tions with the zero cannot be carried on-not to say cannot be even thought of easily—without a symbol of some sort to represent it. Mahāvīrācārya gives, in the very first chapter of his Gaṇita-sāra-saṅgraha, the results of the operations of addition, subtraction, multiplication and division carried on in relation to the zero quantity; and although he is wrong in saying that a quantity, when divided by zero, remains unaltered, and should have said, like Bhāskarācārya, that the quotient in such a case is infinity, still the very mention of operations in relation to zero is enough to show that Mahāvīrācārya must have been aware of some symbolic representation of the zero quantity. Since Brahmagupta, who must have lived at least 150 years before Mahāvīrācārya, mentions in his work the results of operations in relation to the zero quantity, it is not unreasonable to suppose that before his time the zero must have had a symbol to represent it in written calculations. That even Āryabhaṭa knew such a symbol is not at all improbable. It is worthy of note in this connection that in enumerating the nominal numerals in the first chapter of his work, Mahāvīrācārya mentions the names denoting the nine figures from 1 to 9, and then gives in the end the names denoting zero, calling all the ten by the name of saṅkhya: and from this fact also, the inference may well be drawn that the zero had a symbol, and that it was well known that with the aid of the ten digits and the decimal system of notation numerical quantities of all values may be definitely and accurately expressed. What this known zero-symbol was, is, however, a different question.
The labour and attention bestowed upon the study and translation and annotation of the Gaṇita-sāra-saṅgraha