II.—MATHEMATICS.
In Mathematics he was greater
Than Tycho Brahe, or Erra Pater.
BUTLER: iludibras.
IN abstraction and concentration of thought
the Hindus are proverbially happy. Apart from direct testimony on the point, the literature of the Hindus furnishes unmistakable evidence to prove that the ancient Hindus possessed as nishing powers of memory and concentration of thought. Hence all such sciences and branches of study as demand concentration of thought and a highlydeveloped power of abstraction of the mind were highly Cultivated by the Hindus. ie science of mathematics, the most abstract of all sciences, must have had an irresistible fascination for the minds of the Hindu’s. Nor are there proofs wanting to support this sta ement. The most extensive cultivation which astronomy received at the hands of the Hindus is in itself a proof of their high proficiency in mathematics. The high antiquity of Hindu astronomy is an argument in support of a still greater antiquity of their mathematics. That the Hindus were selected by nature to excel all other nations in mathematics, is proved by her revealing to them the foundation of all mathematics. It has been admitted by all competent authorities that the Hindus were the inventors of the numerals. The great German critic, Schlegel, says that the Hindus invented “ the decimal cyphers, the honour of which, next to letters the most important of human discoveries, has, with the common consent of historical authorities, been ascribed to the Hindus.”1
Prof. Macdonell says: “In science, too, the debt of Europe to India has been considerable. There is, in the first place, the great fact that the Indians invented the numerical figures used all over the world. The influence which the decimal system of reckoning dependent on those figures has had not only on mathematics but on the progress of civilization in general, can hardly be overestimated. During the eighth and ninth centuries the Indians became the teachers in arithmetic and algebra of the Arabs, and through them of the nations of the West. Thus, though we call the latter science by an Arabic name, it is a gift we owe to India.”‘
Sir M. Monier Williams says: “From them (Hindus) the Arabs received not only their first conceptions of algebraic analysis, but also those numerical symbols and decimal notations now current everywhere in Europe, and which have rendered untold service to the progress of arithmetical scienee.”3 Says Manning: “To whatever cyclopdia, journal or essay we refer, we uniformly find our numerals traced to India and the Arabs recognised as the medium through which they were introduced into Europe.”4 Sir W. W. Hunter also says: “To them (the Hindus) we owe the invention of the numerical symbols on the decimal scale, The Indian figures 1 to 9 being abbreviated forms of initial letters of he numerals themselves, and the zero, or 0,
1Schlegel’s History of Literature, p, 123. History Sanskrit Literature, p. 424.
3 Indian W isdorn,, p. 124.
}Ancient ancl,Me.liteval India, Vol. I, p.
representing the first letter of the Sanskrit word for empty (stinya). The Arabs borrowed them from the Hindus, and transmitted them to Europe.”‘
Professor Weber says: “It is to them (the Hindus) also that we owe the ingenious invention of the numerical symbols, which in like manner passed from them to the Arabs, and from these again to European scholars. By these latter, who were the disciples of the Arabs, ,frequent allusion is made to the Indians and uniformly in terms of high esteem; and one Sanskrit word even (uchcha) has passed into the Latin translations of Arabian astronomers.” 2
Professor Wilson says: “Even Delambre concedes their claim to the invention of numerical cyphers.”
ARITHMETIC.
Mrs. Manning says: “Compared with other ancient nations, the Hindus were peculiarly strong in, all the branches of arithmetic.”3 Professor Weber, after declaring that the Arabs were disciples of the Hindus, says: “The same thing (i.e., the Arabs borrowed from the ‘ Hindus) took place also in regard to algebra and arithmetic in particular, in both of which it appears the Hindus attained, quite independently, to a high degree of proficiency.” Sir W. W. Hunter also says that the Hindus attained a very high proficiency in arithmetic and algebra independently of any foreign influence.”4
liteperial Gazetteer, p. 219. ‘,India.”
2 Weber’s Indian Literature, p. 256.
3 Ancient and Medimval India, Vol. I, p 374, 4linperial Gazetteer, “ India,” p. 219.
The English mathematician, Prof. Wallace, says “ The Lilavati treats of arithmetic, and contains not only the common rules of that science, but the application of these to various questions of interest, barter, mixtures, combinations, permutations, sums of progression, indeterminate problems, and mensuration of surfaces and solids. The rules are found to be exact and nearly as simple as in the present state of analytical investigation. The numerical results are readily deduced, and if they be compared with the earliest specimens of Greek calculation, the advantages of the decimal notation are placed in a striking light.”‘ It may, however, be mentioned that Lilavati, of which Professor Wallace speaks, is a comparatively modern manual of arithmetic; and to judge of the merits of Hindu arithmetic from this book is to judge of the merits of English arithmetic from Chambers’ manual of arithmetic.
It may be added that the enormous extent to which numerical calculation goes in India, and the possession by the Hindus of by far the largest table of calculation, are in themselves proofs of the superior cultivation of the science of arithmetic by the Hindus.
GEOMETRY.
The ancient Hindus have always been celebrated for the remarkable progress they made in geometry. Professor Wallace says: “However ancient a book may
I Edinburgh Review, Vol. 29, p. 147.
be in which a system of trigonometry occurs, we may be assured it was not written in the infancy of the science. Geometry must have been known in Tndia long before the writing of the Surya Siddhanta,” which is supposed by the Europeans to have been written before 2,000 B.C.2
Profesor Wallace says :, “ Surya Siddhanta contains a rational system of trigonometry, which differs entirely from that first known in Greece or Arabia, In fact it is founded on a geometrical theorem, which was not known to the geometricians of Europe before the time of Vieta, about two hundred years ago. And it employs the sines of arcs, a thing unknown to the Greeks, who used the chords of double arcs. The invention of sines has been attributed to the Arabs, but it is possible that they may have received this improvement in trigonome try as well as the numerical characters from India.”3
Mr. Elphinstone says: “In the Surya Siddhanta is contained ^a system of trigonometry which not only goes far beyond anything known to the Greeks, but involves theorems which were not discovered in Europe till two centuries ago.”
Professor Wallace, says: “In expressing the radius of a circle in parts of the circumference, the Hindus are quite singular. Ptolemy and the Greek mathematicians in their division of the radius preserved no reference to the circumference. The use of sines, as it was unknown to the Greeks, forms a difference between theirs and the Indian trigonometry. Their rule for the computation
1 Mill’s India, Vol. II, p. 150.
2See Mill’s India, Vol. II, p. 3, footnote. 3Edinburgh Eneyelopzedia, “ Geometry,” p. 191. 4ilistory of India, p. 129.
of the lines is a considerable refinement in science first practiced by the mathematician, Briggs.”1
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 theil3rah mans, 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 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 tip 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);
1 Mill’s India, Vol. II, p. 150. 2iheogony of the Hindus, T. v.
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 rightangled 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 covers the area of a triangle when its three sides are known. This does not seem to hare 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,2 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 Clayius, 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
‘Edinburgh Encyclopaedia, “Geometry,” p. 191. 2Journal of the Asiatic Society of Bengal, 1875, p. 227. 3 See History ofHindu Chemistry, Vol. 1, p, xxir; intrEV. 
one unit to the radius and circumference. This proportion, which is confirmed by the,most approved labours of Euro7 ,peans, was not known out of India. until modern times.”‘
