Geometry

The Shulva Sutras explain a large number of simple geometrical constructions — constructions of squares, rectangles, parallelograms and trapezium. These and others involve the following theorems

• The diagonal of a rectangle divides it into two equal parts.
• The diagonals of a rectangle bisect each other and the opposite areas are equal.
• The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.
• A rectangle and a parallelogram on the same base and between the same parallels are equal in area.
• The diagonals of a rhombus bisect each other at right angles.
• The famous theorem known after the name of Pythagoras.
• Properties of similar rectilinear figures.

These cover roughly the first two books and the sixth book of Euclid. How these theorems were actually obtained is a matter for which no definite answer is available. The Baudhayana Shulva sutra gives the construction of geometric shapes such as squares and rectangles. It also gives sometimes approximate, geometric area-preserving transformations from one geometric shape to another. These include transforming a square into a rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a circle, and transforming a circle into a square. In these texts approximations, such as the transformation of a circle into a square, appear side by side with more accurate statements.

Pythagoras theorem before Pythagoras

“Pythagorean theorem” is the rule for the lengths of sides of right triangles (now written as a²+b²=c²) . The sutras contain discussion and non-axiomatic demonstrations of cases of the Pythagorean theorem and Pythagorean triples. It is also implied and cases presented in the earlier work of Apastamba and Baudhayana, although there is no consensus on whether or not Apastamba’s rule is derived from Mesopotamia.

In Shulva Sutra, it was expressed as “the cord stretched in the diagonal of a rectangle produces both areas which the cords forming the longer and the shorter side of a rectangle produce separately.” (Baudhayana 74). In other words, the sum of the squares of the two different sides of the rectangle equals the square of its diagonal. (“Squares” probably meant the area of actual squares drawn on the three sides of the triangle formed.). In Baudhayana, the other rules related to this right angle theorem are given as follows:

• 1.9. The diagonal of a square produces double the area [of the square].
• 1.12. The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal the area [of the square] produced by the diagonal.
• 1.13. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.

“Pythagorean triplets”, sets of side lengths that can be expressed as integers (for example 3,4,5 and 5,12,13), were used as a sort of proof of that proposition, and were also used to make right angles. Pythagorean triples are found in Apastamba’s rules for altar construction. They were used for the construction of right angles. The complete list is:

• (3,4,5)
• (5,12,13)
• (8,15,17)
• (7,24,25)
• (12,35,37).

However, since these triples are easily derived from an old Babylonian rule, Mesopotamian influence is not unlikely.

Carl B Boyer says “we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the Shulva sutras is not unlikely. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides, but this form of the Pythagorean theorem also may have been derived from Mesopotamia. [...] So conjectural are the origin and period of the Shulva sutras that we cannot tell whether or not the rules are related to early Egyptian surveying or to the later Greek problem of alter doubling. They are variously dated within an interval of almost a thousand years stretching from the eighth century B.C. to the second century of our era.” [ref: Boyer (1991). "China and India". p. 207.]

The Satapatha Brahmana and the Taittiriya Samhita were probably also aware of the Pythagoras theorem. Seidenberg (1983) argued that either “Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source”.[6] Seidenberg suggested that this source might be Sumerian and may predate 1700 BC.

We see various applications of this theorem in Shulva Sutras for different geometrical constructions. For example- construction of a square equal (in area) to the sum, difference, of two given squares, or to a rectangle, or to the sum of n squares. These construction unconditionally involve application of algebraic identities, such as (a±b)²=a²+b²±2ab, a²-b²=(a+b)(a-b), ab=((a+b)/2)²-((a-b)/2)², and na²=((n+1)/2)2a²-((n-1)/2)2a² etc.

value of pi

Another of their most famous problems is that of getting a square and circle to be equal in area. Two authors used very different constructions (neither of which actually uses pi) which both lead to a value of pi very close to 3.08831 (pi is actually about 3.14159). What’s remarkable is the similarity of the two values, but one must wonder how this particular value was reached. (It might be noted that only one of the authors showed knowledge that his method was an approximation.)

As an example, the statement of circling the square is given in Baudhayana as:

2.9. If it is desired to transform a square into a circle, [a cord of length] half the diagonal [of the square] is stretched from the centre to the east [a part of it lying outside the eastern side of the square]; with one-third [of the part lying outside] added to the remainder [of the half diagonal], the [required] circle is drawn.

Fig: Circling a square

and the statement of squaring the circle is given as:

2.10. To transform a circle into a square, the diameter is divided into eight parts; one [such] part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth [of the part left] less the eighth [of the sixth part].

2.11. Alternatively, divide [the diameter] into fifteen parts and reduce it by two of them; this gives the approximate side of the square [desired].

Kim Plofker say, “The “circulature” and quadrature techniques in 2.9 and 2.10, the first of which is illustrated in figure 4.4, imply what we would call a value of π of 3.088, [...] The quadrature in 2.11, on the other hand, suggests that π = 3.004 (where s = 2r·13/15), which is already considered only “approximate.” In 2.12, the ratio of a square’s diagonal to its side (our √2is considered to be 1 + 1/3 + 1/(3·4) – 1/(3·4·34) = 1.4142 .” [ref: Plofker, Kim (2007). p. 391.&392]

arithmetic

The essentially arithmetical background of the Shulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Shulvas.

Along with the “simple” mathematics involved in altar-building, there were very complicated problems to decide what math to use where. One of these problems can be understood just by the fact that the most basic altar was in the form of a falcon! However, all of the bricks retained a square, rectangular, triangular, or circular shape. So the precise dimensions and area of every brick had to be calculated, after the specific design was decided upon. Usually, the total area was 7-1/2 units, because that was convenient for the falcon shape. But to make matters exponentially “worse”, when two or more altars were made for the same purpose, subsequent ones usually had to be one unit bigger than the last; and the areas of each brick had to stay proportional. As you can infer, this was one heck of a math problem! And it had to be done twice for each altar, because a different design was needed for alternating layers, so that spaces between bricks weren’t directly over each other.[Ref: Thibaut, George. Mathematics in the Making in Ancient India. Calcutta: K.P. Bagchi, 1984.]

Fractions

We meet with fractions like 3/8 (Thri Ashtama), 2/7 (Dwi Saptama), 3/4 (Chaturbhagona). These are not unit fractions only, as were used in ancient Egypt, Babylonia and China. Apasthamba gives the area of a square of side 1-1/2 purushas as 2-1/4, and that of a square of side 2-1/2 as 6-1/4.. If the area is 7-1/9 sq. purushas, the side of the square is 2-2/3 (Bodhayana).

Surds

Surds of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 = trikarani, √1/3=triteeya karani, √1/7=saptama karani, √18 = ashtadasa karani.

area of – a trapezium

The shape of the Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2, 36√2 prakramas. Its area is stated to be 1944 prakramas (sq. is to be understood).

Area=36√2 x 1/2*(24√2 + 30√2) = 1944

This indicates a knowledge of the method of finding the area of – a trapezium, and simple operations on surds.

approximation to √2 and irrational numbers

Altar construction also led to an estimation of the square root of 2 as found in three of the sutras. In the Baudhayana sutra it appears as:

2.12. The measure is to be increased by its third and this [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side is the measure].[9]

This approximation to √2 occurs also in Shulvas Apasthamba and Katyayana.

√2 = 1 + 1/3 + 1/(3*4) – 1/(3*4*34)=577/408

This gives √2 = 1.4142156 , whereas the true value is 1.414213. The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area.

It is also interesting to note that three approximations of √2 are given.

[latex]\sqrt 2 = 7 / 5[/latex]
[latex]\sqrt 2 = 17 / 12[/latex]
[latex]\sqrt 2 = 577 / 408[/latex]

Now the continued fraction for √2 = 1 + 1/2 + 1/2+ …

The third, fourth and eight convergent of this are exactly the approximations given above. This gives no clue to the method used in Shulva Sutras, but the coincidence is noteworthy.

The Shulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.

One conjecture about how such an approximation was obtained is that it was taken by the formula:

√(a²+r)≈a+{r/(2a)}-[{r/(2a)²}/{2(a+r/(2a))}] with a = 4 / 3 and r = 2 / 9. which is an approximation that follows a rule given by the twelfth century Muslim mathematician Al-Hassar. The result is correct to 5 decimal places. This formula is also similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900-1600 BCE): √2 = 1 + 24/60 + 51/(60)² – 10 /(60)³=1.41421297 which expresses √2 in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding). Indeed an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula: √x≈√(x-1)+1/(2√(x-1)) for large values of x, which bases itself on the non-recursive identity √(a²+r) ≈ a+r/(2a) for values of r extremely small relative to a.

The Manava Shulva gives the following:

40² + 40² = 56²
4² + 4² = (5-2/3)²
36² + 90² = 97²
5² + 6² = (7-5/6)²

The above facts make it clear that the Indians were the first to use irrational numbers. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing segments of length √2 AB, √3 AB,√5 AB, etc. But no rational approximations to √2, √3 etc., are found in Greek mathematics, nor are there any problems involving arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Shulva Sutras are about two or three centuries prior to Pythagoras.

Conclusion

We all know that Euclid’s geometry is based upon certain axioms and postulates, and the proofs involve a strict logical application of these. The logical methods of Greek geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Sulvas are not formal mathematical treatises. They are only adjuncts to certain religious works.

Further Readings

· Seidenberg, A. 1983. “The Geometry of the Vedic Rituals.” In The Vedic Ritual of the Fire Altar. Ed. Frits Staal. Berkeley: Asian Humanities Press.
· Sen, S.N., and A.K. Bag. 1983. The Sulbasutras. New Delhi: Indian National Science Academy.
· A Seidenberg, The Origin of Mathematics in Archives for History of Exact Science
· Plofker, Kim (2007). “Mathematics in India”. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 9780691114859.
· Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. ISBN 0471543977.
· Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823.
· Cooke, Roger (2005), written at New York, The History of Mathematics: A Brief Course, Wiley-Interscience, 632 pages, ISBN 0471444596
· Joseph, G. G. 2000. The Crest of the Peacock: The Non-European Roots of Mathematics. Princeton University Press. 416 pages. ISBN 0691006598. page 229.
· [ref: Thibaut, George. Mathematics in the Making in Ancient India. Calcutta: K.P. Bagchi, 1984]

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