Elements of South-Indian Palæography, from the Fourth to the Seventeenth Century A.D. c6.14

Language: English

c6.14.1

It therefore appears that, neglecting all possibilities, in favour of which evidence does not exist, (such as the simultaneous existence of the more modern system of notation with the older in the fifth century A. D. or even earlier), the only possible conclusion is that, the great improvement of using numerals with a value according to position, and consequently the use of the cipher, first occurred in Central India about 500 A. D. Now though the inscriptions fail us as yet for this period, the acuteness of Woepcke detected some evidence in the works of the astronomers who lived in India during the fifth and following centuries. These are: Āryabhaṭa who himself tells us that he was born in 476 A. D.¹, Varāha Mihira who died in 587 A. D.², Brahmagupta who lived about 600, and Bhaṭṭotpala who lived about 1000 also of our era. All these writers composed their treatises in metre, and to suit the exigencies of the strict limits thus imposed on them, the three last were obliged to express the terms of their calculations by words, and these not the usual ones, but by symbolical words denoting natural objects, and in a conventional way, (as here used) also numerals. This peculiar system (which will be fully explained further on in this chapter) implies value by position, and also has words which express *indirectly* the cipher³. This same system is also used in the Sūryasiddhānta which is of very uncertain date in its actual form.

c6.14.2

It is thus perfectly clear that the Indians knew of numerals with a value according to position in the sixth century A. D., but the system of Āryabhaṭa which is totally different to the one described, *appears* to render improbable the assumption that he also about 500 A. D. found this system in common use, though he was acquainted with it⁴. He himself uses the successive vowels of the Sanskrit alphabet to express place, and thus his system agrees in principle with the Tamil notation; **a**, **ā** and **i** corresponding in value with the Tamil signs for ‘ten’, ‘hundred’ and ‘thousand.’ Woepcke, however, considers that Āryabhaṭa invented this notation to suit his style of composition in verse, and that the system of notation by words with value according to position was ‘probably anterior to Āryabhaṭa’⁵. If it had been then in common use, would not Āryabhaṭa have used it? Beyond the end of the fifth century there is, therefore, nothing to indicate the use of the cipher; for the high orders of numerals (equivalent to billions, trillions, etc.) first noticed by Profr. Weber⁶ do not necessarily imply anything of the kind. An illustrious French mathematician and a more eminent French philosopher have shown that this invention may have occurred (as was the case) in the middle ages of Europe spontaneously¹);

c6.14.3

it must also have occurred independently in India, but as the facts stand at present, it is impossible to connect India and Europe in the transmission of this particular invention from the first country to the other. As it is not proved to have been known in India before about 500 A. D., it is impossible to see how it can have been transmitted from thence to Europe before the rise of the Arabs, for direct communication ceased about the fourth century A. D.²), and in Europe, at all events, the little intellectual activity that was displayed ran in entirely different courses during the sixth, seventh, and eighth centuries. Nor is there the least trace of the use of such numerals during this period. The Arabic numerals now in use certainly came from India, but numerals with value according to position and the cipher were already in use in Europe (by the Neo-Pythagoreans) before they were adopted³). If the derivation of the numeral figures from the initial letters of the Sanskrit words denoting the respective numbers be given up, there is nothing left to show where the figures were first used, by the Pythagoreans in Europe or the astronomers in India. The assumption that the last was the case, but which (as I have already said) an examination of the earliest forms of the numerals preserved in inscriptions will prove to be impossible, is the foundation of the theory that Europe is indebted to India in this respect;

c6.14.4

in fact Woepcke chiefly relies on it⁴), but this assumption was made by J. Prinsep about 1838, or long before the Cave numerals were explained. The resemblance between the Neo-Pythagorean numerals in their cursive form on the one hand, and the India Cave numerals and the forms derived from them on the other, is too striking not to be noticed; but this fact does not warrant a presumption that one is borrowed from the other; more probably, both are from a common source. The question of what might have been the common original of the Neo-Pythagorean and Cave numerals is one for the decision of which evidence must yet be discovered, but a provisional hypothesis may be allowed.