eastern Persia, or in
China, or on the plains of Mesopotamia. We are quite in the dark as to
these early steps; but as to their development in India, the approximate
period of the rise of their essential feature of place value, their
introduction into the Arab civilization, and their spread to the West, we
have more or less definite information. When, therefore, we consider the
rise of the numerals in the land of the Sindhu,[51] it must be understood
that it is only the large movement that is meant, and that there must
further be considered the numerous possible sources outside of India itself
and long anterior to the first prominent appearance of the number symbols.
No one attempts to examine any detail in the history of ancient India
without being struck with the great dearth of reliable material.[52] So
little sympathy have the people with any save those of their own caste that
a general literature is wholly lacking, and it is only in the observations
of strangers that any all-round view of scientific progress is to be found.
There is evidence that primary schools {15} existed in earliest times, and
of the seventy-two recognized sciences writing and arithmetic were the most
prized.[53] In the Vedic period, say from 2000 to 1400 B.C., there was the
same attention to astronomy that was found in the earlier civilizations of
Babylon, China, and Egypt, a fact attested by the Vedas themselves.[54]
Such advance in science presupposes a fair knowledge of calculation, but of
the manner of calculating we are quite ignorant and probably always shall
be. One of the Buddhist sacred books, the _Lalitavistara_, relates that
when the B[=o]dhisattva[55] was of age to marry, the father of Gopa, his
intended bride, demanded an examination of the five hundred suitors, the
subjects including arithmetic, writing, the lute, and archery. Having
vanquished his rivals in all else, he is matched against Arjuna the great
arithmetician and is asked to express numbers greater than 100 kotis.[56]
In reply he gave a scheme of number names as high as 10^{53}, adding that
he could proceed as far as 10^{421},[57] all of which suggests the system
of Archimedes and the unsettled question of the indebtedness of the West to
the East in the realm of ancient mathematics.[58] Sir Edwin Arnold, {16} in
_The Light of Asia_, does not mention this part of the contest, but he
speaks of Buddha's training at the hands of the learned Vi[s.]vamitra:
"And Viswamitra said, 'It is enough,
Let us to numbers. After me repeat
Your numeration till we reach the lakh,[59]
One, two, three, four, to ten, and then by tens
To hundreds, thousands.' After him the child
Named digits, decads, centuries, nor paused,
The round lakh reached, but softly murmured on,
Then comes the k[=o]ti, nahut, ninnahut,
Khamba, viskhamba, abab, attata,
To kumuds, gundhikas, and utpalas,
By pundar[=i]kas into padumas,
Which last is how you count the utmost grains
Of Hastagiri ground to finest dust;[60]
But beyond that a numeration is,
The K[=a]tha, used to count the stars of night,
The K[=o]ti-K[=a]tha, for the ocean drops;
Ingga, the calculus of circulars;
Sarvanikchepa, by the which you deal
With all the sands of Gunga, till we come
To Antah-Kalpas, where the unit is
The sands of the ten crore Gungas. If one seeks
More comprehensive scale, th' arithmic mounts
By the Asankya, which is the tale
Of all the drops that in ten thousand years
Would fall on all the worlds by daily rain;
Thence unto Maha Kalpas, by the which
The gods compute their future and their past.'"
{17}
Thereupon Vi[s.]vamitra [=A]c[=a]rya[61] expresses his approval of the
task, and asks to hear the "measure of the line" as far as y[=o]jana, the
longest measure bearing name. This given, Buddha adds:
... "'And master! if it please,
I shall recite how many sun-motes lie
From end to end within a y[=o]jana.'
Thereat, with instant skill, the little prince
Pronounced the total of the atoms true.
But Viswamitra heard it on his face
Prostrate before the boy; 'For thou,' he cried,
'Art Teacher of thy teachers--thou, not I,
Art G[=u]r[=u].'"
It is needless to say that this is far from being history. And yet it puts
in charming rhythm only what the ancient _Lalitavistara_ relates of the
number-series of the Buddha's time. While it extends beyond all reason,
nevertheless it reveals a condition that would have been impossible unless
arithmetic had attained a considerable degree of advancement.
To this pre-Christian period belong also the _Ved[=a][.n]gas_, or "limbs
for supporting the Veda," part of that great branch of Hindu literature
known as _Sm[r.]iti_ (recollection), that which was to be handed down by
tradition. Of these the sixth is known as _Jyoti[s.]a_ (astronomy), a short
treatise of only thirty-six verses, written not earlier than 300 B.C., and
affording us some knowledge of the extent of number work in that
period.[62] The Hindus {18} also speak of eighteen ancient Siddh[=a]ntas or
astronomical works, which, though mostly lost, confirm this evidence.[63]
As to authentic histories, however, there exist in India none relating to
the period before the Mohammedan era (622 A.D.). About all that we know of
the earlier civilization is what we glean from the two great epics, the
Mah[=a]bh[=a]rata[64] and the R[=a]m[=a]yana, from coins, and from a few
inscriptions.[65]
It is with this unsatisfactory material, then, that we have to deal in
searching for the early history of the Hindu-Arabic numerals, and the fact
that many unsolved problems exist and will continue to exist is no longer
strange when we consider the conditions. It is rather surprising that so
much has been discovered within a century, than that we are so uncertain as
to origins and dates and the early spread of the system. The probability
being that writing was not introduced into India before the close of the
fourth century B.C., and literature existing only in spoken form prior to
that period,[66] the number work was doubtless that of all primitive
peoples, palpable, merely a matter of placing sticks or cowries or pebbles
on the ground, of marking a sand-covered board, or of cutting notches or
tying cords as is still done in parts of Southern India to-day.[67]
{19}
The early Hindu numerals[68] may be classified into three great groups, (1)
the Kharo[s.][t.]h[=i], (2) the Br[=a]hm[=i], and (3) the word and letter
forms; and these will be considered in order.
The Kharo[s.][t.]h[=i] numerals are found in inscriptions formerly known as
Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandh[=a]ra,
now eastern Afghanistan and northern Punjab. The alphabet of the language
is found in inscriptions dating from the fourth century B.C. to the third
century A.D., and from the fact that the words are written from right to
left it is assumed to be of Semitic origin. No numerals, however, have been
found in the earliest of these inscriptions, number-names probably having
been written out in words as was the custom with many ancient peoples. Not
until the time of the powerful King A['s]oka, in the third century B.C., do
numerals appear in any inscriptions thus far discovered; and then only in
the primitive form of marks, quite as they would be found in Egypt, Greece,
Rome, or in {20} various other parts of the world. These A['s]oka[69]
inscriptions, some thirty in all, are found in widely separated parts of
India, often on columns, and are in the various vernaculars that were
familiar to the people. Two are in the Kharo[s.][t.]h[=i] characters, and
the rest in some form of Br[=a]hm[=i]. In the Kharo[s.][t.]h[=i]
inscriptions only four numerals have been found, and these are merely
vertical marks for one, two, four, and five, thus:
| || ||| ||||
In the so-called ['S]aka inscriptions, possibly of the first century B.C.,
more numerals are found, and in more highly developed form, the
right-to-left system appearing, together with evidences of three different
scales of counting,--four, ten, and twenty. The numerals of this period are
as follows:
[Illustration]
There are several noteworthy points to be observed in studying this system.
In the first place, it is probably not as early as that shown in the
N[=a]n[=a] Gh[=a]t forms hereafter given, although the inscriptions
themselves at N[=a]n[=a] Gh[=a]t are later than those of the A['s]oka
period. The {21} four is to this system what the X was to the Roman,
probably a canceling of three marks as a workman does to-day for five, or a
laying of one stick across three others. The ten has never been
satisfactorily explained. It is similar to the A of the Kharo[s.][t.]h[=i]
alphabet, but we have no knowledge as to why it was chosen. The twenty is
evidently a ligature of two tens, and this in turn suggested a kind of
radix, so that ninety was probably written in a way reminding one of the
quatre-vingt-dix of the French. The hundred is unexplained, although it
resembles the letter _ta_ or _tra_ of the Br[=a]hm[=i] alphabet with 1
before (to the right of) it. The two hundred is only a variant of the
symbol for hundred, with two vertical marks.[70]
This system has many points of similarity with the Nabatean numerals[71] in
use in the first centuries of the Christian era. The cross is here used for
four, and the Kharo[s.][t.]h[=i] form is employed for twenty. In addition
to this there is a trace of an analogous use of a scale of twenty. While
the symbol for 100 is quite different, the method of forming the other
hundreds is the same. The correspondence seems to be too marked to be
wholly accidental.
It is not in the Kharo[s.][t.]h[=i] numerals, therefore, that we can hope
to find the origin of those used by us, and we turn to the second of the
Indian types, the Br[=a]hm[=i] characters. The alphabet attributed to
Brahm[=a] is the oldest of the several known in India, and was used from
the earliest historic times. There are various theories of its origin, {22}
none of which has as yet any wide acceptance,[72] although the problem
offers hope of solution in due time. The numerals are not as old as the
alphabet, or at least they have not as yet been found in inscriptions
earlier than those in which the edicts of A['s]oka appear, some of these
having been incised in Br[=a]hm[=i] as well as Kharo[s.][t.]h[=i]. As
already stated, the older writers probably wrote the numbers in words, as
seems to have been the case in the earliest Pali writings of Ceylon.[73]
The following numerals are, as far as known, the only ones to appear in the
A['s]oka edicts:[74]
[Illustration]
These fragments from the third century B.C., crude and unsatisfactory as
they are, are the undoubted early forms from which our present system
developed. They next appear in the second century B.C. in some inscriptions
in the cave on the top of the N[=a]n[=a] Gh[=a]t hill, about seventy-five
miles from Poona in central India. These inscriptions may be memorials of
the early Andhra dynasty of southern India, but their chief interest lies
in the numerals which they contain.
The cave was made as a resting-place for travelers ascending the hill,
which lies on the road from Kaly[=a]na to Junar. It seems to have been cut
out by a descendant {23} of King ['S][=a]tav[=a]hana,[75] for inside the
wall opposite the entrance are representations of the members of his
family, much defaced, but with the names still legible. It would seem that
the excavation was made by order of a king named Vedisiri, and "the
inscription contains a list of gifts made on the occasion of the
performance of several _yagnas_ or religious sacrifices," and numerals are
to be seen in no less than thirty places.[76]
There is considerable dispute as to what numerals are really found in these
inscriptions, owing to the difficulty of deciphering them; but the
following, which have been copied from a rubbing, are probably number
forms:[77]
[Illustration]
The inscription itself, so important as containing the earliest
considerable Hindu numeral system connected with our own, is of sufficient
interest to warrant reproducing part of it in facsimile, as is done on page
24.
{24}
[Illustration]
The next very noteworthy evidence of the numerals, and this quite complete
as will be seen, is found in certain other cave inscriptions dating back to
the first or second century A.D. In these, the Nasik[78] cave inscriptions,
the forms are as follows:
[Illustration]
From this time on, until the decimal system finally adopted the first nine
characters and replaced the rest of the Br[=a]hm[=i] notation by adding the
zero, the progress of these forms is well marked. It is therefore well to
present synoptically the best-known specimens that have come down to us,
and this is done in the table on page 25.[79]
{25}
TABLE SHOWING THE PROGRESS OF NUMBER FORMS IN INDIA
NUMERALS 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000
A['s]oka[80] [Illustration]
['S]aka[81] [Illustration]
A['s]oka[82] [Illustration]
N[=a]gar[=i][83] [Illustration]
Nasik[84] [Illustration]
K[s.]atrapa[85] [Illustration]
Ku[s.]ana [86] [Illustration]
Gupta[87] [Illustration]
Valhab[=i][88] [Illustration]
Nepal [89] [Illustration]
Kali[.n]ga[90] [Illustration]
V[=a]k[=a][t.]aka[91] [Illustration]
[Most of these numerals are given by Buehler, loc. cit., Tafel IX.]
{26} With respect to these numerals it should first be noted that no zero
appears in the table, and as a matter of fact none existed in any of the
cases cited. It was therefore impossible to have any place value, and the
numbers like twenty, thirty, and other multiples of ten, one hundred, and
so on, required separate symbols except where they were written out in
words. The ancient Hindus had no less than twenty of these symbols,[92] a
number that was afterward greatly increased. The following are examples of
their method of indicating certain numbers between one hundred and one
thousand:
[93] [Numerals] for 174
[94] [Numerals] for 191
[95] [Numerals] for 269
[96] [Numerals] for 252
[97] [Numerals] for 400
[98] [Numerals] for 356
{27}
To these may be added the following numerals below one hundred, similar to
those in the table:
[Numerals][99] for 90
[Numerals][100] for 70
We have thus far spoken of the Kharo[s.][t.]h[=i] and Br[=a]hm[=i]
numerals, and it remains to mention the third type, the word and letter
forms. These are, however, so closely connected with the perfecting of the
system by the invention of the zero that they are more appropriately
considered in the next chapter, particularly as they have little relation
to the problem of the origin of the forms known as the Arabic.
Having now examined types of the early forms it is appropriate to turn our
attention to the question of their origin. As to the first three there is
no question. The [1 vertical stroke] or [1 horizontal stroke] is simply one
stroke, or one stick laid down by the computer. The [2 vertical strokes] or
[2 horizontal strokes] represents two strokes or two sticks, and so for the
[3 vertical strokes] and [3 horizontal strokes]. From some primitive [2
vertical strokes] came the two of Egypt, of Rome, of early Greece, and of
various other civilizations. It appears in the three Egyptian numeral
systems in the following forms:
Hieroglyphic [2 vertical strokes]
Hieratic [Hieratic 2]
Demotic [Demotic 2]
The last of these is merely a cursive form as in the Arabic [Arabic 2],
which becomes our 2 if tipped through a right angle. From some primitive [2
horizontal strokes] came the Chinese {28} symbol, which is practically
identical with the symbols found commonly in India from 150 B.C. to 700
A.D. In the cursive form it becomes [2 horizontal strokes joined], and this
was frequently used for two in Germany until the 18th century. It finally
went into the modern form 2, and the [3 horizontal strokes] in the same way
became our 3.
There is, however, considerable ground for interesting speculation with
respect to these first three numerals. The earliest Hindu forms were
perpendicular. In the N[=a]n[=a] Gh[=a]t inscriptions they are vertical.
But long before either the A['s]oka or the N[=a]n[=a] Gh[=a]t inscriptions
the Chinese were using the horizontal forms for the first three numerals,
but a vertical arrangement for four.[101] Now where did China get these
forms? Surely not from India, for she had them, as her monuments and
literature[102] show, long before the Hindus knew them. The tradition is
that China brought her civilization around the north of Tibet, from
Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of
Turkestan. Now what numerals did Mesopotamia use? The Babylonian system,
simple in its general principles but very complicated in many of its
details, is now well known.[103] In particular, one, two, and three were
represented by vertical arrow-heads. Why, then, did the Chinese write {29}
theirs horizontally? The problem now takes a new interest when we find that
these Babylonian forms were not the primitive ones of this region, but that
the early Sumerian forms were horizontal.[104]
What interpretation shall be given to these facts? Shall we say that it was
mere accident that one people wrote "one" vertically and that another wrote
it horizontally? This may be the case; but it may also be the case that the
tribal migrations that ended in the Mongol invasion of China started from
the Euphrates while yet the Sumerian civilization was prominent, or from
some common source in Turkestan, and that they carried to the East the
primitive numerals of their ancient home, the first three, these being all
that the people as a whole knew or needed. It is equally possible that
these three horizontal forms represent primitive stick-laying, the most
natural position of a stick placed in front of a calculator being the
horizontal one. When, however, the cuneiform writing developed more fully,
the vertical form may have been proved the easier to make, so that by the
time the migrations to the West began these were in use, and from them came
the upright forms of Egypt, Greece, Rome, and other Mediterranean lands,
and those of A['s]oka's time in India. After A['s]oka, and perhaps among
the merchants of earlier centuries, the horizontal forms may have come down
into India from China, thus giving those of the N[=a]n[=a] Gh[=a]t cave and
of later inscriptions. This is in the realm of speculation, but it is not
improbable that further epigraphical studies may confirm the hypothesis.
{30}
As to the numerals above three there have been very many conjectures. The
figure one of the Demotic looks like the one of the Sanskrit, the two
(reversed) like that of the Arabic, the four has some resemblance to that
in the Nasik caves, the five (reversed) to that on the K[s.]atrapa coins,
the nine to that of the Ku[s.]ana inscriptions, and other points of
similarity have been imagined. Some have traced resemblance between the
Hieratic five and seven and those of the Indian inscriptions. There have
not, therefore, been wanting those who asserted an Egyptian origin for
these numerals.[105] There has already been mentioned the fact that the
Kharo[s.][t.]h[=i] numerals were formerly known as Bactrian, Indo-Bactrian,
and Aryan. Cunningham[106] was the first to suggest that these numerals
were derived from the alphabet of the Bactrian civilization of Eastern
Persia, perhaps a thousand years before our era, and in this he was
supported by the scholarly work of Sir E. Clive Bayley,[107] who in turn
was followed by Canon Taylor.[108] The resemblance has not proved
convincing, however, and Bayley's drawings {31} have been criticized as
being affected by his theory. The following is part of the hypothesis:[109]
_Numeral_ _Hindu_ _Bactrian_ _Sanskrit_
4 [Symbol] [Symbol] = ch chatur, Lat. quattuor
5 [Symbol] [Symbol] = p pancha, Gk. [Greek:p/ente]
6 [Symbol] [Symbol] = s [s.]a[s.]
7 [Symbol] [Symbol] = [s.] sapta
( the s and [s.] are interchanged as occasionally in N. W. India)
Buehler[110] rejects this hypothesis, stating that in four cases (four, six,
seven, and ten) the facts are absolutely against it.
While the relation to ancient Bactrian forms has been generally doubted, it
is agreed that most of the numerals resemble Br[=a]hm[=i] letters, and we
would naturally expect them to be initials.[111] But, knowing the ancient
pronunciation of most of the number names,[112] we find this not to be the
case. We next fall back upon the hypothesis {32} that they represent the
order of letters[113] in the ancient alphabet. From what we know of this
order, however, there seems also no basis for this assumption. We have,
therefore, to confess that we are not certain that the numerals were
alphabetic at all, and if they were alphabetic we have no evidence at
present as to the basis of selection. The later forms may possibly have
been alphabetical expressions of certain syllables called _ak[s.]aras_,
which possessed in Sanskrit fixed numerical values,[114] but this is
equally uncertain with the rest. Bayley also thought[115] that some of the
forms were Phoenician, as notably the use of a circle for twenty, but the
resemblance is in general too remote to be convincing.
There is also some slight possibility that Chinese influence is to be seen
in certain of the early forms of Hindu numerals.[116]
{33}
More absurd is the hypothesis of a Greek origin, supposedly supported by
derivation of the current symbols from the first nine letters of the Greek
alphabet.[117] This difficult feat is accomplished by twisting some of the
letters, cutting off, adding on, and effecting other changes to make the
letters fit the theory. This peculiar theory was first set up by
Dasypodius[118] (Conrad Rauhfuss), and was later elaborated by Huet.[119]
{34}
A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is
given by Kircher in his work[120] on number mysticism. He quotes from
Abenragel,[121] giving the Arabic and a Latin translation[122] and stating
that the ordinary Arabic forms are derived from sectors of a circle,
[circle].
Out of all these conflicting theories, and from all the resemblances seen
or imagined between the numerals of the West and those of the East, what
conclusions are we prepared to draw as the evidence now stands? Probably
none that is satisfactory. Indeed, upon the evidence at {35} hand we might
properly feel that everything points to the numerals as being substantially
indigenous to India. And why should this not be the case? If the king
Srong-tsan-Gampo (639 A.D.), the founder of Lh[=a]sa,[123] could have set
about to devise a new alphabet for Tibet, and if the Siamese, and the
Singhalese, and the Burmese, and other peoples in the East, could have
created alphabets of their own, why should not the numerals also have been
fashioned by some temple school, or some king, or some merchant guild? By
way of illustration, there are shown in the table on page 36 certain
systems of the East, and while a few resemblances are evident, it is also
evident that the creators of each system endeavored to find original forms
that should not be found in other systems. This, then, would seem to be a
fair interpretation of the evidence. A human mind cannot readily create
simple forms that are absolutely new; what it fashions will naturally
resemble what other minds have fashioned, or what it has known through
hearsay or through sight. A circle is one of the world's common stock of
figures, and that it should mean twenty in Phoenicia and in India is hardly
more surprising than that it signified ten at one time in Babylon.[124] It
is therefore quite probable that an extraneous origin cannot be found for
the very sufficient reason that none exists.
Of absolute nonsense about the origin of the symbols which we use much has
been written. Conjectures, {36} however, without any historical evidence
for support, have no place in a serious discussion of the gradual evolution
of the present numeral forms.[125]
TABLE OF CERTAIN EASTERN SYSTEMS
Siam [Illustration: numerals]
Burma[126] [Illustration: numerals]
Malabar[127] [Illustration: numerals]
Tibet[128] [Illustration: numerals]
Ceylon[129] [Illustration: numerals]
Malayalam[129] [Illustration: numerals]
{37}
We may summarize this chapter by saying that no one knows what suggested
certain of the early numeral forms used in India. The origin of some is
evident, but the origin of others will probably never be known. There is no
reason why they should not have been invented by some priest or teacher or
guild, by the order of some king, or as part of the mysticism of some
temple. Whatever the origin, they were no better than scores of other
ancient systems and no better than the present Chinese system when written
without the zero, and there would never have been any chance of their
triumphal progress westward had it not been for this relatively late
symbol. There could hardly be demanded a stronger proof of the Hindu origin
of the character for zero than this, and to it further reference will be
made in Chapter IV.
* * * * *
{38}