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Hindu Achievements in Exact Science: A Study in the History of Scientific Development c9: III. Geometry

Hindu Achievements in Exact Science: A Study in the History of Scientific Development c9

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III. Geometry

c9.2

THE earliest geometry of the Hindus is to be found in the "Sulva-sutras" 65 of Baudhayana and Apastamba. In these treatises, which form parts of the Vedic literature, we get the application of mathematical knowledge to the exigencies of religious life, sacrifices, rituals, construction of altars, etc.

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At this stage Hindu geometry was quite independent of Greek influence. The following are some of the problems,³³ which were solved by the mathematicians of the Vedic cycle:

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1. The so-called Pythagorean theorem: The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
2. Construction of squares equal to the sum or difference of two squares;
3. Conversion of oblongs into squares, and *vice versa*;
4. Drawing of a perpendicular to a given straight line at a given point of it;
5. Construction of lengths equal to quadratic surds: The approximate value of √()
6. Circling of squares;
7. Squaring of circles,—"that rock upon which so many reputations have been destroyed," both in the East and West. The earliest Hindus got π = 3.0044;
8. Construction of successive larger squares from smaller ones by addition;
9. Determination of the area of a trapezium, of an isosceles trapezium, at any rate, when the lengths of its parallel sides and the distance between them are known.

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The oldest geometrical efforts of the Hindus were not entirely empiric. They doubtless "reasoned out all or most of their discoveries."⁴ These could not have been inspired by the Greeks.³³

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We find Aryabhata (A.D. 476) solving the following among other problems, viz., the determination of:

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1. The area of a triangle;
2. The area of a circle;
3. The area of a trapezium;
4. The distance of the point of intersection of the diagonals of a trapezium from either of the parallel sides;
5. The length of the radius of a circle.

c9.8

Aryabhata gave also the accurate value of π (= 62832/20000) and the area of the circle as π r². The Saracens learnt this from the Hindus. Probably Yakub (eighth century) was the first to get it when the astronomical tables were imported to Bagdad from India. The correct value of π was not known in Europe before Purbach (1423–61).

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At this stage also Hindu geometricians were not indebted to the Greeks. Their independence is thus argued by Mitra:

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"Euclid and his school never meddled with *logistics* which was practically abandoned as hopeless after the time of Apollonius, while the Indian mathematician's turn of mind was nothing if it was not directed to practical computations. The fact that the Indians took the chord of a smaller circular arc as equivalent in length to the arc—a step which no sane Greek mathematician with a free conscience would have even dreamt of taking—ought to settle once for all the question of the dependence of Indian geometry on Greek geometry."

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Fresh contributions ³³ to geometry were made by Brahmagupta (598–660); viz., those relating to

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1. The construction of right-angled triangles with rational sides;
2. The various properties of right-angled triangles;
3. The area of a cyclic quadrilateral;
4. The properties of isosceles trapezium;
5. The properties of cyclic quadrilateral;
6. The properties of circles; Brahmagupta gave the rules (1) for finding the diameter of a circle when the height and chord of a segment of it are given, and (2) for finding the area of a segment of a circle. The first rule in the form given by the Hindu was not known in Greece. Musa (830) learnt both these rules from Brahmagupta's works.
7. The volume of a cone as one-third the volume of the cylinder;
8. The volume of a pyramid as one-third the volume of the prism;
9. The volume of a cavity of uniform bore (prismatic or cylindrical).

c9.13

Bhaskara (1114) summarized and methodized the results of all previous investigators, e.g., Lata, Aryabhata, Lalla (499), Varahamihira (505), Brahmagupta, Shreedhara (853), Mahavira (850), Aryabhata the Younger (970), and Utpala (970).

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Bhaskara took care to explain that though Aryabhata and others knew the exact value of π, yet some later mathematicians took approximate values only for convenience of calculation. "It is not that they did not know." Thus Brahmagupta took π=3 roughly (or √10 closely) "for lessening the labor of calculation."

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Among Bhaskara's original contributions may be mentioned the fact that he gave two proofs of the so-called Pythagorean theorem. One of them was "unknown in Europe till Wallis (1616–1793) rediscovered it."⁴

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It must be admitted that though Hindu geometricians achieved much the same results as the Greek, they did not attain the excellence of Euclid (*c* 306–293 b.c.) in method and system.

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