ng 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