Indian Mathematics/Indian Mathematical Works—

by George Rusby Kaye

V.

19. We have, in the above notes, given in outline the historically important matters relating to Indian mathematics. For points of detail the works mentioned in the annexed bibliography should be consulted; but we here briefly indicate the other contents of the Indian works, and in the following sections we shall refer to certain topics that have achieved a somewhat fictitious importance, to the personalities of the Indian mathematicians and to the relations between the mathematics of the Chinese, the Arabs and the Indians.

20. Besides the subjects already mentioned Brahmagupta deals very briefly with the ordinary arithmetical operations, square and cube-root, rule of three, etc,; interest, mixtures of metals, arithmetical progressions, sums of the squares of natural numbers; geometry as already described but also including elementary notions of the circle; elementary mensuration of solids, shadow problems, negative and positive qualities, cipher, surds, simple algebraic identities; indeterminate equations of the first and second degree, which occupy the greater portion of the work, and simple equations of the first and second degrees which receive comparatively but little attention.

Mahāvīra's work is fuller but more elementary on the whole. The ordinary operations are treated with more completeness and geometrical progressions are introduced; many problems on indeterminates are given but no mention is made of the 'cyclic method' and it contains no formal algebra. It is the only Indian work that deals with ellipses (inaccurately).

The only extant work by S'rīdhara is like Mahāvīra's but shorter; but he is quoted as having dealt with quadratic equations, etc.

Bhāskara's Līlāvatī is based on S'rīdhara's work and, besides the topics already mentioned, deals with combinations, while his Vīja-ganita, being a more systematic exposition of the algebraical topics dealt with by Brahmagupta, is the most complete of the Indian algebras.

After the time of Bhāskara (born A.D. 1114) no Indian mathematical work of historical value or interest is known. Even before his time deterioration had set in and although a "college" was founded to perpetuate the teaching of Bhāskara it, apparently, took an astrological bias.

21. 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. In the Bakhshāli Ms., a cross is used in place of the dot as the latter in the Sārada script is employed to indicate cipher or nought.

The first mention of special terms to represent unknown quantities occurs in Bhāskara's Vīja-ganita which was written in the twelfth century of our era. Bhāskara says: "As many as (yāvat tāvat) and the colours 'black (kālaka), blue (nīlaka), yellow (pītaka) and red (lohitaka)' and others besides these have been selected by ancient teachers^[1] for names of values of unknown quantities."

The term yāvat tāvat is understandable and so is the use of colours but the conjunction is not easy to understand. 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. Diophantus had only one symbol for the unknown and if the use of yāvat tāvat were of Diophantine origin the Indians would have had to look elsewhere for terms for the other unknowns. With reference to the origin of the use of colours for this purpose we may point out that the very early Chinese used calculating pieces of two colours to represent positive and negative numbers.

As neither the Greeks nor the Indians used any sign for addition they had to introduce some expression to distinguish the absolute term from the variable terms. The Greeks used M° an abbreviation for monádes or 'units' while the Indians used rū for rūpa, a unit.

The commoner abbreviations used by the Indians are as follows:—

yā	for	yāvat tāvat, the first unknown.
kā	"	kālaka, the second unknown.
rū	"	rūpa, the absolute quantity.
va	"	varga, a square.
gha	"	ghana, a cube.
ka	"	karana, a surd.

It is hardly appropriate to discuss Sanskrit mathematical terminology in detail here but it will not be out of place to mention a few other terms. To denote the fourth power varga varga is used but it occurs only once within our period. In more modern times varga ghana ghāta^[2] denoted the fifth power, varga ghana, the sixth and so on.

Certain Greek terms are used, e.g., jāmitra (Gk. diámetron), kendra (Gk. kentron), trikona (Gk. trigonon), lipta (Gk. leptē), harija (Gk. ' orízōn), dramma (Gk. drachmē), dīnāra (Gk. dēnārion), etc. Many of these terms, however, are borrowed from Indian astrological works which contain a considerable number of Greek terms such as Hridroga (Gk. (udrochoos) Pārthona (Gk. Parthénos), āpoklima (Gk. apóklima), etc., etc.

The curious may compare pārśva 'a rib,' 'side' with the Greek pleura; koti which primarily means a claw or horn but is used for the perpendicular side of a triangle, with kāthetos; jātya which means 'legitimate,' 'genuine,' but is used to denote a right-angled triangle with orthogōnia; and so on.

Śridhara's Trisátika.

(From the copy used by Colebrook

India Office Catalogue 520e.)

Bakhshāli Ms.

We conclude this section with a few illustrations transliterated from Sanskrit manuscripts.

	Indian Forms.			Equivalents.	References.
1.	yā  6 rū 300 yā 10 rū $\scriptstyle {1{\dot {0}}0}$			$\scriptstyle {6x+300=10x-100}$	V. 104.
2.	yāca 18 yā 0 rū  0 yāva 16 yā 9 rū 18			$\scriptstyle {18x^{2}+0x+0=16x^{2}+9x+18}$	Y. 133.
3.	yā ra ra 1 yā va $\scriptstyle {\dot {2}}$ yā 400 rū 0 yā ra ra 0 yā va 0 yā  0  rū 9999			$\scriptstyle {x^{4}-2x^{2}+400x+0=0.x^{4}+0.x^{2}+0.x+9999}$	V. 138.
4.	yā 197 kā $\scriptstyle {1{\dot {6}}44}$ nī $\scriptstyle {\dot {1}}$ rū 0 yā   0 kā 0    nī 0 rū 6302			$\scriptstyle {197x-1644y-z+0=0.x+0.y+0.z+6302}$	Br. xviii, 55.
5.	ka 6 ka 5 ka 2 ka 3			$\scriptstyle {{\sqrt {6}}+{\sqrt {5}}+{\sqrt {2}}+{\sqrt {3}}}$	V. 37.
6.	${\begin{array}{|c|c|c|c|}\hline \scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}\\\hline \scriptstyle {4}&\scriptstyle {3}&\scriptstyle {6}&\scriptstyle {12}\\\hline \end{array}}$			$\scriptstyle {{\frac {1}{4}}+{\frac {1}{3}}+{\frac {1}{6}}+{\frac {1}{12}}}$	S. 7.
7.	${\begin{array}{|c c|c c c|c c c c|}\hline \scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}\\\scriptstyle {1}&\scriptstyle {2}&\scriptstyle {1}&\scriptstyle {2}&\scriptstyle {3}&\scriptstyle {1}&\scriptstyle {2}&\scriptstyle {3}&\scriptstyle {5}\\\hline \end{array}}$			$\scriptstyle {1\times {\frac {1}{2}}+1\times {\frac {1}{2}}\times {\frac {1}{3}}+1\times {\frac {1}{2}}\times {\frac {1}{3}}\times {\frac {1}{5}}}$
8.	${\begin{array}{|r|r|c|c c|}\hline \scriptstyle {10}&\scriptstyle {163}&\scriptstyle {4}&\scriptstyle {pha}&\scriptstyle {163}\\\scriptstyle {1}&\scriptstyle {60}&\scriptstyle {1}&&\scriptstyle {150}\\\hline \end{array}}$			$\scriptstyle {10:{\overset {163}{60}}::4:{\overset {163}{150}}}$	Bk. 27.
9.	${\begin{array}{|r|}\hline \scriptstyle {1}\\\scriptstyle {1}\\\scriptstyle {1}\\\scriptstyle {\dot {2}}\\\scriptstyle {2}\\\scriptstyle {\dot {9}}\\\scriptstyle {1}\\\scriptstyle {\dot {4}}\\\scriptstyle {6}\\\scriptstyle {\dot {10}}\\\hline \end{array}}$	10.	${\begin{array}{|r l|}\hline \scriptstyle {40}&\\\scriptstyle {1}&\\\scriptstyle {1}&\\\scriptstyle {3}&+\\\scriptstyle {1}&\\\scriptstyle {4}&+\\\scriptstyle {1}&\\\scriptstyle {5}&+\\\hline \end{array}}$	9. $\scriptstyle {(1-{\frac {1}{2}})(1-{\frac {2}{9}})(1-{\frac {1}{4}})(1-{\frac {6}{10}})}$ 10. $\scriptstyle {40(1-{\frac {1}{3}})(1-{\frac {1}{4}})(1-{\frac {1}{5}})}$	L. 53. Bk. 25.

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1. Not Indians.
2. Ghāta=the product.