Tim 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.”
iElphinstone’s History of India, p. 180. India, Vol. II, p. 151, Wilson’s note,
IVIr. Colebrooke, says: “They (the Hindus) under stood 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 indeterriiinate 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.”‘ “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.”2
“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
1Colebrooke’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.”‘
Professor Mother Williams says: “To the Hindus is due the invention of algebra and geometry and their application to astronomy.”8
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) bat he and his successors press hard upon the discoveries of algebraists who lived almost in our own time.”9 “ It is with a feeling of respectful admiration that Mr. Colebrooke alludes to ancient Sanskrit treatises on algebra, arithmetic and men suration.”10
In the Edinburgh Review (Vol. XXI, p. 372) is a striking history of a problem (to find x, so that ax + 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,
‘Ancient and Medimval 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.
who arrives exactly at the point before attained by B hashkaracharya.”
Another occurs in the same Review (Volume XXIX, p. 153), where it is stated, from Mr. Colebrooke that a particular solution given by Bhashkeracharya is exactly the same as that hit on by Lord Brounker in 1657; and that the general solution of the same problem was unsuccessfully attempted by Euler and only accomplished by De la Grange in 1767 A.D.; although it had been as comm pletely givenby Brahmagupta.
“But,” says Mr. Elphinstone, “the superiority of the Hindus over the Greek algebraists is scarcely so conspicuous in their discoveries as in the excellence of their method, which is altogether dissimilar to that of Diaphantus (Strachey’s Bija Ganita quoted in the “Edinburgh Review,” Vol. XXI, pp. 374, 375), and in the perfection of their algorithm (Colebrooke’s Hindu Algebra quoted in the E. R. Vol. XXIX, p. 162).
One of their most favourite processes (that called cattaca) was not known in Europe till published by Bachet de Mezeriac, about the year 1624, and is virtually the same as that explained by Euler (Edinburgh Review, Vol. XXIX, p. 151). Their application of algebra to astronomical investigations and geometrical demonstrations is also an invention of their own; and their manner of conducting it is even now entitled to admiration ‘
lElphiustone’s India, p. 131. Bhashkaracharya wrote the celebrated book “Siddhanta Siromani,” containing treatises on algebra and Arithmetic. His division of a circle is remarkable for its minute analysis, which is as follows :-
60 Vikala (Seconds) = A Kala (Minutes).
60 Kala = A Bbaga (Degree),
30 Bhaga . = A Rasi (Sign).
12 Rasi A Bhagana (Revolution).
(Colebrooke, quoted by Professor Wallace; and Edinburgh Review, Vol. XXIX, p. 158).
Speaking of the Hindu treatises on algebra, arithmetic, and mensuration, Mr. Colebrooke says: “It is not hoped that in the actual advanced condition of the analytical art they will add to its resources and throw new light on the mathematical science in any. other respect than as concerns its history, but had an earlier version of these treatises been completed, had they been translated and given to the public when the notice of mathematicians was first drawn to the attainments of the Hindus in astronomy and in sciences connected with it, some additions would have been then made to the means and resources of algebra, for the general solution of problems, by methods which have been re-invented or have been perfected in the last age.”11
It is thus evident from what Mr. Colebrooke shows that the Hindu literature even in its degenerate state, and when so few works are extant, contains mathematical works that show an advance in the science in no way behind the latest European achievements.
As an instance of the remarkable and extensive practice and cultivation of mathematics in India, may be cited the case of a problem from Lalita Vistar. Mons. Woepcke,2 indeed, is of opinion that the account in the
It may, however, be said that in some quarters the genuineness of the independent solution of the problems mentioned above, and the discovery of methods similar to those of the Hindus by modern Europeans have been doubted, and such doubts may well be excused, considering the extensive intercourse that has existed between India and Europe for a long time past.
7-Mem Surla propagation des chiffres Indiens, Paris, 1863, pp. 75-91,
Lalita Vistara of the problem solved by Buddha on the occasion of his marriage examination, relative to the number of atoms in the length of a Yojana, is the basis of the “Arenarius” of the celebrated scientist Archimedes.
The credit of the discovery of the principle of differential calculus is generally claimed by the Europeans. But it is remarkable that a similar method existed in India ages ago. Bhashkeracharya, one of the world’s greatest mathematicians, has referred to it. Following, however, in the footsteps of his Hindu predecessors he does not expound the method fully, but only gives an outline of it.
Mr. Spottiswoode says: “It must be admitted that the penetration shown by Bhashkeracharya in his analysis is in the highest degree remarkable that the formula which he establishes, and his method, bear more than a mere resemblance—they bear a strong analogy—to the corresponding process in modern mathematical astronomy; and that the majority of scientific persons will learn with surprise the existence of such a method in the writings of so distant a period and so remote a.region “1
Mr. Lethbridge says: “Bhashkeracharya is said to have discovered a mathematical process very nearly resembling the differential calculus of modern European mathematicians.” 2
3 J. R. A. S., Vol. 2.;’. VII.
2Selloor History of India, Appendix A, p.