9,-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 man

wo by "the twins" (yama), "hands" (kara), "eyes" (nayana), etc.; four by "oceans," five by "senses" (vi?aya) or "arrows" (the five arrows of Kāmadēva); six by "seasons" or "flavors"; seven by "mountain" (aga), and so on.[130

to left.[131] The period of invention of this system is uncertain. The first trace seems to be in the ?rautasūtra of Kātyāyana and Lā?yāyana.[132] It was certainly known to Varāha-Mihira (d. 587),[133] for he used it in the B?hat-Sa?hitā.[134] It has a

alphabetic numerals that sprang up in southern India. In this we have

4 5 6

gh ? c

?h ? t

h b

v ?

ters, as shown in the preceding table, and thus it could the more eas

1 5

yan me

ven in a commentary on the Rigveda, representing the number of days that had elapsed from the beginning of the Kaliyuga. Burn

ad its rise in southern India. In this the thirty-four consonants when followed by a (as ka ... la) designate the num

(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 Nānā Ghāt 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

8,443,682,155, may be considered as the Hindus wrote and read it, an

g, 8 billion, 443 mill

ō?is, 3 prayutas, 6 lak?as, 8 ayu

thousand thousand and forty-three thousand thousand, and six hundred thou

ur thousand three hundred sixty-eight myriad

nd 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

(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

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 fr

in which he frequently uses ?ūnya in speaking of numerals, so that it has been thought that he was referr

ewhat related to the question at issue, that Varāha-Mihira u

ich the place value is used) are plays upon, or variations of, position arith

nd 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 élite of the mathematici

s: 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] Bühler[162] gives the copper-plate Gurjara inscription of Cedi-sa?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?vat 872. So F. Kielhorn in the Epigraphia Indica[165] gives a Pathari pillar inscription of Parabala, dated Vikrama-sa?vat 917, which corresponds to 861 A.D., and refers also to another copper-plate inscription dated Vikrama-sa?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?vat 715 (798 A.D.), being incised on a pilaste

ncient 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 w

ispalensis,[170] who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While th

The Kashmirian Atharva-Veda[172], of which the manuscript used is certainly four hundred years old. Similar forms are found in a manus

the modern Sanskrit and Ara

skr

ab

sed with P

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