VIII.
26. Chinese Mathematics.—There appears to be abundant evidence of an intimate connection between Indian and Chinese mathematics. A number of Indian embassies to China and Chinese visits to India are recorded in the fourth and succeeding centuries. The records of these visits are not generally found in Indian works and our knowledge of them in most cases comes from Chinese authorities, and there is no record in Indian works that would lead us to suppose that the Hindus were in any way indebted to China for mathematical knowledge. But, as pointed out before, this silence on the part of the Hindus is characteristic, and must on no account be taken as an indication of lack of influence. We have now before us a fairly complete account of Chinese mathematics[1] which appears to prove a very close connection between the two countries. This connection is briefly illustrated in the following notes.
The earliest Chinese work that deals with mathematical questions is said to be of the 12th century B.C. and it records an acquaintance with the Pythagorean theorem. Perhaps the most celebrated Chinese mathematical work is the Chin-chang Suan-shū or "Arithmetic in Nine Sections" which was composed at least as early as the second century B.C. while Chang T'sang"s commentary on it is known to have been written in A.D. 263. The "Nine Sections" is far more complete than any Indian work prior to Brahmagupta (A.D. 628) and in some respects is in advance of that writer. It treats of fractions, percentage, partnership, extraction of square and cube-roots, mensuration of plane figures and solids, problems involving equations of the first and second degree. Of particular interest to us are the following: The area of a segment of a circle, where 'c' is the chord and 'a' the perpendicular, which actually occurs in Mahāvīra's work; in the problems dealing with the evaluation of roots, partial fractions with unit numerators are used (cf. paragraphs 5 and 7 above); the diameter of a sphere, which possibly accounts for Āryabhata's strange rule; the volume of the cone which is given by all the Indians; and the correct volume for a truncated pyramid which is reproduced by Brahmagupta and Srīdhara. One section deals with right angled triangles and gives a number of problems like the following:
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"There is a bamboo 10 feet high, the upper end of which being broken reaches to the ground 3 feet from the stem. What is the height of the break?" 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. Repeatedly divided by 3 the remainder is 2; by 5 the remainder is 3, and by 7 the remainder is 2. What will be the number?" re-appears in Indian works of the 7th and 9th centuries. The earliest Indian example is given by Brahmagupta and is: "What number divided by 6 has a remainder 5, and divided by 5 has a remainder of 4 and by 4 a remainder of 3, and by 3 a remainder of 2?" Mahāvīra has similar examples.
In the 3rd century the Sea Island Arithmetical Classic was written. Its distinctive problems concern the measurement of the distance of an island from the shore, and the solution given occurs in Āryabhata's Gaņita some two centuries later. The Wu-t'sao written before the 6th century appears to indicate some deterioration. It contains the erroneous rule for areas given by Brahmagupta and Mahāvīra. 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. If then we buy with 100 pieces 100 of them what will be their respective numbers?"
No mention of this problem is made by Brahmagupta, but it occurs in Mahāvīra and Bhāskara in the following form: "Five doves are to be had for 3 drammas, "7 cranes for 5, 9 geese for 7 and 3 peacocks for 9. Bring 100 of these birds for 100 drammas for the prince's gratification." It is noteworthy that this problem was also very fully treated by Abū Kāmil (Shogâ) in the 9th century, and in Europe in the middle ages it acquired considerable celebrity.
Enough has been said to show that there existed a very considerable intimacy between the mathematics of the Indians and Chinese; and assuming that the chronology is roughly correct, the distinct priority of the Chinese mathematics is established. 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. 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.
Muhammad b. 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. 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; and it is now well known that the development of mathematics among the Arabs was largely, if not wholly, independent of Indian influence and that, on the other hand, Indian writers on mathematics later than Brahmagupta were possibly influenced considerably by the Arabs. Alberuni early in the 11th century wrote: 'You mostly find that even the so-called scientific theorems of the Hindus are in a state of utter confusion, devoid of any logical order……since they cannot raise themselves to the methods of strictly scientific deduction……I began to show them the elements on which this science rests, to point out to them some rules of logical deduction and the scientific method of all mathematics, etc.'
The fact is that in the time of el-Māmūn (A.D. 772) a certain Indian astronomical work (or certain works) was translated into Arabic. On this basis it was assumed that the Arabic astronomy and mathematics was wholly of Indian origin, while the fact that Indian works were translated is really only evidence of the intellectual spirit then prevailing in Baghdad. No one can deny that Āryabhata and Brahmagupta preceded M. b. Mūsā[2] but the fact remains that there is not the slightest resemblance between the previous Indian works and those of M. b. Mūsā. The point was somewhat obscured by the publication in Europe of an arithmetical treatise by M. b. Mūsā under the title Algoritmi de Numero Indorum. As is well known the term India did not in mediæval times necessarily denote the India of to-day and despite the title there is nothing really Indian in the work. Indeed its contents prove conclusively that it is not of Indian origin. The same remarks apply to several other mediæval works.
28. From the time of M. b. Mūsā onwards the Muhammadan mathematicians made remarkable progress. To illustrate this fact we need only mention a few of their distinguished writers and their works on mathematics. 'Tābit b. Qorra b. Merwān (826–901) wrote on Euclid, the Almagest, the arithmetic of Nicomachus, the right-angle triangle the parabola, magic squares, amicable numbers, etc. Qostā b. Lūka el-Ba'albeki (died c. A.D. 912) translated Diophantus and wrote on the sphere and cylinder, the rule of two errors, etc. El-Battāni (M. b. Gabir b. Sinān, A.D. 877–919) wrote a commentary on Ptolemy and made notable advances in trigonometry. Abū Kāmil Shogā b. Aslam (c. 850–930) wrote on algebra and geometry, the pentagon and decagon, the rule of two errors, etc. Abū '1-Wefā el-Būzgāni, born in A.D. 940, wrote commentaries on Euclid, Diophantus, Hipparchus. and M. b. Mūsā, works on arithmetic, on the circle and sphere, etc.. etc. Abū Sa 'id, el-sigzi (Ahmed b. M. b. Abdelgalil, A.D., 951–1024) wrote on the trisection of an angle, the sphere, the intersection of the parabola and hyperbola, the Lemmata of Archimedes, conic sections, the hyperbola and its asymptotes, etc., etc. Abū Bekr. el-Karchi (M. b. el-Hasan, 1016 A.D.) wrote on arithmetic and indeterminate equations after Diophantus. Alberuni (M. b. Ahmed, Abū'l-Rihān el-Bīrunī) was born in A.D. 973 and besides works on history, geography, chronology and astronomy wrote on mathematics generally, and in particular on tangents, the chords of the circle, etc. Omār b. Ibrāhim el-Chaijāmi, the celebrated poet, was born about A.D. 1046 and died in A.D. 1123 a few years after Bhāskara was born. He Wrote an algebra in which he deals with cubic equations, a commentary on the difficulties in the postulates of Euclid: on mixtures of metals; and on arithmetical difficulties.
This very brief and incomplete resumé of Arabic mathematical works written during the period intervening between the time of Brahmagupta and Bhāskara indicates at least considerable intellectual activity and a great advance on the Indian works of the period in all branches of mathematics except, perhaps, indeterminate equations.