[The following extract from Herschel's article "Mathematics" in David Brewster's Edinburgh Encyclopaedia (Philadelphia, 1832) is reprinted because it contains facts little known and arguments too good to be ignored. At the time when the article appeared, Colebrooke's great translation of the standard Hindu works of Algebra was still fresh in the public mind. (London, 1817.) Albert J. Edmunds.]
So early as the latter part of the tenth century (A. D. 980) Gerbert, having learned of the Moors in Spain their system of arithmetic, had imparted it to his countrymen the French, whence it rapidly spread over Europe, and continues in use to the present moment. The Moors and Arabs, by their own unanimous avowal, derived this admirable invention from the Hindus who, there is good reason to believe, were in possession of it at least from the time of Pythagoras. The story of this philosopher's visit to the Brahmins is well known, and a suspicion may be entertained that his time there was better employed than in picking up the ridiculous doctrine of the transmigration of souls. Boetius relates the singular fact of a system of arithmetical characters and numeration employed among the pythagoreans, which he transcribes, and which bears a striking resemblance, almost amounting to identity, with those now in use, whose origin we know to be Hindu. The discovery (generally so considered) of the property of the right-angled triangle by the same philosopher, is a remarkable coincidence. This was known ten centuries before to the Chinese, if we may credit the respectable testimony of Gaubil. It was well known to the earliest Hindu writers of whom we have any knowledge, and who appear to have derived it from a source of much more remote antiquity. It is scarce conceivable that a Greek invention, of such extreme convenience as the decimal arithmetic, should have been treated with such neglect, remaining confined to the knowledge of a few speculative men, till, from being communicated as a mystery, it was at last preserved but as a curiosity ; but the aversion of that people to foreign habits will easily account for this, on the supposition of its Hindu origin.
An abstract truth, however, is of no country, and would be received with rapture, from whatever quarter, by men already advanced enough to appreciate its value. We are then strongly inclined to conclude that in the latter as well as in the former instance Pythagoras may have acted only the part of a faithful reporter of foreign knowledge, though the reverse hypothesis, viz., that the first impulse was given to Hindu science at this period by the Greek philosopher, might certainly be maintained.
However this may be, the great question as to the origin of algebra, which has been the cause of so much speculation, seems at length, by the enlightened researches which have of late been made in Hindu literature, nearly decided in favor of that nation. It will be proper to state, as briefly as is consistent with perspicuity, the grounds of this conclusion. The earliest Hindu writer on algebra, of whom any certain or even traditional knowledge has reached us, is the astronomer Aryabhatta who, from various circumstances, is concluded to have written so early as the fifth century.
It is true, the work of Diophantus takes the precedence of this in point of antiquity by about a hundred years, nor is it at all intended to deprive the Greek author of the merit of independent invention. Indeed the comparison of the state of knowledge in the two countries at the periods we speak of, is decidedly favorable to the independence of their views. By what we know of the Hindu author it appears that he was in possession of a general artifice of a very refined description (called in Sanskrit the kattaka, or "pulveriser") for the resolution of all indeterminate problems of the first degree,
and also of the method of resolving equations with several unknown quantities. It is very unlikely that these methods should have arisen at once or been the work of one man, especially as they are delivered incidentally in a work on astronomy., Now, of the latter of them we are not sure that Diophantus had any knowledge, as, although he resolves questions with more than one condition, he
always contrives, by some ingenious substitution, to avoid this difficulty. Of the former he was certainly ignorant. His arithmetic, indeed, though full of ingenious artifices for treating particular
problems, yet lays down no general methods whatever, and indicates a state of knowledge so far inferior to that of the Hindu writer that no supposed communication with India about the third or
fourth century would at all account for the phenomena. But there is yet stronger evidence. The Brahma-siddhanta, the work of Brahmagupta, a Hindu astronomer at the beginning of the seventh
century, contains a general method for the resolution of indeterminate problems of the second degree: an investigation which actually baffled the skill of every modern analyst till the time of La Grange's solution, not excepting the all-inventive Euler himself. This is a matter of a deeper dye.
The Greeks cannot for a moment be thought of as the authors of this capital discovery ; and centuries of patient thought and many successive efforts of invention must have prepared the way to it in the country where it did originate. It marks the maturity and vigor of mathematical knowledge, while the very work of Brahmagupta, in which it is delivered, contains internal evidence that in his time geometry at least was on the decline. For example, he mentions several properties of quadrilaterals as general which are only true of quadrilaterals inscribed in a circle. The discoverer of these properties (which are of considerable difficulty) could not have been ignorant of this limitation, which enters as an essential element in their demonstration. Brahmagupta, then, in this instance retailed, without fully comprehending, the knowledge of his predecessors. When the stationary character of Hindu intellect is taken into the account, we shall see reason to conclude that all we now possess of Hindu science is but part of a system, perhaps of much greater extent, which existed at a very remote period, even antecedent to the earliest dawn of science among the Greeks, and might authorize as well the visits of sages as the curiosity of conquerors.