The testimony and opinions of men like Buehler, Kielhorn, V. A. Smith, Bhandarkar, and Thibaut are entitled to the most serious consideration. As authorities on ancient 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 with a place value--that it took place in India as early as the {48} sixth century A.D.--must stand unless new evidence of great weight can be submitted to the contrary. Many early writers remarked upon the diversity of Indian numeral forms. Al-B[=i]r[=u]n[=i] was probably the first; noteworthy is also Johannes Hispalensis,[170] who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While the chief authority for this is Buehler,[171] several specimens are given which are not found in his work and which are of unusual interest. The ['S][=a]rad[=a] forms given in the table use the circle as a symbol for 1 and the dot for zero. They are taken from the paging and text of _The Kashmirian Atharva-Veda_[172], of which the manuscript used is certainly four hundred years old. Similar forms are found in a manuscript belonging to the University of Tuebingen. Two other series presented are from Tibetan books in the library of one of the authors. For purposes of comparison the modern Sanskrit and Arabic numeral forms are added. Sanskrit, [Illustration] Arabic, [Illustration] {49} NUMERALS USED WITH PLACE VALUE 1 2 3 4 5 6 7 8 9 0 a[173] [Illustration] b[174] [Illustration] c[175] [Illustration] d[176] [Illustration] e[177] [Illustration] f[178] [Illustration] g[179] [Illustration] h[180] [Illustration] i[180] [Illustration] j[181] [Illustration] k[181] [Illustration] l[182] [Illustration] m[183] [Illustration] n[184] [Illustration] * * * * * {51} CHAPTER IV THE SYMBOL ZERO What has been said of the improved Hindu system with a place value does not touch directly the origin of a symbol for zero, although it assumes that such a symbol exists. The importance of such a sign, the fact that it is a prerequisite to a place-value system, and the further fact that without it the Hindu-Arabic numerals would never have dominated the computation system of the western world, make it proper to devote a chapter to its origin and history.