CHAPTER II

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 Brāhmaṇas), and partly philosophical (the Upanishads). Our especial interest is

in the Sūtras, 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 Brāhmaṇas, and the Sūtras, between 1500 B.C. and 800 B.C., thus agreeing with Bürk[[47]] who holds that the knowledge of the Pythagorean theorem revealed in the Sūtras goes back to the eighth century B.C.

The importance of the Sūtras 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

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

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 Bōdhisattva[[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 1053, adding that he could proceed as far as 10421,[[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,

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 Viṣvamitra:

"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 kōti, nahut, ninnahut,

Khamba, viskhamba, abab, attata,

To kumuds, gundhikas, and utpalas,

By pundarīkas 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 Kātha, used to count the stars of night,

The Kōti-Kātha, 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.'"

Thereupon Viṣvamitra Ācārya[[61]] expresses his approval of the task, and asks to hear the "measure of the line" as far as yōjana, 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 yōjana.'

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 Gūrū.'"

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 Vedāṅgas, or "limbs for supporting the Veda," part of that great branch of Hindu literature known as Smṛiti (recollection), that which was to be handed down by tradition. Of these the sixth is known as Jyotiṣa (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

also speak of eighteen ancient Siddhāntas 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 Mahābhārata[[64]] and the Rāmāyana, 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]]

The early Hindu numerals[[68]] may be classified into three great groups, (1) the Kharoṣṭhī, (2) the Brāhmī, and (3) the word and letter forms; and these will be considered in order.

The Kharoṣṭhī numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandhāra, 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 Aśoka, 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

various other parts of the world. These Aśoka[[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 Kharoṣṭhī characters, and the rest in some form of Brāhmī. In the Kharoṣṭhī inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus:

In the so-called Śaka 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:

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 Nānā Ghāt forms hereafter given, although the inscriptions themselves at Nānā Ghāt are later than those of the Aśoka period. The

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 Kharoṣṭhī 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 Brāhmī 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 Kharoṣṭhī 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 Kharoṣṭhī 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 Brāhmī characters. The alphabet attributed to Brahmā is the oldest of the several known in India, and was used from the earliest historic times. There are various theories of its origin,

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 Aśoka appear, some of these having been incised in Brāhmī as well as Kharoṣṭhī. 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 Aśoka edicts:[[74]]

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 Nānā Ghāt 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 Kalyāna to Junar. It seems to have been cut out by a descendant

of King Śātavāhana,[[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]]

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.

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:

From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Brāhmī 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]]