LATER HINDU FORMS, WITH A PLACE VALUE
Before speaking of the perfected Hindu numerals with the zero and the place
value, it is necessary to consider the third system mentioned on page
19,--the word and letter forms. The use of words with place value began at
least as early as the 6th century of the Christian era. In many of the
manuals of astronomy and mathematics, and often in other works in
mentioning dates, numbers are represented by the names of certain objects
or ideas. For example, zero is represented by "the void" (_['s][=u]nya_),
or "heaven-space" (_ambara [=a]k[=a]['s]a_); one by "stick" (_rupa_),
"moon" (_indu ['s]a['s]in_), "earth" (_bh[=u]_), "beginning" (_[=a]di_),
"Brahma," or, in general, by anything markedly unique; two by "the twins"
(_yama_), "hands" (_kara_), "eyes" (_nayana_), etc.; four by "oceans," five
by "senses" (_vi[s.]aya_) or "arrows" (the five arrows of K[=a]mad[=e]va);
six by "seasons" or "flavors"; seven by "mountain" (_aga_), and so on.[130]
These names, accommodating themselves to the verse in which scientific
works were written, had the additional advantage of not admitting, as did
the figures, easy alteration, since any change would tend to disturb the
meter.
{39}
As an example of this system, the date "['S]aka Sa[m.]vat, 867" (A.D. 945
or 946), is given by "_giri-ra[s.]a-vasu_," meaning "the mountains"
(seven), "the flavors" (six), and the gods "_Vasu_" of which there were
eight. In reading the date these are read from right to left.[131] The
period of invention of this system is uncertain. The first trace seems to
be in the _['S]rautas[=u]tra_ of K[=a]ty[=a]yana and
L[=a][t.]y[=a]yana.[132] It was certainly known to Var[=a]ha-Mihira (d.
587),[133] for he used it in the _B[r.]hat-Sa[m.]hit[=a]._[134] It has also
been asserted[135] that [=A]ryabha[t.]a (c. 500 A.D.) was familiar with
this system, but there is nothing to prove the statement.[136] The earliest
epigraphical examples of the system are found in the Bayang (Cambodia)
inscriptions of 604 and 624 A.D.[137]
Mention should also be made, in this connection, of a curious system of
alphabetic numerals that sprang up in southern India. In this we have the
numerals represented by the letters as given in the following table:
1 2 3 4 5 6 7 8 9 0
k kh g gh [.n] c ch j jh n
[t.] [t.]h [d.] [d.]h [n.] t th d th n
p ph b bh m
y r l v ['s] [s.] s h l
{40}
By this plan a numeral might be represented by any one of several letters,
as shown in the preceding table, and thus it could the more easily be
formed into a word for mnemonic purposes. For example, the word
2 3 1 5 6 5 1
_kha_ _gont_ _yan_ _me_ _[s.]a_ _m[=a]_ _pa_
has the value 1,565,132, reading from right to left.[138] This, the oldest
specimen (1184 A.D.) known of this notation, is given in a commentary on
the Rigveda, representing the number of days that had elapsed from the
beginning of the Kaliyuga. Burnell[139] states that this system is even yet
in use for remembering rules to calculate horoscopes, and for astronomical
tables.
A second system of this kind is still used in the pagination of manuscripts
in Ceylon, Siam, and Burma, having also had its rise in southern India. In
this the thirty-four consonants when followed by _a_ (as _ka_ ... _la_)
designate the numbers 1-34; by _[=a]_ (as _k[=a]_ ... _l[=a]_), those from
35 to 68; by _i_ (_ki_ ... _li_), those from 69 to 102, inclusive; and so
on.[140]
As already stated, however, the Hindu system as thus far described was no
improvement upon many others of the ancients, such as those used by the
Greeks and the Hebrews. Having no zero, it was impracticable to designate
the tens, hundreds, and other units of higher order by the same symbols
used for the units from one to nine. In other words, there was no
possibility of place value without some further improvement. So the
N[=a]n[=a] Gh[=a]t {41} symbols required the writing of "thousand seven
twenty-four" about like T 7, tw, 4 in modern symbols, instead of 7024, in
which the seven of the thousands, the two of the tens (concealed in the
word twenty, being originally "twain of tens," the _-ty_ signifying ten),
and the four of the units are given as spoken and the order of the unit
(tens, hundreds, etc.) is given by the place. To complete the system only
the zero was needed; but it was probably eight centuries after the
N[=a]n[=a] Gh[=a]t inscriptions were cut, before this important symbol
appeared; and not until a considerably later period did it become well
known. Who it was to whom the invention is due, or where he lived, or even
in what century, will probably always remain a mystery.[141] It is possible
that one of the forms of ancient abacus suggested to some Hindu astronomer
or mathematician the use of a symbol to stand for the vacant line when the
counters were removed. It is well established that in different parts of
India the names of the higher powers took different forms, even the order
being interchanged.[142] Nevertheless, as the significance of the name of
the unit was given by the order in reading, these variations did not lead
to error. Indeed the variation itself may have necessitated the
introduction of a word to signify a vacant place or lacking unit, with the
ultimate introduction of a zero symbol for this word.
To enable us to appreciate the force of this argument a large number,
8,443,682,155, may be considered as the Hindus wrote and read it, and then,
by way of contrast, as the Greeks and Arabs would have read it.
{42}
_Modern American reading_, 8 billion, 443 million, 682 thousand, 155.
_Hindu_, 8 padmas, 4 vyarbudas, 4 k[=o][t.]is, 3 prayutas, 6 lak[s.]as, 8
ayutas, 2 sahasra, 1 ['s]ata, 5 da['s]an, 5.
_Arabic and early German_, eight thousand thousand thousand and four
hundred thousand thousand and forty-three thousand thousand, and six
hundred thousand and eighty-two thousand and one hundred fifty-five (or
five and fifty).
_Greek_, eighty-four myriads of myriads and four thousand three hundred
sixty-eight myriads and two thousand and one hundred fifty-five.
As Woepcke[143] pointed out, the reading of numbers of this kind shows that
the notation adopted by the Hindus tended to bring out the place idea. No
other language than the Sanskrit has made such consistent application, in
numeration, of the decimal system of numbers. The introduction of myriads
as in the Greek, and thousands as in Arabic and in modern numeration, is
really a step away from a decimal scheme. So in the numbers below one
hundred, in English, eleven and twelve are out of harmony with the rest of
the -teens, while the naming of all the numbers between ten and twenty is
not analogous to the naming of the numbers above twenty. To conform to our
written system we should have ten-one, ten-two, ten-three, and so on, as we
have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the
units, however, preceding the tens and hundreds. Nor did any other ancient
people carry the numeration as far as did the Hindus.[144]
{43}
When the _a[.n]kapalli_,[145] the decimal-place system of writing numbers,
was perfected, the tenth symbol was called the _['s][=u]nyabindu_,
generally shortened to _['s][=u]nya_ (the void). Brockhaus[146] has well
said that if there was any invention for which the Hindus, by all their
philosophy and religion, were well fitted, it was the invention of a symbol
for zero. This making of nothingness the crux of a tremendous achievement
was a step in complete harmony with the genius of the Hindu.
It is generally thought that this _['s][=u]nya_ as a symbol was not used
before about 500 A.D., although some writers have placed it earlier.[147]
Since [=A]ryabha[t.]a gives our common method of extracting roots, it would
seem that he may have known a decimal notation,[148] although he did not
use the characters from which our numerals are derived.[149] Moreover, he
frequently speaks of the {44} void.[150] If he refers to a symbol this
would put the zero as far back as 500 A.D., but of course he may have
referred merely to the concept of nothingness.
A little later, but also in the sixth century, Var[=a]ha-Mihira[151] wrote
a work entitled _B[r.]hat Sa[m.]hit[=a]_[152] in which he frequently uses
_['s][=u]nya_ in speaking of numerals, so that it has been thought that he
was referring to a definite symbol. This, of course, would add to the
probability that [=A]ryabha[t.]a was doing the same.
It should also be mentioned as a matter of interest, and somewhat related
to the question at issue, that Var[=a]ha-Mihira used the word-system with
place value[153] as explained above.
The first kind of alphabetic numerals and also the word-system (in both of
which the place value is used) are plays upon, or variations of, position
arithmetic, which would be most likely to occur in the country of its
origin.[154]
At the opening of the next century (c. 620 A.D.) B[=a][n.]a[155] wrote of
Subandhus's _V[=a]savadatt[=a]_ as a celebrated work, {45} and mentioned
that the stars dotting the sky are here compared with zeros, these being
points as in the modern Arabic system. On the other hand, a strong argument
against any Hindu knowledge of the symbol zero at this time is the fact
that about 700 A.D. the Arabs overran the province of Sind and thus had an
opportunity of knowing the common methods used there for writing numbers.
And yet, when they received the complete system in 776 they looked upon it
as something new.[156] Such evidence is not conclusive, but it tends to
show that the complete system was probably not in common use in India at
the beginning of the eighth century. On the other hand, we must bear in
mind the fact that a traveler in Germany in the year 1700 would probably
have heard or seen nothing of decimal fractions, although these were
perfected a century before that date. The elite of the mathematicians may
have known the zero even in [=A]ryabha[t.]a's time, while the merchants and
the common people may not have grasped the significance of the novelty
until a long time after. On the whole, the evidence seems to point to the
west coast of India as the region where the complete system was first
seen.[157] As mentioned above, traces of the numeral words with place
value, which do not, however, absolutely require a decimal place-system of
symbols, are found very early in Cambodia, as well as in India.
Concerning the earliest epigraphical instances of the use of the nine
symbols, plus the zero, with place value, there {46} is some question.
Colebrooke[158] in 1807 warned against the possibility of forgery in many
of the ancient copper-plate land grants. On this account Fleet, in the
_Indian Antiquary_,[159] discusses at length this phase of the work of the
epigraphists in India, holding that many of these forgeries were made about
the end of the eleventh century. Colebrooke[160] takes a more rational view
of these forgeries than does Kaye, who seems to hold that they tend to
invalidate the whole Indian hypothesis. "But even where that may be
suspected, the historical uses of a monument fabricated so much nearer to
the times to which it assumes to belong, will not be entirely superseded.
The necessity of rendering the forged grant credible would compel a
fabricator to adhere to history, and conform to established notions: and
the tradition, which prevailed in his time, and by which he must be guided,
would probably be so much nearer to the truth, as it was less remote from
the period which it concerned."[161] Buehler[162] gives the copper-plate
Gurjara inscription of Cedi-sa[m.]vat 346 (595 A.D.) as the oldest
epigraphical use of the numerals[163] "in which the symbols correspond to
the alphabet numerals of the period and the place." Vincent A. Smith[164]
quotes a stone inscription of 815 A.D., dated Sa[m.]vat 872. So F. Kielhorn
in the _Epigraphia Indica_[165] gives a Pathari pillar inscription of
Parabala, dated Vikrama-sa[m.]vat 917, which corresponds to 861 A.D., {47}
and refers also to another copper-plate inscription dated Vikrama-sa[m.]vat
813 (756 A.D.). The inscription quoted by V. A. Smith above is that given
by D. R. Bhandarkar,[166] and another is given by the same writer as of
date Saka-sa[m.]vat 715 (798 A.D.), being incised on a pilaster.
Kielhorn[167] also gives two copper-plate inscriptions of the time of
Mahendrapala of Kanauj, Valhab[=i]-sa[m.]vat 574 (893 A.D.) and
Vikrama-sa[m.]vat 956 (899 A.D.). That there should be any inscriptions of
date as early even as 750 A.D., would tend to show that the system was at
least a century older. As will be shown in the further development, it was
more than two centuries after the introduction of the numerals into Europe
that they appeared there upon coins and inscriptions. While Thibaut[168]
does not consider it necessary to quote any specific instances of the use
of the numerals, he states that traces are found from 590 A.D. on. "That
the system now in use by all civilized nations is of Hindu origin cannot be
doubted; no other nation has any claim upon its discovery, especially since
the references to the origin of the system which are found in the nations
of western Asia point unanimously towards India."[169]
The testimony and opinions of men like Buehler, Kielhorn, V. A. Smith,
Bhandarkar, and Thibaut are entitled to the most serious consideration. As
authorities on ancient Indian epigraphy no others rank higher. Their work
is accepted by Indian scholars the world over, and their united judgment as
to the rise of the system with a place value--that it took place in India
as early as the {48} sixth century A.D.--must stand unless new evidence of
great weight can be submitted to the contrary.
Many early writers remarked upon the diversity of Indian numeral forms.
Al-B[=i]r[=u]n[=i] was probably the first; noteworthy is also Johannes
Hispalensis,[170] who gives the variant forms for seven and four. We insert
on p. 49 a table of numerals used with place value. While the chief
authority for this is Buehler,[171] several specimens are given which are
not found in his work and which are of unusual interest.
The ['S][=a]rad[=a] forms given in the table use the circle as a symbol for
1 and the dot for zero. They are taken from the paging and text of _The
Kashmirian Atharva-Veda_[172], of which the manuscript used is certainly
four hundred years old. Similar forms are found in a manuscript belonging
to the University of Tuebingen. Two other series presented are from Tibetan
books in the library of one of the authors.
For purposes of comparison the modern Sanskrit and Arabic numeral forms are
added.
Sanskrit, [Illustration]
Arabic, [Illustration]
{49}
NUMERALS USED WITH PLACE VALUE
1 2 3 4 5 6 7 8 9 0
a[173] [Illustration]
b[174] [Illustration]
c[175] [Illustration]
d[176] [Illustration]
e[177] [Illustration]
f[178] [Illustration]
g[179] [Illustration]
h[180] [Illustration]
i[180] [Illustration]
j[181] [Illustration]
k[181] [Illustration]
l[182] [Illustration]
m[183] [Illustration]
n[184] [Illustration]
* * * * *
{51}