The Hindu-Arabic Numerals Part 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 a.s.serted 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)

Buhler[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 p.r.o.nunciation 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 a.s.sumption. 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 ill.u.s.tration, 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 [Ill.u.s.tration: numerals]

Burma[126] [Ill.u.s.tration: numerals]

Malabar[127] [Ill.u.s.tration: numerals]

Tibet[128] [Ill.u.s.tration: numerals]

Ceylon[129] [Ill.u.s.tration: numerals]

Malayalam[129] [Ill.u.s.tration: 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}

CHAPTER III

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-s.p.a.ce" (_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 G.o.ds "_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 a.s.serted[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 ma.n.u.scripts 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 a.n.a.logous 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]