George Rusby Kaye

“ He is rude enough to compare Āryabhata to a worm which, eating the wood, by chance describes certain characters in it, without intending to draw them. "He, however, who knows these things thoroughly stands opposite to Āryabhata, Śrīshena and Vishnuchandra like the lion against gazelles. They are not capable of letting him see their faces." ”

“ The Pañcha Siddhāntikā contains material of considerable mathematical interest and from the historical point of view of a value not surpassed by that of any later Indian works. The mathematical section of the Pauliśa Siddhānta is perhaps of the most interest and may be considered to contain the essence of Indian trigonometry. It is as follows:—
" (1) The square-root of the tenth part of the square of the circumference, which comprises 360 parts, is the diameter. Having assumed the four parts of a circle the sine of the eighth part of a sign [is to be found] . ”

“ According to the Hindus the modern place-value system of arithmetical notation is of divine origin. This led the early orientalists to believe that, at any rate, the system had been in use in India from time immemorial; but an examination of the facts shows that the early notations in use were not place-value ones and that the modern place-value system was not introduced until comparatively modern times. ”

“ History presents him in no other light than a mathematician of a country most distant from Greece and contiguous to India......Not having taken algebra from the Greeks, he must either have invented it himself or taken it from the Indians.' As a matter of fact his algebra shows, as pointed out by Rodet, no sign of Indian influence and is practically wholly based upon Greek knowledge ”

“ Although it may be possible to offer only conjectures as to the actual route by which any particular class of Greek knowledge reached India, the fact remains that during the period under consideration the intellectual influence of Greek on India was considerable. It is evident not only in the mathematical work of the Indians but also in sculpture, architecture, coinage, astronomy, astrology, &c. ”

“ If a bamboo measuring thirty-two hastas and standing upon level ground be broken in one place by the force of the wind and the tip of it meet the ground at sixteen hastas, say, mathematician, at how many hastas from the root it is broken? ”

“ Brahmagupta gives a fairly complete set of rules dealing with the cyclic quadrilateral and either he or the mathematician from whom he obtained his material had a definite end in view—the construction of a cyclic quadrilateral with rational elements.—The commentators did not fully appreciate the theorems, some of which are given in the works of Mahāvīra and S'rīdhara; and by the time of Bhāskara they had ceased to be understood. Bhāskara indeed condemns them outright as unsound. "How can a person" he says "neither specifying one of the perpendiculars, nor either of the diagonals, ask the rest? ”

“ The use of two such diverse types as yāvat tāvat and kālaka (generally abbreviated to yā and kā) in one system suggests the possibility of a mixed origin. It is possible that the former is connected with Diophantus' definition of the unknown quantity, plēthos monádon aoriston, i.e., 'an undefined (or unlimited) number of units.' To pass from 'an unlimited number' to 'as many as' requires little imagination. ”

“ Further, at no period did the Indians develop any real theory of geometry, and a comparatively modern Indian work denies the possibility of any proof of the Pythagorean theorem other than experience.
The fanciful suggestion of Bürk that possibly Pythagoras obtained his geometrical knowledge from India is not supported by any actual evidence. ”

“ In the 16th century, A.D., Hindu tradition ascribes the invention of the nine figures with the device of places to make them suffice for all numbers 'to the beneficent Creator of the universe'; and this was accepted as evidence of the very great antiquity of the system! ”

“ The Indian method of stating examples—particularly those involving algebraic equations—are of sufficient interest to be recorded here. The early works were rhetorical and not symbolical at all and even in modern times the nearest approach to a symbolic algebra consists of abbreviations of special terms. The only real symbol employed is the negative sign of operation, which is usually a dot placed above or at the side of the quantity affected. ”

“ On the other hand Brahmagupta gives more advanced developments of indeterminate equations than occurs in the Chinese works of his period, and it is not until after Bhāskara that Ch'in Chu-sheo recorded (in A.D. 1247) the celebrated t'ai-yen ch'in-yi-shu or process of indeterminate analysis, which is, however, attributed to I'-hsing nearly six centuries earlier. The Chinese had maintained intellectual intercourse with India since the first century and had translated many Indian (Buddhistic) works. ”

“ The orientalists who exploited Indian history and literature about a century ago were not always perfect in their methods of investigation and consequently promulgated many errors. Gradually, however, sounder methods have obtained and we are now able to see the facts in more correct perspective. ”

“ Angles are not dealt with at all;
(3) There is no mention of parallels and no theory of proportion;
(4) Traditional inaccuracies are not uncommon;
(5) A gradual decline in geometrical knowledge is noticeable. On the other hand, we have the following noteworthy rules relating to cyclic quadrilaterals—where x and y are the diagonals of the cyclic quadrilateral (a, b, c, d) . This (ii) is sometimes designated as 'Brahmagupta's theorem'.
16. The absence of definitions and indifference to logical order sufficiently differentiate the Indian geometry from that of the early Greeks ”

“ The questions (a) whether the Indians of this period had completely realised the generality of the Pythagorean theorem, and (b) whether they had a sound notion of the irrational have been much discussed; but the ritualists who composed the S'ulvasūtras were not interested in the Pythagorean theorem beyond their own actual wants, and it is quite certain that even as late as the 12th century no Indian mathematician gives evidence of a complete understanding of the irrational. ”

“ Other noteworthy points are the rules relating to volumes of solids which contain some remarkable inaccuracies, e.g., the volume of a pyramid is given as half the product of the height and the base; the volume of a sphere is stated to be the product of the area of a circle (of the same radius as the sphere) and the root of this area, or . Similar errors were not uncommon in later Indian works. ”

“ Mūsā, flourished after Brahmagupta so that the Arabs could not have been the intermediaries between the Greeks and Indians. Indeed their chronological position has misled certain writers to the erroneous conclusion that they obtained their elements of mathematics from the Indians. ”

“ Mūsā el-Chowārezmi (A.D. 782) is the earliest Arabic writer on mathematics of note and his best known work is the Algebra. The early orientalists appear to have been somewhat prejudiced against Arabic scholarship for, apparently without examination, they ascribed an Indian origin to M. b. Mūsā's work. The argument used was as follows: "There is nothing in history," wrote Cossali, and Colebrooke repeated it, 'respecting Muhammad ben Mūsā individually, which favours the opinion that he took from the Greeks, the algebra which he taught to the Muhammadans. ”

“ On the other hand, Brahmagupta praises Puliśa for what he does, since he does not differ from the book Smriti.' Again, speaking of Varāhamihira, S'rīshena, Āryabhata and Vishnuchandra, Brahmagupta says: 'If a man declares these things illusory he stands outside the generally acknowledged dogma, and that is not allowed.'
Of Varāhamihira. Alberuni writes: 'In former times, the Hindus used to acknowledge that the progress of science due to the Greeks is much more important than that which is due to themselves. ”

“ According to orthodox Hindu tradition the Sūrya Siddhānta, the most important Indian astronomical work, was composed over two million years ago! Bailly, towards the end of the eighteenth century, considered that Indian astronomy had been founded on accurate observations made thousands of years before the Christian era. Laplace, basing his arguments on figures given by Bailly considered that some 3,000 years B.C. the Indian astronomers had recorded actual observations of the planets correct to one second ”

“ On the other hand it is held that there is no sound evidence of the employment in India of a place-value system earlier than about the ninth century. The suggestion of 'divine origin' indicates nothing but historical ignorance ”

“ The quantity of rubies without flaw, sapphires, and pearls belonging to one person, is five, eight and seven respectively. The number of like gems belonging to another is seven, nine and six. One has ninety, the other sixty-two rupees. They are equally rich. Tell me quickly, then, intelligent friend, who art conversant with algebra, the prices, of each sort of gem?
Answer—14, 1, 1, etc. V. 105 & 156.
[Bhāskara 'assumes' relative values.]
32. The horses belonging to these four persons respectively are five, three, six and eight; the camels belonging to them are two, seven, four and one ”

“ From the triangles (3, 4, 5) and (5, 12, 13) a commentator obtains the quadrilateral (39, 60, 52, 25) , with diagonals 63 and 56, etc. He also introduces a proof of Ptolemy's theorem and in doing this follows Diophantus (iii, 19) in constructing from triangles (a, b, c) and new triangles and and uses the actual examples given by Diophantus, namely (39, 52, 65) and (25, 60, 65) .
18. An examination of the Greek mathematics of the period immediately anterior to the Indian period with which we are now dealing shows that geometrical knowledge was in a state of decay. ”

“ They (unlike their Indian friends) generally give the source of their information and acknowledge their indebtedness with becoming courtesy. From the 7th century Indian scholars were occasionally employed on the Chinese Astronomical Board. Mr. Yoshio Mikami states that there is no evidence of Indian influence on Chinese mathematics. On the other hand he says "the discoveries made in China may have touched the eyes of Hindoo scholars."
27. Arabic Mathematics.—It has often been assumed, with very little justification, that the Arabs owed their knowledge of mathematics to the Hindus. ”

“ Vincent Smith refers "to the cumulative proof that the remarkable intellectual and artistic output of the Gupta period was produced in large measure by reason of the contact between the civilization of India and that of the Roman Empire;" and research is almost daily adding to such proof. ”

“ Neither the value itself nor this form of fraction occurs in any later Indian work.
There is one other point connected with the Pythagorean theorem to be noted, viz., the occurrence of an indication of the formation of a square by the successive addition of gnomons. The text relating to this is as follows:
"Two hundred and twenty-five of these bricks constitute the sevenfold agni with aratni and pradesa."
"To these sixty-four more are to be added. With these bricks a square is formed. The side of the square consists of sixteen bricks. ”

“ The arithmetic of Chang-Ch'iu-chien written in the 6th century contains a great deal of matter that may have been the basis of the later Indian works. Indeed the later Indian works seem to bear a much closer resemblance to Chang's arithmetic than they do to any earlier Indian work.
The problem of "the hundred hens" is of considerable interest. Chang gives the following example: "A cock costs 5 pieces of money, a hen 3 pieces and 3 chickens 1 piece. ”

“ From a tree a hundred hastas high, a monkey descended and went to a pond two hundred hastas distant, while another monkey, jumping a certain height off the tree, proceeded quickly diagonally to the same spot. If the space travelled by them be equal, tell me quickly, learned man, the height of the leap, if thou hast diligently studied calculation?
Answer—50. L. 155; V. 126.
41. The man who travels to the east moves at the rate of 2 yojanas, and the other man who travels northward moves at the rate of 3 yojanas. The latter having journeyed for 5 days turns to move along the hypotenuse. ”

“ In spite, however, of the progress made in historical research there are still many errors current, of which, besides those already touched upon, the following may be cited as examples: (a) The proof by "casting out nines" is not of Indian origin and occurs in no Indian work before the 12th century ”

“ Whereas on the one hand the Indians have been credited with the invention of the sine function, on the other there is no evidence to show that they were acquainted with even the most elementary theorems (as such) relating to angles. ”

“ In connection with (1) and (2) the Pythagorean theorem is stated quite generally. It is illustrated by a number of examples which may be summarised thus:
Kātyāyana gives no such rational examples but gives (with Āpastamba and Baudhāyana) the hypotenuse corresponding to sides equal to the side and diagonal of a square, i.e., the triangle , and he alone gives , and . There is no indication that the S'ulvasūtra rational examples were obtained from any general rule. Incidentally is given an arithmetical value of the diagonal of a square which may be represented by ”

“ The principal multiplied by its time and divided by the interest, and the quotient being multiplied by the factor less one is the time. The sum of principal and interest divided by unity added to the interest on unity is the principal. ”

“ This occurs in every Indian work after the 6th century. The problem about two travellers meeting on the hypotenuse of a right-angled triangle occurs some ten centuries later in exactly the same form in Mahāvīra's work.
The Sun-Tsū Suan-ching is an arithmetical treatise of about the first century. It indulges in big numbers and elaborate tables like those contained in Mahāvīra's work; it gives a clear explanation of square-root and it contains examples of indeterminate equations of the first degree. The example: "There are certain things whose number is unknown. ”