href="@public@vhost@g@gutenberg@html@files@22599@[email protected]#Nt_57" id="NtA_57" class="pginternal" tag="{http://www.w3.org/1999/xhtml}a">^[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:

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:

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