of the lines is a considerable refinement in science first practiced by the mathematician, Briggs." Count Bjornstjerna says: "We find in Ayeen Akbari, a journal of the Emperor Akbar, that the Hindus of former times assumed the diameter of a circle to be to its periphery as 1,250 to 3,927. The ratio of 1,250 to 3,927 is a very close approximation to the quadrature of a circle, and differs very little from that given by Metius of 113 to 355. In order to obtain the result thus found by the Brahmans, even in the most elementary and simplest way, it is necessary to inscribe in a circle a polygon of 768 sides, an operation, which cannot be performed arithmetically without the knowledge of some peculiar properties of this curved line, and at least an extraction of the square root of the ninth power, each to ten places of decimals. The Greeks and Arabs have not given anything so anything so approximate.”2 It is thus clearly seen that the Greeks and the Arabs apart, even the Europeans have but very recently advanced far enough to come into line with the Hindus in their knowledge of this branch of mathematics. Professor Wallace says: "The researches of the learned have brought to light astronomical tables in India which must have been constructed by the principles of geometry, but the period at which they have been framed has by no means been completely ascertained. Some are of opinion that they have been framed from observation made at a very remote period, not less than 3,000 years before the Christian era (this has been conclusively proved by Mons. Bailly); Mill's India, Vol. II, p. 150. Theogony of the Hindus, p. 37. and if this opinion be well founded, the science of geometry must have been cultivated in India to a considerable extent long before the period assigned to its origin in the West; so that many elementary propositions may have been brought from India to Greece." He adds: "In geometry there is much deserving of attention. We have here the celebrated proposition that the square on the hypotenuse of a right-angled triangle is equal to the squares on the sides containing the right angle and other propositions, which form part of the system of modern geometry. There is one remarkable proposition, namely, that which discovers the area of a triangle when its three sides are known. This does not seem to have been known to the ancient Greek geometers." The Sulva Sutras, however, date from about the eighth century B.C., and Dr. Thibaut has shown that the geometrical theorem of the 47th proposition, Book I, which tradition ascribes to Pythagoras, was solved by the Hindus at least two centuries earlier, thus confirming the conclusion of V. Schroeder that the Greek philosopher owed his inspiration to India.3 Mr. Elphinstone says: "Their geometrical skill is shown among other forms. by their demonstrations of various properties of triangles, especially one which expresses the area in the terms of the three sides, and was unknown in Europe till published by Clavius, and by their knowledge of the proportions of the radius to the circumference of a circle, which they express in a mode peculiar to themselves, by applying one measure and 1 Edinburgh Encyclopædia, "Geometry," p. 191. 2 Journal of the Asiatic Society of Bengal, 1875, p. 227. one unit to the radius and circumference. This proportion, which is confirmed by the most approved labours of Europeans, was not known out of India until modern times." ALGEBRA. THE Hindus have been especially successful in the cultivation of algebra. Professor Wallace says: "In algebra the Hindus understood well the arithmetic of surd roots, and the general resolution of equations of the second degree, which it is not clear that Diaphantus knew, that they attained a general solution of indeterminate problems of the first degree, which it is certain Diaphantus had not attained, and a method of deriving a multitude of answers to problems of the second degree, when one solution was discovered by trial, which is as near an approach to a general solution as was made until the time of La Grange." Professor Wallace concludes by adopting the opinion of Playfair on this subject, "that before an author could think of embodying a treatise on algebra in the heart of a system of astronomy, and turning the researches of the one science to the purposes of the other, both must have been in such a state of advancement as the lapse of several ages and many repeated efforts of inventors were required to produce." "This," says Professor Wilson, "is unanswerable evidence in favour of the antiquity, originality, and advance of the Hindu mathematical science.' 1 Elphinstone's History of India, p. 130. 112 Mr. Colebrooke says: "They (the Hindus) understood well the arithmetic of surd roots; they were aware of the infinite quotient resulting from the division of finite quantities by cipher; they knew the general resolution of equations of the second degree, and had touched upon those of higher denomination, resolving them in the simplest cases, and in those in which the solution happens to be practicable by the method which serves for quadratics; they had attained a general solution of indeterminate problems of the first degree; they had arrived at a method for deriving a multitude of solutions of answers to problems of the second degree from a single answer found tentatively."1 "And this," says Colebrooke in conclusion " was as near an approach to a general solution of such problems as was made until the days of La Grange.' 112 "Equally decided is the evidence," says Manning, "that this excellence in algebraic analysis was attained in India independent of foreign aid.” Mr. Colebrooke says: "No doubt is entertained of the source from which it was received immediately by modern Europeans. The Arabs were mediately or immediately our instructors in this study." Mrs. Manning says: "The Arabs were not in general inventors but recipients. Subsequent observation has confirmed this view; for not only did algebra in an advanced state exist in India prior to the earliest disclosure of it by the Arabians to modern Europe, but 1 Colebrooke's Miscellaneous Essays, Vol. II, p. 419. 2 Colebrooke's Miscellaneous Essays, Vol. II, pp. 416-418. For the points in which Hindu algebra is more advanced than the Greek, see Colebrooke, p. 16. the names by which the numerals have become known to us are of Sanskrit origin."1 Professor Monier Williams says: "To the Hindus is due the invention of algebra and geometry and their application to astronomy."2 Comparing the Hindus and the Greeks, as regards their knowledge of algebra, Mr. Elphinstone says: "There is no question of the superiority of the Hindus over their rivals in the perfection to which they brought the science. Not only is Aryabhatta superior to Diaphantus (as is shown by his knowlege of the resolution of equations involving several unknown quantities, and in a general method of resolving all indeterminate problems of at least the first degree) but he and his successors press hard upon the discoveries of algebraists who lived almost in our own time."3 "It is with a feeling of respectful admiration that Mr. Colebrooke alludes to ancient Sanskrit treatises on algebra, arithmetic and mensuration."4 In the Edinburgh Review (Vol. XXI, p. 372) is a striking history of a problem (to find x, so that ax2 + b shall be a square number.) The first step towards a solution is made by Diaphantus, it was extended by Fermat, and sent as a defiance to the English algebraists in the seventeenth century, but was only carried to its full extent by the celebrated mathematician Euler, 1 Ancient and Medieval India, Vol. II, p. 375, "Mr. Colebrooke has fully shown that algebra had attained the highest perfection it ever reached in India before it was ever known to the Arabians. Whatever the Arabs possessed in common with the Hindus, there are good grounds to believe that they derived from the Hindus."-Elphinstone's India, p.133. 2 Indian Wisdom, p. 185. 3Elphinstone's India, p. 131. 4 Manning's Ancient and Medieval India, Vol. I, p. 374 |